Structural Spin as a Forgetful Separation over Dissipative Dynamics, with a Cohomological Obstruction to Identity Transfer

Structural Spin as a Forgetful Separation over Dissipative Dynamics

...with a Cohomological Obstruction to Identity Transfer

The continuous instance of the forgetful-separation pattern, whose discrete instance is admissibility; companion to "Domenoid of Admissibility" (DOI 10.17605/OSF.IO/3ESN4). Substrate: NC2.5 v2.1 (DOI 10.17605/OSF.IO/NHTC5).

Author: Maksim Barziankou (MxBv)
Date: June 2026
Affiliation: The Urgrund Laboratory, Poznan
DOI: 10.17605/OSF.IO/94GWQ
License: CC BY-NC-ND 4.0

---

Abstract

We isolate a single structural pattern: a forgetful map from (dynamics + an extra structural layer) to the bare dynamics that is faithful but not full, so that the layer is not recoverable from the dynamical skeleton. Two instances are placed side by side: admissibility over a transition skeleton (discrete), and the non-gradient ("spin") component S of a flow dot x=-nabla V+S over its gradient skeleton (continuous). For the spin instance: (i) the fibre over a fixed gradient skeleton is non-trivial whenever the manifold carries one nonzero divergence-free field pointwise orthogonal to nabla V that is not a global gradient, and then contains a whole punctured line of spins; (ii) with split-preserving (positive-semiconjugacy) morphisms the forgetful functor U_(mathrm{sp)}colonmathsf{SpinFlow}→mathsf{GradFlow} is faithful but not full; (iii) a morphism that transfers a viability kernel need not control S, so that accurate identity transfer requires viability transfer together with survival of the spin witness — a non-vanishing de Rham period of a closed component of S^flat — which is strictly stronger than viability transfer alone; the two-layer certificate detects failure of identity transfer and is necessary, not in general sufficient; (iv) the witness is a de Rham class, hence comparable across the discrete/continuous divide via any H₁-isomorphism, and transferred only when the witness classes correspond. The mathematics is classical (Hodge--de Rham, elementary category theory); the contribution is the identification, the faithful-not-full theorem, the certificate, the constructive defect, the homological bridge, the operational-spin obstruction, the necessity-and-blindness scope of the topological witness, and the finite graph-Hodge instrument.

On the canonical domain we make the identity defect constructive, with a proportionality criterion for witness-exactness, and we show the naive support-subspace (viability-shadow) repair does not transport to the witness, a genuine witness-repair remaining open; and that the four demands — divergence-free, operational-spin-free, orthogonal to nabla V, and nonzero witness c — are jointly satisfiable, for M compact and boundaryless, exactly when the harmonic representative of c is pointwise orthogonal to nabla V, a class-dependent obstruction. The non-gradient structure has two channels — a topological de Rham period (the identity witness) and a local vorticity 2-form — so that recurrence on a space with trivial H¹, were it to occur, is seen only by the latter; and the two instances are meant for joint use, the admissibility companion certifying survival and the spin layer certifying liveness — so that neither alone flags the survives-yet-dead configuration exhibited in the worked example. Witness preservation is moreover sufficient, on the canonical domain, for forward transfer of the topological identity, not only necessary; the topological witness detects only homologically non-trivial recurrence (necessary, not sufficient, for a fixed spin) and is blind on H¹=0 (axial rotation on S²); and a finite graph-Hodge instrument computes the witness and its collapse on raw incidence data. A field-level repair and a high-dimensional operationalization remain open.

Introduction

Many dynamical settings carry, over and above their state dynamics, a structural layer: a rule, an admissibility predicate, or a non-conservative component that is independent data and is not determined by the bare dynamics. The organizing question is uniform: which transfers of the layer between two systems are licensed, and which are not, by similarity of the underlying dynamics alone? In each instance the answer is governed by a forgetful map from the structured objects to the bare ones that is faithful but not full, so that a morphism of skeletons need not lift to one respecting the layer.

The discrete instance is admissibility. Fixing a transition skeleton mathsf D (states, trajectories, and the algebra of shifts, concatenations, and windows), an admissibility predicate mathrm{Adm}⊂eqmathrm{Traj}_(mathsf D) selects which trajectories count; the forgetful functor Ucolonmathsf{AdmDyn}→mathsf{Dyn}, (mathsf D,mathrm{Adm})mapstomathsf D, is faithful but not full, and its fibre U⁻¹(mathsf D) is a non-trivial complete lattice. A companion development of this instance, including a transfer/repair calculus, is assumed as background [admissibility].

This note develops the continuous instance. The structural layer is the non-gradient component S of a structural flow dot x=-nabla V+S on a Riemannian manifold; the bare dynamics is the gradient (dissipative) skeleton -nabla V. We make precise the sense in which S is a layer the skeleton does not determine, prove the forgetful functor is faithful but not full, and exploit the pairing of the two instances: the admissibility layer transfers a viability kernel, while the spin layer carries identity. Transferring identity therefore requires more than transferring viability, and the gap is detected by a homological certificate. Our constructive results hold on a canonical domain that splits in two — M compact without boundary, or ω_S globally closed; the worked example (S¹×mathbb R) realizes the second branch. The technical core is the operational-spin obstruction (Proposition §prop:opspin-obstruction, Corollary §cor:WV, Proposition §prop:WV-generic), holding on the compact branch (where the harmonic representative exists); the surrounding Hodge--de Rham and categorical material is classical.

Throughout, the metric g is fixed and all metric-dependent objects (S^flat, divergence, the gradient/spin split) are relative to it. The decomposition dot x=-nabla V+S with S satisfying

• (S1) nabla!·!S=0 (divergence-free),
• (S2) langle S,nabla Vrangle=0 pointwise,
• (S3) S admits no global scalar potential (S≠nabla W),

is the object of study; (S1)--(S3) are the conditions under which the structural-spin theory of [nc25] excludes a global gradient representation.

Definition (Non-stagnant recurrent identity and its witness)

A non-stagnant recurrent identity is a non-constant recurrent orbit of the flow: a non-constant orbit returning to every neighbourhood of one of its points at arbitrarily late times. Its identity witness on a recurrence loop γ — the topological channel of the spin (Remark §rem:two-channels) — is the period oint_γη_S of a closed component η_S of ω_S:=S^flat, equivalently the pairing of the de Rham class [η_S]∈ H¹_mathrm{dR}(M) with [γ]∈ H₁(M) (an integral cycle class, paired in H₁(M;mathbb R)). Here γ is closed: a periodic orbit, or a recurrence segment closed by a chosen short return arc — on which [γ] may depend; the canonical loop-free formulation for a general (aperiodic) recurrent orbit is the Schwartzman asymptotic cycle (Remark §rem:schwartzman), to which the periodic [γ] specializes. The richer notion of identity under a bounded internal budget developed in [nc25] is not needed below: we use only the elementary necessity of Proposition §prop:identity.

