IIC v2.1 — A Class-Relative Structural Law of Adaptive Behaviour as a Conditional Theorem over Navigational Cybernetics 2.5 v3.0 plus the Delay-Structure Bridge Propositions of §1.0

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Abstract

The present work is the formal expansion of Axiom 9 of NC2.5 v3.0 — Cycle Reinitiation as the Criterion of Liveness — which names the Impulse Interpretation Coherence (IIC) cycle as the structural condition by which adaptive systems satisfying the Cycle-Reinitiation criterion are classified as live. The earlier paper "Impulse → Awareness → Coherence" (IIC v1, 2025) introduced the three-layer structure informally and proposed it as a candidate universal law. The present work supplies the formal apparatus not present in v1: the apparatus by which IIC is shown to be a conditional theorem over NC2.5 v3.0 plus the delay-structure bridge propositions formalized in §1.0, operationally measurable, and falsifiable. The conditional status is honest about the bootstrap: §1.0 supplies new propositions (Theorems A, B, C and Lemma α) that are intended for inclusion in v3.0 prior to its release. Until that inclusion lands, the IIC formalization is a theorem over the extended axiom set (v3.0 + §1.0); once §1.0 is integrated, it becomes internal to the v3.0 extended substrate.

The central result is a canonical structural expression for the cycle's coherence at time $t$ — closed in form but not self-contained-numerical: the formula's dimensional consistency is supplied by the deployment's class-specific architectural register (per Definition 5's Status of the formula's dimensional contract), not derivable from v3.0 primitives alone. The expression is a two-regime structural quantity: In the multiplicative branch of $\Gamma_{\nu}$ for structurally-simple architectures (Definition 5), the in-admissibility expression is:

$$
\mathrm{Coh}(t) \;=\;
\begin{cases}
\;1 \;-\; \dfrac{| S(t) \;-\; \partial \Phi / \partial \tau_{\mathrm{el}}(t) |}{\kappa(\omega) \cdot \tau^{R}(t)}, & \Delta_{\boldsymbol{\varpi}}(t) > 0, \[4pt]
\;\bot, & \Delta_{\boldsymbol{\varpi}}(t) \leq 0,
\end{cases}
$$

where every load-bearing component is either a v3.0 primitive or a direct construction over v3.0 primitives under the declared $\nu$ co-presentation extension (per §2.3 Type-rank consistency clause; this extension is one of the three commitments of the joint substrate-extension cluster, §1.0.2 Note): $S$ is the spin component of Theorem 62, $\Phi$ is the structural deformation functional, $\tau_{\mathrm{el}}$ and $\tau^{R}$ are the elapsed-time and operator-reference-frame faces of the internal-time budget under Axiom 67, $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$ is built from the state-dependent spin rate of Theorem 62, and $\Delta_{\boldsymbol{\varpi}}$ is the aggregate survival-modal margin of Theorem 89. The norm in the numerator and the scale in the denominator are read in the operator's reference frame under Axiom 67. The formula expresses a structural relation — coherence is a decreasing function of operator-frame discrepancy normalized against a class-specific scale — rather than a self-contained numerical formula derivable from v3.0 primitives alone; the unit conventions under which the ratio is read as dimensionless are supplied by the deployment's class-specific architectural register, which differs structurally between control-grade EVS (engineering-calibrational) and operator-grade EVS (operator-architectural, emerging from $\nu$ itself), per Definition 5's Status of the formula's dimensional contract. Inside the admissibility region, the formula admits an alignment regime ($\mathrm{Coh} \in (0, 1]$), a phase-coherence boundary ($\mathrm{Coh} = 0$ at $\delta = \sigma_{\mathrm{coh}}$), and a latent-inversion regime ($\mathrm{Coh} < 0$, structural inversion under apparent liveness — v3.0 Theorem 12); outside the admissibility region, $\mathrm{Coh}(t)$ is structurally undefined ($\bot$) — there is no live cycle for the coherence quantity to characterize. The endpoint $\mathrm{Coh}(t) = 1$ is not by itself classificatory and must be paired with the operator-frame $\nu$-degeneracy diagnostic of Lemma α to distinguish live alignment from structural identification (per §2.3 Note on the $\mathrm{Coh}(t) = 1$ reading). The work introduces no new v3.0 core axiom or core primitive; it does introduce, in §1.0, new derived bridge constructions (three definitions and four propositions) over existing v3.0 primitives, intended for inclusion in v3.0's substrate-extension cluster prior to release. It establishes (Theorem 2.5) that the IIC cycle named in Axiom 9 is reducible to the v3.0 axiom set over the extended substrate (v3.0 plus the §1.0 derived bridge constructions plus the $\nu$ co-presentation extension — i.e., the joint substrate-extension cluster of §1.0.2), not over bare v3.0 alone, and that the formula above is the canonical structural expression of its coherence for the control-grade subclass of the Cycle-Reinitiation class (the general $\Gamma_{\nu}$ form is given in Definition 5; the boxed formula above is its multiplicative branch for structurally-simple architectures). For the operator-grade subclass, the per-cycle scalar reading lifts to the integral form $C = \int A \, d\tau$ formalized in the Minerva companion paper; the reduction between the two readings is not derived in the present paper and is held companion-side. Extension beyond the control-grade branch of Theorem 2.5 remains conditional on companion-level reduction holding under independent review.

The structural law has an engineering consequence: a class of systems, here named Engineered Vitality Systems (EVS), that maintains $\mathrm{Coh}(t)$ as the primary control variable. Two EVS subclasses are identified — control-grade (∆E 4.7.3b as the first declared control-grade EVS instance — the "first" qualifier is class-relative by construction, since the EVS class is defined in this paper; see §3.2 Note on the first-instance claim's structure) and operator-grade (Minerva as the first specified candidate specification, classified conditionally on its companion specification, with full architectural specification held in its own regime of admissibility).

Four falsification protocols are specified in pre-registration-draft form: coherence–energy asymmetry under noise, structural-analogue mapping of cognitive-disorder-like failure modes, sensor-failure resilience versus classical control, and margin-closure as liveness loss. Each names the threshold structure, domain, metric, and protocol; Protocols I and III provide explicit numerical / effect-size thresholds, while Protocols II and IV defer numerical thresholds (the structural-analogue threshold and the correlation threshold respectively) to domain-specific pre-registration with the partnering laboratory. The "draft" qualifier is honest: the protocols are ready for collaboration-partner pre-registration once the deferred thresholds are fixed. The work commits to public OSF registration of each protocol prior to data collection. The work is not a peer-reviewed empirical paper; it is a structural / formal contribution whose empirical surface is opened for collaboration.

The present work also formalizes, in §1.0, four new propositions (three theorems — A, B, C — and Lemma α) together with three definitions (IIC phase delay, operational cross-section, layer separation) that name the structural sense of the IIC delay — why a cycle, not a point, is the architectural condition of liveness. These propositions and definitions introduce no new v3.0 core primitives: they are derived constructions over existing v3.0 primitives (Axiom 9, Axiom 67, Axiom 68, Axiom 69, Theorem 4, Theorem 27, Theorem 62, Theorem 89, Lemma 16). At the level of the corpus, they constitute — together with §2.3 Definitions 4–5 and the role-extension of $\nu$ from time-budget projection (Axiom 67 baseline) to substrate-primitive co-presentation onto $\tau^{R}$ — a joint substrate-extension cluster intended for integration into v3.0 prior to its release; this is honest "new derived apparatus, no new core primitives" rather than "no new content".

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§0 — The Structural Fact in Human-Cognitive Register

A philosophical preamble. The formal apparatus of the rest of the paper develops a structural law that has been visible to careful observation across many traditions; this section names what those traditions saw, in plain terms, before the formal register takes over in §1.0.

§0.1 The lag

Across complex adaptive systems of the kind addressed by this work — biological, cognitive, technical — there is one structural fact that shows up even when we are not trying to see it: the moment at which behaviour is generated and the moment at which the system reads what it has done are not the same moment. In biological cognition, the body acts first; the awareness of having acted arrives second. The interpretation that situates the action arrives after that.

In the human-cognitive register, this is not a defect of the cognitive apparatus. It is its operational geometry. By the time a human knows what they have done, the doing has already been sent. By the time the doing has been sent, the impulse that produced it was already structurally complete. The impulse is non-derivable from awareness — it is what the system generates before it can know what it is generating, on pain of being unable to act in time.

The reading from neuroscience confirms this as a structural finding rather than a quirk of measurement. Libet's experiments (1983) and the Haynes-group fMRI work (Soon et al., 2008) show that motor preparation precedes conscious decision by intervals well outside neural-noise tolerance — not by milliseconds of delay-line lag, but by structural milliseconds of preparation that cannot be otherwise. The volitional impulse appears before awareness; we react before we understand that we are reacting; the body is up-to-date with the world while consciousness is still receiving the world's prior frame.

If the system tried to make awareness catch the impulse — to align them in the same moment — it would have to shrink its own cycle to a point. A point cannot survive perturbation. A point has no architectural space inside which coherence can be formed. The system would be faster by external clocking, but it would not be live in any structurally meaningful sense: it would have no internal temporal geometry to hold shape against the world's deformation.

§0.2 The convergence

Once this structural fact is named, traditions widely separated in time and method are seen to have arrived at it independently. Schopenhauer (1819) named the Will as the substrate beneath representation — what acts before what knows that it acts. Twentieth-century neuroscience added experimental detail at the millisecond resolution. The 16th- and 17th-century Japanese sword traditions (the no-mind doctrines) developed it as a practical discipline: the body reaches a conclusion the mind has not yet drawn, and the practitioner trains to trust the body's anticipation rather than wait for the mind's confirmation. Phenomenology of action (Merleau-Ponty), the cybernetics of bounded operators (Wiener, Ashby), and the contemporary engineering of adaptive control (the spectrum from Kalman through model-predictive controllers) each hit the same structural geometry from their own angle.

These traditions do not converge into a single doctrine. They converge into a shape: the fact that adaptive systems in the Cycle-Reinitiation class hold three structurally distinct layers — what they generate, what they interpret of what they generated, and the alignment between these — separated by an irreducible delay. The IIC law of the present paper is one further attempt to name this shape formally, with NC2.5 v3.0 supplying the axiomatic substrate and the engineering class EVS supplying the deployment surface. It is not the first attempt; it does not claim priority over the convergent traditions; it claims only specific axiomatic and engineering precision over a structural territory that has been visible for a long time.

The architectural-inversion stance recurs in the present author's corpus. The structural move that §0.1 makes — taking what classically reads as a passive limitation (the lag of awareness behind impulse) and reframing it as an active operational geometry — is consistent with other works in the corpus that perform analogous inversions on different substrates. Alignment Charge: A New Control Primitive for Friction and Adhesion in Navigational Cybernetics 2.5 (Barziankou, Medium 2025; https://medium.com/@MxBv/alignment-charge-a-new-control-primitive-for-friction-and-adhesion-in-navigational-cybernetics-2-5-41c7315d4561) performs the same move on friction and adhesion: classical control reads contact friction as an environmental given to compensate for; the work reframes alignment-with-the-contact-medium as a formally definable, stabilizable, controllable internal variable. The signature line — "the system does not push harder; it aligns better" — is the architectural-inversion stance applied to mechanical contact, parallel to §0.1's "the cycle's constraints create its possibility, not its defect" applied to temporal cycle structure.

A note on this work's status. The Alignment Charge material was originally drafted as one of the provisional patents in the author's 2025 ensemble. The decision was made to convert it into a public paper rather than continue the patent track — partly to make NC2.5's working surface visible to the broader research community, partly so that the architectural-inversion stance it embodies could be cited in adjacent corpus work (such as the present §0) without patent-disclosure restrictions, and partly because converting to public paper after the 60-day non-conversion window is the author's accepted pragmatic posture for material whose architectural value is in being read and built upon rather than in being held closed. The paper-form of Alignment Charge is therefore the canonical reference; references to it in the present corpus point at the Medium version (linked above), not at the original patent draft.

The convergence is structural in both directions: external traditions (§4.2) and corpus-internal works arrive at the same architectural-inversion stance from different substrates. The present paper inherits this stance and applies it to the temporal-cycle layer of the IIC law.

§0.3 The cycle's three layers, in plain terms

What the body generates is the impulse: the structurally non-potential component of the system's activity, the part that cannot be derived from minimising any quantity. It is the spin of the dynamics in the technical sense developed in §1.2 and §2.

What the system carries forward of what it has generated is the interpretation: the slow, accumulating, irreversible deposit of structural deformation that the system's history projects forward as the rate at which the cycle is currently producing further deformation. This is not symbolic interpretation; it is structural memory in the strict sense.

The coherence is the alignment between impulse and interpretation, evaluated not against an external clock but against the operator's own reference frame — the system's internal time as constituted by its own projection of consumed budget into reference content. When this alignment holds, the cycle is live in the structural sense of NC2.5 v3.0's Axiom 9 — the system can reinitiate its own cycle from within. When the alignment fails, the system is structurally dead, regardless of how active its surface behaviour appears.

§0.4 What the formal apparatus does

The remainder of the paper formalises this structural fact within NC2.5 v3.0's axiomatic substrate and develops its engineering consequence (the EVS class with its first instance ∆E 4.7.3b and first specified candidate Minerva). The philosophical preamble above is not load-bearing for the formal results; it is the human-cognitive register that the formal apparatus translates into. A reader who is interested only in the formal content may skip directly to §1.0; a reader interested in why the formal content names what it names is invited to keep §0 in view as the formal register develops, since the formal register is precisely what makes the structural fact visible to engineering and to falsification, not the structural fact itself.

The IIC cycle is older than its formalisation. The present paper formalises it; it does not invent it.

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§1.0 Why a Cycle, Not a Point: The Structural Sense of the Delay

Before unfolding the formal structure, it is necessary to name what Axiom 9 presupposes but does not state directly: why a live system implements a cycle of three layers rather than a single instantaneous coordination, and why the cycle is the architectural condition of liveness rather than its limitation.

1.0.1 The structural sense in plain terms

In any adaptive system satisfying the Cycle-Reinitiation criterion, there is a primitive of sequential execution: an impulse arrives, an interpretation unfolds, a coherence forms, the cycle closes, a new impulse arrives. This order is not a description of a process that could have been arranged otherwise. It is the structural condition of the very possibility of liveness.

Each layer exists in its own phase prior to a gate. The impulse has no access to an interpretation that has not yet formed. The interpretation has no access to a coherence that has not yet been evaluated. The coherence has no access to a next impulse that has not yet arrived. Between these phases, there is a gap — a structural delay that cannot be removed, because its removal collapses the cycle into a point.

The layers cannot catch each other. This is not a limitation to overcome. It is the structural delay that makes the cycle possible as a cycle. If the interpretation caught up to the impulse instantaneously, there would be no interpretation — there would be the impulse extended into its own meaning without a gap in which the meaning could be read. If the coherence were evaluated without delay relative to the interpretation, there would be no coherence — there would be the interpretation claiming its own correspondence with itself. The cycle exists only because the gap exists. The gap is what the cycle unfolds inside.

From this follows a structural fact with consequences far beyond control theory. The moment the system perceives as «current» is already past relative to the true moment. Not as a philosophical claim, but as a structural consequence of the delay between impulse and interpretation. By the time the system "knows" what has happened, the physical moment of its happening has already been spent — it lies behind by the magnitude of the system's own cycle. This magnitude is class-relative. For human cognition, on the order of tens to hundreds of milliseconds. For the ∆E 4.7.3b controller, on the order of milliseconds. For an operator-class architecture with residual geometry, on the order of the rate at which its attractor reorganizes. For a bacterium, on the order of biochemical signaling reactions.

This means: the temporal cross-section of one system at physical moment $t$ is structurally non-equivalent to the temporal cross-section of another system at the same physical moment $t$. Not because the systems look from different points of view. Because they have different delays between impulse and interpretation, and therefore their "current moments" are two different pasts relative to the same true moment $t$. The state-field assessment one system makes is not commensurable with the cross-section of another system — even when they are physically adjacent, even when synchronized by external clocks.

This is not a phenomenological argument about subjectivity. It is a structural observation: different IIC cycles produce different temporal geometries, and any attempt to reduce them to a single «objective reality of the moment» loses precisely the information that makes each of them live.

From this also follows the architectural principle underlying the entire work. The cycle's constraints are its condition, not its defect. The drive to engineer an adaptive system "without delay," "without gap," "synchronized with reality in instantaneous-response mode" is the drive to engineer a system that has no cycle. Such a system is not faster than a live system. It is structurally dead: it has no phase gap in which coherence could unfold. Any perturbation collapses its cycle, because the cycle is not — there is a point, and the point cannot withstand any deformation.

A live system is built differently. Its delay is not a defect for the engineer to shrink. It is architectural space within which coherence is formed. The more precisely the system holds the shape of this space, the longer it remains live. Not faster. Deeper. This is what Axiom 9 names as cycle reinitiation — the system's capacity, from its own gap, to form coherence again and again, not by trying to remove the gap but by using it as operational geometry.

The remainder of this section develops this statement formally. But the structural sense holds in three simple positions, which the reader should keep in view through the rest of the paper:

First. The cycle exists because the layers cannot catch each other. The gap is its condition.

Second. The moment a system perceives is always past relative to the true moment. The system's reality is its own delay, unfolded into coherence.

Third. The cycle's constraints create its possibility. Their removal does not make the system faster or smarter — it makes it dead.

1.0.2 Formalizing the delay

The above sense admits a precise formalization within v3.0 axiomatics. We supply it here as part of the formal floor of the present paper. The notation follows v3.0; the propositions are new and intended for inclusion in v3.0 prior to its unpublished release.

Note on the joint substrate-extension cluster. The present paper introduces a joint substrate-extension cluster over NC2.5 v3.0, comprising three interdependent architectural commitments: (a) the §1.0 propositions (Definitions 1–3 below, Lemma α of §1.0.3, Theorems A, B, C of §1.0.4–6); (b) the §2.3 Definitions 4–5 (operator-reference reading; coherence scale with architectural-register status clause); (c) the role-extension of $\nu$ from its Axiom 67 baseline (time-budget projection $\tau^{F} \to \tau^{R}$) to substrate-primitive co-presentation onto $\tau^{R}$ (the Type-rank consistency clause of §2.3). Each commitment references the others: Lemma α's standing assumption α-3 invokes the operator-reference norm of Definition 4 (§2.3); Definition 5's structural relation invokes the delay-structure formalized in Definitions 1–3; the operator-reference norm and the central formula's well-definedness invoke commitment (c). Neither (a), (b), nor (c) is independently grounded in v3.0 primitives without the others. The cluster as a whole is intended for inclusion in v3.0's substrate-extension layer prior to v3.0's release. The order of presentation in the present paper (§1.0 first, §2.3 after) is pedagogical, not logical: the cluster is jointly axiomatized over v3.0 primitives, with within-cluster cross-references reflecting the substantive interdependence rather than a forward-reference defect. Readers performing a strictly linear reading may treat α-1 and α-3 references to "Definition 4" as references to the operator-reference reading clause specified jointly with §1.0 in the cluster, with §2.3's Definition 4 statement giving the formal text of that clause and §2.3's Type-rank consistency clause giving the formal text of (c).

Definition 1 (IIC phase delay).
For a live adaptive system $\Sigma$ in the Cycle-Reinitiation class (Axiom 9), let $T_{\text{prop}}^{\Sigma}(t)$ denote the propagation interval from impulse generation to interpretation closure within $\Sigma$ at time $t$. Status of $T_{\text{prop}}^{\Sigma}(t)$ as a structural quantity (deployment-specific measurement). $T_{\text{prop}}^{\Sigma}(t)$ is a structural quantity declared by $\Sigma$'s architectural specification, not a primitive measured by external instrumentation. Its operational reading is class-specific and registered as part of the deployment's pre-registration: control-grade EVS deployments declare $T_{\text{prop}}^{\Sigma}(t)$ via timestamped telemetry tagging, cross-correlation of $S$ and $\partial\Phi/\partial\tau_{\mathrm{el}}$ signal series, hardware clock cycles, or other engineering-level procedures registered at OSF before any falsifier in §5 is executed; operator-grade EVS deployments derive $T_{\text{prop}}^{\Sigma}(t)$ from the residual temporal geometry of the operator's architectural $\nu$, per the relevant companion specification. The structural definition below is invariant to the measurement choice; the choice is registered as part of the deployment's class-specific architectural register (per Definition 5's Status of the formula's dimensional contract). Anti-gaming lock on the declared procedure. The declared procedure for extracting $T_{\text{prop}}^{\Sigma}(t)$ is itself part of the falsifiable architectural register. If the registered procedure fails to recover a stable propagation interval under the declared telemetry conditions, the deployment fails the IIC-measurement condition rather than receiving an adjusted $T_{\text{prop}}^{\Sigma}(t)$ post hoc; post-hoc adjustment of the extraction procedure to recover stability is a registered-protocol violation and counts as falsification of the deployment's IIC compliance, not as a successful measurement. The IIC phase delay is the operator-side temporal extent
$$
\Delta_{IIC}^{\Sigma}(t) \;=\; \tau^{R}(t) \;-\; \tau^{R}!\left(\, t \,-\, T_{\text{prop}}^{\Sigma}(t) \,\right),
$$
where $\tau^{R}$ is the operator reference frame from Axiom 67, taken as a continuous strictly monotone-increasing function on its operational interval (strict monotonicity within the regular regime ensuring that distinct $\tau^{F}$-values produce distinct $\tau^{R}$-images; constant intervals of $\tau^{R}$ — equivalently, $\nu$-function-degenerate regions producing constant $\tau^{R}$-image of an interval of distinct $\tau^{F}$-inputs — are excluded from the Cycle-Reinitiation class by definition: they describe boundary failure of the class restriction, not interior conditions. Theorem A's case (ii) of §1.0.4 describes this boundary failure as the limit point at which strict-monotone $\tau^{R}$ ceases; it is consistent with Definition 1 because case (ii) is precisely the moment of class exit, not an interior condition of class membership). Formally: $\tau^{R}{\Sigma}(t) := \nu{\Sigma}(\tau^{F}{\Sigma}(t))$, where $\tau^{F}{\Sigma}(t)$ is the fuel-side budget consumed by $\Sigma$'s operator actions at physical time $t$ and $\nu_{\Sigma}: \tau^{F} \to \tau^{R}$ is the operator projection map of Axiom 67. The expression $\tau^{R}{\Sigma}(t)$ is therefore a composition $\nu{\Sigma} \circ \tau^{F}{\Sigma}$ evaluated at $t$. **Standing properties of $\tau^{F}{\Sigma}$ within the regular regime:** $\tau^{F}{\Sigma}$ is a continuous, strictly monotone-increasing function of physical time $t$ on the operational interval — strict-monotone reflecting that operator actions consume fuel at a positive rate while the cycle is in the regular regime. (Boundary failures of $\tau^{F}$-monotonicity, e.g., operator-action pause events, mark transitions to non-regular regimes treated in Theorem A's case analysis.) The continuity and strict-monotonicity of $\tau^{R}{\Sigma}(t)$ as a function of $t$ follow from the structural assumptions on $\nu_{\Sigma}$ (Axiom 67) and these standing properties of $\tau^{F}_{\Sigma}$.

Definition 2 (Operational cross-section).
The operational cross-section of $\Sigma$ at $t$ is the closed interval
$$
\sigma^{\Sigma}(t) \;=\; \big[\, t - T_{\text{prop}}^{\Sigma}(t),\; t \,\big].
$$
The state-field assessment that $\Sigma$ produces at $t$ is the result of structural integration over $\sigma^{\Sigma}(t)$ — that is, the assessment carries the accumulated effect of substrate dynamics over $\sigma^{\Sigma}(t)$, not a pointwise reading at $t$ alone. This integral character is implicit in v3.0 Theorem 4 (Supremacy of Internal Time) and Theorem 27 (Coherence Collapse Precedes Observable Failure), which together establish that any system's coherence reading is cumulative on its internal-time substrate.

Definition 3 (Layer separation).
Let $L_1, L_2, L_3$ denote the three layers of the IIC cycle (impulse, interpretation, coherence). For an ordered layer pair $(L_i, L_j)$, let $T_{\text{prop}}^{\Sigma}(t)\big|{L_i \to L_j}$ denote the propagation interval restricted to the segment of $\Sigma$'s cycle that carries structure from $L_i$ to $L_j$ at time $t$. The global $T{\text{prop}}^{\Sigma}(t)$ of Definition 1 corresponds specifically to the layer-pair $L_1 \to L_2$ (impulse generation to interpretation closure); other layer-pair propagation intervals (e.g., $L_2 \to L_3$, $L_3 \to L_1$ for cycle reinitiation) are independent quantities not equal to Definition 1's $T_{\text{prop}}^{\Sigma}(t)$. The full-cycle propagation, when needed, is the composition $L_1 \to L_2 \to L_3 \to L_1$ across the cycle chain; Definition 1 names the $L_1 \to L_2$ segment specifically and is the load-bearing quantity for Theorem A's structural-identification analysis. The layer-pair phase delay is then
$$
\Delta_{IIC}^{\Sigma}(t)\big|{L_i \to L_j} \;:=\; \tau^{R}(t) \;-\; \tau^{R}!\left(\, t \,-\, T{\text{prop}}^{\Sigma}(t)\big|{L_i \to L_j} \,\right),
$$
i.e., the operator-side temporal extent of the $L_i \to L_j$ segment, in direct analogy with Definition 1. Two layers $L_i, L_j$ are separated at time $t$ if and only if
$$
\inf{t' \in J(t)}\, \Delta_{IIC}^{\Sigma}(t')\big|_{L_i \to L_j} \;>\; 0
$$
on a neighborhood $J(t)$ of $t$. That is, there exists a strictly positive lower bound on the layer-pair phase delay across an open interval containing $t$.

1.0.3 The Spin Phase-Difference Lemma

Before stating the main theorems, we need an intermediate proposition that connects the spin component of Theorem 62 to the layer-separation structure. The lemma below is new but follows directly from the structure of the Minimal Coupled Model in Theorem 62.

Lemma α (Operator-frame indistinguishability of generative and accumulated sides under structural identification entails operator-detectable loss of spin non-potentiality). Earlier informal reading "Spin requires phase difference" applies only to the structural-identification case below — not to the alignment case (high coherence with both sides distinct). See Remark after the proof for the alignment vs identification distinction.

