Identity Does Not Drift

Channel Switching, the Two Ledgers, and the Parallel Tunnel — a Structural Note in NC2.5

Navigational Cybernetics 2.5, The Urgrund Lab, MxBv 2026

Maksim Barziankou (MxBv)

July 2026 · Poznań

Contact: [e-mail]

LinkedIn: https://www.linkedin.com/in/maxbarzenkov

License: CC BY-NC-ND 4.0

This work DOI: 10.17605/OSF.IO/4NMTW
Axiomatic core anchor: NC2.5 v2.1, DOI 10.17605/OSF.IO/NHTC5

Formal apparatus: ONTOΣ XV, bundle DOI 10.17605/OSF.IO/EAUD5

Website: https://petronus.eu

One work in the 130+ work corpus of Navigational Cybernetics 2.5. The formal core assembles from the published transport machinery of ONTOΣ XV; the perceptual and biological readings are Illustration-level throughout.

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"The budget wears out. The witness does not: it rides unchanged through the tunnel, or changes by an event".
— MxBv, July 2026

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0. Register (read first)

This note makes four formal statements and one architectural claim, and it keeps their registers apart.

• [STRUCTURAL — derived] The Switching Lemma (§4) and the Quantised-Change Proposition (§5.1) are derived within the finite-state transport apparatus imported from the ONTOΣ XV companion (bundle DOI 10.17605/OSF.IO/EAUD5): transport verdicts (its Proposition 4.3), carrier restrictions (its Lemma 4.9 and Definition 4.10), the declared drift class (its Definition 2.9), and maintenance quantisation (its Proposition 3.11). Both statements are short, and that is the point: their content lives in the typing, not in the derivation — exactly as XV's own derived-restriction lemma carries its content in the image-preservation condition.
• [STRUCTURAL — derived + search-backed existence] The Protected-Covector Proposition (§5.2) is elementary finite linear algebra over the induced homology actions. Its non-tunnel, noncommuting example and the Order-Sensitivity Corollary (§5.3) are independently replayed from an exhaustive certificate in a declared bounded model. The search exhausts all 544 admissible records and establishes existence and minimum total image-word length only inside that model; it is not empirical evidence and is not imported from XV.
• [ARCHITECTURAL — declared] The claim that capacity-limited perception is organised as a switching schedule over channels, governed by a controller whose action is witness-silent (§2, §6), is a declared architecture of the class this note names — not a theorem about brains, spiders, or devices. It carries its falsification surface in §7.
• [ILLUSTRATION] The two-eye observation (§1), the salticid spider (§6), and the momentum-computing bit swap (§5) are illustrations. The cited empirical observations motivate structural analogies only; no claim is made that any biological or physical system satisfies this note's formal typing, and nothing here is deployment-witnessed.

Throughout, "identity" is read in the imported formal sense: the topological witness class [η] on a declared finite recurrence carrier — ONTOΣ XV §1, inheriting XIV (bundle DOI 10.17605/OSF.IO/KAGMH). Its period pairing is a numerical witness observable, not identity itself. The distinction matters: under a non-tunnelled transport a non-zero period may change sign or magnitude while the class remains witness-bearing; zero pairing is the annihilating verdict for that declared class. In this note, drift means budget-style wear — monotone accumulation through admissible increments with no positive minimum step. Proposition 5.1 excludes that form of sub-quantum wear from the witness observable; it does not claim that every non-tunnelled transport leaves the period numerically fixed. Proposition 5.2 identifies a weaker, witness-relative route to exact constancy: a non-tunnel alphabet may share a fixed covector even while acting non-trivially on the rest of the carrier. The tunnel remains the stronger map-level condition. Nothing here is a psychological or phenomenal claim.

1. The Observation

Lie on your side, close to a large dog, and look past it at the far wall. The left eye, below the fur line, sees fur — close, dense, holding attention near. The right eye, above it, sees a clean horizon. Close the right eye: only fur. Close the left: only horizon. Open both: the horizon holds, and on it lies a translucent mirage of fur — not an average of the two scenes, but one scene elected as load-bearing with the conflicting channel rendered over it, half-admitted.

Vision science knows the ingredients of this picture well — binocular rivalry, interocular suppression, the constructed cyclopean scene. This note takes from the observation only its structural skeleton, which does not depend on the physiology: there are multiple channels; they are not fused symmetrically; something elects, weights, suppresses, and switches; and through all of it, the one who is looking remains continuously one. That last clause is a phenomenological motivation, not the identity notion formalised below. The channels flicker, alternate, contradict each other, get suppressed and readmitted; the formal question is whether an independently declared witness can remain outside that traffic.

The question this note answers structurally: what must an architecture satisfy for that to be possible — for a system to time-share a capacity-limited arsenal of channels, paying real accumulating costs, without its identity ever entering the traffic?