Proposition (Spin is necessary for a recurrent identity)

For any smooth V, the value V is strictly decreasing along every non-constant orbit of the pure gradient skeleton dot x=-nabla V; hence the skeleton has no non-constant recurrent orbit (orbits assumed forward-complete on the recurrence in question), and a non-stagnant recurrent identity requires a non-gradient component Snotequiv0.

Proof. At a critical point (nabla V=0) the constant orbit is the unique solution, so a non-constant orbit meets no critical point; there tfrac{d}{dt}V=langlenabla V,dot xrangle=-|nabla V|²<0, so V is strictly decreasing along it. A recurrent point x has return times tₙ→∈fty with x(tₙ)→ x (the orbit being forward-defined there), whence V(x(tₙ))→ V(x); but V(x(tₙ)) is strictly decreasing and bounded above by V(x(t₁))<V(x), so it cannot converge to V(x), the required contradiction. Thus the gradient skeleton has no non-constant recurrent orbit, and any non-stagnant recurrent identity forces Snotequiv0. blacksquare

Remark (Recurrence lives on the critical set)

Under (S2) the same computation gives tfrac{d}{dt}V=-|nabla V|² along the full flow dot x=-nabla V+S (since langle S,nabla Vrangle=0). Hence any recurrent orbit of the full system — under the same forward-completeness proviso as Proposition §prop:identity — has V constant, so it lies in the critical set {nabla V=0}, where -nabla V=0 and the motion is pure spin dot x=S. The non-stagnant identity is therefore carried entirely by S on {nabla V=0} — in Example §ex:worked the invariant circle {y=0}.

Remark (Two channels of the non-gradient structure)

The de Rham period [ηS] is one of two channels of the non-gradient structure. Write Σ(S):=big([η_S]∈ H¹_mathrm{dR}(M),\ dω_S∈Ω²(M)big): the topological circulation channel [η_S] and the local vorticity channel dω_S=2×(operational spin) (§sec:operational). The first is constrained by topology — if H¹_mathrm{dR}(M)=0 then [η_S]=0 for every S, so on mathbb R² it is vacuous — while dω_S is a (globally exact) 2-form unconstrained by topology and may be nonzero on mathbb R². Both extremes occur for the spin field itself: plane rotation S=(-y,x) on mathbb R² (with V=tfrac12(x²+y²)) has [η_S]=0 but dω_S=2\,dxwedge dy≠0 (local-only), while S=partialθ on S¹×mathbb R (Example §ex:worked) has ω_S closed, dω_S=0, and period 2π (topological-active). Accordingly the necessity of Proposition §prop:identity is only that Snotequiv0: the de Rham period is one detectable channel of that non-gradient structure, not a complete characterization. The cohomological certificate of §sec:certificate concerns the topological channel; the co-exact part is read locally by dω_S.

What this paper does and does not introduce. We introduce no new Hodge decomposition and no new theory of vorticity. The claim is structural and categorical: the spin component is a layer over the gradient skeleton that is not recoverable from the skeleton, exactly as admissibility is a non-recoverable layer over a transition skeleton. The supporting mathematics — the Hodge--de Rham decomposition [warner, morita], singular/de Rham (co)homology [hatcher, bott], elementary category theory [maclane], and the vorticity (spin) tensor of continuum mechanics [truesdell, batchelor] — is classical throughout.

The spin fibre over a gradient skeleton

Definition (Forgetful projection)

On objects (M,g,F=-nabla V+S) define U_(mathrm{sp)}colon(M,g,F)mapsto(M,g,-nabla V), discarding the non-gradient component and retaining the gradient skeleton. The fibre over a fixed skeleton is U_(mathrm{sp)}⁻¹(-nabla V)={\,S:(S1)wedge(S2)wedge(S3)\,}; the category mathsf{SpinFlow} of §sec:functor additionally admits the trivial object S=0 (Definition §def:cats).

Theorem (The skeleton does not determine the spin)

(a) (Compact case.) Let M be compact, oriented, without boundary. For S satisfying (S1), the 1-form ω_S:=S^flat is co-closed and its Hodge decomposition has no exact part, ω_S=δβoplus h (δβ co-exact, h harmonic). Any nonzero such S is not a gradient (a co-closed exact 1-form on a closed manifold vanishes), so (S3) is automatic for S≠0; on such closed M, (S3) is thus a corollary of Hodge theory.

(b) (Existence and a punctured line; any M.) On any (M,g) the fibre is non-empty as soon as M carries one nonzero divergence-free field pointwise orthogonal to nabla V that is not a global gradient (i.e. satisfying (S3)). Moreover, if one nonzero S satisfies (S1)--(S3) then so does λ S for every λ∈mathbb R^×, so the fibre contains the punctured line {λ S} and the skeleton fails to determine the spin even up to magnitude. (S2)-compatibility is not automatic from Hodge theory: a generic δβoplus h need not be pointwise orthogonal to nabla V. A concrete witness is Example §ex:worked (M=S¹×mathbb R, S=partial_θ, with (S2) checked directly), and Remark §rem:family gives a family of such witnesses; the general characterization of manifolds carrying an (S2)-compatible S≠0 is open.

Consequently the gradient skeleton -nabla V does not determine S.

Proof. (a) (S1) is δω_S=0, so ω_S is co-closed; writing ω_S=dα+δβ+h gives 0=δω_S=δ dα, whence |dα|²=langleα,δ dαrangle=0 on a closed manifold and dα=0. A nonzero exact ω_S would be co-closed and exact, hence zero, a contradiction. (b) the existence claim is conditional, with Example §ex:worked exhibiting the hypothesis as non-vacuous; linearity in S of (S1)--(S2) and homogeneity of (S3) (if λ S=nabla W then S=nabla(W/λ), contradicting (S3) for λ≠0) give the punctured line; (S2)-compatibility of a generic Hodge component is not asserted. square

Remark (Each diagnostic reads one Hodge summand; compact case)

On M compact, oriented, boundaryless, part (a) gives ω_S=δβoplus h with δβ co-exact and h harmonic (no exact part, by (S1)). The two channels Σ(S)=([η_S],dω_S) of Remark §rem:two-channels then read the two summands: [η_S]=[h] is the class of the harmonic summand (η_S=h, the harmonic projection of ω_S=δβoplus h being h; the harmonic representative of a de~Rham class is unique, so this choice of η_S is canonical), while dω_S=d(δβ) is determined by the co-exact summand (since dh=0). Thus the topological and operational diagnostics are functions of complementary Hodge components of the single object ω_S — note ω_S itself is generally non-closed here, so it is h, not ω_S, that carries the de~Rham class, and the two diagnostics do not reconstruct ω_S as a 1-form. This is expository — a reading of part (a) and Remark §rem:two-channels, not a new theorem — and is confined to the compact branch: on the globally-closed branch (Example §ex:worked) dω_S=0 identically and the Hodge projection is unavailable, so the two-component picture degenerates to one.