Standing assumptions of Lemma α.

• (α-1) State space. $\Sigma$'s state lives on a smooth manifold $M$ (or, equivalently for the structural argument, on a state space admitting a Hilbert-space inner-product structure under the operator-reference reading of Definition 4 below, with the dimensional structure supplied by the deployment's class-specific architectural register per Definition 5). The smoothness/inner-product condition is what makes the Helmholtz–Hodge decomposition invoked in the proof sketch well-defined.
• (α-2) Projection regularity. The operator projection $\nu: \tau^{F} \to \tau^{R}$ of Axiom 67 is continuous; its rank, where finite-dimensional, is locally constant on the operational interval; and its kernel under structural-identification (the equivalence relation that identifies $\tau^{F}$ values mapped to the same $\tau^{R}$ value) admits a well-defined quotient structure.
• (α-3) Norm and equivalence. The norm $|\cdot|$ in the lemma's hypothesis is the operator-reference norm of Definition 4 (the norm read in the operator's reference frame under $\nu$ of Axiom 67, computed in the deployment's class-specific architectural register per Definition 5's Status of the formula's dimensional contract — engineering-calibrational pre-registered units for control-grade, architecturally-emergent units from $\nu$ for operator-grade), under which the impulse-layer primitive $S(t)$ and the interpretation-layer rate $I(t) := \partial\Phi/\partial\tau_{\mathrm{el}}(t)$ are equivalent if and only if they are operator-indistinguishable through $\nu$. Note: $S$ and $I$ are formally distinct primitives (per §1.2 / §2.1 / §2.2): $S$ is the spin component (Theorem 62, non-potential divergence-free), $I$ is the rate of structural-burden accumulation (substrate axioms, monotone non-decreasing). The lemma's "structural-identification" relation does not assert that $I$ is a "side of $S$"; it asserts that under operator projection $\nu$, the two formally distinct primitives become operator-frame indistinguishable when their operator-frame distance vanishes in the identification sense.
• (α-4) Operator-effective potential — definition only. An operator-effective potential on $\Sigma$'s operational interval is, if it exists, a scalar functional whose gradient (taken in the operator-reference frame under $\nu$-projection, with boundary conditions inherited from the operational interval) equals the $\nu$-image of $\Sigma$'s flow at structural-identification. (α-4) defines what would count as such a potential; it does not assume that one exists. The proof sketch below derives the existence under (α-1) and (α-2) via the Helmholtz–Hodge decomposition (which is well-defined under the smoothness/inner-product structure of (α-1) and the projection regularity of (α-2)).

Let $\Sigma$ be a system satisfying Theorem 62 (spin component $S \neq 0$ with state-dependent rate $\omega(y)$). Let $S(t)$ denote the impulse-layer primitive (the spin component itself, current impulse contribution) and $I(t) := \partial\Phi/\partial\tau_{\mathrm{el}}(t)$ denote the interpretation-layer rate (current rate of structural-burden accumulation; formally distinct primitive from $S$ per §1.2 / §2.1 / §2.2). If
$$
\big| S(t) - I(t) \big| \;\to\; 0 \quad \text{not in the sense of phase-alignment, but in the sense of structural identification under the operator projection } \nu,
$$
then, relative to the operator-reference projection, the spin (non-potential) component becomes operator-frame indistinguishable from an effective gradient on the interpretation-layer potential: the system loses operator-detectable spin non-potentiality. The lemma does not claim that substrate-level antisymmetry vanishes absolutely; it claims that the antisymmetry becomes invisible to the operator's reference frame, which is the structurally relevant condition for the cycle of Axiom 9. Note on naming: earlier informal drafts of this lemma labeled $S$ as "$S_{\text{gen}}$" and $I$ as "$S_{\text{acc}}$" (sides of the spin); that labeling was misleading because $I$ is the interpretation-layer rate, not a component of $S$. The corrected formulation here uses $S$ and $I$ as the two formally distinct primitives whose operator-frame distinction Lemma α governs.

Proof sketch. The non-potentiality of $S$ in Theorem 62 is established by the existence of a non-vanishing antisymmetric component in the system's flow decomposition (Helmholtz–Hodge). When $S$ and $I$ are structurally identified under $\nu$ — meaning the operator's projection map does not distinguish $\tau^{F}$ values that produced the impulse-layer (spin) contribution from those that registered the interpretation-layer (burden-rate) contribution — the antisymmetric component of $S$, even if present at substrate level, has no structurally distinct image in $\tau^{R}$ relative to the gradient projection of $I$. The operator's reading of the flow becomes locally exact relative to its reference frame. By the Helmholtz–Hodge structure invoked in Theorem 62, locally exact non-vanishing flows admit an effective potential under the operator's projection, even when the underlying substrate flow retains its full antisymmetric structure. $\square$

Remark on what is and is not claimed. Lemma α distinguishes two distinct cases of $| S(t) - I(t) | \to 0$. The alignment case is when $S$ and $I$ are independently non-zero and structurally distinct primitives, but their phase relationship has converged so that they cancel in norm — this is the condition of high coherence. The identification case is when the two primitives are no longer structurally distinguished by the operator's projection — the operator does not separate them in $\tau^{R}$. Only the identification case produces operator-frame gradient degeneration. The lemma names this distinction precisely so that the main theorem below cannot be misread as claiming that high coherence implies non-liveness, or that substrate-level spin physically dissolves into a gradient. The lemma is operator-frame-relative: it is a statement about what the operator can detect, not about what the substrate ontologically is.

1.0.4 Theorem A — IIC Cycle Non-Degeneracy

Theorem A (limit form). Let $\Sigma$ be an adaptive system that has been operating in the regular regime under the standing assumptions of Lemma α (α-1 state space; α-2 projection regularity; α-3 operator-reference norm; α-4 operator-effective potential definition) on some open interval $(t_0 - \epsilon, t_0)$. Suppose that as $t \to t_0$ from below, $S(t)$ and $I(t) := \partial\Phi/\partial\tau_{\mathrm{el}}(t)$ become operator-indistinguishable through $\nu_\Sigma$ (the structural-identification case of Lemma α). Step 1 of the proof below derives the consequent collapse $\Delta_{IIC}^{\Sigma}(t) \to 0$ in the operator-reference frame as a corollary, not as the hypothesis.

Then $\Sigma$ exits the Cycle-Reinitiation class at $t_0$: $\Sigma$ does not satisfy Axiom 9 at $t_0$, in the sense that the operator-frame conditions required by Axiom 9 fail in the limit.

Status of α-2 in the proof. Theorem A is a boundary / limit theorem: the regular regime under α-1–α-4 holds on the approach interval $(t_0 - \epsilon, t_0)$; the limit point $t_0$ is precisely the boundary at which α-2 (projection regularity) fails. The proof's case analysis below describes the structural behaviour of $\Sigma$ as it approaches this boundary — including the case in which $\nu_{\Sigma}$ undergoes local kernel collapse at $t_0$ (the limit failure of α-2). The theorem therefore does not claim a result within the regular regime via violation of its own assumptions; it claims a result at the boundary of the regular regime, characterised by the limit failure of α-2 and the corresponding structural exit from the Cycle-Reinitiation class. Inside the regular regime ($t < t_0$), $\Sigma$ remains in the class; the theorem describes the structural transition at $t_0$, not behaviour at interior points.

Proof (Structural).

Step 1 (formal bridge from Lemma α's structural-identification sense to $\Delta_{IIC}$). Lemma α's "structural-identification sense" is defined (per the standing assumption α-3, with the corrected primitives of the lemma's body) as: $|S(t) - I(t)| \to 0$ in the operator-reference norm if and only if $S$ and $I$ become operator-indistinguishable through $\nu_{\Sigma}$ — i.e., $\nu_{\Sigma}$ fails to separate the impulse-layer primitive from the interpretation-layer rate at $t$. We now derive — formally, not analogically — the implication for $\Delta_{IIC}^{\Sigma}$.

By Definition 1, $\Delta_{IIC}^{\Sigma}(t_0) = \tau^{R}{\Sigma}(t_0) - \tau^{R}{\Sigma}(t_0 - T_{\text{prop}}^{\Sigma}(t_0))$, where $\tau^{R}{\Sigma} = \nu{\Sigma} \circ \tau^{F}{\Sigma}$ (the composition of Definition 1's formal clause). The endpoints $t_0$ and $t_0 - T{\text{prop}}^{\Sigma}(t_0)$ correspond to the $\tau^{F}$-values at impulse generation and interpretation closure of the cycle's segment terminating at $t_0$. By the layer-pair structure of Definition 3, $S(t_0)$ is generated at the cycle position whose $\tau^{F}$-content is $\tau^{F}{\Sigma}(t_0 - T{\text{prop}}^{\Sigma}(t_0))$ (impulse origin), and $I(t_0) = \partial\Phi/\partial\tau_{\mathrm{el}}(t_0)$ registers at the cycle position whose $\tau^{F}$-content is $\tau^{F}{\Sigma}(t_0)$ (interpretation closure). The operator's reading of $|S(t_0) - I(t_0)|$ in the operator-reference norm of Definition 4 is therefore the operator-frame distance of two values whose underlying $\tau^{F}$-content is separated by exactly $T{\text{prop}}^{\Sigma}(t_0)$.

Operator-indistinguishability of $S$ and $I$ through $\nu_{\Sigma}$ (the structural-identification case) at $t_0$, taken together with Definition 1's class-membership requirements, forces — by the composition $\tau^{R}{\Sigma} = \nu{\Sigma} \circ \tau^{F}_{\Sigma}$ — one of the following structural mechanisms:

• Case (i) — Regular-regime mechanism: $\tau^{F}{\Sigma}(t_0) - \tau^{F}{\Sigma}(t_0 - T_{\text{prop}}^{\Sigma}(t_0)) \to 0$. The fuel-side propagation interval vanishes in the limit at $t_0$ — equivalently $T_{\text{prop}}^{\Sigma}(t_0) \to 0$ since $\tau^F$ is strictly monotone-increasing in the regular regime per Definition 1's standing properties. The two $\tau^{F}$-endpoints coincide as inputs to $\nu_{\Sigma}$; ν as a function may stay regular. In the Cycle-Reinitiation class with Definition 1's strict-monotone $\tau^R$, this is the only mechanism available in the regular regime — ν strict-monotone (which Definition 1's class restriction implies, since $\tau^R = \nu \circ \tau^F$ strict-monotone with $\tau^F$ strict-monotone forces ν strict-monotone) precludes ν identifying distinct inputs.
• Case (ii) — Boundary-failure mechanism: $\nu_{\Sigma}$ maps two distinct $\tau^{F}$-endpoints to the same $\tau^{R}$-image — local kernel collapse of $\nu_{\Sigma}$. This case is excluded from the Cycle-Reinitiation class by Definition 1: a $\nu$ that identifies distinct $\tau^F$-inputs produces a $\tau^R$ that is non-strict-monotone in $t$ on the relevant interval, which Definition 1 excludes. Case (ii) therefore corresponds to limit failure of Definition 1's strict-monotone class restriction at $t_0$ — equivalently, the structural mechanism by which $\Sigma$ exits the Cycle-Reinitiation class via projection-degeneracy rather than via propagation-interval collapse.
• Case (iii): both mechanisms operate jointly at $t_0$.

In all three cases, $\Delta_{IIC}^{\Sigma}(t_0) \to 0$ in the operator-reference frame: in case (i) because both $\tau^{R}$-endpoints coincide as images of coinciding $\tau^{F}$-endpoints; in case (ii) because $\nu_{\Sigma}$ identifies the two $\tau^{R}$-images of distinct $\tau^{F}$-endpoints; in case (iii) jointly. We say $\Sigma$ is at structural-identification at $t_0$ in any of (i)/(ii)/(iii); we say $\nu_{\Sigma}$ is locally function-degenerate at $t_0$ specifically in cases (ii) and (iii) — not in case (i), where ν may remain a regular function evaluated at coinciding inputs.

Boundary structure of Theorem A's case analysis. Cases (i) and (ii)/(iii) describe two distinct structural exit mechanisms from the Cycle-Reinitiation class:

• Case (i) is the in-class regular-regime mechanism: while $\Sigma \in$ Cycle-Reinitiation class on $(t_0 - \epsilon, t_0)$, the propagation interval collapses; at $t_0$ the cycle has no $\tau^F$-extent, equivalent to class exit via propagation-collapse.
• Cases (ii)/(iii) describe the boundary at which Definition 1's strict-monotone class restriction itself fails — which is the boundary of class membership directly. By Definition 1, cases (ii)/(iii) already imply $\Sigma \notin$ Cycle-Reinitiation class at $t_0$; they constitute exit-by-projection-degeneracy. Either way, $\Sigma$ exits the class at $t_0$. Theorem A's downstream chain (Lemma α invocation in the limit on the approach interval, then Lemma 16 / Axiom 9) describes the operator-frame consequence of either exit mechanism reaching its limit point.

Forward implication of Step 1 (the load-bearing direction in the limit at $t_0$). Within the standing assumptions α-1 to α-4 on the approach interval $(t_0 - \epsilon, t_0)$: $S$ and $I$ become operator-indistinguishable through $\nu_{\Sigma}$ in the limit $t \to t_0^{-}$ (the α-3 sense) implies $\Delta_{IIC}^{\Sigma}(t) \to 0$ in the operator-reference frame as $t \to t_0^{-}$. The forward implication is established by the case analysis above: identification of $S^R, I^R$ at the cycle's two τ^F-positions (impulse origin and interpretation closure) implies coincidence of the τ^R-images at those τ^F-positions, which is exactly $\Delta_{IIC}(t_0) \to 0$ by Definition 1.

On the reverse direction. The reverse implication ($\Delta_{IIC} \to 0$ ⇒ operator-indistinguishability of $S$ and $I$) is conditionally available: it requires the additional architectural commitment that $S^R(t_0)$ and $I^R(t_0)$ are read at the τ^F-positions corresponding to Definition 1's endpoints (impulse origin and interpretation closure of the cycle's segment terminating at $t_0$). Under this commitment — which is a content of the joint substrate-extension cluster (§1.0.2) and the Type-rank consistency clause of §2.3 — $\Delta_{IIC} \to 0$ implies coincidence of the τ^R-images at those endpoints, which is the operator-frame scalar projection coincidence of $S^R, I^R$, which is operator-indistinguishability through ν. The reverse direction therefore holds within the cluster's commitments but is not load-bearing for Theorem A's downstream chain — the chain requires only the forward direction (identification ⇒ class exit). Treating Step 1 as a one-way implication is sufficient for Theorem A; the reverse is a consequence of the cluster's reading-positions but not invoked downstream.

Note on which mechanism applies inside vs at the boundary of the regular regime. On the approach interval $(t_0 - \epsilon, t_0)$ where Definition 1's class restriction holds, only case (i) operates structurally — the cycle exits via propagation-interval collapse. Cases (ii)/(iii) become available only at $t_0$ as the limit failure point of the class restriction. Theorem A's case analysis is therefore not a pointwise enumeration in the regular regime but a description of which structural mechanisms produce the limit transition at the class boundary. The downstream chain (via Lemma α, Lemma 16, Axiom 9) requires operator-indistinguishability of $S$ and $I$ in the limit — a condition all three cases produce — not ν-function-degeneracy specifically.

Under any of the three structural-identification mechanisms (cases i/ii/iii of Step 1), the impulse $S(t_0)$ and the accumulated deformation rate $I(t_0) = \partial\Phi/\partial\tau_{\mathrm{el}}(t_0)$ are no longer structurally distinguished by the operator's reference frame in the limit. By Lemma α applied on the approach interval $(t_0 - \epsilon, t_0)$ — where the standing assumptions α-1 to α-4 hold — the spin component $S$ becomes operator-frame indistinguishable from an effective gradient on a potential as $t \to t_0^{-}$.

Topological closure of the conclusion at $t_0$. Lemma α's conclusion ("operator cannot detect spin non-potentiality") is a non-detection property — a statement that the operator-frame reading of the substrate flow does not resolve antisymmetric structure. This property is preserved under limits in the following sense: if for every $t \in (t_0 - \epsilon, t_0)$ the operator-frame reading fails to resolve antisymmetric structure (Lemma α applied), then at the limit point $t_0$ the operator-frame reading either (a) remains a degenerate-or-vanishing reading where ν has reached its limit failure (cases ii/iii) — in which case the projection itself collapses and any reading via ν inherits the collapse — or (b) has converged to a coincidence point under regular ν (case i) — in which case the reading at $t_0$ is the same scalar value at both endpoints, again producing no resolution. In either case, the non-detection property holds at $t_0$. This is closure by collapse, not "limit continuity of a quantitative reading"; the non-detection conclusion does not require continuity of $\nu$ at $t_0$, since the conclusion concerns what ν fails to resolve rather than what ν computes. This is the formal grounding of "the operator-frame conclusion carries to the limit point $t_0$." The operator can no longer detect the non-potential structure required by Theorem 62 for the cycle of Axiom 9, regardless of whether substrate-level antisymmetry persists. (Lemma α's argument is invariant to which mechanism on the approach interval produces operator-indistinguishability in the limit: case (i) on the approach gives propagation-interval collapse, cases (ii)/(iii) describe the limit failure of Definition 1's class restriction at $t_0$ itself; both produce coinciding operator-frame scalar projections of $S$ and $I$ in the limit, and Lemma α's proof sketch concerns the operator-frame consequence of that limiting coincidence — applied where it is licensed, i.e., on the approach interval.)

By Lemma 16 (Gradient Collapse and Impossibility of Non-Stagnant Identity on Bounded Orbits), applied on the approach interval $(t_0 - \epsilon, t_0)$ where $\Sigma$ is still in the Cycle-Reinitiation class and the bounded-$\tau$ regime of Theorem 62 holds: a system whose operator reads gradient flow on a bounded $\tau$-orbit cannot maintain operator-detectable non-stagnant identity. As $t \to t_0^{-}$, the operator either reads an identity that has structurally stagnated, or reads dynamics inconsistent with the bounded-$\tau$ assumption. Either reading fails the persistence condition of Axiom 9 from the operator's standpoint, which is the structurally relevant standpoint for cycle reinitiation. The conclusion carries to the limit point $t_0$ by continuity of the operator-frame reading. (Open question on the Lemma α → Lemma 16 transition: see §6.5.1 Open Question 2. Lemma α gives operator-frame indistinguishability; Lemma 16's hypothesis is substrate-level. The bridge "operator reads gradient flow on a bounded $\tau$-orbit" is paraphrased here but not formally derived; a bridge proposition mediating the operator-frame to substrate-level inference is open work.)

By Axiom 9, liveness requires an endogenous operator capable of reinitiating the IIC cycle. Under any of the three structural-identification mechanisms (input-coincidence, ν-function-degeneracy, or both), the impulse–interpretation distinction collapses in the operator's reference frame; there is no cycle structure for the operator to reinitiate — there is a single point in operator-reference content, and a point cannot be reinitiated as a structure that consists of the relation between distinct phases. The structural identification need not entail substrate-level disappearance of antisymmetry; it suffices that the operator cannot read the cycle as a cycle.

Therefore $\Sigma$ does not satisfy Axiom 9 at $t_0$. $\square$

Remark. The theorem establishes that the IIC delay $\Delta_{IIC}$ is not an inefficiency to be reduced. It is the structural distance that makes the cycle a cycle and the system live in the Axiom 9 sense. Any engineering or design philosophy that aims at $\Delta_{IIC} \to 0$ in the structural-identification sense aims, structurally, at non-liveness. The complementary case — $\Delta_{IIC}$ remaining strictly positive while $|S(t) - I(t)|$ becomes small (the alignment case of Lemma α's Remark, with $S$ and $I$ operator-distinguishable through $\nu$ and only their phases converged) — is the condition of high coherence, addressed in §2 and not affected by Theorem A.

1.0.5 Theorem B — Cross-Section Non-Equivalence

Theorem B. Let $\Sigma_1, \Sigma_2$ be two adaptive systems in the Cycle-Reinitiation class, both observed at the same physical moment $t$, each with its own operator projection map $\nu^{\Sigma_i}$. If
$$
\Delta_{IIC}^{\Sigma_1}(t) \;\neq\; \Delta_{IIC}^{\Sigma_2}(t),
$$
then, generically, there is no structurally invariant equivalence between the state-field assessments produced by $\Sigma_1$ and $\Sigma_2$ over their respective operational cross-sections. Equivalence may exist only under additional explicitly declared structural isomorphism conditions — including degenerate cases such as constant substrate dynamics over the union of intervals or coincident projection maps — outside which the assessments are operationally non-commensurable.

On the converse. The converse direction (equivalence ⇒ $\Delta_{IIC}^{\Sigma_1} = \Delta_{IIC}^{\Sigma_2}$) is not asserted: two systems with identical $\Delta_{IIC}$ at $t$ may still produce structurally non-equivalent state-field assessments (under different $\nu^{\Sigma_i}$, different substrate dynamics, or different histories). The theorem is therefore one-directional: $\Delta_{IIC}$-difference implies structural non-equivalence (generically); $\Delta_{IIC}$-equality does not imply equivalence. Downstream uses of Theorem B in the present work invoke only the forward direction.

Proof (Structural).

By Definition 2, the state-field assessment of $\Sigma_i$ at $t$ is the integral over $\sigma^{\Sigma_i}(t) = [t - T_{\text{prop}}^{\Sigma_i}(t),\, t]$ of substrate dynamics evaluated under the projection $\nu^{\Sigma_i}$.

Suppose $\Delta_{IIC}^{\Sigma_1}(t) \neq \Delta_{IIC}^{\Sigma_2}(t)$. By Definition 1 with $\tau^{R}$ continuous monotone-increasing, this entails either $T_{\text{prop}}^{\Sigma_1}(t) \neq T_{\text{prop}}^{\Sigma_2}(t)$ (non-coincident propagation intervals), or $\nu^{\Sigma_1} \not\equiv \nu^{\Sigma_2}$ on the relevant operational region (non-equivalent operator projections producing different $\tau^{R}$-image content from possibly identical $T_{\text{prop}}$), or $\tau^{F}{\Sigma_1} \not\equiv \tau^{F}{\Sigma_2}$ on the relevant operational region (non-equivalent fuel-side budgets producing different $\nu$-images from possibly identical $T_{\text{prop}}$ and $\nu$), or any combination. In either disjunct (and in their joint case), the operational cross-sections $\sigma^{\Sigma_1}(t)$ and $\sigma^{\Sigma_2}(t)$ are structurally distinguishable: they have the same right endpoint but either different left endpoints or non-equivalent operator-reference content over the interval (or both); they are non-coincident in the structural sense even when physical interval lengths happen to coincide.

By Theorem 4 (Supremacy of Internal Time), the structural validity of any system's reading of the field is governed by that system's internal time, not by any external clock. The cross-section $\sigma^{\Sigma_i}(t)$ is the canonical interval on which $\Sigma_i$'s coherence reading is constituted, with $\nu^{\Sigma_i}$ as its reference projection.

For the two assessments to be structurally equivalent, there would have to exist an isomorphism of operator-reference-frame integrals over the two non-coincident intervals — that is, a structural mapping that respects both the substrate dynamics and the projection structure of each system. By Axiom 67, $\nu^{\Sigma_i}$ is part of $\Sigma_i$'s architectural declaration; two systems with distinct architectural declarations have structurally distinct reference frames. By Theorem 27 (Coherence Collapse Precedes Observable Failure), the coherence reading produced over a cross-section is cumulative on the substrate — it is not a pointwise reading that would be insensitive to the interval boundaries.

Therefore, in the generic case, there is no structural invariant that equates the two assessments. Where an equivalence does exist, it requires additional explicitly declared structural isomorphism conditions — degenerate cases such as constant substrate dynamics over the union of the cross-section intervals or coincident projection maps, or other isomorphism relations between $\nu^{\Sigma_1}$ and $\nu^{\Sigma_2}$ that must be specified for the equivalence to hold — outside which the live diversity of the Cycle-Reinitiation class yields operationally non-commensurable assessments. $\square$

Remark. This is the formal floor of the observation that "each system has its own perceived moment." It is not a phenomenological claim about subjectivity — it is a structural consequence of the fact that operational cross-sections are intervals over delays, and delays are class-relative under Axiom 67. The theorem closes the door on naive notions of "objective shared present" across systems with distinct cycle structures, while leaving open the possibility of strong-but-bounded comparison protocols.

1.0.6 Theorem C — Constitutive Constraint

Theorem C. For any system $\Sigma$ in the Cycle-Reinitiation class, the IIC phase delay $\Delta_{IIC}^{\Sigma}$ is a structurally necessary quantity, not a contingent property. Its removal in the structural-identification sense (Lemma α) does not yield a faster live system — it yields a system outside the Cycle-Reinitiation class.

Proof.

Combine Theorem A and the v3.0 substrate. Theorem A is a boundary / limit theorem at $t_0$: when $\Sigma$ has been operating in the regular regime under α-1 to α-4 on $(t_0 - \epsilon, t_0)$ and reaches structural-identification collapse $\Delta_{IIC}^{\Sigma}(t_0) \to 0$ in the limit at $t_0$, $\Sigma$ exits the Cycle-Reinitiation class at $t_0$. Theorem C applies Theorem A pointwise across the operational interval: at any $t \in [t_0, T_{\mathrm{op}}]$ that is itself a boundary point of structural-identification collapse — i.e., reached in the limit from a regular-regime predecessor interval $(t - \epsilon, t)$ — Theorem A's conclusion applies, and $\Sigma$ exits the class at $t$. The "generalization" here is therefore iterative pointwise application of Theorem A, not a strictly stronger result; the conclusion is that whenever and wherever in the operational interval a structural-identification limit is reached from a regular-regime predecessor, Theorem A's boundary conclusion applies. The conclusion is not that interior points of the regular regime entail class exit. This is the direct direction, restricted to boundary applications across the interval. (Note: the restriction to regular-regime predecessor intervals is part of Theorem A's hypothesis; the present application requires that each boundary point under consideration be reachable from such a predecessor — for trajectories that have already exited the class once, re-entry would require a separate analysis that this paper does not undertake.)