2. The Architecture: Capacity Forces Scheduling under Full-Arsenal Coverage

Fix the class. A system in this note's class carries:

• a declared finite identity carrier M — the recurrence carrier of the imported apparatus — with a sealed witness class [η] ∈ H¹(M) and at least one witness-bearing recurrence class (XV §1; the witness is what the topological functor reads, and its period pairing is the witness observable);
• a declared finite family of channel loci C₁, dots, C_N — the perception/reading channels, each budget-metered in the sense of the X+M ledger discipline;
• a declared capacity bound 1 le k < N: at most k channels are active in any window;
• a declared full-arsenal coverage requirement: every channel must be active in at least one window;
• a switching schedule: a window sequence w₁, dots, wₘ with active sets Aⱼ ⊂eq {1, dots, N}, lvert Aⱼ rvert le k, bigcupⱼ₌₁ᵐ Aⱼ = {1, dots, N}, and transition events t₁, dots, tₘ₋₁ between consecutive windows.

Because k < N, no single window can meet the coverage requirement. Any full-arsenal execution therefore needs at least lceil N/k rceil non-empty windows and at least two distinct active sets: under declared coverage, capacity forces scheduling. Without the coverage requirement, k < N would show only that simultaneous full-arsenal reading is impossible; a system could ignore some channels forever. The schedule is not an imperfection of this architecture — it is the architecture's answer to capacity plus coverage. What the schedule costs, and what it does not cost, is the substance of the next three sections.

3. The Two Ledgers

The imported apparatus keeps two books, and they behave differently under the schedule.

The budget ledger drifts. Window by window, the active channels accrue burden: prefix-difference increments, monotone, irreversible, never refunded — the declared drift class of XV Definition 2.9 is the discipline of this accrual, with its sealed per-window bands and its forced-crossing horizon. Alternating channels does not stop the drift; it only distributes it. The system as a whole ages through its schedule. This is the ledger on which fatigue, depletion, and forced exhaustion live.

The witness ledger does not wear. The witness is a class, not a level. Under a transport event, its pairing on a declared recurrence class can remain non-zero (witness-preserving for that class) or vanish (annihilating for that class); across a declared family, the partial verdict records which members retain non-zero pairing and which do not (XV Proposition 4.3 and Remark 4.4). A non-tunnelled transport may also change the sign or magnitude of a non-zero period — for example 3 → -3 or 3 → 9 — without annihilating that declared witness-bearing class. Exact numerical constancy is guaranteed either carrier-wide by the tunnel (§4) or witness-relatively by a verified common fixed covector (§5.2); it is not the definition of preservation in general. What the admissible lattice-valued evolution excludes is budget-style wear: below one quantum there is no permitted increment to accumulate (§5, under the integral or common-denominator rational witness data declared there). On the budget axis, arbitrarily small losses may accumulate into exhaustion; on the witness axis, the observable repeats or changes by an event-sized lattice step, and zero marks annihilation for the declared class.

The night formulation that this note fixes: budget can wear by arbitrarily small steps; identity cannot. In the tunnel it is preserved exactly. A non-tunnel alphabet may also preserve one witness exactly when its generators share that fixed covector; otherwise the witness observable may jump, and the witness-bearing verdict must be read anew.

4. The Switching Lemma

Definition 4.1 (Parallel-tunnel condition). A transition event tⱼ of a switching schedule is tunnelled iff its carrier restriction to the identity carrier M is the identity graph map — the switch reconfigures active channel loci and their control data only, and touches neither the states nor the recurrence edges of M. (This is a typing condition in the register of XV Definition 4.10: it must be declared, and it is checkable on declared finite data.) In the companion's finite representation, the declaration is total: every carrier vertex and every source edge must appear. An edge collapse is represented explicitly by an empty image path; omission is not a second spelling of identity or collapse.

The parallel-tunnel condition types only the witness coordinate. A separate question is whether a trajectory admissible in one window remains admissible after the switch. Domenoid of Admissibility (DOI 10.17605/OSF.IO/3ESN4) establishes why this cannot be inferred from the forgotten dynamics: the functor U:mathsf{AdmDyn}→mathsf{Dyn} is faithful but not full, so even an identity map of a dynamic skeleton need not preserve independently supplied admissibility data. If windows are assigned objects (mathsf{D}ⱼ,mathrm{Adm}ⱼ) and a switch induces τⱼ:mathsf{D}ⱼ→mathsf{D}ⱼ₊₁, then a cross-window admissibility claim carries the independent weak-morphism obligation mathrm{Adm}ⱼ⊂eqτⱼ^*(mathrm{Adm}ⱼ₊₁). A switch may therefore be tunnelled yet fail admissibility transfer, preserve admissibility yet be non-tunnelled, satisfy both, or satisfy neither. Lemma 4.2 uses only the tunnel coordinate; the Domenoid supplies neither witness-preservation nor quantisation proof.