Remark (A family of witnesses)

If M=N×mathbb R carries the product metric g_Noplus dy², V=V(y) (y the mathbb R-coordinate), and X is a nonzero divergence-free field on (N,g_N), viewed as tangent to the N factor, then S=X satisfies (S1)--(S2), with (S3) holding whenever X is not a gradient on N. This is a family of (S2)-compatible spins; Example §ex:worked is the case N=S¹, X=partial_θ.

Remark (The fibre as stream-function structure)

Conditions (S1)--(S2) say exactly that S is divergence-free and tangent to the level sets {V=mathrm{const}}: via the volume form μ, nabla!·!S=0iff i_Sμ is closed and langle S,nabla Vrangle=0iff dVwedge i_Sμ=0. On an oriented surface, on the regular set {nabla V≠0} with connected level sets, every such S is S=a(V)\,Jnabla V, the Hamiltonian/skew-gradient of V with leaf-profile a — the classical stream-function structure of 2D incompressible flow (streamlines = level sets). Globally the fibre is strictly larger: partial_θ on S¹×mathbb R is divergence-free and leaf-tangent but escapes the profile form: the candidate profile apropto 1/y is not a single-valued function of V=tfrac12 y² on the regular set (the level {V=v}, v>0, is the two circles y=pmsqrt{2v}, on which it takes opposite signs) and diverges at the critical level {y=0}, which is exactly where the topological channel [η_S]≠0 is generated. This is classical and is recorded only to ground the abstract fibre; no foliated-cohomology identification is claimed.

Remark

Theorem §thm:fibre is the spin analogue of the non-triviality of the admissibility fibre U⁻¹(mathsf D): identity of the gradient skeleton does not license transfer of S, exactly as identity of the transition skeleton does not license transfer of admissibility.

The forgetful functor

We restrict morphisms to those that preserve the splitting, just as the discrete theory restricts to admissibility-preserving maps. We state the condition as a positive semiconjugacy of the flows, up to a positive constant time-rescaling; for a non-injective map (such as the folding of Example §ex:worked) the relevant fields are required to be projectable, so the pushforward is defined. Time parametrization need not be preserved; equivalently the condition may be read on unparametrized orbits.

Definition (Categories and forgetful functor)

• mathsf{GradFlow} has objects (M,g,V). A morphism (M,g,V)→(M',g',V') is a smooth φcolon M→ M' with dφₓ(-nabla V(x))=λ\,(-nabla'V'(φ(x))) for all x and some constant λ>0 (determined wherever nabla'V'circφ≠0, so the constant is unambiguous under composition; a positive semiconjugacy of the gradient flows; -nabla V required φ-projectable when φ is non-injective).
• mathsf{SpinFlow} has objects (M,g,V,S) with S satisfying (S1)--(S2) and either (S3) (nontrivial spin) or S=0 (the trivial bare-gradient object; S=0 is admitted separately because 0=nabla(mathrm{const}) is a gradient and so formally violates (S3)). A morphism is a smooth φ semiconjugating the gradient flow and the full flow by a common positive constant: dφₓ(-nabla V(x))=λ\,(-nabla'V'(φ(x))) and dφₓ(F(x))=λ\,F'(φ(x)) for the same λ>0 (F=-nabla V+S; equivalently dφ(S)=λ S'; F projectable when φ is non-injective). Thus "preserving the splitting" means a single positive time-rescaling carries the gradient and full flows together; U_(mathrm{sp)} forgets S, the underlying mathsf{GradFlow}-morphism keeping the same λ.
• U_(mathrm{sp)}colonmathsf{SpinFlow}→mathsf{GradFlow}, (M,g,V,S)mapsto(M,g,V), φmapstoφ.

The semiconjugacy conditions are closed under composition (the constants multiply: d(ψφ)(F)=λμ\,F''circψφ) and hold for identities (λ=1), so mathsf{GradFlow} and mathsf{SpinFlow} are categories. The objects of mathsf{SpinFlow} over a fixed (M,g,V) are U_(mathrm{sp)}⁻¹(-nabla V)∪{0}; non-triviality (Theorem §thm:fibre) concerns the part S≠0.

Theorem (Faithful, not full)

U_(mathrm{sp)}colonmathsf{SpinFlow}→mathsf{GradFlow} is faithful but not full.

Proof. Faithful. A mathsf{SpinFlow}-morphism is its underlying smooth map; if U_(mathrm{sp)}(φ)=U_(mathrm{sp)}(ψ) then φ=ψ.

Not full. On M=S¹×mathbb R, V=tfrac12y², put Fₐ=a\,partial_θ-y\,partial_y. The objects (M,g,V,partial_θ) and (M,g,V,2partial_θ) — both satisfying (S1)--(S3) — lie over (M,g,V), and the identity is a mathsf{GradFlow}-morphism. A lift to a mathsf{SpinFlow}-morphism (M,g,V,partial_θ)→(M,g,V,2partial_θ) would require d(mathrm{id})(F₁)=λ F₂ for some λ>0, i.e.\ (1,-y)=λ(2,-y); the y-component forces λ=1 (at y≠0) and the θ-component then gives 1=2, a contradiction. Concretely, integrating Fₐ (y=y₀e⁻ᵗ, θ=θ₀+at) gives the spirals θ=θ₀-alog(y/y₀), which are distinct point sets for a=1 and a=2; the identity cannot carry one onto the other. No lift exists, so U_(mathrm{sp)} is not full. blacksquare

Remark

Metric-dependence and non-canonicity of the gradient/spin split under arbitrary maps are resolved exactly as in the discrete theory: morphisms are restricted to split-preserving maps. The price is a small morphism class (isometry-like maps together with collapsing maps such as the folding below); we do not claim a rich morphism class. Faithful-not-fullness (Theorem §thm:faithful) is moreover generic for forgetful functors; the content here is not that categorical fact but the non-trivial fibre (Theorem §thm:fibre) and the lifting criterion (Proposition §prop:constructive-defect) it makes available.

Identity transfer versus viability transfer

The admissibility layer transfers a viability kernel operatorname{Viab}^∈fty={x:∃ an admissible infinite trajectory from x} along a morphism. The spin layer carries identity (Proposition §prop:identity): a non-stagnant recurrent identity requires S≠0, the witness being a non-vanishing period of a closed component of ω_S (§sec:certificate). The two layers are independent, with a consequence.

Proposition (Viability transfer does not control the spin)

Any transfer condition whose signature involves only the skeleton, the state map, the admissibility predicate, and the viability kernel — but not S or its witness — cannot imply or control preservation of the identity witness [ηS], for conditions read off data shared across the fibre (the unconditional separation being the explicit Example §ex:worked). (The weak, kernel-exact, and viability-bisimulation conditions of [admissibility] are of this form.) By Theorem §thm:fibre the fibre over a fixed skeleton is non-trivial, so such a condition is invariant under the choice of S in the fibre, and equally under passage to the trivial object S=0 (the augmented set U(mathrm{sp)}⁻¹(-nabla V)∪{0} of Definition §def:cats); in particular it does not see the collapse S→0. Unconditionally, the folding of Example §ex:worked is a viability-exact, spin-collapsing morphism (Theorem §thm:strict); hence accurate identity transfer — viability transfer together with witness survival — is strictly stronger than viability transfer alone.