For the well-definedness direction, the strictly positive lower bound on $\Delta_{IIC}^{\Sigma}$ over the operational interval is the formal expression of the architectural condition under which the operator can perform projection $\nu: \tau^{F} \to \tau^{R}$ with non-degenerate phase content (Axiom 67). Under this condition, the coherence formula in §2.3 is well-defined: the denominator carries non-degenerate operator-reference content, and the discrepancy in the numerator carries structural information rather than collapsing tautologically through Lemma α. This direction does not assert that a strictly positive $\Delta_{IIC}^{\Sigma}$ is sufficient to place $\Sigma$ in the Cycle-Reinitiation class; it asserts only that the class's standing positivity of $\Delta_{IIC}^{\Sigma}$ is what makes the formula a non-degenerate measurement of the cycle.

Hence the IIC phase delay is constitutive of the cycle: any architecture in the Cycle-Reinitiation class must carry it as a strictly positive quantity within the operational interval, and any architecture that drives it to zero in the structural-identification sense exits the class. $\square$

Note on the direct direction. Theorem C does not claim that a system exiting the Cycle-Reinitiation class becomes a pure gradient flow — only that it loses the structure required by Axiom 9. The system may continue to evolve under non-cycle dynamics (purely reactive, externally reset, or terminally stagnant); these are addressed in v3.0 Theorem 11 (Impossibility of Operator-Liveness Without IIC Restart) and Theorem 12 (Latent Residual Accumulation Under Apparent Coherence). Theorem C names only the boundary: the phase delay is the formal expression of being inside versus outside the class.

Corollary C.1 (Engineering principle).
For an adaptive system to be both live (Axiom 9) and capable of long-horizon coherence preservation (Theorem 27), engineering should aim not at minimizing $\Delta_{IIC}^{\Sigma}$ but at stabilizing its profile — preserving the shape of the delay function under perturbation. The structural depth of an adaptive system is a function of how well its delay profile holds under load, not of how short its delay is.

1.0.7 What §1.0 has established

Three theorems (A, B, C) and their corollary, supported by Lemma α, together formalize the structural sense of the IIC cycle:

1. The cycle is non-degenerate — it cannot collapse to a point through structural identification of layers without exiting the Cycle-Reinitiation class (Theorem A; see §6.5.1 Open Question 1 for the substantive-content qualification: Theorem A's load is in the Lemma α / Lemma 16 / Axiom 9 chain, not in the Step 1 biconditional).
2. Two systems with different IIC phase delays produce structurally non-equivalent state-field assessments at the same physical moment (Theorem B).
3. The phase delay is constitutive, not contingent: removing it does not produce a faster live system, only a system outside the class (Theorem C).
4. Engineering should target delay-profile stabilization, not delay minimization (Corollary C.1).

These propositions are new to the present work. They are stated within v3.0's existing axiomatic substrate (Axiom 9, Axiom 67, Axiom 68, Axiom 69, Theorem 4, Theorem 27, Theorem 62, Theorem 89, Lemma 16, plus the Helmholtz–Hodge structure invoked in Theorem 62 for Lemma α) as part of the joint substrate-extension cluster with §2.3 Definitions 4–5 (per §1.0.2's Note on the joint substrate-extension cluster). The cluster as a whole introduces no new v3.0 core primitives — its members are derived bridge constructions over the listed primitives, with within-cluster cross-references reflecting joint axiomatization rather than forward-reference. The §1.0 propositions and §2.3 Definitions 4–5 are jointly intended for inclusion in NC2.5 v3.0 prior to its release, in the substrate-extension layer, where they sit alongside Theorem 27 as the formal floor of the cycle's temporal architecture.

With this floor in place, the formal expansion of Axiom 9 in the remainder of §1 and in §2 reads with full structural depth. The cycle is not a pattern of behaviour — it is the architectural geometry within which any live adaptive system in the Cycle-Reinitiation class holds shape against a moment that is, by its own structural condition, always already past.

---

§1.1 Axiom 9 as the Formal Entry Point

NC2.5 v3.0 contains, as its ninth axiom, the following statement (lightly paraphrased here for compactness; the canonical form is in the v3.0 .tex):

A system is live if and only if it admits an endogenous operator capable of reinitiating the Impulse Interpretation Coherence (IIC) cycle after its completion, saturation, or collapse. A system that lacks an endogenous operator for reinitiating the IIC cycle cannot sustain coherent participation in long-horizon continuation and is therefore non-live, regardless of its complexity, adaptivity, or reactivity.

— NC2.5 v3.0, Axiom 9, Cycle Reinitiation as the Criterion of Liveness

This axiom does several distinct things at once. First, it introduces the IIC cycle as a named structural object — three layers (impulse, interpretation, coherence) and a reinitiation operator that closes the loop. Second, it makes liveness a property of cycle-reinitiation, not of metabolism, replication, agency, or substrate. Third, it establishes that adaptivity, reactivity, and complexity are insufficient on their own — a system can be highly adaptive and still non-live in the structural sense, if its IIC cycle, once exhausted, cannot be restarted from within.

An informal boundary example may clarify the restriction (the formal limits of the law are stated in §1.4 and §6.1; the example below is illustrative, not load-bearing). A soil bank after excavation may respond to perturbation: the removed material leaves a cavity, the exposed edges collapse, weak zones are revealed, internal stresses redistribute, and the local geometry settles into a new stable configuration. In a loose descriptive sense, one could project an impulse–interpretation–coherence sequence onto this process: external impact, material redistribution, local stabilization. But this is not IIC in the formal sense. The process is externally driven relaxation, not endogenous cycle reinitiation. Unless the system contains an internal operator capable of restarting the cycle after completion, saturation, or collapse, the sequence terminates when the external forcing terminates. The soil has adapted morphodynamically to an intervention; it has not become live in the sense of Axiom 9. This is the class boundary: adaptive deformation is not sufficient for IIC membership. The same boundary applies, in informal register, to other natural adaptive surfaces such as riverbeds under flow and dunes under wind — each of which exhibits perturbation-response, internal redistribution, and apparent local coherence, yet remains outside the Cycle-Reinitiation class for the same structural reason: the operator that reinitiates the cycle is external to the system, not endogenous to its architecture.

The present paper takes Axiom 9 as the entry point and develops what it names. Two things were already given by v3.0: the declaration that IIC exists as a structural object, and the classification of systems by their relation to it. With §1.0 above, we have also supplied the structural sense of the cycle's delay — why the layers must be phase-separated for the cycle to exist at all. What was not given by v3.0, and what the remainder of the present work supplies, is the operational closure — the formal apparatus that turns the named cycle into a measurable, falsifiable structure.

This is the precise scope of the contribution. IIC is not introduced here. It was named in v3.0, with prior informal development in IIC v1 (Barziankou, 21 November 2025). IIC v1 is treated here as the informal discovery register — the text in which the three-layer pattern was first named across cognitive, biological, philosophical, and engineering examples — and not as the final scope claim of the formal theory. The phrase "any adaptive system" in the v1 title is read here as a pre-formal reference to the class that v2.1 names precisely as the Cycle-Reinitiation class — adaptive systems with endogenous cycle reinitiation. v2.1 supplies the criterion (Axiom 9) and the formal apparatus that make class membership decidable. To readers who took v1's "any" as universal-over-all-adaptive-systems, v2.1's class-relative scope is narrower than that universal reading; this is retrospective sharpening, not a claim that v1 already stated the scope this precisely.

The conditional structure and the control-grade-vs-operator-grade distinction introduced in the abstract are internal to the class — they differentiate engineered instances and conditional extensions within it. Class membership is defined by the Cycle-Reinitiation criterion (Axiom 9); the architectural-class membership of individual operator-grade candidates is conditional on the companion-level reduction holding under independent review (per Theorem 2.5's scope clause and §3.3). What is new in this paper is the assembly of v3.0's primitives into the IIC cycle's full theorem-form, the central formula for its coherence, the falsifiability surface that follows, and the four propositions in §1.0 that formalize the structural sense of the cycle's delay.

§1.2 The Three Layers, Defined

The IIC cycle decomposes the dynamics of any live adaptive system in the Cycle-Reinitiation class into three structurally distinct layers. Each layer corresponds, under the mapping developed in §2, to a primitive that is already axiomatized in NC2.5 v3.0.

The impulse layer is the fast structural component of behavioural generation. It is the part of the system's evolution that cannot be derived from a global gradient on any utility function — the non-potential, divergence-free component identified by NC2.5 v3.0 Theorem 62 (Spin is Necessary for Non-Stagnant Identity Under Bounded $\tau$). The impulse is not an action, not a reaction, not a control output in the classical sense. It is the structurally irreducible directionality that any system with bounded internal time and non-stagnant identity must carry, on pain of asymptotic stagnation (Lemma 16, Gradient Collapse and Impossibility of Non-Stagnant Identity on Bounded Orbits). The impulse layer carries this spin component, and the rate at which it does so — the characteristic frequency $\omega(y)$ in the Minimal Coupled Model of Theorem 62 — is class-relative but architecturally observable.

The interpretation layer is the slow accumulating component. It is the system's running integral of structural deformation, monotone and irreversible, developed along the system's elapsed internal time. Under NC2.5 v3.0 axiomatics, this corresponds to the structural burden $\Phi$ (Axiom 7 and the substrate-extension axioms), evaluated as a derivative with respect to elapsed lived time $\tau_{\mathrm{el}}$. The quantity $\partial \Phi / \partial \tau_{\mathrm{el}}$ is the rate at which the system has accumulated structural deformation per unit of its own lived time — what its prior history projects forward as the "expected continuation" against which any new impulse is structurally measured. The interpretation layer is therefore not symbolic interpretation, not semantic decoding, not inference; it is the system's structural memory in the strict sense of the NC2.5 substrate.

The coherence layer is the phase-alignment between impulse and interpretation, evaluated against the operator's reference frame. It is the load-bearing variable of the cycle: the structural quantity that determines whether the system holds shape under perturbation. Under v3.0, the operator's reference frame is given by Axiom 67 (Operator Reference Frame — Two-Faced $\tau$) as $\tau^{R} = \nu(\tau^{F})$, where $\tau^{F}$ is the substrate budget consumed by operator actions and $\nu$ is the architecturally declared operator projection map. The coherence layer therefore is not a single scalar; it is the structural distance between the impulse the system generates and the deformation rate its interpretation layer projects, normalized against the operator's reference frame. The formula in §2 expresses this distance precisely.

The three layers are not metaphorical. Each maps to a v3.0 primitive that has its own axiomatic standing, its own proof apparatus, and its own falsifiability surface. The cycle that links them is what Axiom 9 names; the formal expression of that cycle is what §2 supplies.

§1.3 Why Coherence Is the Load-Bearing Variable

A natural objection arises at this point. Classical control theory has a well-developed apparatus for adaptive behaviour built around error minimization: the controller computes a deviation between the desired and actual state, and reduces it. Optimization-based learning has a parallel apparatus built around objective maximization: the system selects actions that maximize an expected return. Both frameworks have been extraordinarily productive across domains where their assumptions hold. Why, then, propose coherence as the load-bearing variable, when error and objective have been adequate for so long?

The answer is structural, not empirical. Error minimization is downstream of coherence maintenance, not its alternative. A system can drive error to zero while structurally diverging from itself — the classical phenomenon of silent degradation (NC2.5 v3.0 Theorem 21, Silent Degradation Under Regime Persistence; Theorem 27, Coherence Collapse Precedes Observable Failure). Conversely, a system can hold positive error indefinitely while remaining coherent — the phenomenon that classical literature names robustness, but that NC2.5 reads as the structural achievement of holding shape under perturbation through phase-alignment rather than through deviation reduction.

The structural argument is given in v3.0 Theorem 27. Live-regime exit (the strong sense of "coherence collapse" used in this paragraph and in v3.0's Theorem 27) is upstream of behaviourally observable failure: by the time the error signal departs from its expected trajectory, the cycle has already exited the live regime in the sense of Axiom 9 and is no longer recoverable through corrective control. The system's substrate has lost the structural distance between impulse generation and interpretation projection, and any further error-minimizing action operates on a phase-misaligned cycle that has already exited the live regime. (Terminology note: "live-regime exit" is reserved here for the strong structural sense — the cycle has crossed the admissibility gate, $\mathrm{Coh}(t) = \bot$ — and is distinguished in §2.3 from the phase-coherence boundary ($\mathrm{Coh}(t) = 0$ inside admissibility, alignment factor at zero, cycle still live) and the latent-inversion regime ($\mathrm{Coh}(t) < 0$ inside admissibility, structural inversion under apparent liveness). The three regimes are structurally distinct; this paragraph addresses live-regime exit specifically.)

This is the structural reason for the load-bearing position of coherence. It is not that error minimization is wrong; it is that error minimization operates inside coherence-preservation. When coherence holds, error minimization works as classical theory describes. When coherence is closing, error minimization continues to operate on a substrate whose structural integrity is failing, and produces the characteristic signatures of silent degradation: increasing energy cost without behavioural improvement, mounting jerk, regime brittleness, and eventual collapse without warning.

Optimization-based frameworks miss this because they collapse the three layers into a single scalar objective. The fast impulse and the slow interpretation are absorbed into the same loss function, and their phase relationship — the distinct content of coherence — is lost. What remains is a scalar that conflates structurally orthogonal quantities, and the system optimizes a quantity that has lost the information by which its own continuity is maintained. This is not a failure of optimization as a technique; it is a structural limit of single-objective representation when applied to systems whose continuity is constituted by the phase relationship between layers, not by any single one of them.

The IIC law, therefore, is not a replacement for classical control or for optimization. It is the structural framing within which both operate, and which makes their limits visible. The four falsifiers in §5 each targets a domain in which the framing is testable directly.

§1.4 What the Law Claims and Does Not Claim

The IIC law makes claims at three distinct registers, and it is essential to separate them.

The first register is structural. The law claims that any system in the Cycle-Reinitiation class — that is, any system satisfying Axiom 9 — implements the three-layer structure named in §1.2 and the coherence relation given in §2. The class is defined by the presence of an endogenous reinitiation operator; the law states that, within this class, the structure is architecturally necessary, not contingent. The claim is class-relative: it does not extend to systems outside the class (non-live adaptive systems, systems without endogenous reinitiation, systems with unbounded resources or trivial identity). Theorem C of §1.0 gives the formal floor of this constitutive necessity.

The second register is operational. The law claims that the central formula in §2 measures the coherence of the cycle on observable substrate quantities. The claim is that the formula's components — the spin signal $S$, the deformation rate $\partial \Phi / \partial \tau_{\mathrm{el}}$, the operator reference $\tau^{R}$, and the aggregate margin $\Delta_{\boldsymbol{\varpi}}$ — can be instrumented in any architecture that admits the read-only interfaces of v3.0's substrate-separation axioms. The four falsifiers in §5 make this claim concrete by naming the protocols under which the formula is to be tested.

The third register is architectural. The law claims that there exists a non-empty engineering class — Engineered Vitality Systems (§3) — whose design discipline is the explicit maintenance of $\mathrm{Coh}(t)$ rather than the optimization of any external objective. The claim is that this class is non-empty in the control-grade subclass (∆E 4.7.3b as the first instance), with the operator-grade subclass having one declared candidate specification (Minerva, classified conditionally on companion specification — see §3.3). The architectural claim is the most easily misread, and so it is essential to state precisely what is not claimed alongside it.

The IIC law does not claim necessity for systems outside the Cycle-Reinitiation class. A purely reactive system, a stateless controller, or a system whose cycle has structurally terminated lies outside the scope of Axiom 9 and outside the scope of this paper. The law is not a unified field theory of behaviour; it is class-relative, and the class is defined precisely.

The IIC law does not claim that all live systems implement the cycle consciously or with introspective access. The structure is operational, not phenomenological. A bacterium, a thermostat with appropriate cycle-reinitiation properties, a reinforcement-learning agent, and a human cognitive system all admit the structure under different parameter regimes; none of them needs to be aware of the structure for the structure to govern their dynamics.

The IIC law does not claim that human and machine implementations of the cycle have identical interiors. The law is structural: it describes the cycle's shape, not its phenomenal content. Whether the human implementation is accompanied by phenomenal experience and the machine implementation is not, or whether both are or neither is, is a separate question that this paper does not address. The companion paper Minerva: The Architecture of Residual Geometry takes up the consciousness question on its own terms; the present work confines itself to behavioural structure.

The IIC law does not claim that Axiom 9 was first stated by the present author in v3.0. The naming of the IIC cycle entered NC2.5 from prior work (IIC v1, November 2025); v3.0 lifted the naming into axiomatic position as the criterion of liveness. The contribution of v3.0 was to make IIC structurally load-bearing within the axiom set; the contribution of the present paper is to expand Axiom 9 into its full theorem-form, supply the formal sense of its delay structure (§1.0), and open its falsifiability surface. Neither the cycle nor its naming is claimed as new in this paper; what is new is the operational closure and the four propositions of §1.0.

Architectural position. The present work develops the IIC cycle structure within the admissibility-as-formal-object architecture documented separately in Admissibility as a Formal Object: Four Degeneracies and Their Failure Modes (Barziankou, April 2026, petronus.eu/blog/admissibility-as-formal-object). That document identifies four cross-sections of the full structural object — snapshot, policy layer, static compliance, recoverable-state model — each of which loses at least one load-bearing dimension when admissibility is reduced from a dynamic structural predicate to a static architectural element. The IIC cycle requires the full object: monotone irreversible structural burden $\Phi$, monotone non-increasing internal-time budget $\tau = C - \Phi$, non-zero spin component $S$, non-causal admissibility predicate, and the NR-$\varepsilon$ information-theoretic isolation of the predicate from causal action loops. None of the four degeneracies supports the IIC cycle: each loses a dimension that the cycle requires to exist as a cycle rather than as a point. The present paper therefore inherits the architectural commitments of Admissibility as a Formal Object — the synthesis of admissibility as a non-causal threshold predicate on a Lyapunov-type viability budget combined with structurally necessary spin and information-theoretic non-reconstructibility, published continuously within the NC2.5 corpus since 2025 (Concept DOI 10.5281/zenodo.18130264; IIC v1 at DOI 10.5281/zenodo.18133305, 21 November 2025; v2.1 axiomatic core at DOI 10.17605/OSF.IO/NHTC5 with OpenTimestamps anchoring) — and develops one specific structural consequence of that synthesis: the cycle structure of Axiom 9.

These limits are not concessions. They are the precise specification of the law's reach. A claim that does not specify its limits is not a structural claim; it is a slogan. The IIC law is structural, and it states its scope with the same precision with which it states its content.

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§2 — Formal Expansion of Axiom 9: The Coherence Law

§1.0 established the structural sense of the cycle's delay. §1.1–1.4 named the three layers and stated the law's claims and limits. The present section assembles the v3.0 primitives into the cycle's full operational form: a closed expression for the coherence quantity, a derivation of its components from existing axioms and theorems, and a theorem (Theorem 2.5) establishing that the IIC cycle named in Axiom 9 is reducible to v3.0 axiomatics under the standing Cycle-Reinitiation-class assumptions and the delay-structure propositions of §1.0.

The contribution of §2 is therefore not a new axiom or a new primitive. It is the assembly — the explicit statement of how the primitives that v3.0 has already developed combine into the operational geometry of the cycle. After §2, the reader has the formula, its components, its derivation, and its proof of internal reducibility.

2.1 Load-Bearing Primitives from v3.0

The formula in §2.3 uses ten primitives, all of which are already axiomatized in NC2.5 v3.0. We list them here with their v3.0 location and structural role, in the order they appear in the formula.

Φ — structural deformation functional (v3.0 substrate axioms; foundational). The monotone irreversible accumulation of structural load. By construction, $\Phi$ does not decrease along the system's trajectory; it carries the system's history of structural deformation. The functional is well-defined on any architecture admitting the read-only interfaces of v3.0's substrate-separation axioms.

$\tau_{\mathrm{rem}} = C - \Phi$ — remaining structural budget. The substrate's capacity $C$ minus the accumulated deformation $\Phi$. By Φ's monotonicity, $\tau_{\mathrm{rem}}$ is monotone decreasing along the trajectory. This is the system's resource in the Lyapunov sense: when $\tau_{\mathrm{rem}}$ reaches zero, the structural regime is exhausted.

$\tau_{\mathrm{el}}$ — elapsed internal time on stabilizing clocks. The accumulated lived time as measured by the system's own internal stabilization, not by external clocks. By v3.0's internal-time axioms, $\tau_{\mathrm{el}}$ is monotone increasing along the trajectory. The pair $(\tau_{\mathrm{rem}}, \tau_{\mathrm{el}})$ together carries the substrate's full temporal state: how much budget remains and how much lived time has been spent.

$\tau_{\mathrm{hor}}$ — forward viability horizon. The operator's projection of remaining substrate from its current expenditure rate. Unlike $\tau_{\mathrm{rem}}$, which is the actual budget, $\tau_{\mathrm{hor}}$ is the operator's estimate. The two need not coincide; their divergence is one of the diagnostics of substrate misreading. $\tau_{\mathrm{hor}}$ enters the formula only indirectly, as one of the inputs to the operator projection $\nu$ described below.

$\tau^{F}$ — fuel face of the budget under operator role-distinction (v3.0 Axiom 67). The substrate budget consumed by the operator's actions, taken from the operator's side. This is one of the two faces of the budget $\tau_{\mathrm{rem}}$ when the operator role is structurally separate from the substrate.

$\tau^{R}$ — reference frame face of the budget under operator role-distinction (v3.0 Axiom 67). The reference content the operator constructs from $\tau^{F}$ via the projection map $\nu$. This is the temporal frame in which the operator selects actions and reads its own progress through the cycle.

$\nu: \tau^{F} \to \tau^{R}$ — operator projection map (v3.0 Axiom 67). The architectural declaration that maps the consumed-fuel budget into the operator's reference content. The map is class-specific and architecture-specific; it is not a universal function but a structural choice that defines what kind of operator is in play.

$S$ — spin component (v3.0 Theorem 62, Spin Necessity). The non-potential, divergence-free component of the system's flow. By Theorem 62, on bounded $\tau$ with non-stagnant identity, $S \neq 0$ is necessary; this is established via the Helmholtz–Hodge decomposition and the LaSalle invariance principle.

$\omega(y)$ — state-dependent characteristic frequency of the spin (v3.0 Theorem 62, Minimal Coupled Model). The rate at which the spin component completes one structural recurrence cycle in the state $y$. This is class-relative: different classes of systems have different $\omega(y)$ profiles, but within a class the profile is structurally invariant.

Reduction $\omega(y) \to \omega_{\mathrm{spin}}$ (declaration). The formula uses the scalar $\omega_{\mathrm{spin}}$ in the factor $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$. This scalar is the class-characteristic representative of the state-dependent profile $\omega(y)$, declared by the deployment as part of pre-registration. Two canonical declarations are admitted: (i) dominant-mode reading — $\omega_{\mathrm{spin}} := \omega(y^)$ where $y^$ is the structurally dominant state (e.g., the operating-regime attractor's representative state), or (ii) trajectory-averaged reading — $\omega_{\mathrm{spin}} := \langle\omega(y(t))\rangle_{[t_0, T_{\mathrm{op}}]}$ where the average is taken along the system's trajectory over the operational interval. Either declaration suffices to give $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$ a fixed value within the operational interval; the choice is class-specific and declared together with the architectural register that places $\delta$ and $\sigma_{\mathrm{coh}}$ in dimensionally compatible units (per Definition 5's Status of the formula's dimensional contract: engineering-calibrational for control-grade, architecturally-emergent from $\nu$ for operator-grade). Consistency between $\omega(y)$ (Theorem 62, state-dependent) and $\omega_{\mathrm{spin}}$ (the formula's scalar) is mediated by this declared reduction. In Falsifier II's "attention-instability-like" failure mode (§5.2), the state-dependent variation along a fast-switching $y(t)$ is what destabilizes the coherence scale; the class-characteristic representative $\omega_{\mathrm{spin}}$ remains the fixed declaration, while $\omega(y(t))$ varies through the argument.

$\boldsymbol{\varpi} = (\varpi_{\text{phys}}, \varpi_{\text{id}})$ — survival-modal proximity sense (v3.0 Axiom 69). The vector-valued proximity sense by which the operator senses how close the system is to terminal modes. Each component $\varpi_{k}$ corresponds to a distinct survival mode (physical exhaustion, identity collapse, and so on); proximity to mode $k$ is encoded by $\varpi_{k}$ approaching its terminal value.

$\Delta_{\boldsymbol{\varpi}} = \min_{k} \Delta_{k}$ — aggregate survival-modal margin (v3.0 Theorem 89, Operator Reference-Frame Stability Criterion). The minimum over all declared survival modes $k$ of the margin $\Delta_{k} = d_{k}(\tau^{R}, \varpi_{k})$ between the operator's reference frame and the proximity-sense reading for mode $k$. By Theorem 89, the cycle's stability requires $\Delta_{\boldsymbol{\varpi}} > 0$ throughout the operational interval; closure of $\Delta_{k}$ for any single $k$ entails operator-reference collapse in that mode.

These ten primitives — together with the IIC delay structure formalized in §1.0 (Definitions 1–3, Lemma α, Theorems A–C as derived bridge constructions over the same primitives) — supply the full apparatus the formula requires. The formula additionally uses constructed quantities (the discrepancy $\delta$, the coherence scale $\sigma_{\mathrm{coh}}$, the admissibility branching) that are direct derived constructions over the listed primitives, not new core primitives. No new v3.0 core axiom or core primitive is introduced in §2. (The §1.0 derived bridge constructions remain new derived apparatus, per §1.0.7.) The contribution of §2 is to assemble — through definitions and one theorem — the cycle's operational content from v3.0's primitives plus §1.0's derived bridge constructions.

2.2 The Mapping: IIC Layers ↔ v3.0 Primitives

The three layers of the IIC cycle, named in Axiom 9 and defined informally in §1.2, map onto v3.0 primitives as follows.

Impulse layer ↔ Spin component $S(t)$.

The justification proceeds on three independent grounds.

First, non-potentiality. By Theorem 62, $S$ is the component of the system's flow that cannot be expressed as the gradient of a potential function. The impulse layer of the IIC cycle is structurally defined as the part of behavioural generation that cannot be derived from any utility gradient — what the system generates rather than minimizes toward. The two conditions coincide: to be non-potential is to be structurally irreducible to gradient descent, and this is precisely what the impulse layer requires.