Structural Spin as a Forgetful Separation over Dissipative Dynamics (DOI 10.17605/OSF.IO/94GWQ) supplies the continuous analogue of this independence. On its canonical domain, a viability-exact morphism may still annihilate the de Rham witness; with a=[ηS] and b=φ^*[η(S')], its topological identity defect is ker bsetminusker a, empty exactly when ker b⊂eqker a, equivalently a=λ b. This preserves non-vanishing, not period values: λ is free. The tunnel guarantee used below is stronger in what it asserts — its identity restriction and sealed witness force exact constancy, not merely an empty identity defect. Structural Spin supplies neither the finite switching derivation nor Proposition 5.1's lattice hypothesis.

Lemma 4.2 (Switching preserves identity). Let a switching schedule have every transition tunnelled. Then the composite transport across the schedule is witness-preserving, and the transported witness period is constant across all windows: for every witness-bearing recurrence class [γ],

langle [η], [γ] rangle_(w₁) = langle [η], [γ] rangle_(w₂) = dots = langle [η], [γ] rangle_(wₘ),

regardless of the budget increments accrued in the switched channels, the choice of active sets, and the length of the schedule.

Proof. With the witness held sealed across transitions (§2), a tunnelled transition's carrier restriction is the identity graph map on M, so its induced pushforward on cycles and its induced action on H¹(M) are the identity; the pairing langle [η], [γ] rangle is therefore invariant at each step, and the composite of identities is the identity (the composition calculus of XV, applied in its trivial case). Budget increments live on the channel loci and enter the prefix-difference functionals of the drift class, which by declaration take no argument on M: the two ledgers do not share a variable. blacksquare

The proof is three lines, and honestly so: like XV's derived-restriction lemma, the entire content sits in the condition. What the lemma buys is the separation: it is not that switching is free — the budget ledger pays for every window — but that the payment is confined by typing to the ledger that can absorb it. A system with a tunnelled schedule uses its whole arsenal in alternation, ages while doing so, and carries its identity through untouched. The tunnel is "parallel" in exactly this sense: the witness rides a locus that the switching traffic never enters.

What the lemma does not say. It does not say switches are automatically tunnelled — the condition is declared, and §7 gives the falsifier for a declaration that lies. It does not say the controller is free — the controller's own actions are budget-charged like everything else (§6). And it does not say identity survives everything: a transition whose carrier restriction is not trivial is an ordinary transport and receives an ordinary verdict, including annihilation (XV Proposition 4.11(b): the pinch) and non-zero transformations whose period changes sign or magnitude. Proposition 5.2 shows that a non-tunnel alphabet can nevertheless guarantee exact constancy for one sealed covector when every generator fixes it. Thus the tunnel is a strong carrier-level sufficient condition, not a necessary condition for witness-relative exact constancy.

5. Witness Change: Quantisation and Protected Alphabets

5.1 The Quantised-Change Proposition, and the Physical Analogy

Proposition 5.1 (The witness observable is quantised; no sub-quantum accumulation). Fix a finite identity carrier M and a sealed witness [η] with integral witness data — an integer-valued cochain representative, the hypothesis of XV Proposition 3.11 — and declared recurrence classes that are integral cycles (automatic in the finite-state reading). Over any switching schedule whose transitions are drawn from a declared finite alphabet of transport events, each with a carrier-coherent restriction in the sense of XV Definition 4.10, the witness observable — the transported period langle [η], φ_*[γ] rangle — takes values in the lattice Z; rational witness data with common denominator q rescale the lattice to tfrac{1}{q}Z. Consequently no admissible evolution of the class produces monotone sub-quantum accumulation on the witness ledger: between consecutive windows the observable either repeats or jumps by at least one lattice quantum — never by less. Integrality, or common-denominator rationality, is the load-bearing hypothesis; the finite carrier makes each restriction audit finite, while the declared finite alphabet makes the event catalogue auditable, but neither is what forbids the drift — the reachable value set need not be finite (a degree-d self-map multiplies periods indefinitely), and the quantisation is lattice membership, not finiteness. Continuous or sub-quantum identity erosion is untypeable in the class; discrete lattice-step transformations are permitted and are not called drift here.

Proof. A carrier-coherent restriction is a graph map, and its pushforward carries integer cycles to integer cycles; the declared recurrence classes are integral, so every class reached across a schedule is an integral cycle. Pairing an integer-valued representative of the sealed [η] against an integral cycle is a signed integer sum, hence an integer: the observable lies in Z at every window (in tfrac{1}{q}Z for denominator-q rational data). Any two of its values are therefore equal or differ by at least the lattice quantum, and a monotone accumulation with steps strictly below the quantum would require values outside the lattice. blacksquare

This is the same structural fact that ONTOΣ XV's quantisation result (its Proposition 3.11) reads on the cost side — integral witness data force costs onto a lattice with a positive floor — here read on the survival side: the witness observable is a lattice observable. Budget variables admit wear-like accumulation; the witness observable can only repeat or move by a lattice step. The corpus's two axes differ not merely in what they track but in the kind of change they admit.