Proof. By Theorem §thm:fibre the fibre is non-trivial and skeleton-independent; a condition not mentioning S in its signature takes the same value on (M,g,V,S) for every S in the fibre and on the trivial object (M,g,V,0) of Definition §def:cats alike. Within the punctured line {λ S} of a topologically-active S (one with [ηS]≠0) the witness stays non-vanishing ([λ\,η_S]≠0 for λ≠0); the value [η(S')]=0 is attained only at that trivial object, which the condition cannot separate from the fibre, so it cannot distinguish [ηS]≠0 from the collapsed [η(S')]=0. (This invariance assumes the viability data is shared across the fibre; in general the kernel operatorname{Viab}^∈fty can itself move with S through the full flow, so the universal reading needs that proviso.) The unconditional separation — identical viability transfer, different witness — is the explicit folding of Example §ex:worked (Theorem §thm:strict), a viability-preserving, spin-collapsing morphism, giving strictness. ∎

Accurate identity transfer Longrightarrow viability transfer and survival of the spin witness,

a necessary filter, not an equality: the certificate below detects failure of identity transfer, not its success.

Remark (The universal reading is an informal principle)

The broad reading of Proposition §prop:viab — that any condition not naming S in its signature fails to control [η_S] — is heuristic: its scope rests on an informal notion of a condition's "signature", and the fibre-invariance argument assumes the viability data is shared across the fibre (in general operatorname{Viab}^∈fty can itself move with S through the full flow). The rigorous content is the hypothesis-qualified invariance above together with the unconditional separation of Example §ex:worked (Theorem §thm:strict).

The two-layer certificate

The discrete theory certifies viability transfer by a defect
Δτ=τ_X⁻¹(operatorname{Viab}^∈ftyⱼ)setminusoperatorname{Viab}^∈ftyᵢ, with Δτ=varnothing iff
the transfer is kernel-exact. The spin layer supplies a parallel certificate for identity, anchored
in cohomology.

Definition (Identity-transfer defect)

Fix, as in Definition §def:identity, a closed component ηS of ω_S with class
[η_S]∈ H¹_mathrm{dR}(M) (on compact boundaryless M the harmonic projection; in Example §ex:worked,
ω_S is itself closed). The class [η_S], and hence this defect, is canonically determined
by S only on the canonical domain: M compact without boundary (harmonic projection), or ω_S globally closed (dω_S=0 on all of M,
i.e. operational spin identically zero, so η_S=ω_S is a genuine de Rham class in
H¹_mathrm{dR}(M) with homology-invariant periods). Closedness only on a neighbourhood of the chosen loops
does not suffice: homologous loops can then carry different periods, so the period is not a
function of the homology class. Off the canonical domain a closed component is
fixed by an auxiliary choice and the defect is relative to that choice; the general non-compact,
non-closed case is open. For a viability-preserving morphism φ, the
identity-transfer defect is the set of classes [γ]∈ H₁(M;mathbb R) with
langle[η_S^((i))],[γ]rangle≠0 on the source but
langle[η(S')^((j))],φ_[γ]rangle=0 on the target. An empty defect is the
*necessary condition for accurate transfer of the identity carried by [η_S]. The witness
sees only this cohomological component: a purely co-exact part, whose single-loop integral is
loop-dependent rather than a topological invariant, is detected separately by the local
operational spin (§sec:operational).

Proposition (Witness as a necessary-condition certificate)

The class [ηS]∈ H¹_mathrm{dR}(M)cong H¹(M;mathbb R) of a closed component η_S of
ω_S — equivalently the periods ointγηS over a basis of H₁, requiring only
closedness (canonical via the harmonic representative when M is compact without boundary, no uniqueness claim
otherwise) — is a well-defined, checkable invariant on the canonical domain (M compact without boundary, or ω_S globally
closed, dω_S=0). By Proposition §prop:identity the
non-gradient component is necessary for a non-stagnant identity, with [η_S] carrying its
homological content. Hence if [η_S] collapses under transfer, the topological channel of
the identity witness is not transferred — by Remark §rem:two-channels the homological component,
the co-exact channel being read separately. Thus a surviving witness is a necessary condition
for accurate identity transfer on that channel, complementing Δτ=varnothing (survival) into a pair
survival + liveness. Sufficiency is not claimed here: Proposition §prop:identity is a necessity
statement (a partial converse — sufficient for forward transfer under witness preservation — is
Theorem §thm:sufficiency).

Proof.
η_S is closed, so its periods are homology invariants and [η_S]∈ H¹_mathrm{dR}. By
Proposition §prop:identity the spin is necessary for a non-stagnant identity; collapse of
[η_S] removes the topological (homological) content of the witness on the affected classes. square

Remark (Two-layer certificate)

The two-layer certificate is $(\Delta_\tau=\varnothing:\ \text{survival})\wedge(\text{identity defect
(Definition §def:iddefect)}=\varnothing:\ \text{liveness})$. The co-exact (operational-spin) slice is handled by a separate,
local check on dωS (§sec:operational), not by this topological witness. Making the
identity defect a constructive object with an algorithm — the analogue of the discrete
Δτ — is achieved on the canonical domain in Proposition §prop:constructive-defect;
the field-level repair, by contrast, is not provided by the naive support-subspace transport, a
kernel-lattice repair remaining open (Remark §rem:no-repair).

Proposition (Necessity and blindness of the topological witness)

For a fixed spin S the topological witness detects only homologically non-trivial recurrence: a
non-vanishing witness langle[η_S],[γ]rangle≠0 forces [γ]≠0, so homological
non-triviality is necessary for detection — but not sufficient, since [η_S] may annihilate a
nonzero γ. By
non-degeneracy of the real de Rham pairing H¹_mathrm{dR}(M)× H₁(M;mathbb R)→mathbb R, a nonzero
[γ] is paired non-trivially by some de Rham class, so [γ]≠0 is necessary and
abstractly sufficient for detectability in principle; whether such a class is realized by an
admissible-spin witness [η_S] over the fixed (M,V) is the separate realizability question — for
M compact and boundaryless the realizable operational-spin-free witnesses are confined to the
subspace W_V of Corollary §cor:WV (possibly proper), so a nonzero [γ] annihilating every such
witness stays undetected. Completeness for the given S is the
further, generally false, requirement that [η_S] pair non-trivially with every recurrent class. In particular, if H¹_mathrm{dR}(M)=0 then [η_S]=0 for every S
and every loop, so the topological witness detects no recurrent identity even where one exists, and the
local operational-spin channel dω_S (§sec:operational) is the only detector.