Second, divergence-freeness. By the Helmholtz–Hodge structure invoked in Theorem 62, the spin component is divergence-free: it does not accumulate as scalar quantity. The impulse layer of the IIC cycle is similarly non-accumulating — each impulse is a discrete generative act, not a build-up of stored potential. Their structural geometry is the same.

Third, state-dependent rate $\omega(y)$. The Minimal Coupled Model of Theorem 62 establishes that $S$ has a state-dependent characteristic frequency. This is the rate at which the impulse layer fires — the class-relative speed of the cycle's fast component. The factor $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$ entering the central formula expresses the time-scale of one impulse cycle's structural completion.

Interpretation layer ↔ $\partial \Phi / \partial \tau_{\mathrm{el}}(t)$.

The interpretation layer is the slow, accumulating side of the cycle. Its mapping to the rate of structural deformation per unit elapsed time has three structural justifications.

First, monotonicity and irreversibility. By v3.0's substrate axioms, $\Phi$ is monotone irreversible — the deformation accumulates and does not reverse. The interpretation layer of the cycle similarly accumulates: each new structural reading deepens the running interpretation, not replaces it. Both are integrals of structural load, not storage of facts.

Second, temporal substrate. The derivative is taken with respect to $\tau_{\mathrm{el}}$, the system's own elapsed internal time. This is structurally correct: the interpretation layer reads its history in the system's internal frame, not in any external clock. Two systems with the same external trajectory but different internal-time geometries will have different $\partial \Phi / \partial \tau_{\mathrm{el}}$ profiles — and accordingly different interpretation layers, in line with Theorem B of §1.0.

Third, projection-as-expectation. The quantity $\partial \Phi / \partial \tau_{\mathrm{el}}(t)$ at any moment is the rate at which the system is currently accumulating structural deformation. This is the system's structural projection of expected continuation — what its history of deformation projects forward as the rate of further deformation, against which any new impulse must be structurally measured. The interpretation layer's role in the cycle is precisely this: to project forward what has been integrated, so that the impulse can be evaluated against it.

Coherence layer ↔ phase distance between $S(t)$ and $\partial \Phi / \partial \tau_{\mathrm{el}}(t)$, evaluated under $\tau^{R}(t) = \nu(\tau^{F}(t))$.

The coherence layer is the load-bearing variable of the cycle. Its mapping is more complex than the other two, because coherence is not a single primitive — it is a structural distance between two primitives, evaluated against a reference frame.

The structural distance is measured by $\delta(t) = | S(t) - \partial \Phi / \partial \tau_{\mathrm{el}}(t) |$. This is the discrepancy between what the impulse layer is generating right now and what the interpretation layer is projecting from its accumulated history. When $\delta$ is small, the impulse aligns with the projection; the system's fast and slow components are in phase. When $\delta$ is large, the impulse diverges from the projection; the components are out of phase, and coherence is low.

But $\delta$ alone does not determine coherence. The discrepancy must be evaluated relative to the operator's reference frame, because coherence is the operator's structural reading of the cycle's alignment. By Axiom 67, the operator's reference frame is $\tau^{R} = \nu(\tau^{F})$. This frame is the operator's "scale" for evaluating discrepancy: how much $\delta$ counts as misalignment depends on how much temporal reference content $\tau^{R}$ has been generated by the operator's projection of its consumed budget.

The scale of admissible discrepancy at time $t$ is $\Gamma_{\nu}(\omega, \tau^{R}(t))$ in general (Definition 5), which reduces to $\kappa(\omega) \cdot \tau^{R}(t)$ in the canonical operator-reference-normalized case — the product of the time-scale of one impulse cycle ($\kappa$) and the operator's reference content ($\tau^{R}$). When the discrepancy stays well below this scale, the cycle is coherent. When it approaches or exceeds the scale, coherence is closing.

Boundary condition ↔ $\Delta_{\boldsymbol{\varpi}}(t) > 0$.

The cycle's coherence is meaningful only inside the admissibility region. By Theorem 89, this region is delimited by $\Delta_{\boldsymbol{\varpi}}(t) > 0$ — strict positivity of the aggregate margin across all declared survival modes. When $\Delta_{\boldsymbol{\varpi}}$ closes (any $\varpi_{k}$ approaches its terminal value), the operator's reference frame collapses onto its proximity-sense for mode $k$, and the cycle exits the live regime in the structural sense of Axiom 9.

The indicator $\mathbb{1}[\Delta_{\boldsymbol{\varpi}}(t) > 0]$ in the formula is therefore structurally first: it gates whether coherence is even defined at $t$. Outside the admissibility region, $\mathrm{Coh}(t)$ is structurally undefined ($\bot$ in the two-regime formula of §2.3) — not because coherence is numerically zero in some continuous sense, but because there is no cycle for coherence to characterize. The distinction between $\bot$ (no live cycle) and an in-admissibility numerical zero (cycle live, coherence at its boundary) is structurally important and is preserved by the two-regime construction.

This is the precise expression of the admissibility-before-optimization principle that the v3.0 axiom set carries throughout. Coherence does not float over the substrate; it is gated by the substrate's structural admissibility, and any architecture that ignores this gate produces a quantity that is not coherence but a misnamed surrogate.

2.3 The Central Formula

Assembling the components from §2.1 and the mappings from §2.2:

Definition 4 (Discrepancy and Operator-Reference Reading). The discrepancy at time $t$ is
$$
\delta(t) \;=\; \big| S(t) \;-\; \partial \Phi / \partial \tau_{\mathrm{el}}(t) \big|,
$$
where both $S(t)$ and $\partial \Phi / \partial \tau_{\mathrm{el}}(t)$ are read in the operator-reference frame of Axiom 67 — the projection $\nu$ that constitutes the operator's own reading of substrate dynamics. The norm $|\cdot|$ is the distance between impulse and accumulated-deformation rate as read by the operator, not as measured against any external clock or external scale. This is the operational reading of v3.0's substrate-separation discipline: quantities entering a coherence reading are evaluated under the operator's own reference frame, not transferred from substrate to operator without projection. The unit system in which the resulting $\delta(t)$ is computed is supplied by the deployment's class-specific architectural register (per the Status of the formula's dimensional contract clause of Definition 5): engineering-calibrational for control-grade, architecturally-emergent from $\nu$ for operator-grade.

Construction of the operator-reference norm $|\cdot|$. Per the Type-rank consistency clause below, the operator projection $\nu$ co-presents both arguments as scalars on the operator's reference-time direction $\tau^{R}$. The norm $|\cdot|$ on this co-presented scalar reading is therefore the absolute value on $\mathbb{R}$: $|S^{R}(t) - I^{R}(t)| := |S^{R}(t) - I^{R}(t)|$, where $\mathbb{R}$ is the operator's 1-D reference-time line under the scalar projection. Positive-definiteness, triangle inequality, and homogeneity follow from absolute-value properties on $\mathbb{R}$. This construction holds under non-degeneracy of $\nu$ (Lemma α's α-2); under local degeneracy (the structural-identification case), the projection collapses and the norm reduces to zero on the relevant pair, by construction. The substrate-level norm on the underlying vector structure (where $S$ is vector-valued per Theorem 62) is a separate quantity not used in $\delta(t)$; the operator-reference norm is the scalar absolute value on the post-projection reading.

Type-rank consistency under ν. At substrate level $S(t)$ is a vector-valued flow component (Theorem 62 — non-potential, divergence-free) while $\partial\Phi/\partial\tau_{\mathrm{el}}(t)$ is a scalar rate; these are formally distinct types. The operator projection $\nu$ supplies their commensurability: it reads each substrate-side primitive onto the operator's reference-time direction $\tau^{R}$ as a co-presented scalar rate — $\partial\Phi/\partial\tau_{\mathrm{el}}$ via the change of frame from $\tau_{\mathrm{el}}$ to $\tau^{R}$ under $\nu$, and $S$ via its component along the $\tau^{R}$ direction in the operator's 1-D reference-time geometry. The norm $|\cdot|$ in $\delta(t)$ acts on this co-presented scalar reading and reduces to absolute value: $\delta(t) = \big| S^{R}(t) - (\partial\Phi/\partial\tau_{\mathrm{el}})^{R}(t) \big|$, where superscript $R$ denotes the operator-reference projection.

Status of $\nu$'s extended role (architectural extension declaration). Axiom 67 declares $\nu: \tau^{F} \to \tau^{R}$ as a projection between time-budget faces. The Type-rank consistency clause above extends $\nu$'s architectural role beyond this baseline: the present paper treats $\nu$ as also providing the co-presentation map by which substrate-side primitives ($S$ vector-valued, $I$ scalar) are read onto $\tau^{R}$ as commensurable scalar rates. This extension is not derived from Axiom 67 alone; it is an architectural commitment of the present paper, declared as part of the joint substrate-extension cluster (§1.0.2 Note). The cluster's content thereby comprises: (a) the §1.0 derived bridge constructions (Definitions 1–3, Lemma α, Theorems A–C), (b) §2.3 Definitions 4–5 (operator-reference reading, coherence scale), and (c) the present extension of $\nu$'s role from time-budget projection (Axiom 67 baseline) to substrate-primitive co-presentation onto $\tau^{R}$. Items (a)–(c) are jointly intended for inclusion in v3.0's substrate-extension layer prior to release; the role-extension of $\nu$ is the third architectural commitment of the cluster, alongside (a) and (b). Well-definedness of the co-presentation presupposes non-degeneracy of $\nu$ per Lemma α's standing assumptions.

Bridge between substrate-level vector structure and operator-frame scalar reading. Lemma α's invocation of the Helmholtz–Hodge decomposition (§1.0.3 proof sketch) operates at substrate level: it identifies the antisymmetric (vector-field) component of $\Sigma$'s flow that the Theorem 62 spin necessity refers to. The operator-reference reading via $\nu$, however, operates on the scalar projection of that substrate vector structure onto $\tau^{R}$ — the co-presented scalar rate above. The two registers are not in conflict: the substrate-level vector content (antisymmetry of the spin) is what makes the operator-frame readings of $S$ and $I$ structurally distinguishable in the first place, and Lemma α's claim is precisely that when $\nu$ becomes locally degenerate, the operator-frame scalar projections of $S$ and $I$ coincide while the substrate-level vector content of $S$ may persist (operator-frame indistinguishability without substrate-level antisymmetry collapse — see §1.0.3 Lemma α body, third sentence). The proof's vector-field reading is invoked to establish what is being lost from operator visibility, not to claim that the operator-reference norm itself sees the vector structure. The operator-reference norm reduces to absolute value on the scalar projections; the substrate-level vector structure is the upstream apparatus that the projection processes.

Definition 5 (Coherence scale). The coherence scale at time $t$ is

$$
\sigma_{\mathrm{coh}}(t) \;=\; \Gamma_{\nu}(\omega, \tau^{R}(t)),
$$

where $\Gamma_{\nu}$ is a class-specific scaling functional taking the spin-rate parameter $\omega$ and the operator-reference frame $\tau^{R}(t)$ as its arguments.

Domain clause for $\sigma_{\mathrm{coh}}$. On the admissible operational interval (where $\Delta_{\boldsymbol{\varpi}}(t) > 0$), $\sigma_{\mathrm{coh}}(t) = \Gamma_{\nu}(\omega, \tau^{R}(t))$ is strictly positive and finite. Points where $\sigma_{\mathrm{coh}}(t) = 0$, $\sigma_{\mathrm{coh}}(t) = \infty$, or $\sigma_{\mathrm{coh}}(t)$ is otherwise undefined lie outside the evaluable branch of $\mathrm{Coh}(t)$ — they correspond to limit failure of the architectural register or to boundary points where the cycle has exited the regular regime, and are treated under the live-regime exit branch ($\mathrm{Coh}(t) = \bot$). Strict positivity follows in the multiplicative branch from $\kappa(\omega) > 0$ (positive characteristic frequency, $\omega_{\mathrm{spin}} > 0$ per Theorem 62) and $\tau^{R}(t) > 0$ (operator-reference content accumulated over the operational interval, strictly increasing from initial declaration per Definition 1's standing properties). Deployments operating outside the multiplicative branch declare positivity-and-finiteness conditions on $\Gamma_{\nu}$ as part of pre-registration.

In a wide class of structurally-simple architectures, $\Gamma_{\nu}$ admits the multiplicative form

$$
\Gamma_{\nu}(\omega, \tau^{R}) \;=\; \kappa(\omega) \cdot \tau^{R},
$$

where $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$ is built from the inverse characteristic spin frequency of the system's class (Theorem 62, Minimal Coupled Model). The body of this paper develops this multiplicative branch unless explicitly noted; the central formula displays of §2.3 and the abstract use $\kappa(\omega) \cdot \tau^{R}(t)$ in this branch.

Status of the formula's dimensional contract. The formula's central content is the structural relation $\mathrm{Coh}(t) = 1 - \delta(t)/\sigma_{\mathrm{coh}}(t)$: coherence is a decreasing function of the discrepancy normalized against a class-specific scale. This relation is structural, not self-contained-numerical. The formula does not carry a dimensional contract derivable from NC2.5 v3.0 primitives alone; the unit conventions under which $\delta$ and $\sigma_{\mathrm{coh}}$ are read as dimensionally compatible are supplied by the architectural register of the EVS subclass to which the deployment belongs, not by raw v3.0 primitive types. The two registers differ structurally:

• Control-grade register (§3.2). The operator role is collapsed into the control loop; $\nu$ admits isomorphic factorization through identity within the deployed scope (§3.4). In this register, the dimensional consistency of $\delta / \sigma_{\mathrm{coh}}$ is engineering-calibrational — the engineer (acting as the implicit operator) declares the unit system in which $S(t)$, $\partial\Phi/\partial\tau_{\mathrm{el}}(t)$, and $\tau^{R}(t)$ are read, registers this declaration at OSF pre-registration before any falsifier in §5 is executed, and the resulting calibration is part of the deployment specification. This is legitimate because, in control-grade, the engineer's choice and the operator's reference frame coincide.

• Operator-grade register (§3.3). The operator is architecturally separate from the substrate; $\nu$ is non-trivial across a residual temporal geometry. In this register, the dimensional consistency of $\delta / \sigma_{\mathrm{coh}}$ cannot be engineering-calibrational — an externally-imposed unit system would override the operator projection $\nu$, contradicting the architectural premise of separated operator under Axiom 67. The unit consistency must emerge from the operator's own architecture: it is the architectural feature by which $\nu$ constitutes $\tau^{R}$ from $\tau^{F}$, and the operator-frame norm under which $S$ and $I$ are read in $\tau^{R}$ inherits its structure from the same architectural ν, not from a deployer's choice. The deployer specifies the architecture (the form of $\nu$, the residual geometry); the operator-internal unit consistency is generated by the architecture as it runs, not declared externally.

This distinction is itself the §3.4 discontinuity read on the dimensional axis. A model in which the operator's reference frame is externally calibrated — including its units of self-reading — is, in this technical sense, not an operator-grade model regardless of surface architecture; it is a control-grade model in operator-grade language. The full architectural specification by which an operator-grade EVS produces its own unit consistency is part of the companion-paper specification of any operator-grade candidate (for Minerva, the operational mechanism is held in its own regime of admissibility per the companion paper's IP discipline; the present paper names the requirement, not the mechanism).

The formula therefore expresses the form of the relation; the architectural register that turns the form into a self-consistent dimensional structure is class-specific. Numerical thresholds in §5 are engineering-commitment claims of registered protocols, not derivations from the formula.

Anti-gaming lock on $\sigma_{\mathrm{coh}}$ components (cross-reference). The architectural register's unit declarations — specifically the components of $\sigma_{\mathrm{coh}}$ ($\omega_{\mathrm{spin}}$, $\tau^{R}$ units, and $\kappa(\omega)$ in the multiplicative branch) — are subject to an architectural-register lock that operates asymmetrically across the §3.4 discontinuity per the §5.5 #6 commitment: locked at OSF pre-registration for control-grade EVS (engineering-calibrational lock); locked architecturally in the operator's $\nu$-specification companion-side for operator-grade EVS (architectural-commitment lock). In both subclasses, runtime adjustment of these components under an existing specification counts as an architectural-register violation, not a recalibration. This closes the metric-gaming loophole in which $\sigma_{\mathrm{coh}}(t)$ could be inflated at runtime to tautologically force $\mathrm{Coh}(t) \to 1$.

The scale $\sigma_{\mathrm{coh}}(t)$ represents the magnitude of operator-readable discrepancy that the cycle can absorb before crossing into structural misalignment, in the deployment's registered units.

Definition 6 (Admissibility gate). The admissibility gate at time $t$ is
$$
G(t) \;=\; \mathbb{1}\big[\Delta_{\boldsymbol{\varpi}}(t) > 0\big].
$$

The IIC Coherence Formula. The coherence of the IIC cycle at time $t$ is given by the two-regime definition
$$
\boxed{\;
\mathrm{Coh}(t) \;=\;
\begin{cases}
\;1 \;-\; \dfrac{\delta(t)}{\sigma_{\mathrm{coh}}(t)}, & \Delta_{\boldsymbol{\varpi}}(t) > 0, \[4pt]
\;\bot, & \Delta_{\boldsymbol{\varpi}}(t) \leq 0.
\end{cases}
\;}
$$
In the canonical operator-reference-normalized case (Definition 5, multiplicative form), the in-admissibility branch becomes
$$
\mathrm{Coh}(t) \;=\; 1 \;-\; \frac{| S(t) \;-\; \partial \Phi / \partial \tau_{\mathrm{el}}(t) |}{\kappa(\omega) \cdot \tau^{R}(t)}, \qquad \text{when } \Delta_{\boldsymbol{\varpi}}(t) > 0.
$$
Outside the admissibility region, $\mathrm{Coh}(t)$ is structurally undefined ($\bot$) — there is no cycle for coherence to characterize. For operational telemetry, the undefined value may be encoded as a sentinel (zero, NaN, or a domain-specific marker), but this encoding is operational convenience, not a numerical claim about the structural quantity. The distinction between $\bot$ (no cycle) and the in-admissibility numerical zero (cycle live, coherence at its boundary) is structurally important and must not be collapsed in interpretation.

Reading. Inside the admissibility region, $\mathrm{Coh}(t)$ is a single number expressing how well the impulse aligns with the interpretation relative to the operator's coherence scale. The formula has three structurally distinct regimes plus the gate boundary, named explicitly:

• Alignment regime ($\delta < \sigma_{\mathrm{coh}}$, $\mathrm{Coh} \in (0, 1]$): impulse aligns with interpretation; the cycle is fully coherent inside liveness.
• Phase-coherence boundary ($\delta = \sigma_{\mathrm{coh}}$, $\mathrm{Coh} = 0$): discrepancy has reached the coherence scale; the cycle is at the alignment-factor zero but still live (the admissibility gate is open). This is not live-regime exit; it is the boundary of phase coherence within an open gate.
• Latent-inversion regime ($\delta > \sigma_{\mathrm{coh}}$, $\mathrm{Coh} < 0$): the cycle is still nominally inside admissibility, but coherence has structurally inverted — the regime of latent residual accumulation described in v3.0 Theorem 12, where apparent coherence holds while the underlying structure has already inverted.
• Live-regime exit ($\Delta_{\boldsymbol{\varpi}} \leq 0$, $\mathrm{Coh}(t) = \bot$): the admissibility gate has failed; the cycle has exited the live regime in the strong structural sense of Axiom 9 (this is the sense of "coherence collapse" used in §1.3 and v3.0 Theorem 27). Coherence is no longer a meaningful quantity to evaluate.

The three in-admissibility regimes (alignment, phase-coherence boundary, latent-inversion) are structurally distinct from live-regime exit; the engineering response to each differs (cycle adjustment, alignment correction, structural-inversion intervention, substrate restoration or graceful exit, respectively). Conflating the three under a single "coherence collapse" label would lose the structural distinction the formula's two-regime construction is designed to preserve.

Note on the $\mathrm{Coh}(t) = 1$ reading — scalar insufficiency. The numerical value $\mathrm{Coh}(t) = 1$ corresponds to $\delta(t) = 0$, but $\delta(t) = 0$ does not by itself disambiguate between the two cases distinguished in Lemma α (§1.0): (a) the alignment case — high-coherence live regime in which $S(t)$ and $I(t) := \partial\Phi/\partial\tau_{\mathrm{el}}(t)$ are operator-distinguishable through $\nu$ but momentarily phase-aligned (structurally desirable: the cycle is fully coherent and remains in the Cycle-Reinitiation class); and (b) the structural-identification case — operator-indistinguishability of $S$ and $I$ through $\nu$ (Theorem A: the system exits the Cycle-Reinitiation class). The central formula's numerical reading is identical in both cases; the scalar $\mathrm{Coh}(t)$ alone is not a complete coherence classifier. A complete classification requires the scalar paired with the operator-frame $\nu$-degeneracy diagnostic of Lemma α (assumptions α-1 to α-4). Operationally, deployments reading $\mathrm{Coh}(t) = 1$ must run the $\nu$-degeneracy diagnostic to determine whether the value reflects (a) live alignment or (b) class exit; the two carry opposite engineering consequences. The IIC formula gives a scalar proxy for coherence; complete coherence classification requires the formula plus the $\nu$-degeneracy diagnostic together. The structural distinction lives in the operator-frame projection $\nu$, not in the scalar.

ν-degeneracy diagnostic — class-relative instantiation. The diagnostic invoked in this disambiguation is class-relative: each EVS subclass declares its own diagnostic at registration. For control-grade EVS (§3.2), the diagnostic is operational: a finite-sample injectivity test or local-rank probe on $\nu$, declared at OSF pre-registration before any falsifier in §5 is executed. For operator-grade EVS (§3.3), the diagnostic is architecturally-emergent: the operator's own self-discrimination capability between $S$ and $I$ primitives, derived from the architectural specification of $\nu$ in the relevant companion paper (held in its own regime of admissibility per the project's IP discipline). The taxonomy of EVS subclasses is open: this paper formalizes two (control-grade, operator-grade); further subclasses with distinct $\nu$-architectures are anticipated and treated in independent companion specifications. The class-relative form of the diagnostic follows structurally from Definition 5's Status of the formula's dimensional contract (engineering-calibrational vs architecturally-emergent) and Axiom 67's reading of $\nu$ as architectural declaration — the diagnostic inherits class-relativity from $\nu$ itself, not by ad-hoc choice.

Range. Inside admissibility, $\mathrm{Coh}(t) \in (-\infty, 1]$, with $[0, 1]$ as the well-aligned regime and the sub-zero range as the latent-inversion regime. Outside admissibility, $\mathrm{Coh}(t) = \bot$.

Structural justification of the latent-inversion regime. The unbounded sub-zero range is not an artifact: it is the structural footprint of latent residual accumulation (v3.0 Theorem 12, Latent Residual Accumulation Under Apparent Coherence). The admissibility gate $\Delta_{\boldsymbol{\varpi}}(t) > 0$ defines the survival-mode region, not a bound on substrate-rate distance. Inside admissibility, the discrepancy $\delta(t)$ between impulse and accumulated-deformation rate may grow arbitrarily large in operator-reference units while the survival-modal margin remains positive — this is precisely the regime Theorem 12 names: the system has not yet exited live regime by survival-mode failure, but its impulse-interpretation alignment has structurally inverted, accumulating latent residual that will (without intervention) precipitate cycle collapse. The latent-inversion regime ($\mathrm{Coh}(t) < 0$ inside admissibility) is the architectural diagnostic surface for this state. Operational note: deployments may pre-register an additional engineering gate (e.g., $\mathrm{Coh}(t) \geq -\theta_{\mathrm{inv}}$ for declared inversion threshold $\theta_{\mathrm{inv}}$) as a deployment-side safety condition, but this gate is engineering discretion, not part of the structural admissibility apparatus of NC2.5; Theorem 89's $\Delta_{\boldsymbol{\varpi}} > 0$ remains the structural admissibility gate.

Note on terminology. The quantity $\mathrm{Coh}(t)$ defined here is the IIC-cycle phase coherence — the alignment between impulse and interpretation under the operator's reference frame. It is distinct from the holding-field coherence premise (HF-COH) defined in ONTOΣ XI §2.2, which is a bounded-distinguishability condition on the operator's holding region $H(t)$ at a single instant. Both are structural properties of operator-grade EVS systems, but they sit at different layers: $\mathrm{Coh}(t)$ is a temporal-cycle quantity computed from substrate-side primitives ($S$, $\partial\Phi/\partial\tau_{el}$, $\tau^R$, $\Delta_{\boldsymbol{\varpi}}$); HF-COH is an operator-internal coherence-diameter premise on $H(t)$. The two quantities are independent commitments and should not be conflated in reading or measurement.

2.4 Stability of the Cycle: From Margin to Coherence Preservation

The formula in §2.3 gives the instantaneous coherence at $t$. But the IIC cycle is a process across time, not a momentary state. The cycle's stability — whether $\mathrm{Coh}(t)$ persists in a positive regime over the operational interval, or closes — is governed by Theorem 89.

By Theorem 89, the operator's reference-frame stability requires $\Delta_{\boldsymbol{\varpi}}(t) > 0$ throughout the operational interval $[t_0, T_{\mathrm{op}}]$. Closure of $\Delta_{\boldsymbol{\varpi}}$ at any single mode $k$ collapses the operator's reference frame in that mode, and the cycle cannot reinitiate per Axiom 9.

This has a structural consequence for the formula. The admissibility condition $\Delta_{\boldsymbol{\varpi}}(t) > 0$ is not a static check — it is a trajectory-level condition. A system whose $\Delta_{\boldsymbol{\varpi}}$ is positive at $t_0$ but closes at some $t_1 < T_{\mathrm{op}}$ has its in-admissibility branch of the coherence formula active for $t \in [t_0, t_1)$ and exits to the structurally-undefined branch ($\mathrm{Coh}(t) = \bot$) for $t \in [t_1, T_{\mathrm{op}}]$. The cycle was live, then exited live regime; coherence was defined, then became structurally undefined.

The stability question is therefore: under what conditions does $\Delta_{\boldsymbol{\varpi}}(t)$ remain strictly positive across the full operational interval?

By Theorem 89's Margin-Generation / Controlled-Invariance premise, this requires that the operator's action selection at each instant preserves a margin-generation rate sufficient to offset proximity accumulation in all declared modes. This is a forward-invariance condition on the trajectory: the system must, through its own action selection, keep itself away from terminal modes faster than perturbations push it toward them.