The continuous witness of Structural Spin is formulated over real de Rham cohomology and needs no integrality for its non-vanishing transfer test. It therefore supplies no quantisation premise here: the positive lattice floor remains wholly dependent on the integral or common-denominator rational data imported from XV.

The distinction is visible in three finite cases carried by the harness. A tunnelled identity map gives 3 → 3: exact preservation. A non-tunnelled but carrier-coherent reflection or degree-3 map gives 3 → -3 or 3 → 9: the numerical observable transforms by a lattice jump while remaining non-zero. A collapse gives 3 → 0: annihilation for that declared witness-bearing class. Quantisation separates wear from events; it does not collapse every event into the binary phrase "unchanged or dead".

[ILLUSTRATION] The physical analogy. K. J. Ray and J. P. Crutchfield's Gigahertz Sub-Landauer Momentum Computing offers a close physical analogy, not a verified instance of Definition 4.1. In physically calibrated Langevin simulations of a superconducting circuit, the coarse positional memory states cease to determine the future during a bit swap; dynamically relevant information is carried transiently in momentum and recovered at readout. That resembles the tunnel's geometry: a monitored channel can become insufficient while another degree of freedom carries what matters through the transition. But the study does not declare this note's recurrence carrier, sealed witness, identity graph map, or tunnel audit, so the correspondence stops at the analogy. Its cost accounting also depends on a controller boundary outside the scored device; applying the note's ledger language would require that boundary to be declared and sealed before any stronger identification.

5.2 The Protected-Covector Proposition

Let a declared finite event alphabet mathcal{A}={T₁,dots,Tᵣ} act on H₁(M;Z), with cycles written as columns, and let the induced dual maps act on H¹(M;Q). Its common fixed-covector space is

operatorname{Fix}^*(mathcal{A})
=bigcapᵢ₌₁ʳker(Tᵢ^*-I)
={[η]:[η]Tᵢ=[η] for every i}.

This is a witness-relative condition. It does not require any Tᵢ to be the identity on H₁(M), still less the underlying graph map to be the identity on M.

Proposition 5.2 (Protected-covector criterion). A sealed covector [η] has exactly constant transported period on every integral homology class, and therefore on every declared integral recurrence class, under every finite schedule word over mathcal{A} iff [η]∈operatorname{Fix}^*(mathcal{A}).

Proof. If [η]Tᵢ=[η] for each generator, induction on word length gives [η]T_(iₛ)·s T_(i₁)=[η], hence the same pairing on every cycle. Conversely, exact equality for every schedule word includes every word of length one; equality on every integral homology class then gives [η]Tᵢ=[η] for each generator. blacksquare

A protected covector is called emergent relative to a declared finite search when the search is given a local map language or event family but no covector, and the non-zero intersection above is produced as an output. This is algebraic emergence inside the declared model, not a claim of spontaneous physical or biological emergence.

The companion search supplies a minimum constructive certificate on the one-vertex bouquet with loop basis (x,y). It enumerates all 21 signed edge words of length at most two, all 21²=441 raw graph maps, their 169 distinct induced integer matrices, and the 16 non-identity unimodular events with one-dimensional individual fixed-covector space. It then exhausts every eligible ordered generator pair, critical event, protected driver, and canonical primitive cycle of coordinate radius two. Exactly 544 records satisfy the protected-pair, collapse, and order-effect predicates. Canonical representatives are selected first by total image-word length and then by the declared signed-word order. The global minimum inside this bounded model has total image-word length seven:

• A: (xmapsto x,\ ymapsto y⁻¹), with column matrix T_A=[(1,0)^T,(0,-1)^T] and det T_A=-1.
• B: (xmapsto xy,\ ymapsto y), with column matrix T_B=[(1,1)^T,(0,1)^T] and det T_B=+1.
• C: (xmapsto y,\ ymapsto x), with column matrix T_C=[(0,1)^T,(1,0)^T] and det T_C=-1.

Neither A nor B is tunnelled. They do not commute, since

AB=begin{pmatrix}1&0\\-1&-1end{pmatrix}
≠
begin{pmatrix}1&0\\1&-1end{pmatrix}=BA,

yet their common fixed-covector space is exactly operatorname{span}(1,0). Therefore every word over {A,B} preserves the period of [η]=(1,0) on every recurrence class, despite non-trivial and order-dependent action on the complementary homology coordinate. Event C is individually unimodular and has fixed covector (1,1), but

operatorname{Fix}^*({A,B,C})={0}.

One locally valid event can therefore destroy a globally protected channel. The exhaustive search discovers these maps and [η] rather than receiving them as fixtures; the independent replay reconstructs every matrix from its signed paths, the fixed-space dimensions, both products, all 544 admissible record keys, the global minimum score, and the deterministic tie-break winner.

5.3 The Order-Sensitivity Corollary

Corollary 5.3 (The event multiset does not determine the witness verdict). Outside a protected alphabet, two schedules with the same event multiset, capacity, coverage, and budget increments can have different final witness verdicts.