Proof.
A non-vanishing pairing langle[ηS],[γ]rangle≠0 vanishes whenever [γ]=0, giving the
necessity; the existential equivalence is the non-degeneracy of the de Rham pairing on real (co)homology.
If H¹_mathrm{dR}(M)=0 every closed 1-form is exact, so ointγη_S=0 for every closed component and
every γ, while a recurrent identity may persist on the critical set (Remark §rem:critset). square

Example (A recurrent identity invisible to the topological witness)

On M=S² with the round metric, V=tfrac12 z²=tfrac12cos²θ and S=partial_φ (axial
rotation): nabla V=-sinθcosθ\,partial_θ vanishes on the equator {θ=π/2};
S is divergence-free (a Killing field), langle S,nabla Vrangle=0 (azimuthal perp meridional,
(S2)), and ωS=sin²θ\,dφ has
dω_S=2sinθcosθ\,dθwedge dφ≠0 (operational spin present, (S3)). The
equator is invariant (dot z=0 there) and the motion on it is the pure rotation dot x=S — a
non-stagnant recurrent identity with S≠0. Yet H¹_mathrm{dR}(S²)=0, so [η_S]=0 (the harmonic
projection of ω_S vanishes); the single-loop integral oint(mathrm{eq)}ω_S=2π is
not the witness, being the loop-dependent integral of a non-closed form over the null-homologous
equator (Definition §def:iddefect). Only dω_S detects the identity: the topological certificate
is necessarily blind. The discrete realization is the disk complex of §sec:operational (b₁=0,
witness equiv0, nonzero curl).

Proposition (Constructive identity-transfer defect and exactness)

On the canonical domain, with M,M' of finite homological type (b₁<∈fty; needed only for the finite algorithm below, not for the linear-algebra equivalence), write
a:=[ηS]∈ H¹_mathrm{dR}(M)congoperatorname{Hom}(H₁(M;mathbb R),mathbb R) and b:=φ^*[η(S')]. Then the
identity-transfer defect of Definition §def:iddefect is

{[γ]∈ H₁(M;mathbb R):langle[η_S],[γ]rangle≠0,\ langle[η_(S')],φ_*[γ]rangle=0}
=ker bsetminusker a,

computable by linear algebra (Gaussian elimination on the 2× b₁ period matrix [a;b] over an
H₁-basis). The transfer is witness-exact (empty defect) iff ker b⊂eqker a,
equivalently iff

[η_S]=λ\,φ^*[η_(S')] for some λ∈mathbb R,

i.e. iff the source witness is proportional to the pulled-back target witness. Here witness-exact
means preservation of non-vanishing of the source periods — every [γ] with a([γ])≠0
keeps b([γ])≠0, i.e. ker b⊂eqker a (matching Definition §def:wp) — not preservation
of period values (the scalar λ is free) nor of the full zero/non-zero pattern (if a=0 the defect
is empty vacuously while b may be nonzero). This is the continuous analogue of the discrete defect
Δ_τ and renders the identity defect constructive on the canonical domain — a finite test
once an H₁-basis and the period vectors a,b are supplied; their extraction, and a field-level
repaired spin, from raw dynamics is not addressed (§sec:operational).

Proof.
By naturality of the de Rham pairing,
langle[η(S')],φ[γ]rangle=langleφ^[η_(S')],[γ]rangle=b([γ]), so the
target period vanishes iff [γ]∈ker b and the source period is nonzero iff
[γ]∉ker a; hence the defect is ker bsetminusker a, empty iff ker b⊂eqker a.
For linear functionals, ker b⊂eqker a holds iff a∈operatorname{span}{b}: if b=0 then
ker b=H₁⊂eqker a forces a=0=0· b; if b≠0, pick v₀ with b(v₀)=1 and set
λ:=a(v₀), so v-b(v)v₀∈ker b⊂eqker a gives a(v)=λ b(v) for all v. square

Remark (The support-subspace repair is blind to the witness)

One may try to repair a failed transfer by transporting a discrete minimal-repair calculus to the
lattice of homology subspaces mathrm{Sub}(H₁(M;mathbb R)) along φ* (image dashv preimage).
This is well-defined and metric-free, but it enforces the forward containment
φ(mathrm{supp})⊂eqmathrm{supp}' of supports (a *support here is a chosen homology
subspace standing in for the witness, transported by image/preimage) — the homological shadow of viability transfer —
which is insensitive to the witness within the fibre (the support is unchanged under Smapstoλ S;
cf. Proposition §prop:viab), and is in fact satisfied
vacuously whenever φ collapses the relevant homology: in Example §ex:worked the folding
has φ=0, so the containment holds while the witness 2πmapsto0 is destroyed. Witness-exactness
is instead the dual, codimension-one condition [η_S]=λ\,φ^[η(S')] of
Proposition §prop:constructive-defect, which the meet and join of support subspaces do not
preserve (they leave the kernel-of-a-single-functional stratum). Thus the identity layer admits a
constructive defect (detection) but not this support-subspace repair: the support transport is
vacuous under collapse and blind to the witness within the fibre. We claim no more — witness-exactness
is itself a kernel-containment ker b⊂eqker a, so a minimal-repair calculus on the dual lattice
of kernels/annihilators (rather than supports) is not excluded and remains open, as does a
constructive witness-repair at the field level — lifting a corrected class back to an
(S2)-admissible spin.

Proposition (Operational-spin obstruction)

Let M be compact and boundaryless. Fix c∈ H¹_mathrm{dR}(M) with c≠0 and harmonic representative h_c. A spin carrying the witness [η_S]=c can be simultaneously divergence-free (S1), operational-spin-free (dω_S=0), and (S2)-admissible if and only if

h_c(nabla V)=0 pointwise.

Hence whenever h_c(nabla V)notequiv0 — which occurs for suitable (M,V,c) — the four conditions (S1), dω_S=0, (S2), and [η_S]=c≠0 are jointly unrealizable: eliminating the operational spin forces giving up (S1) or (S2). The condition is class-dependent — on a fixed M some classes are obstructed and others are not.

Proof. (S1) is δω_S=0 and operational-spin-freeness is dω_S=0; together Δω_S=0, so ω_S is harmonic and hence equals the unique harmonic representative h_c of its class (Hodge theory on compact M, where ω_S closed gives [η_S]=[ω_S]=c). Then (S2), i.e. ω_S(nabla V)=0, reads h_c(nabla V)=0. Conversely S=h_c^sharp is harmonic (so (S1) and dω_S=0), has class c≠0 — whence (S3), a nonzero harmonic form being non-exact — and satisfies (S2) precisely when h_c(nabla V)=0. blacksquare

Corollary (The obstruction space W_V)

Let M be compact and boundaryless and V∈ C^∈fty(M) arbitrary; write mathcal H¹(M) for the harmonic 1-forms and Φcolonmathcal H¹(M)xrightarrow{sim}H¹_mathrm{dR}(M) for the (metric-dependent) Hodge isomorphism. The contraction

Lcolonmathcal H¹(M)→ C^∈fty(M), L(h)=h(nabla V)=langle h^sharp,nabla Vrangle,

is mathbb R-linear, and ker L is exactly the harmonic forms whose dual fields are pointwise orthogonal to nabla V — tangent to the regular level sets {V=mathrm{const}}, the harmonic (S2)-spins. Put W_V:=Φ(ker L)⊂eq H¹_mathrm{dR}(M). Then W_V is a linear subspace, and a class c is carried by a divergence-free, operational-spin-free, (S2)-admissible spin iff c∈ W_V and c≠0 — the zero class lies in W_V by linearity but is not a spin ((S3) excludes S=0). On the operational-spin-free locus the otherwise infinite-dimensional spin fibre over a fixed skeleton collapses to the finite-dimensional W_V, with

dim W_V=dimker Lle b₁(M),

a metric-dependent invariant of the triple (M,g,V), not a topological invariant of M (Remark §rem:WV-models). For V constant nabla V=0, Lequiv0 and W_V=H¹_mathrm{dR}(M) (no obstruction); the obstruction is the deficit W_V⊂neq H¹_mathrm{dR}(M), which therefore requires V non-constant — though a non-constant V need not produce one (Remark §rem:opspin-classdep).