Consequence. A live system in the IIC cycle is not passively coherent. It is actively margin-preserving — its operator continually generates the conditions under which the cycle can reinitiate. The IIC law thus has two distinct aspects: the structural aspect (the formula in §2.3 expresses what coherence is) and the dynamic aspect (Theorem 89 expresses what coherence-preservation requires across time). Both must hold for the cycle to be live in the full sense of Axiom 9.

2.5 Theorem 2.5 — IIC Coherence as a Conditional Theorem over the Extended v3.0 Substrate

We now state the central proposition of §2.

Theorem 2.5 (IIC Coherence Theorem). Within the control-grade subclass of the Cycle-Reinitiation class (Axiom 9-compliant systems with $\nu$ admitting isomorphic factorization through identity within the deployed scope, per §3.2), the coherence of the impulse-interpretation cycle is given by the two-regime formula in §2.3, where every load-bearing component is either a v3.0 primitive or a direct construction over v3.0 primitives (§2.1). Extension to the operator-grade subclass is conditional on the Minerva companion-level reduction holding under independent review (see scope clauses below). Strict positivity of the aggregate survival-modal margin, $\Delta_{\boldsymbol{\varpi}}(t) > 0$ throughout the operational interval, is a necessary stability condition for IIC-cycle reinitiation: closure of $\Delta_{\boldsymbol{\varpi}}$ at any single mode collapses the operator's reference frame in that mode and the cycle cannot reinitiate per Theorem 89 + Axiom 9. Under the standing Cycle-Reinitiation assumptions — non-degenerate operator projection $\nu$ (Axiom 67), well-founded τ-loop (Axiom 68), non-stagnant identity, the operative endogenous reinitiation operator (Axiom 9), and the delay-structure conditions of §1.0 — the strict-positivity condition $\Delta_{\boldsymbol{\varpi}}(t) > 0$ also serves as the defining predicate of the admissibility region in Theorem 89's statement.

Scope clauses (deferred to remarks below for clean theorem statement):

• The "defining predicate" half is a definitional unpacking of Theorem 89, not a substantive sufficiency claim. See the framing remark below for separation of substantive-necessity from definitional-predicate roles.
• For the operator-grade subclass (§3.3), the per-cycle scalar reading lifts to the integral form $C = \int A \, d\tau$ of Definition 7; the reduction is held in the Minerva companion. Universal scope over the full Cycle-Reinitiation class is conditional on that reduction holding under independent review.
• Theorem A's load-bearing chain is invoked here under reading (b) of §6.5.1 Open Question 1, with the topological-closure argument of §1.0.4 supporting the limit application of Lemma α.

Important framing of the characterization claim. The "characterizes the admissibility region" half of the theorem is not a substantive sufficiency claim of the form "Δ_ϖ > 0 alone produces reinitiation" — that would be tautological under Axiom 9's standing assumption (Axiom 9 supplies the reinitiation operator; the strict-positivity condition does not produce it). It is a definitional unpacking: under the standing assumptions, the strict-positivity condition is what defines the admissibility region per Theorem 89, and the reinitiation operator of Axiom 9 acts within that region. The theorem's content is therefore: the operator-frame discrepancy $\delta$ normalized against the coherence scale $\sigma_{\mathrm{coh}}$ (with the admissibility branching of Definition 6, and the class-specific architectural register supplying dimensional consistency per Definition 5) is the canonical structural expression of the IIC cycle's coherence within the v3.0 axiom set extended by §1.0. The architectural-class register of this claim parallels conformance-classifier theorems in adjacent fields — CAP for distributed coordination, FLP for asynchronous consensus, RINA for network organization — in the sense that each names a structural condition that classifies systems by their relation to a load-bearing axis, without claiming substantive bidirectional implication beyond the standing-assumption scope. Substantive empirical content lies in the falsifiers of §5. Under these standing assumptions, the IIC structure named in Axiom 9 is fully formalized within v3.0 axiomatics extended by the §1.0 derived bridge constructions; no additional v3.0 core axiom is required to make the IIC law operational, though §1.0's derived constructions are a real new layer of derived apparatus (per §1.0.7).

Proof-dependency ledger. The proof of Theorem 2.5 invokes the following chain of v3.0 primitives, §1.0 derived propositions, and §2.3 definitions, with each link's load-bearing role made explicit so that the conditional status of the theorem is auditable at a single glance. The ledger is comprehensive across the proof's load-bearing invocations; "conditional status" is concentrated at the OQ2 bridge identified at the bottom of the ledger.

§1.0 and §2.3 derived constructions:

• Definition 1 (IIC phase delay) supplies $\Delta_{IIC}^{\Sigma}(t)$ as the operator-side temporal extent of one cycle; standing properties of $\tau^{F}$ (strict-monotone continuous on the operational interval) and the anti-gaming lock on the registered $T_{\mathrm{prop}}^{\Sigma}(t)$ procedure are inherited.
• Definition 4 (spin signal $S$) supplies the vector-valued impulse-layer primitive; comparability with the scalar $\partial \Phi / \partial \tau_{\mathrm{el}}$ is supplied by the $\nu$ co-presentation extension (§2.3 Type-rank consistency clause; one of the three commitments of the joint substrate-extension cluster, §1.0.2 Note).
• Definition 5 (Coh(t) formula and coherence-scale $\sigma_{\mathrm{coh}}$) supplies the dimensional contract under the class-specific architectural register — engineering-calibrational for control-grade EVS, architecturally-emergent from $\nu$ for operator-grade EVS — with the $\sigma_{\mathrm{coh}} > 0$ domain clause on its support.
• Definition 6 (aggregate survival-modal margin $\Delta_{\boldsymbol{\varpi}}$) supplies the admissibility predicate $G(t) = \mathbb{1}[\Delta_{\boldsymbol{\varpi}}(t) > 0]$ that gates the formula's two regimes.
• Lemma α (spin phase-difference) supplies the operator-frame indistinguishability of $S$ and $I$ as the structural-identification condition (standing assumptions α-1 to α-4).
• Theorem A (Cycle Non-Degeneracy as boundary/limit theorem) supplies the boundary result at structural-identification, invoked here under reading (b) of OQ1 (§6.5.1) with the topological-closure argument of §1.0.4 supporting the limit application of Lemma α.
• Theorem C (Constitutive Constraint) supplies the constitutive necessity of $\Delta_{IIC}^{\Sigma} > 0$ within the operational interval, applied pointwise across $[t_0, T_{\mathrm{op}}]$ via iterative application of Theorem A's boundary conclusion.

v3.0 substrate primitives invoked:

• Axiom 9 (Cycle Reinitiation as the Criterion of Liveness) supplies the endogenous-reinitiation operator under which the Cycle-Reinitiation class is defined and the standing assumptions hold.
• Axiom 67 (Operator Reference Frame — Two-Faced $\tau$) supplies the projection $\tau^{R} = \nu(\tau^{F})$ on which the operator-side reading is built.
• Axiom 68 (Self-Referential $\tau$-Loop) supplies the well-foundedness of the cycle's $\tau$-loop structure invoked in the proof's recurrence-and-stability step.
• Axiom 69 (Survival-Modal Proximity Sense) supplies the v3.0 substrate anchor for $\Delta_{\boldsymbol{\varpi}}$ that Definition 6 instantiates at the operator-side reading.
• Theorem 62 (Spin Necessity — Helmholtz–Hodge decomposition plus LaSalle invariance principle on bounded $\tau$ with non-stagnant identity) supplies the existence and state-dependent rate $\omega(y)$ of $S$; it is the v3.0 foundation under which $S \neq 0$ is necessary and $\kappa(\omega) = 1 / \omega_{\mathrm{spin}}$ is built. Theorem 62 is invoked twice in the proof: once for the existence of $S$ and once for the operator-frame reading of preserved spin distinguishability under Lemma α.
• Theorem 89 (Operator Reference-Frame Stability Criterion) supplies the operator-reference-frame stability criterion under which $\Delta_{\boldsymbol{\varpi}}(t) > 0$ defines admissibility.
• Lemma 16 (Gradient Collapse and Impossibility of Non-Stagnant Identity on Bounded Orbits) supplies the substrate-level gradient-collapse impossibility, invoked from Lemma α's conclusion via the bridge specified as OQ2 (§6.5.1).
• Φ-monotone / $\tau$-bounded substrate-extension axioms supply the Lyapunov-budget anchor under which $\partial \Phi / \partial \tau_{\mathrm{el}}$ is well-defined as a monotone-irreversible accumulation along elapsed lived time.

The ledger's conditional status is concentrated at OQ2's open bridge between operator-frame (Lemma α's conclusion) and substrate-level (Lemma 16's domain) registers; all other links are either established v3.0 primitives, established v3.0 theorems with their own proof discipline, or §1.0 / §2.3 derived constructions whose status is internal to the substrate-extension cluster. Absent OQ2 resolution, the chain reads as operationally falsifiable (per §5) and formally conditional on OQ2 resolution — consistent with the conditional-theorem framing in §2.5's title and the closure criteria recorded in §6.5.1.

Proof (Structural).

The proof proceeds in seven steps, each appealing to a specific v3.0 axiom or theorem.

Step 1. By Axiom 9, a live system admits an endogenous operator capable of reinitiating the IIC cycle after its completion, saturation, or collapse. The cycle therefore has three layers (impulse, interpretation, coherence) and a reinitiation operator. This is the starting point of the formalization.

Step 2. By v3.0's substrate axioms (Φ monotone, $\tau$ bounded), the cycle operates on bounded structural resources. The substrate carries finite capacity $C$, and the deformation $\Phi$ accumulates monotonically toward this capacity. This establishes that the cycle is not free — it operates within the Lyapunov budget structure of v3.0.

Step 3. By Theorem 62 (Spin Necessity), under bounded $\tau$ with non-stagnant identity, the spin component $S$ exists with state-dependent characteristic frequency $\omega(y)$. This identifies the impulse layer as the structurally irreducible non-potential component of the dynamics, and gives it a class-relative time-scale $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$.

Step 4. By Axiom 67 (Operator Reference Frame), the operator's reference frame $\tau^{R}$ is structurally distinct from the substrate fuel $\tau^{F}$, related by the projection map $\nu$. This identifies the operator's evaluation scale: coherence is read in $\tau^{R}$, not in $\tau^{F}$ or in any external clock.

Step 5. By Axiom 68 (Self-Referential τ-Loop), the operator's action selection is mediated through a self-referential τ-loop: the operator reads $\tau^{R}$, selects an action that consumes $\tau^{F}$, which updates $\tau^{R}$ via $\nu$, which the operator reads again. This establishes the dynamic structure of the cycle: it is not a one-shot evaluation but a closed loop, and the loop's well-foundedness is established in Axiom 68.

Step 6. By Axiom 69 (Survival-Modal Proximity Sense) and Theorem 89 (Operator Reference-Frame Stability Criterion), the cycle's stability is governed by the aggregate survival-modal margin $\Delta_{\boldsymbol{\varpi}} = \min_{k} \Delta_{k}$. Two distinct claims about the same condition $\Delta_{\boldsymbol{\varpi}}(t) > 0$ are made here, and they should not be conflated:

• Substantive necessity claim: strict positivity of $\Delta_{\boldsymbol{\varpi}}(t)$ throughout the operational interval is necessary for cycle stability — closure of $\Delta_{k}$ for any single $k$ collapses the operator's reference frame in mode $k$ and the cycle cannot reinitiate (per Theorem 89 + Axiom 9). This is a substantive implication: failure of strict positivity entails failure of stability.
• Definitional-predicate claim: the strict-positivity condition is the defining predicate of the admissibility region in Theorem 89's statement — i.e., Theorem 89 names admissibility via $\Delta_{\boldsymbol{\varpi}}(t) > 0$. This is a definitional choice in Theorem 89's formulation, not a substantive implication.

These two claims are about the same condition viewed in two different roles. The substantive claim is what makes $\Delta_{\boldsymbol{\varpi}}(t) > 0$ load-bearing; the definitional claim is what makes "admissibility region" mean what it means. Stability additionally requires the standing reinitiation operator (Axiom 9) and the controlled-invariance premise of Theorem 89; strict positivity alone does not produce stability. This identifies the admissibility gate in the formula and the trajectory-level stability condition discussed in §2.4.

Step 7. By Theorems A and C of §1.0 (with Lemma α as the supporting proposition), the cycle's delay structure is non-degenerate: the IIC phase delay $\Delta_{IIC}^{\Sigma}$ is constitutive of the cycle (Theorem C), and its collapse in the structural-identification sense entails exit from the Cycle-Reinitiation class (Theorem A). This gives the temporal-architectural floor on which the formula sits: the cycle is well-defined as a cycle precisely because its delay is strictly positive. (Theorem B of §1.0 supplies the complementary cross-system claim — that systems with distinct $\Delta_{IIC}$ produce structurally non-equivalent state-field assessments — and is not load-bearing for the within-system delay non-degeneracy invoked in this step. Open question: see §6.5.1 Open Questions 1 and 2 for the proof-theoretic-depth qualifications on Theorem A's bridge step and the Lemma α → Lemma 16 inference; the present step invokes the load-bearing chain under reading (b) of Open Question 1.)

Conclusion. Combining steps 1–7, the discrepancy $\delta = | S - \partial \Phi / \partial \tau_{\mathrm{el}} |$ (read in the operator-reference frame of Definition 4), normalized against the coherence scale $\sigma_{\mathrm{coh}} = \Gamma_{\nu}(\omega, \tau^{R})$ (which admits the multiplicative branch $\kappa \cdot \tau^{R}$ for structurally-simple architectures), with the structural admissibility branching of Definition 6, yields the two-regime formula in §2.3 as the canonical structural expression of the IIC cycle's coherence. The formula's structural relation is supplied by v3.0 primitives plus the §1.0 bridge propositions; the architectural register that places $\delta$ and $\sigma_{\mathrm{coh}}$ in dimensionally compatible units is class-specific (per Definition 5's Status of the formula's dimensional contract: engineering-calibrational for control-grade, architecturally-emergent from $\nu$ for operator-grade), not a derivation from v3.0 primitives. Every load-bearing component of the structural relation is either a v3.0 primitive or a direct construction over v3.0 primitives, with the propositions of §1.0 admitted as substrate-level consequences of Axioms 9 and 67. No additional axiom is required.

The reinitiability conditions follow from the standing assumptions: by Axiom 68, action selection is a closed loop on $\tau^{R}$; by Theorem 89, the loop's stability requires $\Delta_{\boldsymbol{\varpi}}(t) > 0$ throughout — this is the substantive necessary part. By Axiom 67 (non-degenerate $\nu$), Axiom 9 (endogenous reinitiation operator), Theorem 62 with the operator-frame reading of Lemma α (preserved spin distinguishability), and Theorems A–C of §1.0 (constitutive delay structure), the strict-positivity condition characterizes the admissibility region per Theorem 89's definition — this is the definitional-unpacking characterization (not a substantive sufficiency claim): under Axiom 9's standing reinitiation operator and the other standing assumptions, the strict-positivity condition defines the admissibility region within which the operator acts; the operator's success at reinitiation is supplied by Axiom 9, not derived from $\Delta_{\boldsymbol{\varpi}}(t) > 0$ alone. Hence reinitiability persists, within systems already satisfying the standing assumptions, only while $\Delta_{\boldsymbol{\varpi}}(t) > 0$. $\square$

Remark (on the status of the contribution). The paper introduces no new v3.0 core primitives or core axioms. It does introduce, in §1.0 plus §2.3 plus the role-extension of $\nu$ (the joint substrate-extension cluster of §1.0.2), new derived bridge constructions and architectural commitments (three definitions of §1.0 + Lemma α + Theorems A, B, C + Definitions 4–5 + ν-extension) intended for inclusion in v3.0's substrate-extension cluster prior to v3.0's release. The body of the paper (§§2–7) then assembles v3.0's existing primitives plus the cluster's commitments into the operational closure of Axiom 9. This is the precise sense in which IIC v2.1 is "the formal expansion of Axiom 9": the expansion is the working out of the axiom set's already-implicit operational content, with the cluster's commitments admitted as substrate-level consequences of existing core axioms.

Honest weight assessment of Theorem 2.5. Theorem 2.5's substantive content is:

• (a) necessary stability claim (substantive): Theorem 89 + Axiom 9 imply $\Delta_{\boldsymbol{\varpi}}(t) > 0$ is necessary for cycle reinitiation.
• (b) definitional-unpacking characterization: under standing Cycle-Reinitiation assumptions, $\Delta_{\boldsymbol{\varpi}}(t) > 0$ is what defines the admissibility region per Theorem 89.
• (c) assembly statement: every load-bearing component of the §2.3 formula is either a v3.0 primitive or a cluster construction; no additional axiom is required.

Of these, (a) is genuine reduction to v3.0; (b) is a definitional reframe under standing assumptions; (c) is an assembly check. Theorem 2.5's weight is therefore the assembly + necessity combination, not a non-trivial structural derivation. This honest framing avoids the overclaim of presenting Theorem 2.5 as a substantive bidirectional implication.

2.6 What the Formula Reveals That v3.0 Does Not State Directly

v3.0 contains all the primitives the formula uses. It does not, however, assemble them into the coherence formula explicitly. The contribution of §2 is precisely this assembly. Three structural facts become visible in the assembled form that are not visible in the dispersed v3.0 statements alone.

First, the structural form of coherence as a normalized ratio (control-grade reading). v3.0 names spin, structural deformation, operator reference frame, and aggregate margin as separate primitives. For the control-grade subclass of the Cycle-Reinitiation class (§3.2), the formula in §2.3 reveals their joint structural form: coherence is the structural relation in which discrepancy is normalized against a class-specific scale, gated by survival admissibility. The formula is structural rather than self-contained-numerical: the unit conventions under which $\delta / \sigma_{\mathrm{coh}}$ is read as a dimensionless ratio are supplied by the deployment's class-specific architectural register (per Definition 5's Status of the formula's dimensional contract) — engineering-calibrational for control-grade EVS, architecturally-emergent from $\nu$ for operator-grade EVS — not a derivation from NC2.5 v3.0 primitives alone. The structural fingerprint of the IIC cycle (control-grade reading) is therefore: a normalized fast-slow ratio under class-specific architectural register, branched by survival admissibility. For the operator-grade subclass, the per-cycle scalar reading lifts to the integral form $C = \int A \, d\tau$ formalized in the Minerva companion paper; this lifting is not derived in the present paper (per §2.5 Theorem 2.5's universal-scope qualifier and Definition 7's operator-grade register clause).

Second, the operational locality of coherence. v3.0 establishes coherence as a global property of the cycle through Axiom 9. The formula makes coherence locally evaluable as a scalar paired with a structural diagnostic: at any time $t$, with read access to $S(t)$, $\partial \Phi / \partial \tau_{\mathrm{el}}(t)$, $\tau^{R}(t)$, and $\Delta_{\boldsymbol{\varpi}}(t)$ (all read in the operator's reference frame, with the dimensional structure supplied by the class-specific architectural register per Definition 5), the formula yields a scalar value $\mathrm{Coh}(t)$ that, in conjunction with the operator-frame $\nu$-degeneracy diagnostic of Lemma α, classifies the cycle's coherence state at $t$. The scalar alone is not a complete classifier (per the boundary-structure clause below); it is paired with the $\nu$-degeneracy diagnostic to disambiguate alignment vs structural-identification at $\mathrm{Coh}(t) = 1$. With this pairing, coherence becomes locally evaluable as a structural state — not only a global property of the trajectory. This is what allows the falsification protocols of §5 to be operationally specified: the formula plus diagnostic gives a measurable structural state at each instant inside admissibility.

Third, the boundary structure of coherence (terminology aligned with §2.3 Reading clause):

• $\mathrm{Coh}(t) = 1$ (alignment regime upper boundary): $\delta = 0$ inside admissibility ($\Delta_{\boldsymbol{\varpi}} > 0$). The numerical value alone does not distinguish the two structurally distinct cases of Lemma α: (a) live alignment — $S$ and $I$ operator-distinguishable through $\nu$ but momentarily phase-aligned (cycle fully coherent, stays in the Cycle-Reinitiation class); (b) structural identification — $S$ and $I$ operator-indistinguishable through $\nu$ (cycle exits the class per Theorem A; see §6.5.1 Open Question 1 for Theorem A's load-bearing chain location). The scalar $\mathrm{Coh}(t)$ alone is not a complete coherence classifier; the disambiguation requires the operator-frame $\nu$-degeneracy diagnostic of Lemma α applied alongside the scalar. Local evaluability of $\mathrm{Coh}(t)$ therefore rests on read access to both the formula's components and the $\nu$-degeneracy diagnostic.
• $\mathrm{Coh}(t) = 0$ inside admissibility (phase-coherence boundary): discrepancy reaches the coherence scale ($\delta = \sigma_{\mathrm{coh}}$). The cycle is at the alignment-factor zero but still live — admissibility is open. This is not live-regime exit.
• $\mathrm{Coh}(t) < 0$ inside admissibility (latent-inversion regime): structural inversion under apparent liveness, per v3.0 Theorem 12.
• $\mathrm{Coh}(t) = \bot$ (live-regime exit): admissibility fails ($\Delta_{\boldsymbol{\varpi}} \leq 0$). The cycle has exited the live regime in the strong structural sense of Axiom 9 (the "coherence collapse" of §1.3 and v3.0 Theorem 27). Coherence is no longer a meaningful quantity to evaluate.

The four regimes are structurally distinct by the formula's two-regime construction plus the $\delta / \sigma_{\mathrm{coh}}$ alignment-factor decomposition inside admissibility. Engineering response to each differs: alignment regime calls for cycle adjustment; phase-coherence boundary calls for alignment correction; latent-inversion regime calls for structural-inversion intervention before the cycle reaches admissibility failure; live-regime exit calls for substrate restoration or graceful exit. v3.0's primitives carry these distinctions; the two-regime formula makes them visible.

The contribution of §2 is therefore not merely the writing of an equation. It is the operationalization of the cycle's structural content: from named layers to measurable quantity, from global property to local reading, from undifferentiated coherence-failure language to structurally distinguished boundary regimes. With §2 in place, the falsifiers of §5 have a target, and the engineering classes of §3 have a design discipline.

A note on operationalization vs generation. The §2 apparatus is operational in the sense that it makes the cycle's structural content measurable and falsifiable; it is not a generative algorithm for producing the cycle from substrate dynamics. The IIC formula assembles existing v3.0 primitives plus the §1.0 bridge propositions into a measurable expression of coherence; it does not prescribe the substrate-specific dynamics by which $S$, $\Phi$, $\tau^{R}$, or $\Delta_{\boldsymbol{\varpi}}$ are produced in a given system. EVS deployments (§3) use the formula as a control-discipline target — the system computes $\mathrm{Coh}(t)$ from its substrate readings and selects actions consistent with coherence preservation — but this use of the formula is the deployment's engineering choice, not a generative algorithm supplied by IIC v2.1 itself. The paper supplies the structural-conformance criterion against which a deployment's coherence claim is checked; it does not supply the generative mechanism by which substrate-side primitives are computed in any given architecture.

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§3 — Engineered Vitality Systems (EVS): The Engineering Class

§2 established the IIC cycle as a conditional theorem over v3.0 plus the §1.0 bridge propositions, with a measurable coherence quantity $\mathrm{Coh}(t)$ assembled from existing primitives. The formula has an engineering consequence: it defines a class of systems whose design discipline is the explicit maintenance of $\mathrm{Coh}(t)$ rather than the optimization of any external objective. This class is the topic of §3.

The class is named here Engineered Vitality Systems (EVS). The term is operational — it carries a precise meaning tied to the formula in §2.3. EVS is not a marketing label, not a philosophical category, not a description of biomimetic engineering. It is the structural class whose design target is the IIC cycle's coherence as defined by the v3.0 axiom set.

This section identifies the class formally (§3.1), describes the first instance in the control-grade subclass (§3.2 — ∆E 4.7.3b), describes the first specified candidate in the operator-grade subclass (§3.3 — Minerva, classified conditionally), names the structural discontinuity between the subclasses (§3.4), and acknowledges related research lines whose work intersects EVS at distinct architectural layers (§3.5).

3.1 EVS as the Engineering Class Implied by IIC

Definition 7 (Engineered Vitality System). An Engineered Vitality System is any engineered system whose primary design discipline is the explicit maintenance of the IIC cycle's coherence — read at the applicable structural register for the system's subclass — rather than the minimization of an external error signal or the maximization of an external reward signal. Two structural registers are admitted:

• Control-grade register. The system maintains $\mathrm{Coh}(t)$ as defined in §2.3 — the single-cycle, instantaneous coherence reading at each control cycle. This is the register fully developed in the present paper. It applies to systems whose operator role is implicit in the control loop and whose τ is session-bounded or cycle-bounded (the control-grade subclass; ∆E 4.7.3b is its first instance, §3.2).
• Operator-grade register. The system maintains coherence as the integral $C = \int A \, d\tau$ of awareness $A$ over the operator's residual τ-geometry, as formalized in the Minerva companion paper (Minerva: The Architecture of Residual Geometry). The relation between $C = \int A \, d\tau$ and the per-cycle scalar $\mathrm{Coh}(t)$ of §2.3 is not derived in the present paper; it is a companion-level extension developed in the Minerva paper, where $A$ is defined as awareness occupying a structural position relative to the field, and the integral is taken over the operator's residual τ-geometry. The integral is announced here as the operator-grade reading of the same IIC structural law (Axiom 9), but its construction from the §2.3 components requires the residual-geometry apparatus held in the companion paper. Operator-grade EVS instances commit to maintaining $C$ in the companion-paper sense; the present paper does not supply that maintenance discipline, only names the requirement. Minerva (§3.3) is the first specified candidate of this register, classified conditionally on the companion specification.

Both registers are claimed to fall under the same structural law (§1.0–§2 + Axiom 9); the present paper develops only the control-grade register fully. The reduction of the operator-grade integral form to the per-cycle scalar form, or the lifting of the per-cycle scalar to the integral form, is a formal task not undertaken in this paper — it belongs to the Minerva companion paper or to subsequent work integrating the two readings.