Certificate. Use the preceding B and C, [η]=(1,0), and the primitive recurrence class [γ]=(1,-1)^T, whose initial period is 1. In chronological notation, B,C means apply B and then C, so the final cycle is CB[γ]. The exact traces are

`B,C: 1longrightarrow 1longrightarrow 0,

C,B: 1longrightarrow -1longrightarrow -1.`

The first event alone leaves a non-zero pairing in both orders: B[γ] pairs to 1, while C[γ] pairs to -1. Annihilation occurs only after the second event in one order. Both schedules use the multiset {B,C} and the same three-window context N=3, k=1, active sets {0},{1},{2} in the harness's zero-based encoding (the formal sets {1},{2},{3}), and increments 1/3,1/3,1/3; only event order changes. blacksquare

The corollary does not say that all non-tunnel alphabets are order-sensitive, or that a zero pairing makes the cohomology class itself zero. It says that coverage and total budget do not determine the selected recurrence verdict once the schedule leaves the protected alphabet. The change is still an integer event-sized jump, not drift.

6. The Controller

Someone runs the schedule. The architectural claim — declared, not derived — is that the controller of a well-typed switching system is witness-silent: its interventions are transitions of the schedule, and in a tunnelled schedule those interventions never act on the identity carrier. The controller pays — its actions are budget-charged, its windows drift like all windows — but what it spends is capacity, never identity. In the corpus's terms it is the same downward pattern the master operator cuts one level up (ONTOΣ XIII): direction imposed from above, admission cascading down, with the imposing structure not itself a participant in the traffic it directs.

IIC v2.1 (DOI 10.17605/OSF.IO/NYT45) supplies a conditional dynamic typing for the controller: if it lies in the Cycle-Reinitiation class, its scheduling activity is carried by an endogenous impulse–interpretation–coherence cycle that can be reinitiated from within. This explains how a live controller may continue to run the schedule; it does not make its induced transitions witness-silent. IIC liveness/coherence and tunnel preservation are independent predicates: a controller may satisfy either, both, or neither. In particular, the control-grade condition that ν:τ^(F)→τ^(R) admits isomorphic factorisation through identity concerns the operator projection; it is not the graph-map condition φ|M=mathrm{id}_M. Neither mathrm{Coh}(t), Δ(boldsymbol{varpi)}, nor ν is the sealed witness [η] or its period observable, and IIC's open operator-frame-to-substrate bridge (§6.5.1) supplies no missing implication.

[ILLUSTRATION] Jumping-spider vision supplies a narrower functional analogy. The principal eyes combine high acuity with a small field of view and movable retinas; the secondary eyes have wider fields and are especially sensitive to motion. In an eye-tracking experiment, masking the antero-lateral secondary eyes disrupted smooth principal-eye tracking of moving targets, supporting a routing relation in which wide-field detection guides narrow-field inspection. The result establishes neither one fused spider image nor this note's controller, capacity bound, witness carrier, or tunnel condition. It shows only the division-of-labour pattern that motivates the architecture: one channel can guide where another, more discriminating channel looks.

This section also names what the note deliberately leaves open (§8): the controller is itself a subsystem with a carrier. Typing the self-reference — the switch-controller as an identity-bearing structure whose own witness rides some further tunnel — is the recursion the corpus meets everywhere, and it is not resolved here.

7. Falsification Surface

Each register carries its own test, in the sealed-declaration discipline the corpus imports.

Before those tests are read operationally, two evidence channels must be separated. A finite check of the declared carrier restriction φ|_M=mathrm{id}_M, sealed witness data, denominator q, and event graph map is a privileged declaration-side audit. Detection from emissions is a different claim. The Identifiability Bridge (DOI 10.17605/OSF.IO/3F6UJ) supplies a conditional admissibility schema for that second claim: the target must be typed as a Case A internal-trajectory predicate or a Case B design/audit predicate, and an identity-specific Bridge Detectability Premise (BDP)-realising statistic, a non-degenerate ensemble, a predicate-specific NR-window, fixed declared context, and calibration-valid nuisance adjustment must be supplied. Only under such an instantiation could tunnel preservation, annihilation, or off-lattice change be tested from emissions without reconstructing M or [η]. The Bridge supplies neither that witness-specific statistic nor a proof of Lemma 4.2, Propositions 5.1–5.2, or Corollary 5.3.

A declaration-side audit is not a certificate that the declaration is adequate to the claimed real system. A claimed instance therefore carries a separate anti-vacuity correspondence obligation: (i) carrier completeness relative to the identity claim — every state and recurrence edge whose alteration would count against that claim must be represented in M; (ii) witness relevance — the sealed witness family must be justified as reading the claimed identity predicate rather than an easy peripheral invariant; and (iii) budget calibration and provenance — the declared increments must be tied to the resource boundary actually consumed, with the carrier, witness, and budget version held fixed for the audited run. These are membership obligations, not consequences of Lemma 4.2, Propositions 5.1–5.2, or Corollary 5.3. Omitting switch-touched structure from M, selecting an irrelevant witness, or understating burden can make every finite internal check pass while falsifying the real system's membership in the declared class. The harness verifies internal consistency of sealed declarations; it cannot certify their empirical adequacy.