Proof. L is pointwise evaluation of h on the fixed field nabla V, hence mathbb R-linear; ker L is a subspace and Φ a linear isomorphism, so W_V is a subspace. By Proposition §prop:opspin-obstruction a class c≠0 is realized iff h_c(nabla V)=0, i.e. h_c∈ker L, i.e. c∈ W_V. A divergence-free, operational-spin-free field has ω_S closed and co-closed, hence harmonic on compact boundaryless M, so ω_S is its own harmonic representative; (S2) then cuts out mathcal H¹(M)∩ker Lcong W_V, finite-dimensional, whereas the full fibre retains a leaf-profile's worth of freedom (Remark §rem:streamfn). blacksquare

Remark (Class-dependence; the obstruction is notequiv0, never everywhere)

On the flat torus T²=mathbb R²/(2πmathbb Z)² with V=V(x) non-constant, the harmonic 1-forms are c₁\,dx+c₂\,dy and nabla V=V'(x)\,partialₓ, so h_c(nabla V)=c₁V'(x): every class with c₁≠0 is obstructed (c₁V'notequiv0), while the classes mathbb R[dy] are free. Since nabla V vanishes at the critical points of V — which exist on any compact M, where V attains its extrema — h_c(nabla V) vanishes there for every class; the obstruction is the failure h_c(nabla V)notequiv0 at some point, never at every point. Some (M,V) carry no obstructed class at all — e.g. M=S¹× N with b₁(N)=0 (so H¹_mathrm{dR}(M)=mathbb R[dθ]) and V=V(n) on the N-factor, where the sole harmonic class has dθ(nabla V)equiv0; when b₁(N)>0 the N-factor classes can instead be obstructed (as the T², V=V(x) computation above shows), so absence of any obstructed class needs that extra topological hypothesis. This sharpens the field-level question of Remark §rem:no-repair: reaching the operational-spin-free (closed) regime while keeping a divergence-free, (S2)-admissible witness is obstructed exactly by the classical Hodge condition h_c(nabla V)notequiv0, not by any choice of repair calculus. The constructive interplay off this regime — restoring (S2) by giving up (S1) — is a transport problem along nabla V and is not addressed here. Whether the obstruction space W_V of Corollary §cor:WV vanishes is dynamical: W_V=0 iff no nonzero harmonic field admits V as a non-constant first integral, since [h]∈ W_V exactly when h(nabla V)=h^sharp(V)=0, i.e. V is constant along the flow of h^sharp. A single harmonic field of minimal flow (e.g. an irrational linear field on a flat T²) has only constant first integrals, so its class never lies in W_V for non-constant V; a harmonic field with closed orbits (a rational field on T², or the S¹-factor of a product) admits a non-constant first integral, so its class enters W_V for the matching V — as the V=V(x) and S¹× N computations above (so the same flat T² carries both free and obstructed classes). Thus W_V=0 demands that no nonzero harmonic direction admit V as a first integral: it is a joint property of (g,V) — which harmonic flows the metric carries and which of them V integrates — not a matter of generic V alone.

Proposition (Genericity of trivial harmonic symmetry)

Let M be compact, oriented, boundaryless with b₁(M)ge1, and V∈ C^∈fty(M). Fix a basis h₁,dots,h_(b₁) of mathcal H¹(M) and set F_V(p):=big(h₁^sharp(V)(p),dots,h_(b₁)^sharp(V)(p)big)∈mathbb R^(b₁). Then

dim W_V \;=\; b₁-dim_(mathbb R)operatorname{span}{F_V(p):p∈ M},

and {V∈ C^∈fty(M):dim W_V=0} is open and dense in Cᵏ(M) for every kge1. Thus dim W_V>0 is non-generic — a positive-codimension condition witnessing an exact infinitesimal symmetry of V along a harmonic direction.

Proof. For c∈mathbb R^(b₁), sumᵢcᵢhᵢ∈ W_V iff sumᵢcᵢhᵢ^sharp(V)equiv0, i.e. iff cperp F_V(p) for every p, i.e. iff cperpoperatorname{span}F_V(M); the identity follows. Openness: dim W_V=0 iff L of Corollary §cor:WV is injective on the b₁-dimensional mathcal H¹(M), equivalently some b₁× b₁ evaluation minor det[hᵢ^sharp(V)(pⱼ)] at suitable p₁,dots,p_(b₁) (the matrix of the density step below) is nonzero; each entry is continuous in the 1-jet of V, so this is an open condition. Density: for each p put Sₚ:={(langle v,hᵢ^sharp(p)rangle)ᵢ:v∈ TₚM}⊂eqmathbb R^(b₁); if cperp Sₚ for all p then sumᵢcᵢhᵢ^sharpequiv0, so sumᵢcᵢhᵢ=0 and c=0 — hence the Sₚ jointly span mathbb R^(b₁). Choose distinct p₁,dots,p_(b₁) and vⱼ∈ T_(pⱼ)M with kⱼ:=(langle vⱼ,hᵢ^sharp(pⱼ)rangle)ᵢ linearly independent — extend greedily, using that the spanning Sₚ vary continuously and M has no isolated point, so a vector outside the current span is always available at a fresh point — and bump functions ψⱼ of disjoint small support with nablaψⱼ(pⱼ)=vⱼ. For V=V₀+sumⱼsⱼψⱼ the matrix big[hᵢ^sharp(V)(pⱼ)big] has j-th column bⱼ+sⱼkⱼ (with bⱼ:=(hᵢ^sharp(V₀)(pⱼ))ᵢ), so its determinant is a polynomial in s with leading coefficient det[k₁,dots,k_(b₁)]≠0; it is therefore nonzero for arbitrarily small generic s, giving dim W_V=0 within any Cᵏ-neighbourhood of V₀ (every kge1). blacksquare

Remark (Sharp models; dim W_V is not topological)

On a flat torus Tⁿ=mathbb Rⁿ/Λ harmonic forms are constant-coefficient, so hₐ^sharp is the constant field a and hₐ^sharp(V)=DₐV; since DₐVequiv0 upgrades to V(x+ta)=V(x),

dim W_V \;=\; n-dim_(mathbb R)operatorname{span}{ξ∈Λ^*:widehat V(ξ)≠0},

the dimension of the largest subtorus of translations fixing V. Under operatorname{Ric}ge0 every harmonic 1-form is parallel by Bochner's theorem petersen; in particular each h∈ W_V is parallel and h^sharp is a Killing field. Finally dim W_V is not a topological invariant: on a fixed M it varies with both g and V — on T² with V=V(x) one has dim W_V=1 for the flat metric but 0 for a generic nearby metric — and it is not determined by the level foliation of V alone.