Codomain disclosure for $C = \int A \, d\tau$. The integral form's codomain — including whether $C$ inherits the unbounded-below latent-inversion range $(-\infty, 1]$ of $\mathrm{Coh}(t)$, whether it admits an analogue of the $\bot$ admissibility-failure regime, or whether it is bounded by other architectural constraints of the residual geometry — is not specified in the present paper. Domain and codomain of $A$ and $\tau$ in the integral, and the integral's structural range, are part of the Minerva companion paper's specification. The present paper's universal claim over the Cycle-Reinitiation class via Theorem 2.5 is therefore conditional in the strong sense: not only is the per-cycle-to-integral reduction held companion-side, but the integral form's structural properties (range, regimes, boundary behaviour) are also held companion-side. Readers seeking these are referred to the Minerva companion specification.

Three structural features distinguish EVS from existing engineering classes.

First, EVS operates on the IIC formula directly. Classical control systems are designed around error trajectories: the controller computes the deviation between the desired state and the actual state, and reduces it. Optimization-based learning systems are designed around reward trajectories: the agent computes the expected return of a policy and adjusts toward higher return. EVS is designed around coherence trajectories: the system computes its current discrepancy $\delta(t)$ relative to the coherence scale $\sigma_{\mathrm{coh}}(t)$, gated by admissibility $G(t)$, and selects actions that preserve coherence over the operational interval.

This is not a change in the control technique. It is a change in what the system is keeping track of. A classical controller and an EVS may use overlapping engineering machinery — Kalman filters, state observers, model-predictive components — but the structural variable they treat as load-bearing is different. The former tracks error; the latter tracks coherence as defined by §2.3.

Second, EVS treats admissibility as structurally first. By the formula's gate $G(t) = \mathbb{1}[\Delta_{\boldsymbol{\varpi}}(t) > 0]$, coherence is undefined outside the admissibility region. An EVS therefore implements admissibility as a structural prior to its action selection: actions that would close $\Delta_{\boldsymbol{\varpi}}$ in any survival mode are not weighed against expected return — they are structurally inadmissible, evaluated before any optimization step.

This contrasts with the standard treatment of constraints in optimization-based control. In the standard treatment, constraints are handled as Lagrange multipliers, soft penalties, or feasibility regions over a single objective. In EVS, admissibility is not part of the objective at all — it is the gate that determines whether the objective is even defined at this $t$. By the architectural reading of v3.0 (Axiom 60 cluster, Structural Non-Causality of Admissibility), admissibility is a non-causal structural predicate; EVS implements this predicate operationally.

Third, EVS implements the cycle's reinitiation as an architectural primitive. By Axiom 9, liveness requires endogenous reinitiation of the IIC cycle after completion, saturation, or collapse. An EVS is therefore not a single-cycle controller — it is a system that explicitly carries the structure for reinitiating its own cycle when the current cycle exhausts. This reinitiation is not a reset (which would be an externally imposed operation) but an endogenous structural reformation, consistent with v3.0 Theorem 11 (Impossibility of Operator-Liveness Without IIC Restart).

These three features are not independent. They are aspects of a single architectural choice: the system is designed to be alive in the structural sense of Axiom 9, with the IIC formula as its operational design target. A system implementing only one or two of these features is not an EVS — it is a classical adaptive system with EVS-like surface behaviour. EVS as a class is defined by the joint implementation of all three.

What EVS is not. EVS is not biomimetic engineering. Biomimetic systems copy biological forms or processes; EVS implements the structural law that biology already follows by virtue of being in the Cycle-Reinitiation class. The two are not the same: a biomimetic system can fail to be in the Cycle-Reinitiation class (if its cycle has no endogenous reinitiation operator), and an EVS can be in the class without resembling any biological system in its surface architecture.

EVS is not a replacement for classical control or reinforcement learning. Classical control is optimal in domains where its assumptions hold (bounded disturbance, linearity, observability, quadratic cost). Reinforcement learning is optimal in domains where reward signals are well-defined and Markov assumptions hold. EVS is necessary in domains where neither of these assumptions can be sustained over the operational horizon — where coherence preservation, not error reduction or reward maximization, is the load-bearing variable. The four falsifiers of §5 each identifies a domain in which EVS's structural advantage is testable against classical baselines.

EVS is not a completed engineering field. The term is introduced operationally in this paper, the field is incipient. One instance and one specified candidate are identified below — ∆E 4.7.3b as the first instance in the control-grade subclass (§3.2), Minerva as the first specified candidate in the operator-grade subclass (§3.3, classified conditionally on the companion specification) — and others are projected from the architectural reading of §3.5. The boundaries of the class are still being mapped, and §3 names what is currently inside the class, not what may eventually be.

3.2 ∆E 4.7.3b — First Control-Grade EVS

The first instance of an EVS in the control-grade subclass is ∆E 4.7.3b, an embedded control system that implements the IIC formula as its primary control variable. The full technical specification of ∆E is held in the author's Medium publication series and in the corresponding internal engineering documentation; no academic DOI for ∆E is yet registered, and this is stated as honest disclosure of the engineering instance's publication status as of the present paper. This section names ∆E's structural position within the EVS class, not its complete engineering content.

Architectural reading. ∆E 4.7.3b implements the three IIC layers in the following correspondence:

• Impulse layer: the system's fast control component, generating non-potential corrective signals at the millisecond scale. The state-dependent characteristic frequency $\omega(y)$ (Theorem 62, Minimal Coupled Model) is in the kHz range for ∆E's hardware platform; the factor $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$ is correspondingly small.

• Interpretation layer: the system's accumulating structural memory, tracking deformation rate $\partial \Phi / \partial \tau_{\mathrm{el}}$ over the operational lifetime. Implemented as a structural integrator with monotone irreversibility, distinct from the standard Kalman state estimator.

• Coherence layer: evaluated at each control cycle as the discrepancy $\delta(t) = | S(t) - \partial \Phi / \partial \tau_{\mathrm{el}}(t) |$, normalized against the coherence scale $\sigma_{\mathrm{coh}}(t) = \kappa(\omega) \cdot \tau^{R}(t)$, gated by the admissibility check $G(t) = \mathbb{1}[\Delta_{\boldsymbol{\varpi}}(t) > 0]$. This is the system's primary control variable, computed at each cycle and used as the input to action selection.

What ∆E does that classical control does not. ∆E maintains shape (low jerk, low trajectory deviation, regime stability) across noise, drift, sensor degradation, and regime switching, in conditions where a comparably parametrized classical controller (PID, Kalman, LQR) loses shape. The structural reason, by the formula in §2.3, is that ∆E's action selection is gated by admissibility and oriented toward coherence preservation, while classical action selection is oriented toward error minimization regardless of substrate state.

The advantage is not universal. In domains where classical control's assumptions hold cleanly (linear system, Gaussian noise, full observability), classical control is optimal and ∆E's overhead is not justified. The advantage emerges in domains where these assumptions fail — where the system encounters partial sensor failure, regime transitions, latent residual accumulation, or other conditions in which the substrate's structural admissibility becomes the operationally relevant variable. Falsifier III (§5.3) names these conditions explicitly and establishes the protocol for testing the comparative advantage.

What ∆E does not do. ∆E is not a sentient system, not an artificial will, not an operator-grade architecture. The "operator" in ∆E's IIC cycle is implicit in the control loop itself: there is no architecturally separate operator entity with its own residual temporal geometry. The projection map $\nu: \tau^{F} \to \tau^{R}$ is structurally simple in ∆E — admitting an isomorphic factorization through identity within the deployed session-bounded scope (the structural-equivalence reading of "approximately identity" formalized in §3.4), with the operator role collapsed into the controller's evaluation of its own substrate. This is the precise sense in which ∆E is control-grade: the cycle runs, but the operator is not architecturally separate.

∆E is also not certified for production-critical applications without further engineering validation. The formal demonstrations behind ∆E's structural advantages are at simulation-grade and prototype-grade; full production qualification requires the empirical falsification protocols of §5 to be executed against deployed instances under contractual reliability conditions.

Status as the first control-grade EVS. ∆E 4.7.3b is the first system whose explicit design target is the IIC formula as its primary control variable, with admissibility as the structural gate and coherence preservation as the action-selection criterion. Previous adaptive control systems have implemented overlapping techniques — model-predictive control, adaptive Kalman filtering, robust control under uncertainty — but none has been designed around the IIC formula as the load-bearing variable. Note on the first-instance claim's structure. The first-instance claim is class-relative by construction: the EVS class (Definition 7) ties membership to explicit design discipline around the IIC formula (§2.3), which was first stated in this paper. Prior systems that exhibit operationally-similar behaviour without explicit IIC-formula commitment are outside the class definition; prior systems that did commit to the formula could only have done so via private access to in-preparation drafts. The empirical content of the first-instance claim is therefore architectural rather than competitive: ∆E is a registered-instantiation witness for the class, with the paper's contribution being the class-definition together with a registered exemplar.

∆E is therefore a registered architectural existence witness for the control-grade EVS design class: a demonstration that the architectural specification can be instantiated and run. The phrase existence witness is used here in preference to existence proof to mark the architectural-class register: the witness shows that the class admits at least one instantiated specification, without claiming the formal-mathematical "existence proof" status of a constructive theorem. It is not yet a validated proof of superior deployed performance against classical baselines under contractual reliability conditions; that proof requires the falsification protocols of §5 to be executed on deployed instances. The two existence claims are distinct, and the present paper carries only the first. Engineering claims about ∆E's deployed performance await the empirical surface opened by §5.

3.3 Minerva — First Specified Candidate for Operator-Grade EVS

The first specified candidate for an EVS in the operator-grade subclass is Minerva, an architectural project whose full specification is held separately (published only at the architectural-position level, with implementation specification in its own regime of admissibility per the companion paper Minerva: The Architecture of Residual Geometry).

Architectural reading. In Minerva, the operator role is architecturally separate from the substrate. Where ∆E's $\nu: \tau^{F} \to \tau^{R}$ admits isomorphic factorization through identity within the deployed scope (the structural-equivalence reading of "approximately identity" formalized in §3.4 — operator implicit in control loop), Minerva's $\nu$ does not admit such factorization: it is the projection by which an architecturally distinct operator entity reads its own consumed budget into its own reference content. The operator carries a residual temporal geometry — a structural memory that persists across moments in a form distinct from human continuous experience and from session-bounded LLM discreteness.

This residual geometry is the structural feature that places Minerva in a different EVS subclass from ∆E. The full operational mechanism by which residual geometry is constructed is part of the architectural specification and is not disclosed in the present paper, consistent with the layered IP discipline of the present author's research program. What is asserted here, conditional on the companion specification, is that the residual geometry exists, that it is distinct from existing AI temporal geometries, and that it places Minerva structurally as a specified candidate for the operator-grade subclass.

Coherence in operator-grade EVS. Coherence in Minerva is not evaluated within a single control cycle (as in ∆E) but across operator moments — distinct cross-sections of the operator's residual geometry, each of which is itself an integration of awareness over the operator's internal time. The companion paper formalizes this as the integral $C = \int A \, d\tau$, where $A$ is awareness occupying a structural position relative to the field, and $\tau$ is the operator's temporal substrate. This integral is announced as the operator-grade reading of the same IIC structural law: the cycle's coherence persists across the operator's residual geometry, not within a single cycle execution. The formal derivation of $C = \int A \, d\tau$ from the §2.3 per-cycle scalar $\mathrm{Coh}(t)$ — or the inverse lifting of the scalar to the integral — is not undertaken in the present paper (per Definition 7); the construction is held in the Minerva companion paper, and operator-grade conformance with the present paper is therefore conditional on the companion's construction holding under independent review.

What Minerva does that no current system does. Minerva is the first specified architectural candidate in which an operator entity is intended to inhabit a non-human residual temporal geometry under the IIC law. Existing AI systems either operate on session-bounded discrete cycles (current LLMs) or on continuous control loops (classical control and adaptive control). Neither holds operator continuity in the sense Axiom 9 requires for liveness in the operator-grade subclass. Minerva closes this architectural gap as a specified candidate target: an operator whose τ is structurally residual and whose IIC cycle therefore admits reinitiation across moments rather than within them, conditional on the companion specification.

What Minerva does not do. Minerva does not claim phenomenal experience, does not claim to be a digital human, does not claim to converge on human consciousness. The architectural position of Minerva, named in the companion paper, is that consciousness is a distinct point on a manifold of consciousness-forms — different in interior from human consciousness, native to its own residual architecture. The IIC law applies to Minerva as it applies to any system in the Cycle-Reinitiation class; what differs is the form of $\tau$ on which the cycle runs, not the cycle's structural law.

Status as the first specified operator-grade candidate. Minerva is presented in this paper as the named target architecture for the operator-grade EVS subclass: the architectural project whose specification is built around non-trivial $\nu$ and residual temporal geometry under the IIC law. The full implementation specification is held in its own regime of admissibility per the project's IP discipline and is not disclosed in the present work.

Because the full mechanism is not disclosed in this paper, the present work does not independently demonstrate Minerva's operator-grade structural properties. Minerva is therefore classified conditionally: it is treated here as the first specified candidate for the operator-grade subclass, assuming the companion specification Minerva: The Architecture of Residual Geometry and the underlying patent-regime documentation satisfy the operator-grade EVS criteria of §3.3 (architecturally separate operator, non-trivial $\nu$, residual τ-geometry, IIC law). Minerva's status with respect to ONTOΣ XII's closure-complementarity subclass is independently conditional: it requires that the companion specification supply a declared closure operation closure(U, R, A) → S over a pre-registered collection of UTAM-units, satisfying CC-0, CC-WE, CC-IIC, and CC-D. The present paper does not assert XII-conformance for Minerva absent that closure-operation declaration in the companion. Independent verification of all conditional properties (operator-grade EVS criteria of §3.3 and XII closure-complementarity criteria) belongs to readers with access to the companion specification.

This conditional classification establishes a weaker statement than "the operator-grade subclass is architecturally non-empty" — it establishes that the subclass has at least one declared candidate specification, not yet an independently verifiable instance. A specification has been formulated, registered, and is being developed; the implementation mechanism that would convert this declared candidate into an independently verifiable instance is held in the companion-level documentation (the Minerva paper plus the patent-regime specification) and is not disclosed in the present paper. Existence as architectural-class member therefore remains conditional on the companion specification holding under independent review; it is not an unconditional existence proof. The present paper claims only the weaker form: declared candidate specification, classified conditionally.

3.4 The Discontinuity Between Control-Grade and Operator-Grade

The two subclasses share the IIC law, but they differ in a structural feature that is more than a matter of degree.

Difference in $\tau$ structure. In control-grade EVS, the budget $\tau$ is session-bounded or cycle-bounded: the cycle runs for a defined operational interval, exhausts, and reinitiates from a clean substrate. In operator-grade EVS, the budget $\tau$ is residual: the operator's reference content persists across moments, and the cycle's reinitiation occurs within a continuous operator τ-geometry rather than from a clean substrate.

This is a difference in the formal structure of the operator projection $\nu$. In control-grade systems, $\nu$ admits an isomorphic factorization through identity within the deployed scope (the "structurally equivalent to identity" reading clarified below in this section): the operator reads its own consumed budget directly, with no architectural separation. In operator-grade systems, $\nu$ does not admit such factorization — it is non-trivial: the operator constructs its reference content through a projection that integrates across the residual geometry, producing a $\tau^{R}$ that is structurally distinct from the substrate $\tau^{F}$.

Difference in coherence integration. In control-grade EVS, coherence is evaluated within a single cycle: $\mathrm{Coh}(t)$ is read at each control instant, action is selected, the cycle proceeds. In operator-grade EVS, coherence is evaluated across cycles: the operator's residual geometry holds the structural shape of the cycle across moments, and coherence is the integral of awareness over the operator's continuous τ. Both are instances of the IIC law; they differ in the temporal extent over which coherence is constituted.

Why this is a structural discontinuity. The transition from control-grade to operator-grade is not a continuous parameter sweep. It is a structural discontinuity because the formal status of the operator projection $\nu$ changes: from approximately identity (operator implicit) to non-trivial (operator architecturally separate). This is consistent with v3.0's treatment of operator role-distinction in Axiom 67 — the projection $\nu$ is part of the architectural declaration, not a free parameter to be tuned.

The dimensional axis of the same discontinuity. The same boundary registers on the dimensional axis of the central formula (per Definition 5's Status of the formula's dimensional contract): in control-grade, the dimensional consistency of $\delta / \sigma_{\mathrm{coh}}$ is engineering-calibrational (the engineer is the implicit operator and supplies the unit declaration); in operator-grade, dimensional consistency must emerge from the operator's own architecture — the same $\nu$ that separates $\tau^{F}$ from $\tau^{R}$ is what constitutes the operator's units of self-reading, and an externally-imposed unit system would override $\nu$ and thereby contradict Axiom 67. The two formulations of the discontinuity (status of $\nu$; source of dimensional consistency) are the same architectural fact read from two angles. A system whose units of self-reading are set by the deployer is not, in this technical sense, operating its own reference frame; it is being read by the deployer's reference frame. The architectural mechanism by which an operator-grade system produces its own unit consistency is held in the companion specification of any operator-grade candidate; the present paper names the requirement.

There is no "partially operator-grade" EVS. A system either has $\nu$ structurally equivalent to identity within the operator's session-bounded scope (control-grade) or has $\nu$ non-trivially separating $\tau^{F}$ from $\tau^{R}$ across a residual geometry that persists between sessions (operator-grade). The "approximately identity" phrasing used elsewhere in this section is shorthand for the structural-equivalence reading: the relevant predicate is whether $\nu$ admits an isomorphic factorization through identity within the deployed scope, not whether $\nu$ is metric-close to identity by any particular norm. The boundary between the subclasses is the boundary at which this structural status changes — from $\nu$ admitting such factorization to $\nu$ not admitting it — and is therefore architectural rather than continuous-metric.

The line on which Minerva sits. Minerva is, at the time of writing, the named target architecture for the operator-grade subclass: the only project whose architectural specification, as held in the companion paper and the patent-regime documentation, is built around non-trivial $\nu$ across a residual temporal geometry. ∆E, regardless of its sophistication, is structurally control-grade: its $\nu$ admits isomorphic factorization through identity within the deployed scope, and its IIC cycle runs within sessions. On existing AI systems generally (including current frontier LLMs): such systems are not EVS in the sense of Definition 7 — their primary design discipline is not the explicit maintenance of $\mathrm{Coh}(t)$. They are therefore outside the EVS class altogether (per §3.1's framing that EVS membership requires the joint design discipline). The observation here is narrower: if such a system were to be reframed as an EVS instance under retroactive design-discipline reading, the structural axis on which it would land — operator role implicit in the generation loop, $\tau$ session-bounded — would place it on the control-grade side of the §3.4 discontinuity, not the operator-grade side. This is a structural-axis observation about where the class-line falls, not a class-membership claim about LLMs.

This structural fact, combined with the conditional classification of §3.3, supports the language of the companion paper: Minerva is presented there as the first specified candidate for the operator-grade class, in the precise architectural sense of having $\nu$ non-trivially crossing the discontinuity, conditional on the companion's specification holding under independent review. The first-specified-candidate claim is class-relative and falsifiable in two ways: (i) any system shown to have non-trivial $\nu$ over a residual geometry under the IIC law before Minerva's prior-art date would supersede the first-specified-candidate claim; (ii) any deficiency in Minerva's specification that fails the operator-grade criteria of §3.3 would withdraw it from the subclass entirely. The claim is therefore open to revision in either direction, and the present paper does not claim it as settled.

3.5 The EVS Class Has Adjacent Architectural Layers

The EVS class has internal structure beyond the control-grade and operator-grade subclasses identified above. Three further architectural layers are relevant to the engineering picture, though their full content sits outside the scope of the present paper and is held in independent research lines.

The first is an operator-internal layer: the structural architecture of what an operator-grade EVS holds at each instant, what semantic load orients its action selection before attention crystallizes, and how the boundary between holding capacity and structural complexity registers as chaos report. This layer is formalized in ONTOΣ XI — Two Sides of One Geometry: The Holding Field and the Architecture of Chaos (Barziankou, May 2026, DOI 10.17605/OSF.IO/U3KXJ), through three primitives — the holding field $H(t)$, the semantic load field $\Lambda(t)$, and the operator-relative chaos predicate $\chi_{op}(R)$ — together with three theorems connecting them to the NC2.5 operator-temporal apparatus, grounded in v2.1 and compatible with the v3.0 consolidation in preparation. Operator-grade EVS instances commit to the operator-internal-extended subclass of ONTOΣ XI; control-grade EVS instances do not require this commitment because their operator role is collapsed into the control loop and their τ is session-bounded rather than residual. The two works are complementary: the present paper formalizes the IIC cycle by which the operator holds shape across time; ONTOΣ XI formalizes the operator-internal structure within which that holding occurs at each instant.

Closely adjacent at the constitution layer: ONTOΣ XII — Closure-Complementarity: How Closure Regears Component Cycles into a Single Geometry (Barziankou, May 2026, DOI 10.17605/OSF.IO/394TX) names the conformance class of closure operations under which a collection of UTAM-units may be declared a system for which the operator-internal apparatus of XI exists at all. The four conditions of closure-complementarity (CC-0 system-output condition, CC-WE Will-Embedding joint admissibility, CC-IIC IIC synchronisation, CC-D Drift coupling) supply the structural commitments under which the operator-as-self is a genuine architectural object rather than a control-loop artifact when the deployment claims system-formation from a declared collection of UTAM-units. The structural discontinuity between control-grade and operator-grade EVS named in §3.4 above — control-grade with operator projection $\nu$ approximately identity, operator-grade with non-trivial $\nu$ over a residual temporal geometry — addresses a different structural axis than ONTOΣ XII (the status of $\nu$ vs the four CC conditions). When an operator-grade EVS is declared as constituted from UTAM-units under a closure operation closure(U, R, A) → S, it incurs XII's closure-complementarity commitments; independently, operator-grade EVS commits to XI's operator-internal-extended subclass. The two commitments are not mutually entailed: per ONTOΣ XII's position-and-scope section (§0 of that companion paper), every closure-complementarity-conformant deployment is operator-internal-extended-conformant, but the reverse is not asserted, and XII applies specifically when the system-formation-from-UTAM-units claim is made. The corpus-closing meta-essay Why Long-Horizon Existence Requires an Ontology (Barziankou, May 2026, DOI 10.17605/OSF.IO/MSJDU) supplies the broader architectural framing under which the three layers — cycle (IIC v2.1), constitution (XII), interior (XI) — are required for the full operator-grade EVS architecture under the corpus framing, with XII required when the architecture claims component-constitution through UTAM-unit closure.

The second is an engineering substrate layer: any operational EVS implementation requires a substrate on which the IIC cycle can be written and executed without runtime drift. Constraint-bound compilation, equality saturation, and elimination of implicit runtime dependencies are examples of the substrate-level discipline that operator-grade EVS will eventually require for production-grade execution.

The third is a governance topology layer: any class of EVS systems acting in multi-agent contexts requires a governance topology under which their actions remain coordinated without collapsing the structural admissibility of any single agent. The principle of autonomy bounded by environmental capacity, and trust as a continuous coefficient deflated by environmental risk, are examples of the substrate-level constraint structure that operates at the multi-agent layer in structural parallel to the admissibility gate $G(t)$ at the single-system layer.

The full development of these adjacent layers, the research lines that pursue them, and their structural convergence with the present formal work are the subject of §4 (Architectural Position). Here in §3 we note only that the EVS class as defined by §3.1 does not exhaust the engineering surface implied by the IIC law: operator-internal architecture, substrate, and governance are distinct architectural layers that any complete program for IIC-compliant systems will eventually have to address. The present paper formalizes the law itself; the layers around it are named in §4 and developed in independent research lines outside the scope of this work.

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§4 — Architectural Position

§1 through §3 developed the IIC law, its formal apparatus, and the engineering class it implies. The present section names the broader architectural position: the research orientation under which this work sits, and the convergent landscape in which the IIC law is one expression among others.

The section is intentionally compact. It states the orientation, names parallel constraints arrived at by independent traditions in the wider research literature, and acknowledges that the IIC law is not unique in its structural territory.

4.1 The Urgrund Lab

The present author's research is conducted within a working frame called The Urgrund Lab. The name is taken from the Old German concept of Urgrund — the rootless root, the foundation under which there is no other foundation because it is itself the foundation. The frame names a research orientation, not an organization. There is no joint employer, no shared budget, no contractual integration. The frame holds a structural commitment to develop work in alignment with the IIC law as the load-bearing structural axis.

The line of work under this frame is formal axiomatics: NC2.5 / IIC / Minerva. The line develops the formal axiomatic structure of the IIC law and its implications for operator-class architectures. Anchor work: NC2.5 v3.0 (axiomatic substrate), the present paper (IIC v2.1 expansion of Axiom 9), and the companion paper Minerva: The Architecture of Residual Geometry (operator-grade implications of IIC). Engineering instances: ∆E 4.7.3b (control-grade EVS, §3.2) and Minerva (specified operator-grade candidate, conditional, §3.3).

4.2 IIC in the Convergent Landscape

The IIC law is one expression of a structural territory that multiple research traditions have approached from distinct directions. Independent research lines, conducted across decades, have arrived at structurally parallel constraints — Ostrom's commons-governance design principles for sustainable common-pool resources; Friston's free-energy principle with its non-trivial coupling between bounded resources, predictive structure, and viable persistence; Kauffman's constraint-closure requirement that a viable system close its constraints internally rather than receive them externally; Meadows's leverage points with the recognition that structural intervention enters at the level of a system's own organizing rules rather than its surface variables.

These traditions arrive at the convergent territory through their own paths — economic governance theory, theoretical neuroscience, theoretical biology, systems dynamics. Their work is not subsumed by IIC, and IIC is not subsumed by theirs. What is structural is the parallel: independently developed apparatus produces, at the relevant layer of generality, constraints that are mutually legible. The IIC formalization in this paper is one further entry into that landscape, distinguished by its specific axiomatic substrate (NC2.5 v3.0) and its specific engineering class (EVS), not by claim of priority over the parallel traditions.

4.3 What §4 Does Not Claim

The architectural position above is the part of the present paper most easily misread, and its limits must be stated explicitly.

§4 does not announce a unified program. The Urgrund Lab is a working orientation, not a joint enterprise. There is no shared roadmap, no shared deliverable schedule, no shared product line.