• The correspondence declaration (per claimed instance). Supply versioned provenance for the carrier boundary, witness family, channel inventory, capacity and coverage premises, and budget increments, together with a justification of completeness, relevance, and calibration. An omitted identity-relevant state or edge, an irrelevant witness, or an understated budget refutes the system's membership declaration even if every internal map, period, and mutation check passes. Such a failure does not refute the formal statements; a passing internal harness cannot repair a false correspondence.

• The tunnelled declaration (per switch). On the privileged declaration side, a transition declared tunnelled makes a finite checkable claim: the carrier restriction on M is the identity graph map and direct evaluation of the sealed witness gives a constant transported period across the switch. If that audit shows a non-identity restriction or changed period, the deployment has falsified its tunnel declaration — not the lemma. An emission-side detector is admissible only conditionally under the preceding Bridge schema and is not interchangeable with the finite map-and-witness audit. The lemma itself would be falsified only by a counterexample within the typing: a schedule of verified-tunnelled transitions across which the period nonetheless changes. None can exist by §4.

• The admissibility-transfer declaration (per switch, when made). A tunnel declaration does not certify that admissible source trajectories remain admissible after switching. For each claimed cross-window admissibility transfer, declare (mathsf{D}ⱼ,mathrm{Adm}ⱼ) and τⱼ, then audit mathrm{Adm}ⱼ⊂eqτⱼ^*(mathrm{Adm}ⱼ₊₁). Failure refutes the deployment's admissibility typing, not Lemma 4.2. A stronger kernel-exact claim additionally requires Δ_(τⱼ)=varnothing together with weakness; a viability-bisimulation claim additionally requires admissible path-lifting. An empty defect is neither a tunnel certificate nor a witness-period test.

• The continuous spin-realisation claim (when made). If a claimed instance identifies the finite witness with a structural-flow witness, declare the canonical domain, the source class a=[ηS], the pulled-back target class b=φ^*[η(S')], and the map φ, then audit ker bsetminusker a. An empty defect certifies preservation of non-vanishing only; exact tunnel constancy requires equality of the period functionals — in the non-zero proportionality typing, λ=1 rather than arbitrary λ. If H¹=0, the topological test is blind and the local operational-spin channel dω_S must be kept separate. Failure refutes the continuous realisation claim, not Lemma 4.2.

• The protected-alphabet declaration (per event alphabet). Declare the carrier, signed edge images, induced actions Tᵢ, and sealed covector. Audit [η]Tᵢ=[η] for every generator, or equivalently compute operatorname{Fix}^*(mathcal{A}). A changed period under a verified protected word falsifies the declaration. A non-zero common fixed covector is witness-relative and does not certify that any event is tunnelled, admissibility-preserving, or witness-silent on the whole carrier.

• The order-sensitive certificate (per claimed finite instance). Reconstruct the certified matrices from their signed paths, verify the common fixed-space dimensions before and after adjoining C, verify the two event lists have the same multiset, and replay both cycle and period traces. A path/matrix mismatch, a non-zero extended common fixed space, unequal event multisets, or failure of exactly one terminal zero refutes this finite certificate. Passing it establishes neither universal order-sensitivity nor a biological instance.

• The quantised-change claim (per membership declaration). Proposition 5.1 forbids an off-lattice value or a non-zero sub-quantum step on fixed declared data. Observing either in a system claimed to be of the class refutes its membership declaration — the carrier was not fixed, the witness was not sealed, its data lacked the declared common denominator, or a transition's carrier restriction was not the declared graph map — and the discipline requires saying which. A lattice-sized jump such as 3 → -3 or 3 → 9 does not refute the proposition. For an emission-only claim, the detecting statistic must additionally be calibrated to the declared lattice quantum 1/q under the preceding Bridge premises.

• The architectural claim (per system). A claimed instance fails membership in the declared architecture if (a) one measured window reads the full arsenal, contradicting the strict capacity bound; (b) a declared full-arsenal execution leaves any channel uncovered, contradicting the coverage requirement; or (c) a controller-induced switch acts non-trivially on the declared identity carrier, contradicting witness-silence. An IIC-live or high-mathrm{Coh} controller may still fail this test; cycle reinitiation and witness-silence are independent declarations. Conversely, a non-tunnelled transport that changes a non-zero period is not a counterexample to Proposition 5.1; it simply lies outside the tunnelled subclass. A protected non-tunnel alphabet under Proposition 5.2 may preserve one witness exactly while still failing this architecture's stronger witness-silence condition, because its carrier action is non-trivial. These tests separate false capacity, coverage, and tunnel declarations from a valid but different architecture.