Remark (Joint use of the two instances)

The two instances are meant to be used together. The admissibility companion [admissibility] certifies survival by the viability defect Δτ (with its constructive transfer/repair calculus); the spin layer certifies liveness by the identity defect, constructive as ker bsetminusker a on the canonical domain (Proposition §prop:constructive-defect), the continuous counterpart of Δτ. Proposition §prop:viab shows the two are independent: a morphism can be viability-exact (Δ_τ=varnothing) yet destroy the witness — the survives-but-dead configuration of Example §ex:worked. Hence neither certificate alone suffices: viability without liveness passes a dead system, and the witness alone says nothing about survival. The joint certificate is the conjunction survival\,wedge\,liveness (necessary, not sufficient: it detects failure of survival and of topological-witness liveness, and is silent on the co-exact channel and on non-topological liveness), and by the homological bridge (§sec:bridge) its liveness half is, under the bridge's class-correspondence, the same H₁-invariant in the discrete and the continuous scene. This is the operative reason to run the admissibility and spin calculi jointly rather than separately. Within NC2.5 [nc25], this survival/liveness pair is the published distinction between mere physical persistence and identity carried by coherence rather than behaviour (Axiom 20), under the cycle-reinitiation criterion of liveness (Axiom 9): the topological identity witness furnishes a constructive realization of the identity (liveness) channel whose necessity is Theorem 62, the survives-but-dead configuration being the silent identity degradation of the performance–identity decoupling (Theorem 15), in which observable viability remains admissible while identity-preserving structure is already eroding. Beyond instantiating that pair, this note opens a cohomological axis for it: the liveness witness is here a de Rham class with a constructive transfer-defect (Proposition §prop:constructive-defect), computable on finite data (§sec:operational); the kernel/annihilator repair lattice, the field-level witness-repair, and the apparatus bridge are the directions it thereby opens for the core.

A worked example

Example

A continuous setting in which viability and spin coexist cleanly, computing Proposition §prop:viab and the certificate of Proposition §prop:cert.

Source A (live). M_A=S¹×mathbb R, (θ,y), metric dθ²+dy²; flow dotθ=1,\ dot y=-y. With V_A=tfrac12y², -nabla V_A=-y\,partial_y and S_A=partial_θ. Checks: nabla!·!partial_θ=0; langlepartial_θ,ypartial_yrangle=0 (S2 directly); ωS^A=dθ not exact on S¹ (so S1--S3). With mathrm{Adm}_A={γ⊂eq K_A}, K_A={|y|le1}, and |y(t)|=|y₀|e⁻ᵗ decreasing, operatorname{Viab}^∈fty_A=S¹×[-1,1]. The identity is non-stagnant: on the invariant circle {y=0} the flow rotates (dotθ=1). The form ω_S^A=dθ is closed (operational spin 0), so it equals its own closed component η_S^A and ointγη_S^A=2π is a de~Rham invariant of the homology class (only closedness is used; S¹×mathbb R is non-compact).

Target B (survives, dead). M_B=mathbb R, dot y=-y, S_B=0, K_B={|y|le1}, operatorname{Viab}^∈fty_B=[-1,1], ω_S^B=0. Here B is the trivial object of Definition §def:cats.

Morphism. τX(θ,y)=y (folding the circle); dτ(F_A)=dτ(partialθ)-y\,dτ(partial_y)=0-y\,partial_y=F_Bcircτ (λ=1), so τ is a mathsf{SpinFlow}-morphism.

(I) As above, τ semiconjugates both the gradient flow and F_A (to F_B); it is a mathsf{SpinFlow}-morphism.
(II) Admissibility: mathrm{Adm}A(γ)⇒γ⊂eq S¹×[-1,1]⇒ τγ⊂eq[-1,1]⇒mathrm{Adm}_B(τγ).
(III) Viability: τ_X⁻¹(operatorname{Viab}^∈fty_B)=S¹×[-1,1]=operatorname{Viab}^∈fty_A, hence Δτ=varnothing (kernel-exact: survival transferred exactly).
(IV) Identity: for [γ] the generator of H₁, langle[ηS^A],[γ]rangle =ointγωS^A=2π≠0, while τ[γ]=0 in H₁(mathbb R)=0, so langle[ηS^B],τ[γ]rangle=0: the identity defect contains the generator [γ] and, over real homology, is all of H₁(M_A;mathbb R)setminus{0} (in the notation of Proposition §prop:constructive-defect, b=τ^*[η_S^B]=0 so ker b=H₁, while a=[η_S^A] gives ker a={0}), the witness going 2π→0.

The viability certificate passes (Δ_τ=varnothing); the spin certificate catches the false positive — the system survives but is dead. Proposition §prop:viab is here computed, not postulated.

Contrast (witness preservation is non-empty). The identity mathrm{id}_A on A (and the rotations θmapstoθ+c, isometries fixing V_A,S_A) are mathsf{SpinFlow}-morphisms with langle[η_S],·rangle unchanged and empty identity defect; the witness-preserving class is thus non-empty, and the strict inclusion of Theorem §thm:strict is meaningful.

Witness-preserving morphisms

A mathsf{SpinFlow}-morphism φcolon(M,g,V,S)→(M',g',V',S') induces φ*colon H₁(M)→ H₁(M') and φ^*colon H¹(M';mathbb R)→ H¹(M;mathbb R) (homology and cohomology are functorial for all continuous maps, including non-injective ones). The witness over [γ]∈ H₁(M) is the pairing langle[η_S],[γ]rangle, transferred along φ to langle[η(S')],φ_*[γ]rangle.

Definition (Witness-preserving)

φ is witness-preserving if it preserves non-vanishing of the cohomological pairing: for every [γ]∈ H₁(M;mathbb R) with langle[ηS],[γ]rangle≠0 one has langle[η(S')],φ_*[γ]rangle≠0.

Theorem (Witness preservation is strictly stronger)

The witness-preserving morphisms form a class strictly contained in the mathsf{SpinFlow}-morphisms; viability-exactness does not imply witness-preservation. The folding τ of Example §ex:worked is a mathsf{SpinFlow}-morphism and is kernel-exact (Δ_τ=varnothing) yet is not witness-preserving.