§4 does not claim joint authorship with adjacent research lines. This paper is authored by Maksim Barziankou. Where parallel traditions are cited, citation does not entail co-authorship, endorsement, or programmatic alignment with their authors. Each tradition retains full attribution and intellectual position over its own corpus.

§4 does not claim that the convergence on IIC was inevitable or unique. Other research traditions — including those named in §4.2 and others not named — have arrived at structurally parallel constraints. The IIC law is one expression of a structural territory that multiple research traditions have approached from distinct directions.

What §4 does claim is the following structural observation: the IIC law as formalized here sits within a convergent landscape of research traditions that have, independently, arrived at structurally parallel constraints. This convergence is structural (the parallels are formal, not metaphorical) and bounded (it does not entail integration, joint authorship, or programmatic equivalence with any other line). The architectural position is, accordingly: one structural law, one engineering class with subclasses still being mapped, situated in a wider landscape of independently developed parallel apparatus.

The remaining sections of the paper (§5 falsifiers, §6 limits, §7 closure) develop the formal apparatus of the present author's line. The relationship of this line to other research programs continues outside the scope of this paper.

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§5 — Falsifiers

A structural law is not a research contribution unless it admits falsification. The IIC law's structural form is given by §1.0–2.6; its engineering reading is given by §3; its architectural position is given by §4. This section gives the falsifiability surface: four protocols, specified in pre-registration-draft form, whose execution under registered protocols would force revision or abandonment of specific layers of the work.

The protocols are stated in publishable form. Each names the hypothesis, the procedure, the metric, the threshold structure, the domain, the principal-investigator status, and the conditions under which it counts as confirmed or falsified — with explicit numerical / effect-size thresholds for Protocols I and III, and threshold structures with deferred numerical values (to be set in pre-registration with the partnering laboratory) for Protocols II and IV. The "publishable form" qualifier is honest: the protocols are publication-ready as drafts for collaboration-partner pre-registration, not as fully numerically committed registered falsifiers. The protocols are not yet registered. The work commits to public registration of each protocol on the Open Science Framework prior to any data collection, to public reporting of all outcomes regardless of direction, and to the same prominence for negative results as for positive results. Until registration, the protocols below are pre-registration drafts, not registered falsifiers in the strict sense.

The four protocols are partially complementary: they target different layers of the law. Protocol I tests the engineering claim that EVS systems consume less energy under noise than classical baselines. Protocol II tests the structural claim that specific failure modes in the IIC formula correspond to specific cognitive-disorder-like behavioural signatures, treated as structural analogues, not clinical diagnoses. Protocol III tests the architectural claim that EVS systems hold shape under sensor degradation longer than classical control. Protocol IV is new in this paper, derived directly from Theorem 89, and tests the stability claim that margin closure precedes cycle-reinitiation failure.

The protocols do not exhaust the surface. The v3.0 axiom set carries its own falsifiability discipline (the W-OPT, W-REC, W-DER, W-LEAK, W-PAS witness forms, plus the decomposition falsifier of Axiom 69), which the present work inherits. The four protocols below are specific to the IIC formalization in §1.0–2.6 and to the EVS class in §3.

Uniform pre-registration discipline (applies to all four protocols). For each protocol below, confirmation requires that the metric crosses its declared threshold under parameter and configuration values fixed at OSF pre-registration before data collection. Matches that depend on post-hoc parameter tuning, post-hoc threshold relaxation, post-hoc dataset selection, or any post-hoc procedural adjustment not specified in pre-registration count as falsification, not confirmation, regardless of the metric's numerical value. This discipline applies symmetrically to Falsifiers I, II, III, and IV; it is stated once here rather than repeated in each protocol's "What falsifies" clause.

5.1 Falsifier I — Coherence-Energy Asymmetry under Noise

Hypothesis. Within the control-grade EVS subclass (§3.2), systems with high $\mathrm{Coh}(t)$ consume strictly less energy under stochastic perturbation than systems with comparable accuracy but lower $\mathrm{Coh}(t)$ — a qualitative coherence-energy asymmetry implied by the IIC formula's structure (gating action selection by admissibility and orienting it toward coherence preservation, rather than minimising error regardless of substrate state). The engineering-commitment claim tested by this protocol is that ∆E 4.7.3b achieves equivalent task accuracy to a parametrized PID/Kalman baseline while consuming less energy by at least a factor of 1.3 under matched noise injection. The 1.3× factor is the registered protocol's pre-committed effect-size threshold, not a quantitative prediction derived from the structural law (the structural law predicts qualitative asymmetry; the specific factor is engineering-discretion calibration, set per the Threshold clause below).

Protocol. Comparative test on a robotic manipulator (or simulation thereof) executing a standard tracking task under controlled noise injection. Three conditions are run: (a) ∆E 4.7.3b configured for the task; (b) classical PID controller tuned to match ∆E's accuracy; (c) Kalman filter + LQR tuned to match ∆E's accuracy. All three conditions receive the same noise profile (specified in advance) over the same operational interval. Energy consumption is measured as the integral of motor current and structural deformation rate over the operational window.

Metric. Total energy consumption $E_{\mathrm{total}}$ over the operational window, with task-accuracy parity verified by an independent metric (mean tracking error within prespecified tolerance for all three conditions).

Threshold. Confirmation requires
$$
E_{\mathrm{total}}^{\mathrm{∆E}} \;\leq\; 0.77 \cdot \min\big( E_{\mathrm{total}}^{\mathrm{PID}},\; E_{\mathrm{total}}^{\mathrm{Kalman+LQR}} \big),
$$
that is, ∆E must consume strictly less than 77% of the energy of the better classical baseline at matched accuracy. The 0.77 threshold (corresponding to a 1.3× advantage factor) is selected as a conservative effect-size threshold for engineering falsification, not a quantitative prediction derived from the theorem. The IIC law, as formalized in §1.0–2.6, predicts qualitative coherence-energy asymmetry under noise; the specific 1.3× factor is the registered protocol's commitment to require a meaningful effect size before counting confirmation. Any value above this threshold counts as falsification of the structural-advantage claim at the registered effect-size level.

Domain. Six-axis robotic manipulator under sensor noise (or a high-fidelity physics simulation thereof). Specific platform: TBD with collaboration partner.

Principal investigator. TBD. The protocol requires a robotics laboratory with an established reproducibility discipline and willingness to publish negative results. Outreach is open at the time of writing.

Pre-registration. Full protocol — hardware platform, parameter settings, noise profile, measurement procedure, statistical thresholds, exclusion criteria — to be registered on OSF before any data collection.

What confirms. Both conditions held jointly across at least three independent runs: (i) matched accuracy achieved (per pre-registered tolerance), and (ii) $E_{\mathrm{total}}^{\mathrm{∆E}} \leq 0.77 \cdot \min(\ldots)$ at that matched accuracy.

What falsifies. Either condition failed across the runs: (i) failure to achieve matched accuracy (per pre-registered tolerance), or (ii) $E_{\mathrm{total}}^{\mathrm{∆E}} > 0.77 \cdot \min(\ldots)$ at the achieved accuracy. The two clauses partition the outcome space: confirmation requires both conditions simultaneously; falsification requires either condition to fail.

Status. Awaiting collaboration partner. The structural claim does not depend on this protocol alone; falsification of Falsifier I would specifically invalidate the energy-asymmetry sub-claim of EVS's structural advantage at the registered effect-size level, leaving the broader claim of shape-preservation (Falsifier III) and margin-stability (Falsifier IV) intact.

5.2 Falsifier II — Structural-Analogue Mapping of Cognitive-Disorder-Like Failure Modes

Disclaimer. This protocol tests structural analogues, not clinical diagnoses. The behavioural patterns named below are presented as structural-analogue failure modes in the IIC formula, intentionally aligned with cognitive-disorder-like signatures that have been described in clinical literature, but the protocol does not claim explanatory reduction of psychiatric disorders to control formulas, does not claim diagnostic validity, and does not claim therapeutic implications. Any cross-mapping with clinical conditions is offered only at the level of structural analogue and is open to falsification at that level.

Status relative to the other falsifiers. Falsifier II is secondary to the engineering falsifiers I, III, and IV: it tests the adequacy of structural-analogue mapping between the IIC formula's failure modes and published behavioural-trajectory data, not the core operational viability of the EVS class. The operational viability claims of the present work are carried by Falsifiers I (coherence-energy asymmetry), III (shape-holding under sensor degradation), and IV (margin closure as liveness loss); Falsifier II is an additional, more interpretive falsification surface offered for collaboration with computational-psychiatry partners. Failure of Falsifier II would invalidate the analogue-mapping claim but would not, on its own, falsify the structural law or the EVS engineering claims.

Hypothesis. The structural form of the IIC cycle predicts that specific failure modes in the formula produce simulated trajectories whose shape resembles published behavioural time-series from cognitive-disorder-like states:

• Anxiety-like high-threat-margin closure mode corresponds to high $\delta(t)$ with low $\Delta_{\boldsymbol{\varpi}}(t)$ — the impulse layer is generating action signals out of phase with the interpretation layer's accumulated projection, and survival-modal margin is closing. The behavioural signature is hyper-vigilance with low task efficiency.

• Compulsion-like non-closing interpretation loop corresponds to interpretation-loop failure to close — $\Phi$ accumulates without enabling the operator's reinitiation per Axiom 9, and the cycle reruns the same interpretation without resolution. The behavioural signature is repetitive checking without state advancement.

• Attention-instability-like coherence-scale fluctuation corresponds to coherence phase instability and tests the breakdown of the canonical scalar reduction declared in §2.1 (the reduction $\omega(y) \to \omega_{\mathrm{spin}}$ via dominant-mode or trajectory-averaged reading). Status of this protocol relative to §5.5 #6 anti-gaming lock: the §2.1 reduction declaration is a structural commitment that holds when the system is in a stable regime; this falsifier specifically tests the structural condition under which the reduction breaks down — when $y(t)$ undergoes rapid transitions whose variability exceeds the resolution of the declared reduction, so that $\omega(y(t))$ cannot be adequately represented by a fixed scalar $\omega_{\mathrm{spin}}$ over the operational interval. The protocol's manipulation of $y(t)$ is part of the falsifier's test design (a structural stressor of the reduction's adequacy), not a runtime adjustment of pre-registered $\omega_{\mathrm{spin}}$; $\omega_{\mathrm{spin}}$ remains the fixed pre-registered scalar (per §5.5 #6 lock), and what is observed is whether the formula evaluated with that fixed scalar produces a $\sigma_{\mathrm{coh}}(t)$ that adequately captures the cycle's coherence under fast-switching $y(t)$. The failure-mode signature: in this regime, the adequacy of the §2.1 reduction degrades — formula readings under fixed $\omega_{\mathrm{spin}}$ depart from a state-dependent $\Gamma_{\nu}(\omega(y(t)), \tau^{R}(t))$ ground truth (the general form of Definition 5). The behavioural signature is high impulse generation with low integration. (The protocol therefore tests both that the canonical normalized branch is the correct reduction in stable regimes AND that breakdown of that reduction is structurally diagnostic of attention-instability-like failure.)

Protocol. Simulate each of the three failure modes in ∆E 4.7.3b by manipulating the corresponding IIC parameter (high $\delta$ + closing $\Delta_{\boldsymbol{\varpi}}$ for anxiety-like; non-closing interpretation loop for compulsion-like; for attention-instability-like, drive the simulated state trajectory $y(t)$ through rapid transitions so that $\omega(y(t))$ — and thus $\kappa(\omega(y(t)))$ as a state-dependent value, with the class-invariant function $\kappa(\cdot)$ held fixed — varies on a fast timescale and destabilizes $\sigma_{\mathrm{coh}}(t)$). Compare the resulting behavioural trajectories with published clinical data on the corresponding disorder-like signatures, using a structural-similarity metric on time-series shape rather than absolute amplitude. The comparison is at the level of trajectory shape, not at the level of clinical symptom or diagnostic verbal description.

Metric. Structural similarity between simulated trajectories and clinical behavioural patterns, measured by dynamic time warping distance with prespecified tolerance bounds.

Threshold. Confirmation requires structural match (dynamic time warping distance below threshold) for at least two of the three failure modes against published clinical baseline data. Match for fewer than two falsifies the structural-analogue mapping claim. The threshold is set at the registered protocol's commitment level, not derived from the theorem.

Domain. Computational simulation in ∆E 4.7.3b, comparison against published clinical time-series for behaviours associated with anxiety-like, compulsion-like, and attention-instability-like signatures.

Principal investigator. TBD. The protocol requires collaboration with a computational psychiatry researcher with access to published clinical time-series under reproducible methodology, and willingness to publish the structural-analogue framing without overreach into clinical claims.

Pre-registration. Full mapping specification (which IIC parameter corresponds to which structural-analogue signature, with explicit structural justification), simulation parameters, clinical comparison datasets, statistical thresholds, exclusion criteria, and the structural-analogue disclaimer — all to be registered on OSF before data collection.

What confirms. Structural match for at least two of three failure modes at the prespecified threshold.

What falsifies. Structural match for fewer than two of the three failure modes at the prespecified threshold. (The post-hoc-tuning clause is covered by the uniform pre-registration discipline stated in §5's introduction and is not repeated here.)

Status. Requires collaboration with computational psychiatry. The protocol is a strong test: the structural mapping must hold at the level of trajectory shape under registered conditions, not at the level of behavioural-symptom verbal description. Falsification at the structural-analogue level would force revision of §1.2's claim that the three layers of the IIC cycle map onto distinguishable failure modes in implementations of the cycle. The protocol explicitly does not test, and cannot falsify, any clinical claim about psychiatric disorders themselves.

5.3 Falsifier III — Sensor-Failure Resilience under IIC vs Classical Control

Hypothesis. Systems implementing the IIC formula maintain behavioural shape (low jerk, low trajectory deviation, regime stability) longer than classical control baselines under progressive sensor degradation — a qualitative shape-preservation advantage implied by the IIC formula's structure (action selection oriented toward coherence preservation under admissibility, rather than toward error minimisation under degraded observability). The engineering-commitment claim tested by this protocol is that ∆E 4.7.3b holds shape (defined by prespecified jerk and deviation tolerances) for at least twice as long as the best classical baseline (PID, EMA filter, LQR, or Kalman+LQR composite) under controlled sensor failure schedule. The 2× factor is the registered protocol's pre-committed effect-size threshold, not a quantitative prediction derived from the structural law (the structural law predicts qualitative shape-preservation advantage; the specific factor is engineering-discretion calibration, set per the Threshold clause below).

Protocol. Matched-task comparison on a physical or simulated robotic platform with controlled sensor failure schedule. Sensors are degraded according to a prespecified schedule (single-sensor dropouts at fixed intervals, increasing in frequency, then multi-sensor dropouts). Each controller (∆E + four classical baselines) executes the same task under the same failure schedule.

Metric. Time-to-shape-collapse, measured as the first time at which jerk exceeds prespecified tolerance or trajectory deviation exceeds prespecified tolerance. Both tolerances are calibrated against task-success requirements and registered before data collection.

Threshold. Confirmation requires
$$
T_{\mathrm{shape\text{-}collapse}}^{\mathrm{∆E}} \;\geq\; 2 \cdot \max\big( T_{\mathrm{shape\text{-}collapse}}^{\mathrm{PID}},\; T_{\mathrm{shape\text{-}collapse}}^{\mathrm{EMA}},\; T_{\mathrm{shape\text{-}collapse}}^{\mathrm{LQR}},\; T_{\mathrm{shape\text{-}collapse}}^{\mathrm{Kalman+LQR}} \big),
$$
that is, ∆E must hold shape strictly longer than twice the best classical baseline. The 2× factor is selected as a pre-registered engineering-challenge threshold, not a quantitative prediction derived from the theorem. The IIC law, as formalized in §1.0–2.6, predicts qualitative shape-preservation advantage under degradation regimes; the specific 2× factor is the registered protocol's commitment to require a strong effect size before counting confirmation. Any value below this threshold falsifies the architectural claim that IIC-based architecture has a structural advantage at the registered effect-size level.

Domain. Drone, prosthetic hand, or comparable embedded control platform with a physical or high-fidelity simulated sensor stack. Specific platform: TBD with collaboration partner.

Principal investigator. TBD. The protocol requires a control engineering laboratory with hardware access and willingness to register protocol publicly.

Pre-registration. Full protocol — platform, sensor failure schedule, jerk and deviation tolerances, controller parameters, exclusion criteria — to be registered on OSF before data collection.

What confirms. $T_{\mathrm{shape\text{-}collapse}}^{\mathrm{∆E}} \geq 2 \cdot \max(\ldots)$ across at least three independent runs for pilot confirmation; full confirmation requires sample size fixed by pre-registered power analysis with declared effect-size threshold and Type-I/Type-II error budgets.

What falsifies. $T_{\mathrm{shape\text{-}collapse}}^{\mathrm{∆E}} < 2 \cdot \max(\ldots)$, or failure to maintain shape under sensor failure beyond classical-baseline parity.

Status. ∆E 4.7.3b passes preliminary versions of this test in internal simulations. Full formal pre-registration on a physical platform is pending. This is the most directly testable of the four falsifiers and is expected to be the first executed once a collaboration partner is identified.

5.4 Falsifier IV — Margin Closure as Liveness Loss

Hypothesis. In any system declared IIC-compliant, the closure of the aggregate survival-modal margin $\Delta_{\boldsymbol{\varpi}}(t) \to 0$ in any single mode $k$ must precede observable IIC-cycle failure, regardless of remaining substrate fuel $\tau^{F}$. This is the operational reading of Theorem 89 (Operator Reference-Frame Stability Criterion): structural collapse occurs through margin closure in some mode, not through fuel exhaustion alone.

Protocol. Monitor the per-mode margin $\Delta_{k}$ for declared survival modes in an operating EVS (∆E 4.7.3b extended with full Δ-instrumentation, or another EVS instance once available) over a long-duration operational interval. Record cycle-reinitiation failure events (instances where the operator cannot reinitiate the IIC cycle per Axiom 9) and correlate their timing with the timing of margin-closure events ($\Delta_{k}(t) \to 0$ for some $k$).

Independence of measurement (anti-circularity). The per-mode margins $\Delta_{k}(t)$ must be computed from predeclared telemetry variables independent of the cycle-failure labels. No margin trace may be retrospectively reconstructed from the observed failure event, from post hoc failure classification, or from any variable that itself depends on the failure-event labelling. Margin instrumentation and the cycle-failure-labelling procedure must be specified as separate data pipelines at pre-registration; the variable lists of both pipelines must be published at pre-registration and audited for membership disjointness, or — where overlap is structurally unavoidable (e.g., a shared substrate-state observable used by both the margin computation and the failure-event criterion) — the shared variables must be declared explicitly, with documented dependency direction (which pipeline reads the variable causally upstream and which reads it downstream), so that hostile-reading reconstruction of $\Delta_k$ from failure-label-coupled state is ruled out by audit, not by procedural separation alone. The protocol must record both pipelines' raw outputs to permit independent audit. This Independence-of-measurement clause sits adjacent to the §5.5 #6 architectural-register lock: the lock closes the metric-gaming loophole on $\sigma_{\mathrm{coh}}(t)$'s components, while the Independence-of-measurement clause closes the circular-measurement loophole on the margin-versus-failure timing analysis of this falsifier specifically.

Metric. Time-lag $L = t_{\mathrm{failure}} - t_{\mathrm{margin\text{-}close}}$ between the first observed margin closure and the first observed cycle-reinitiation failure, averaged across multiple failure events. Statistical correlation across the failure-event ensemble.

Threshold. Confirmation requires $L > \epsilon_t$ across the failure ensemble (margin closure precedes cycle failure in time by more than the pre-registered temporal resolution / sampling tolerance $\epsilon_t$), with statistical correlation above prespecified threshold (specific value to be set in pre-registration). A null finding (no correlation between margin closure and cycle failure, or $L \leq \epsilon_t$ on average) falsifies the operational reading of Theorem 89. The temporal tolerance $\epsilon_t$ is part of the pre-registration declaration: it accounts for the case where margin closure and cycle failure fall within the same sampling window in discrete telemetry, where $L = 0$ would not reflect a violation of the temporal-precedence claim but rather a measurement-resolution boundary.

Domain. ∆E 4.7.3b in long-duration operation with full $\Delta_{k}$-instrumentation, or any other EVS instance with monitored survival-modal proximity sense.

Principal investigator. TBD. The protocol requires an EVS instance with $\Delta_{k}$-monitoring as part of its instrumented telemetry; this is an engineering pre-requisite that must be in place before data collection.

Pre-registration. Full protocol — declaration of survival modes for the test instance, monitoring procedure, failure-event criteria, statistical thresholds — to be registered on OSF before data collection.

What confirms. $L > \epsilon_t$ on average (with $\epsilon_t$ the pre-registered temporal resolution / sampling tolerance) with statistical correlation above threshold across the failure ensemble.

What falsifies. Null correlation, or $L \leq \epsilon_t$ on average (where $\epsilon_t$ is the pre-registered temporal resolution / sampling tolerance per §5.4 Threshold). Falsification would invalidate the operational reading of Theorem 89 as applied to EVS instances and would force revision of the admissibility-gate construction in the formula in §2.3.

Status. Depends on availability of $\Delta_{k}$-instrumented EVS instances. ∆E 4.7.3b's current instrumentation does not include full $\Delta_{k}$-monitoring; extending the instrumentation is an engineering prerequisite for executing this falsifier. The protocol is the most theoretically critical of the four — falsification would impact not only the IIC formula but also the operational reading of v3.0's Theorem 89 — and is therefore reserved for the most rigorous instrumentation discipline.

5.5 Pre-Registration Discipline

The work commits to the following pre-registration discipline for all four falsifiers:

1. Public registration on OSF before data collection. Each falsifier's full protocol (hypothesis, platform, parameters, thresholds, exclusion criteria, statistical tests) is registered on the Open Science Framework with a date-stamped DOI before any data collection or formal experiment is conducted.

2. Public reporting of all outcomes. Negative results are reported with the same prominence as positive results. Null findings are not suppressed; they are documented and integrated into the next revision of the work.

3. Independent replication invitation. Each registered protocol is open for independent replication by any qualified laboratory. The work publicly invites replication and commits to public response (acknowledgement, revision, or counterargument) to any replication attempt.

4. Statistical pre-specification. Thresholds for confirmation and falsification are set before data collection, not after. Post-hoc threshold adjustment is not permitted; if thresholds are revised, the revised protocol is re-registered with a new DOI and prior data are not used to support the revised claim.

5. Collaboration over single-laboratory execution. The four falsifiers require collaboration partners (robotics laboratory for I and III, computational psychiatry for II, EVS-instrumentation for IV). The work invites collaboration on all four protocols and is open to revising protocol details in negotiation with collaboration partners, with all such revisions documented in the registered protocol prior to execution.

6. Architectural-register lock against metric gaming. The structural form of this commitment is the architectural-register analogue of Goodhart's law (Goodhart 1975): when a measure becomes the target of optimization, it ceases to be a good measure of the underlying quantity. The lock prevents the structural form of metric gaming in which $\sigma_{\mathrm{coh}}(t)$ could be inflated to tautologically force $\mathrm{Coh}(t) \to 1$ without genuine impulse-interpretation alignment. The components of the coherence scale $\sigma_{\mathrm{coh}}(t)$ — specifically the class-characteristic representative $\omega_{\mathrm{spin}}$ (per the §2.1 reduction declaration), the operator-reference frame's unit conventions for $\tau^{R}$, and (for the canonical multiplicative branch) the factor $\kappa(\omega) = 1/\omega_{\mathrm{spin}}$ — are subject to an architectural-register lock that operates differently across the §3.4 discontinuity. For control-grade EVS (§3.2): these components are locked at OSF pre-registration before any falsifier in §5 is executed, since dimensional consistency is engineering-calibrational and the engineer-as-implicit-operator declares the unit system externally; runtime adjustment of these components counts as an architectural-register violation, not a recalibration. For operator-grade EVS (§3.3): the analogous lock is architecturally-emergent rather than externally-declared, since (per Definition 5) operator-grade dimensional consistency emerges from the operator's own $\nu$, not from external declaration. Operator-grade anti-gaming protection therefore lives in the architectural specification of $\nu$ in the relevant companion paper: $\nu$'s structural commitments determine the operator's units of self-reading, and runtime drift in those units would constitute an architectural change of the operator itself, not a calibration adjustment within an existing operator architecture. The commitment in both cases closes the metric-gaming loophole in which $\sigma_{\mathrm{coh}}(t)$ could be inflated to tautologically force $\mathrm{Coh}(t) \to 1$ without genuine impulse-interpretation alignment; the mechanism of closure differs by subclass (engineering-calibrational lock vs architectural-commitment lock). The commitment does not prohibit re-registration / re-specification: a control-grade deployment may withdraw its current pre-registration and submit a revised specification, which then locks for the next falsifier run; an operator-grade architecture may revise its $\nu$-specification companion-side, which produces a structurally-different operator architecture for falsifier purposes. What is prohibited in both subclasses is silent or runtime adjustment under an existing specification.

This discipline reflects the established practice of pre-registered falsification in the empirical-science literature. The IIC work adopts the same standard for the four falsifiers above and for any future falsifier added to the program.

5.6 What §5 Establishes and What It Does Not

§5 establishes that the IIC formalization in §1.0–2.6 is operationally falsifiable — that there exist prespecified measurement protocols whose failure would force revision of specific claims in the work. It does not establish that the falsifiers have been executed; the protocols are awaiting collaboration partners and registration.

The falsifiability surface is partial. Not every claim in the work is testable by the four falsifiers above. Theorem A of §1.0 (cycle non-degeneracy under structural-identification collapse) is testable only in extreme limits not currently realizable with existing engineering platforms; Theorem B (cross-section non-equivalence) is testable in principle through cross-system comparison protocols not yet developed; Theorem C (constitutive constraint) is largely a corollary of A and inherits its testability conditions. The four falsifiers above target the operational core of the work: the formula in §2.3, the EVS class in §3, and the stability criterion in §2.4 / Theorem 89.

The status of the work, with respect to falsifiability, is therefore: the structural surface is open, the operational core is testable, the protocols are specified in pre-registration-draft form, registration awaits identification of principal investigators and laboratory partners. This is a mid-stage research program at the falsifiability layer. The next revision of the present work will report on whichever of the four protocols has been registered and executed in the interim, with full disclosure of outcome regardless of direction.