8. Open Problems

1. The controller's own typing. The self-referential level: specify whether the switch-controller is a control-grade or operator-grade Cycle-Reinitiation system in the IIC sense; declare its own carrier and witness; compose its ν-mediated cycle transitions with the target carrier M; and prove that every scheduled transition restricts to mathrm{id}_M. IIC supplies the candidate liveness layer, not this composition theorem: its mathrm{Coh}(t) and identity-factorisation language are operator-frame notions, its operator-frame-to-substrate bridge remains open, and neither decides whether the controller's witness requires a tunnel one level up or whether the recursion grounds or regresses. The corpus's master-operator and tower layers remain the natural apparatus for that recursion.
2. Continuous carriers. All four formal statements are finite-state. Structural Spin supplies a continuous spin-side de Rham witness, a constructive topological identity defect on its canonical domain, and an H₁ comparison with discrete models; it also shows that eliminating operational spin while retaining a non-zero class is possible only when h_c(nabla V)=0, with realisable classes confined to W_V. It does not supply a functor preserving this note's switching schedule, budget ledger, admissibility selector, or lattice quantum, and an H₁-isomorphism alone does not identify witness classes. Construct the full discrete–continuous bridge and the continuous-time drift/admissibility interfaces rather than treating homological comparability as transfer.
3. Correspondence discipline for biological systems. Construct a versioned correspondence certificate for any actual perceptual system: justify carrier completeness relative to the identity claim, witness relevance and non-vacuity, channel-inventory and coverage completeness, and calibration/provenance of the budget increments against the resource boundary actually consumed. This is the step that would move §1 and §6 from Illustration to instance. No finite internal replay can supply it, and the present note leaves it untouched as its analogue of XV's calibration problem.
4. Scaling in N. The large-N limit: whether routing-dominated architectures (N gg k) admit a derived bound linking schedule complexity, budget drift rate, and the number of tunnels a single carrier can serve.
5. Witness-property identifiability. Instantiate the Identifiability Bridge for the present witness system: type tunnel preservation, annihilation, and off-lattice change as Case A or Case B predicates; exhibit a BDP-realising statistic and a non-degenerate ensemble; declare the predicate-specific NR-window and fixed context; prove calibration validity; and resolve the statistic at the lattice quantum 1/q. No such substrate-specific detection result is supplied here.

6. Joint controller–tunnel–admissibility typing. Construct a combined certificate for each scheduled transition that records (i) the controller's IIC liveness typing, (ii) the tunnel condition φ|_M=mathrm{id}_M, and (iii) the Domenoid weak-morphism obligation mathrm{Adm}ⱼ⊂eqτⱼ^*(mathrm{Adm}ⱼ₊₁); then prove any coupling rather than assuming one coordinate from another. With the transition map fixed, the Domenoid's minimal source-restriction or target-enlargement repairs do not themselves alter its restriction on M, but they can remove source schedules or enlarge the target trajectory language beyond its original selector class. Characterise when full-arsenal coverage and controller reinitiation survive such specification-time repair. No joint composition theorem is supplied here, and the repairs are not licensed as autonomous runtime patches.

7. Protected-channel scaling and robustness. Classify the dimension and arithmetic of operatorname{Fix}^*(mathcal{A}) for larger carriers and longer path alphabets; determine the minimum critical extensions that collapse it; and separate exact shared invariants from approximate numerical near-invariants. The present exhaustive minimum is only for two loops, signed edge words of length at most two, primitive cycles of coordinate radius two, and the stated protected-pair, collapse, and order-effect predicates.

9. Honesty Notes

1. The vision vignette of §1 is a genesis story and a structural skeleton, not data; binocular rivalry has a large literature represented here by only one orientation reference, not reviewed in this note.
2. Lemma 4.2 and Proposition 5.1 are short, and their brevity is disclosed rather than dressed: the mathematical content is the typing (Definition 4.1 and the integral or common-denominator rational witness data); the derivations are direct applications of the imported XV machinery. Proposition 5.2 is likewise elementary finite linear algebra. The non-trivial addition is constructive: an exhaustive search finds a minimum non-tunnel, noncommuting protected alphabet and a critical extension inside its explicitly bounded model.
3. The claim "identity does not drift" uses drift narrowly: wear-like accumulation through arbitrarily small admissible increments. It does not identify the formal witness with one frozen number. For a declared recurrence class, a non-zero transported period remains witness-bearing, zero is annihilating, and a non-tunnelled event may change the non-zero period by a lattice step. A verified tunnel guarantees exact constancy carrier-wide; a verified protected alphabet guarantees it only for its common fixed covectors. Whether any phenomenal or biological identity is of either class is exactly the open correspondence problem of §8.3 — asserted nowhere in this note. A real-system membership claim additionally requires a versioned, non-vacuous correspondence certificate for carrier completeness, witness relevance, channel coverage, and budget calibration/provenance; internal consistency of the finite declarations is not a substitute.
4. The companion harness comprises harness/switch_transport.py, harness/emergence_search.py, harness/emergence_certificate.json, harness/emergence_verify.py, harness/teeth_audit.py, and harness/check_bundle.py. The transport verifier retains its 10/10 fixtures. The search enumerates 21 signed words, 441 raw maps, 169 induced matrices, 16 eligible events, and all 544 admissible records, then emits the global minimum score-seven certificate without receiving its concrete maps, covector, or cycle as inputs. The replay module imports no search code and independently rebuilds the universe, path-to-matrix actions, fixed spaces, noncommutation products, all admissible record keys, the global minimum score, the deterministic tie-break winner, and both order traces; its baseline plus sixteen tamper controls report 17/17. The mutation gate applies 34 one-site transformations to temporary copies of the actual executables: all 33 load-bearing mutations must make the genuine selftest fail at their named evidence markers, while one invariant-preserving reordering must survive. No alternate fixture implementation or in-memory monkeypatch is used. The bundle gate runs the executable battery in normal and optimised modes, checks the certificate, inventory, counts, references, and public-text hygiene, and plants detector defects only in scratch copies of the actual bundle. The harness is stdlib-only, exact-rational where rational data occur, and self-contained; it verifies this note's finite-state statements and nothing else.
5. The word emergent in §5.2 is deliberately local to the search protocol: the protected covector is a global invariant recovered from generator actions and is not an input fixture. The claimed minimality is total image-word length seven over every admissible record in the stated two-loop, length-two word language with primitive-cycle coordinate radius two and the deterministic tie-break, not over all carriers, longer graph maps, or unbounded cycle searches.

10. References

• NC2.5 v2.1 (axiomatic core). DOI 10.17605/OSF.IO/NHTC5.
• ONTOΣ XV — Spin-Channel and Nestability, with its formal companion. Bundle DOI 10.17605/OSF.IO/EAUD5. Repository: NC2.5-ONTOSigma-XV-Spin-Channel-and-Nestability-corpus. (Transport verdicts, carrier restrictions, drift class, quantisation, sealed-declaration discipline — the apparatus assembled here.)
• ONTOΣ XIV — Nested Substrates and the Frame-Relative Derivative. Bundle DOI 10.17605/OSF.IO/KAGMH. Repository: NC2.5-ONTOSigma-XIV-Cross-Layer-Forgetful-Separation-corpus. (Witness functor, recurrence carriers, X+M ledger.)
• ONTOΣ XIII — Master Operator. DOI 10.17605/OSF.IO/FVBMZ. (The downward pattern of §6.)
• K. J. Ray, J. P. Crutchfield. Gigahertz Sub-Landauer Momentum Computing. Phys. Rev. Applied 19, 014049 (2023); arXiv:2202.07122. (The physical analogy of §5 — Illustration register.)
• R. Blake, H. Wilson. Binocular Vision. Vision Research 51 (2011), 754–770. DOI 10.1016/j.visres.2010.10.009. (Binocular integration, rivalry, and suppression — background for §1.)
• E. M. Jakob, R. R. Long, L. A. Harland, D. Jackson, S. D. Carey, A. C. Searles, D. Porter, C. J. Canavesi, N. S. Rolland. Lateral Eyes Direct Principal Eyes as Jumping Spiders Track Objects. Current Biology 28 (2018), R1092–R1093. DOI 10.1016/j.cub.2018.07.065. (The routing analogy of §6.)
• Identifiability Bridge — Conditional Admissibility Theorems for Property Detection under Non-Reconstructibility within Navigational Cybernetics 2.5. DOI 10.17605/OSF.IO/3F6UJ. (Conditional reconstruction-versus-detection schema for the emission-side reading of §7; not a proof source for §§4–5.)
• 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. DOI 10.17605/OSF.IO/NYT45. (Conditional controller/liveness typing for §6; its cycle-coherence quantities and operator projection are not the witness or tunnel map of §§4–5.)
• Domenoid of Admissibility — Operator over Dynamics — A Traceable Structural Program (2025–2026) — and Its Formal Core — the Domenoid of Admissibility. DOI 10.17605/OSF.IO/3ESN4. (Independent admissibility layer, weak-morphism obligation, transfer-regime hierarchy, viability-transfer defect, and specification-time repair for §§4, 7, and 8.6; not a proof source for Lemma 4.2 or Proposition 5.1.)
• Structural Spin — Structural Spin as a Forgetful Separation over Dissipative Dynamics, with a Cohomological Obstruction to Identity Transfer. DOI 10.17605/OSF.IO/94GWQ. (Continuous spin-side witness/liveness separation, constructive identity defect, homological bridge, topological-channel blindness, and operational-spin obstruction for §§4, 5, 7, and 8.2; not a proof source for Lemma 4.2 or Proposition 5.1.)

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Companion code

Companion verification code and the reproducible finite-state harness are available in the public repository: Identity Does Not Drift.