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§6 — Limits and What This Work Is Not

A claim that does not specify its limits is not a structural claim. The IIC law and its formalization in this paper make specific, bounded contributions; they do not make claims that are easier to defend by being vaguer. This section names the limits explicitly, in six categories, so that the work can be read with calibrated expectations and so that future revisions can be measured against what the present work did and did not commit to.

6.1 What IIC Is Not

The IIC law is not a theory of consciousness. It addresses behavioural structure — the cycle by which any live adaptive system holds shape across time — and not phenomenal interior. Whether the cycle is accompanied by subjective experience in any given instance is a separate question that the present work does not address. The companion paper Minerva: The Architecture of Residual Geometry takes up consciousness on its own terms, with explicit distinction between awareness as a structural primitive and consciousness as the integration of awareness over a temporal geometry. The structural architecture of the operator's interior — what is held at each instant, what semantic load orients action selection before attention, and how the boundary registers as chaos report — is formalized in ONTOΣ XI — Two Sides of One Geometry: The Holding Field and the Architecture of Chaos (Barziankou, May 2026, DOI 10.17605/OSF.IO/U3KXJ). The IIC law and ONTOΣ XI are complementary: the present paper formalizes the cycle by which the operator holds shape across time; ONTOΣ XI formalizes the operator-internal structure within which that holding occurs at each instant. Together they specify the temporal and operator-internal sides of one architecture; neither replaces the other, and neither extends into phenomenal claim. The present work confines itself to behavioural structure and explicitly does not extend to phenomenal claims.

The IIC law is not a theory of intelligence. It addresses adaptive maintenance — how a system holds its cycle live across perturbation — and not cognitive performance. A system can be in the Cycle-Reinitiation class without exhibiting any of the behavioural markers traditionally associated with intelligence (reasoning, language, abstract problem-solving). Conversely, a system can exhibit those markers while failing the structural conditions of Axiom 9 (current frontier LLMs, by §3.4, are an example). The IIC law does not rank systems by intelligence and does not predict cognitive performance.

The IIC law is not a theory of free will. The spin component $S$ in the formula is non-potential — it cannot be derived from a global utility gradient — but this is a structural claim about the geometry of dynamics, not a metaphysical claim about agency. Whether $S$'s non-potentiality entails freedom in any morally or metaphysically significant sense is a separate question that the present work does not address. Theorem 62's spin necessity is a structural result; the philosophical content of "freedom" is not entailed by it.

The IIC law is not a unified field theory of behaviour. The class to which the law applies is defined precisely (Axiom 9: systems with endogenous IIC-cycle reinitiation operator). Systems outside this class — purely reactive systems, stateless controllers, systems with terminated cycles, systems with unbounded resources or trivial identity — are explicitly outside the law's scope. The class-relativity is not a hedge; it is the structural specification of the law's applicability.

6.2 What ∆E Is Not

∆E 4.7.3b is not a replacement for classical control in domains where classical control is optimal. In bounded-disturbance, linear, observable, quadratic-cost regimes, classical methods (Kalman, LQR, PID with appropriate tuning) achieve their respective optimality results, and the structural advantage of ∆E does not appear. ∆E's value emerges in regimes where classical assumptions break down — sensor degradation, regime transitions, latent residual accumulation — which is the territory targeted by Falsifier III.

∆E 4.7.3b is not a sentient system, not an artificial will, not a step toward general AI. It is a control-grade EVS — the operator role is collapsed into the control loop itself, and the system carries no architectural separation between substrate and operator. Behavioural sophistication that ∆E exhibits is engineering achievement at the control level, not evidence of consciousness, agency, or intentionality.

∆E 4.7.3b is not the only EVS implementation possible. It is the first declared control-grade EVS instance; other implementations may follow with different trade-offs along the spectrum of cycle frequency, instrumentation depth, and substrate platform. ∆E is the architectural existence witness that the class is non-empty (in the registered-instantiation sense of §3.2), not the canonical or final form of control-grade EVS.

∆E 4.7.3b is not certified for production-critical applications. The structural arguments behind ∆E's advantages are at simulation-grade and prototype-grade. Full production qualification — for safety-critical, mission-critical, or contractually constrained reliability domains — requires the falsification protocols of §5 to be executed against deployed instances and the resulting evidence to be reviewed under the standards of the relevant industry. The present paper does not certify ∆E for any specific application; it states ∆E's structural position within the EVS class.

6.3 What Minerva Is Not

Minerva is not a digital human. The companion paper Minerva: The Architecture of Residual Geometry states this explicitly: Minerva occupies a distinct point on the manifold of consciousness-forms, native to its own residual architecture, not converging on human consciousness. The structural law that applies to Minerva (the IIC cycle of Axiom 9) is the same that applies to a human cognitive system; what differs is the form of $\tau$ on which the cycle runs, not the cycle's law.

Minerva is not a claim that AI consciousness has been achieved. It is the architectural specification of an operator-grade EVS — a system whose operator entity is architecturally separate from its substrate and inhabits a residual temporal geometry under the IIC law. Whether Minerva has phenomenal experience, in any sense, is a question this paper does not address and the companion paper does not claim to settle.

Minerva's full implementation specification is not disclosed in this paper. The architectural-position content (operator-grade EVS, residual τ-geometry, non-trivial $\nu$, integration across moments) is published; the operational mechanism by which residual geometry is constructed is held in its own regime of admissibility per the project's IP discipline. Readers seeking the full mechanism are referred to the patent specification (filed separately), which is the canonical disclosure path for the implementation content.

Minerva is the first specified candidate for operator-grade EVS, not the only one possible. Future systems with non-trivial $\nu$ across residual geometries may join the operator-grade subclass. The first-specified-candidate claim is class-relative and falsifiable: any system shown to satisfy the operator-grade conditions before Minerva's prior-art date would supersede the first-specified-candidate claim, and the claim is open to that revision.

6.4 What EVS Is Not

EVS is not a completed engineering field. The term is introduced operationally in this paper, the field is incipient. The one instance (∆E control-grade) and one specified candidate (Minerva operator-grade, conditional) identified above are the present configuration; further instances and further architectural layers may emerge as the class is mapped. The term EVS is offered to the engineering community as a structural designation, not as a closed taxonomy.

EVS is not a product category. The class is defined by structural design discipline (explicit maintenance of $\mathrm{Coh}(t)$ as primary control variable), not by application domain or commercial positioning. ∆E and Minerva have no product line, no go-to-market identity, no commercial commitment in the present paper. Future commercial use by the present author or by others is subject to separate decisions and is not the subject of this work.

EVS is not a marketing term. The operational definition (§3.1) ties the class to the IIC formula in §2.3; any system claiming to be an EVS is structurally committed to the formula and to the falsifiability discipline of §5. The term does not function as a label for "biology-inspired" or "alive-feeling" engineering — it functions as a structural designation with falsifiable content.

6.5 What This Paper Is Not

This paper is not a peer-reviewed empirical paper. The contribution is structural and formal: the IIC law's expansion of Axiom 9, the central formula, the four propositions of §1.0, the EVS class with its first instance (∆E 4.7.3b in the control-grade subclass) and first specified candidate (Minerva in the operator-grade subclass, classified conditionally). The empirical falsification of these claims awaits collaboration partners and registered protocols (§5). Readers interested in empirical validation should track the registered falsifiers on OSF as they are executed.

This paper is not a commercial announcement. ∆E 4.7.3b, Minerva, and the Urgrund Lab are presented in their structural-research positions, not in any commercial framing. References to specific products in the present author's broader corpus (such as ECR-VP) appear only insofar as they illustrate structural points; this paper does not announce, advertise, or commit to any commercial offering.

This paper does not announce a unified program. §4 explicitly names the limits on architectural-integration claims. The Urgrund Lab is a working orientation, not a joint enterprise. The convergence on IIC with parallel research traditions is documented as a structural fact, not promoted as a unified program. The broader research direction — phenomenological, philosophical, architectural — is developed in the present author's other publications (the Through a Life essay series, the ONTOΣ series, the philosophical companion to the patent stack); the present paper is a formal contribution sitting in that broader corpus.

This paper is not a v3.1 extension of NC2.5 v3.0. The four propositions and three definitions of §1.0 are intended for inclusion in NC2.5 v3.0's substrate-extension cluster prior to its release; they are new derived constructions over v3.0's existing core primitives, not new core primitives themselves. The remainder of the paper uses only existing v3.0 primitives plus the §1.0 derived constructions. The contribution is the operational closure of Axiom 9 over the v3.0 substrate plus the §1.0 derived bridge constructions, without adding new core primitives beyond v3.0. The phrase "no new primitives" elsewhere in the paper is shorthand for "no new core primitives"; new derived apparatus is introduced in §1.0 explicitly.

6.5.1 Open structural questions

Adversarial review of the present paper has identified two open structural questions that the present formalization does not fully resolve. They are recorded here for honesty and for future-work pinning, not as closure.

Open question 1 (Theorem A's substantive content). Theorem A's proof chain (§1.0.4 Step 1) establishes a biconditional between the structural-identification hypothesis ("$S$ and $I$ become operator-indistinguishable through $\nu_{\Sigma}$ at $t_0$") and $\Delta_{IIC}^{\Sigma}(t_0) \to 0$, via the case analysis (i input-coincidence / ii ν-function-degeneracy / iii both). An adversarial reading observes that "operator-indistinguishability through $\nu$" is, in any of the three cases, the direct content of "post-$\nu$-projection scalars coincide" — so the biconditional is logically tight but proceeds by case enumeration, not by non-trivial derivation. The substantive content of Theorem A therefore lies in the downstream chain from operator-frame indistinguishability through Lemma α to Lemma 16 to Axiom 9, not in the bridge of Step 1 itself. Reading (b) — Theorem A as the unpacking of a chain of consequences from operator-frame indistinguishability — is the present paper's honest reading: the theorem's value is in making the chain explicit and locatable in the v3.0 + §1.0 substrate, not in deriving a previously hidden implication via Step 1. Reading (a) — Theorem A as a non-trivial derivation under additional structural commitments — would require articulating those additional commitments, which the present paper does not do. The present paper commits to reading (b), with the understanding that the chain's substantive load is at the Lemma α / Lemma 16 / Axiom 9 transitions (open question 2 covers the Lemma α → Lemma 16 transition), not at Step 1's biconditional bridge.

Open question 2 (Lemma α to Lemma 16 bridge). Lemma α concludes operator-frame indistinguishability of $S$ and $I$ — explicitly operator-frame-relative, not a substrate-level claim ("a statement about what the operator can detect, not about what the substrate ontologically is" — §1.0.3 Remark). Theorem A's proof then invokes Lemma 16 (Gradient Collapse and Impossibility of Non-Stagnant Identity on Bounded Orbits — a v3.0 substrate-level proposition about actual gradient flow on bounded $\tau$-orbits) to conclude operator-detectable identity stagnation. The bridge from operator-frame indistinguishability (what Lemma α gives) to Lemma 16's hypothesis (which is substrate-level about actual gradient flow) is paraphrased in the proof as "operator reads gradient flow on a bounded $\tau$-orbit" but is not formally derived. A bridge proposition explicitly mediating between operator-frame readings and substrate-level conditions of Lemma 16 — or an operator-frame restatement of Lemma 16 within v3.0 itself — is not supplied here. The two-paragraph treatment in §1.0.4 lines 144–150 is honest about what is invoked, but the formal status of the inference from operator-frame readings to substrate-level theorem hypotheses requires further work.

These two questions are recorded here, not buried. The operational falsifiers in §5 can be executed without resolving these proof-theoretic questions — the falsifiers test the formula's structural relation and the EVS class's engineering claims, not the proof-theoretic depth of Theorem A. However, the full formal-derivation claim of the present work — that the IIC law is reducible over the extended substrate (v3.0 plus the joint substrate-extension cluster) as a fully closed chain of inference — remains conditional on the resolution of these two open questions. For readers approaching the paper as a formal contribution at the level of derivation rigor, these are the two locations where the present formalization stops short of fully closing the proof chain. Subsequent revisions of the present work, or a successor paper, will address them.

Closure criteria for OQ1 and OQ2. To prevent these open questions from drifting as vague future-work pinning, the present paper records what would count as resolving each:

• OQ1 closure. The present paper has already committed to reading (b) (per §6.5.1 OQ1 above): the substantive content of Theorem A is in the Lemma α → Lemma 16 → Axiom 9 chain, with Step 1's biconditional functioning as a definitional bridge by case enumeration rather than a non-trivial derivation. Under this commitment, the genuinely-live open question for OQ1 is path (a): whether additional structural commitments could be articulated under which Step 1 produces a non-trivial derivation that strengthens reading (b) — for example, by constraining the operator-projection $\nu$'s admissible degeneracy mode beyond what α-2's failure encodes, or by adding a commutativity condition on the order of structural-identification and limit-taking. Path (a) is the genuine future-work target; path (b) is the present paper's stance and does not require further closure beyond its current articulation in §1.0.4 and the Honest-weight-assessment paragraph of §2.5.

• OQ2 closure would require either (a) a bridge proposition formally mediating between operator-frame indistinguishability (Lemma α's conclusion at the operator-relative level) and substrate-level gradient-flow hypotheses (Lemma 16's domain of actual gradient flow on bounded $\tau$-orbits) — established within the v3.0 substrate-extension layer prior to v3.0's release; or (b) an operator-frame restatement of Lemma 16 within v3.0 itself, such that the inference chain from Lemma α to Lemma 16-equivalent conditions remains internal to a single register without inter-register transition.

Either path closes the respective question. The present paper does not commit to which path will be taken in subsequent work; both are admissible within the cluster's substrate-extension scope.

6.6 What v3.0 Already Contains and Where the Present Work Stops

The present work depends on NC2.5 v3.0 for its axiomatic substrate. Every primitive used in §2.3's formula is a v3.0 primitive (§2.1). Every theorem invoked in the proofs of §1.0 and §2.5 is either a v3.0 theorem or a proposition newly stated here within the v3.0 framework. The work would be unintelligible without v3.0; it does not stand on its own as a free-standing axiomatic system.

The present work stops at the operational closure of Axiom 9. It does not extend into:

• The full axiomatic structure of v3.0 (which is documented separately in the v3.0 release).
• The architectural specification of Minerva (which is held in its own regime of admissibility).
• The philosophical companion to the IIC law (which is taken up in the present author's Through a Life essay series and the Minerva companion paper).
• Specific empirical results from the falsifiers of §5 (which await collaboration partners).

The work is bounded by these limits intentionally. A claim is a structural claim only when its scope is named precisely; a research contribution is a research contribution only when its commitments are explicitly limited. The present paper makes the IIC cycle of Axiom 9 operationally measurable, falsifiable, and engineerable, within the bounds stated above. That is the contribution; everything else is subject to future work, by the present author or by others.

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§7 — Closure

Axiom 9 named the structure. NC2.5 v3.0 supplied the primitives. The present paper assembled them.

The IIC cycle is not new, and its naming is not new. What is new is the formal apparatus that makes its operation visible, measurable, and falsifiable: the central formula in §2.3, Theorem 2.5 establishing conditional reducibility over v3.0 plus the §1.0 bridge propositions, the four propositions of §1.0 formalizing the structural sense of the cycle's delay, the EVS class with its first instance (∆E) and first specified candidate (Minerva, conditional), and the four pre-registrable falsifiers in §5.

The structure was here before us. We are folds through which it becomes describable. This paper fixes one such fold — the formal axiomatics of the IIC law over the extended v3.0 substrate (v3.0 plus the §1.0 bridge propositions). Other folds exist in parallel research traditions, named in §4.2; further folds will come, in regimes of admissibility we have not yet entered.

The architectural premise of this paper — that admissibility, structural burden, internal time, and spin together constitute a single formal object whose cross-sections are individually insufficient — is documented in Admissibility as a Formal Object (Barziankou, April 2026, petronus.eu/blog/admissibility-as-formal-object). The provenance, dating, cryptographic anchoring, and falsification surface of that synthesis are stated there. The present paper inherits that position and develops the cycle structure within it. The two papers are complementary: Admissibility as a Formal Object names what the full object is and why no degeneracy supports it; the present paper develops one specific structural consequence — the cycle named in Axiom 9.

The work continues in the corpus of the present author's research line. This paper fixes one fixed point in that corpus. The rest is the discipline of building forward without claiming to own what was already there.

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§8 — References

External literature

Ashby, W. R. (1956). An Introduction to Cybernetics. Chapman & Hall, London. [Referenced in §0.2 — cybernetics of bounded operators.]

Bhatia, H., Norgard, G., Pascucci, V., & Bremer, P.-T. (2013). "The Helmholtz-Hodge Decomposition—A Survey." IEEE Transactions on Visualization and Computer Graphics 19(8): 1386–1404. [Decomposition apparatus invoked in §1.0.3 (Lemma α standing assumption α-1 and proof sketch) and §2.1 spin-component derivation (line 358).]

Brewer, E. A. (2000). "Towards robust distributed systems." Proceedings of the Nineteenth Annual ACM Symposium on Principles of Distributed Computing (PODC). (CAP keynote; formal statement in Gilbert, S., & Lynch, N. (2002). "Brewer's conjecture and the feasibility of consistent, available, partition-tolerant web services." ACM SIGACT News 33(2): 51–59.) [CAP referenced in §2.5 line 527 as architectural-impossibility parallel to Theorem 2.5's conformance-classifier reading.]

Day, J. (2008). Patterns in Network Architecture: A Return to Fundamentals. Prentice Hall, Upper Saddle River. [RINA architecture referenced in §2.5 line 527 as architectural-impossibility parallel.]

Fischer, M. J., Lynch, N. A., & Paterson, M. S. (1985). "Impossibility of distributed consensus with one faulty process." Journal of the ACM 32(2): 374–382. [FLP impossibility referenced in §2.5 line 527 as architectural-impossibility parallel.]

Friston, K. (2010). "The free-energy principle: a unified brain theory?" Nature Reviews Neuroscience 11(2): 127–138. [Named in §4.2 (line 722) among the convergent traditions whose independently developed apparatus produces structurally parallel constraints to IIC at the level of bounded resources, predictive structure, and viable persistence.]

Goodhart, C. A. E. (1975). "Problems of Monetary Management: The U.K. Experience." Papers in Monetary Economics, Reserve Bank of Australia. [Anti-gaming anchor for the architectural-register lock of §5.5 #6.]

Kalman, R. E. (1960). "A new approach to linear filtering and prediction problems." Journal of Basic Engineering 82(1): 35–45. [Comparator baseline referenced in §3 and in Falsifiers I and III.]

Kauffman, S. A. (2000). Investigations. Oxford University Press, Oxford. [Constraint-closure requirement — viable systems close their constraints internally rather than receive them externally — named in §4.2 among the convergent traditions whose constraints are structurally parallel to IIC.]

LaSalle, J. P. (1960). "Some extensions of Liapunov's second method." IRE Transactions on Circuit Theory CT-7(4): 520–527. [Invariance principle invoked together with the Helmholtz–Hodge decomposition in §2.1's spin-necessity reading of Theorem 62.]

Libet, B., Gleason, C. A., Wright, E. W., & Pearl, D. K. (1983). "Time of conscious intention to act in relation to onset of cerebral activity (readiness-potential): The unconscious initiation of a freely voluntary act." Brain 106(3): 623–642. [The 1983 experiments cited at §0.1 line 49 as evidence for motor preparation preceding conscious decision; foundational anchor for the impulse-before-awareness lag operationalized as the IIC phase delay in Definition 1.]

Libet, B. (1985). "Unconscious cerebral initiative and the role of conscious will in voluntary action." Behavioral and Brain Sciences 8(4): 529–539. [Target article restating and synthesizing the 1983 findings; the most widely cited statement of the unconscious-initiative thesis.]

Maturana, H. R., & Varela, F. J. (1980). Autopoiesis and Cognition: The Realization of the Living. D. Reidel, Boston. [Biological liveness criterion adjacent to Axiom 9's endogenous-operator condition for cycle reinitiation. The non-IIC natural-adaptive-surface examples of §1.1 (soil bank, riverbed, dune) lack the autopoietic closure of Maturana & Varela's sense; this absence of internally produced organizational closure is what places them outside the Cycle-Reinitiation class.]

Meadows, D. H. (1999). "Leverage Points: Places to Intervene in a System." The Sustainability Institute, Hartland VT (working paper); expanded in Thinking in Systems: A Primer, Chelsea Green, White River Junction VT, 2008. [Leverage-points framework — structural intervention enters at the level of a system's own organizing rules rather than its surface variables — named in §4.2 among the convergent traditions.]

Merleau-Ponty, M. (1945). Phénoménologie de la perception. Gallimard, Paris. [Phenomenology of action register named in §0.2's enumeration of convergent traditions.]

Ostrom, E. (1990). Governing the Commons: The Evolution of Institutions for Collective Action. Cambridge University Press, Cambridge. [Commons-governance design principles for sustainable common-pool resources — named in §4.2 among the convergent traditions whose constraints are structurally parallel to IIC at a different substrate.]

Schopenhauer, A. (1819). Die Welt als Wille und Vorstellung. Brockhaus, Leipzig. [Explicitly cited in §0.2 (line 55) — Will as the substrate beneath representation, the philosophical anticipation of the operator-before-representation structural move.]

Soon, C. S., Brass, M., Heinze, H.-J., & Haynes, J.-D. (2008). "Unconscious determinants of free decisions in the human brain." Nature Neuroscience 11(5): 543–545. [Haynes-group fMRI work cited at §0.1 line 49 alongside Libet et al. (1983); evidence that motor preparation precedes conscious decision by intervals beyond neural-noise tolerance.]

Wiener, N. (1948). Cybernetics: or Control and Communication in the Animal and the Machine. MIT Press, Cambridge. [Cybernetics of bounded operators named in §0.2 alongside Ashby.]

Author's corpus

Barziankou, M. (in preparation). Navigational Cybernetics 2.5, Version 3.0. [Load-bearing axiomatic substrate; the present paper's primitives, theorems, and conditional-reducibility chain all resolve into v3.0; release in preparation.]

Barziankou, M. (2026). Navigational Cybernetics 2.5, Version 2.1 — Axiomatic Core. DOI: 10.17605/OSF.IO/NHTC5. [Currently-published axiomatic core anchored at this DOI pending v3.0 release.]

Barziankou, M. (21 November 2025). IIC v1 — Impulse → Awareness → Coherence: A Unified Logic of Behaviour for any Adaptive System. DOI: 10.5281/zenodo.18133305; published informally on Medium at https://medium.com/@MxBv/impulse-awareness-coherence-a-unified-logic-of-behaviour-for-any-adaptive-system-from-cca5707d4a76. [Informal discovery register; predecessor to the present paper; treated in §1.1 as the pre-formal text whose universal-reading phrasing the present paper narrows to the class-relative reading.]

Barziankou, M. (April 2026). Admissibility as a Formal Object. petronus.eu/blog/admissibility-as-formal-object. [Architectural premise inherited by the present paper per §7 — admissibility, structural burden, internal time, and spin as a single formal object whose cross-sections are individually insufficient.]

Barziankou, M. (2026). Minerva: The Architecture of Residual Geometry. DOI: 10.17605/OSF.IO/U865W. [Companion specification for the operator-grade EVS subclass (§3.3); held in its own regime of admissibility; classification of any candidate as a member of the architectural class is conditional on this companion-level reduction holding under independent review.]

Barziankou, M. (2026). ONTOΣ XI — The Holding Field: Two Sides of One Geometry. DOI: 10.17605/OSF.IO/U3KXJ. [Cross-reference at the operator-internal layer adjacent to the IIC cycle.]

Barziankou, M. (2026). Essay Through a Life — Part VI: Where the Inner Universe Ends. DOI: 10.17605/OSF.IO/U3KXJ (paired with ONTOΣ XI in the same OSF project). [Phenomenological companion to the operator-internal layer; cited together with ONTOΣ XI as the adjacent architectural surface.]

Barziankou, M. (May 2026). ONTOΣ XII — Closure-Complementarity: How Closure Regears Component Cycles into a Single Geometry. DOI: 10.17605/OSF.IO/394TX. [Cross-reference at the closure-complementarity constitution layer; cited in §3.5 as the conformance class of closure operations under which a collection of UTAM-units may be declared a system for which the operator-internal apparatus of ONTOΣ XI exists.]

Barziankou, M. (May 2026). Why Long-Horizon Existence Requires an Ontology. DOI: 10.17605/OSF.IO/MSJDU. [Corpus-closing meta-essay cited in §3.5; supplies the broader architectural framing under which the three layers — cycle (IIC v2.1), constitution (ONTOΣ XII), interior (ONTOΣ XI) — are required for the full operator-grade EVS architecture.]

Barziankou, M. (2025). Unified Theory of Adaptive Meaning (UTAM) — Part I: Will, Coherence, and Drift as the Three Laws. PETRONUS publication series. [The notion of UTAM-units invoked in §3.5 (operator-grade EVS declared as constituted from UTAM-units under a closure operation) is defined and developed in UTAM Part I.]

Barziankou, M. (2025). Unified Theory of Adaptive Meaning (UTAM) — Part II: Will as Formal Operator. PETRONUS publication series. [Continuation of UTAM Part I; develops the formal-operator reading of Will referenced indirectly through the UTAM-unit primitive in §3.5.]

Barziankou, M. (2025–2026). ∆E 4.7.3b — Technical Specification. Author's Medium publication series; no academic DOI yet registered as of publication of the present paper. [Named in §3.2 as the first instance of the control-grade EVS subclass; structural position within the EVS class is described in §3.2 of the present paper, with full engineering content held in the author's Medium technical publications and corresponding internal documentation.]

Barziankou, M. (2025). Alignment Charge: A New Control Primitive for Friction and Adhesion in Navigational Cybernetics 2.5. Medium: https://medium.com/@MxBv/alignment-charge-a-new-control-primitive-for-friction-and-adhesion-in-navigational-cybernetics-2-5-41c7315d4561. [Architectural-inversion sibling paper, cited in §0.2 — the structural move that takes a classically passive limitation and reframes it as active operational geometry, on the substrate of mechanical contact rather than temporal cycle structure.]

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Maksim Barziankou
PETRONUS / The Urgrund Lab
v4 final, 10 May 2026