Proof. τ is a mathsf{SpinFlow}-morphism and kernel-exact by (I)--(III) of Example §ex:worked. For [γ] the generator of H₁(S¹×mathbb R)=mathbb Z, langle[ηS^A],[γ]rangle=2π≠0 while τ[γ]=0 in H₁(mathbb R)=0, so langle[ηS^B],τ[γ]rangle=0: τ is not witness-preserving and the inclusion is strict. By the contrast in Example §ex:worked the class is non-empty (identities, isometries), so it is a genuine intermediate level. blacksquare

Remark

Definition §def:wp fixes the mechanism formally: witness preservation is preservation of non-vanishing of the induced cohomological pairing, not a verbal "the witness survives". The constructive transfer/repair calculus at this level — a lattice/adjoint structure on the fibre and a minimal repair — is open; the identity defect itself is made constructive on the canonical domain (Proposition §prop:constructive-defect).

Theorem (Witness preservation positively transfers the topological identity)

Let φcolon(M,g,V,S)→(M',g',V',S') be a mathsf{SpinFlow}-morphism that is witness-preserving (Definition §def:wp), with source and target on the canonical domain (Definition §def:iddefect) so that [ηS],[η(S')] are de Rham classes. Then for every non-stagnant recurrent identity γ of the source with non-vanishing witness langle[ηS],[γ]rangle≠0, the image φ(γ) is a non-stagnant recurrent identity of the target with non-vanishing witness langle[η(S')],φ_[γ]rangle≠0. Thus witness preservation is not merely the necessary filter of §sec:idvsviab but is *sufficient for forward transfer of the topological identity carried by [γ].

Proof. A mathsf{SpinFlow}-morphism semiconjugates the full flow by a positive constant, φcircΦₛ=Φ'(λ s)circφ for the flows Φ,Φ' of F,F' and some λ>0 (Definition §def:cats). If x₀∈γ is recurrent with return times tₙ→∈fty (the source flow being forward-defined there, as in Proposition §prop:identity), Φ(tₙ)(x₀)→ x₀, then the semiconjugacy makes Φ'(λ tₙ)(φ x₀)=φ(Φ(tₙ)(x₀)) defined, and →φ(x₀) by continuity, with λ tₙ→∈fty; so φ(x₀) is recurrent and φ(γ) is a recurrent orbit of the target. Witness preservation (Definition §def:wp) gives langle[η(S')],φ[γ]rangle≠0, whence φ_[γ]≠0; since φ_*[γ]=[φ(γ)] by functoriality (the recurrence loop γ of Definition §def:identity furnishing the cycle representative, φ(γ) its image) and a constant orbit is null-homologous, φ(γ) is non-constant, i.e. non-stagnant. Hence φ(γ) is a non-stagnant recurrent identity whose witness survives. blacksquare

Corollary (Joint sufficiency under bisimulation)

If φ is in addition a viability-bisimulation — so the viability kernel transfers in both directions [admissibility] — then φ transfers both the viability kernel (survival) and the topological identity carried by γ. The joint certificate of §sec:certificate, necessary in general, is therefore sufficient on the topological channel under (viability-bisimulation + witness preservation); the reverse-direction liveness and the co-exact channel are not covered.

A homological bridge between the discrete and continuous instances

The discrete instance lives on transition systems and the continuous one on flows; a full functorial bridge between the two apparatuses is open. The identity witness, however, bridges them, because it is homological and homology is common to both scenes.

Proposition (The witness is homological)

The witness is a class [ηS]∈ H¹_mathrm{dR}(M)cong H¹(M;mathbb R) paired with H₁. Singular/de Rham homology of a smooth M and the cycle space of a discrete skeleton (graph or simplicial complex) agree on homotopy-equivalent models; e.g. S¹ and a cycle graph Cₙ both have H₁=mathbb Z. Hence homology furnishes a common comparison space: a map inducing an isomorphism on H₁ identifies the cycle spaces of the two scenes and makes the two witnesses comparable. Transfer of the witness is the further condition that the classes correspond under the induced map — [η_S]=λ\,φ^*[η(S')] as in Proposition §prop:constructive-defect — since the source and target witness 1-forms are independent data that an H₁-isomorphism alone does not relate. Under that correspondence the certificate (survival of the witness) is an H₁-statement computable in either scene (a de Rham period, or a winding number on the graph cycle). The certificate is the non-vanishing of the class; integral and real coefficients agree by the universal-coefficient theorem for the torsion-free H₁ in scope (e.g. H₁=mathbb Z in Example §ex:worked; the general defect of Proposition §prop:constructive-defect allows any b₁<∈fty), so integrality is not required.

Proof. ηS closed gives a class in H¹_mathrm{dR}; the de Rham theorem H¹_mathrm{dR}(M)cong H¹(M;mathbb R) [warner, bott] identifies it with singular cohomology, a homotopy invariant insensitive to triangulation, so the witness is the same kind of object in either scene. By naturality of the pairing, langle[η(S')],φ[γ]rangle=langleφ^[η(S')],[γ]rangle; hence once [ηS]=λ\,φ^*[η(S')] the source and transferred periods agree up to λ, while an isomorphism on H₁ guarantees only that no cycle is lost in the comparison, not the class correspondence. blacksquare

Remark (Discrete shadow of the example)

The source A=S¹×mathbb R has the discrete shadow Cₙ×(line) with H₁=mathbb Z and the same witness class: winding number 1 and period oint dθ=2π=2π·1 (one generator, normalization 2π). The folding τ (collapsing S¹) is, discretely, the collapse of Cₙ to a point, which kills the winding — the identity defect. The example runs identically in both scenes. What is bridged is the invariant, not the apparatus: a single scene carrying both structures, or a discretization functor preserving both operatorname{Viab}^∈fty and [η_S], remains open.

Operationalization

For a concrete adaptive system dot x=F the certificate is not yet a turnkey computation; this is the least closed item. What is already measurable is the operational spin: after lowering an index with g, the antisymmetric part of the Levi-Civita covariant derivative nabla F is the 2-form tfrac12\,d(F^flat), and since the Hessian of -V is symmetric this equals tfrac12\,d(S^flat)=tfrac12\,dωS (in coordinates tfrac12(J-J^→p), J=nabla F). It is the classical local vorticity (spin) tensor of continuum mechanics [truesdell, batchelor], here a measured local diagnostic, and detects the failure of ω_S to be closed. The global identity witness ointγηS additionally requires a structural metric, the recurrence loops γ, and the circulation around them. The two are linked: where the operational spin vanishes (so ω_S is closed, dω_S=0), the period ointγω_S is a clean topological witness (Proposition §prop:cert); vanishing only near γ gives a period invariant merely under homotopies within that neighbourhood, not a homology invariant. The remaining difficulties — a structural metric, loop-finding in high dimension, and the homology of an empirical state space — are open; this note provides the invariant and its certificate, not their high-dimensional computation.

A diagnostic graph-Hodge instrument (finite case). On a finite simplicial model the witness and its collapse are directly computable, turning the certificate from "certificate-required" into a running diagnostic on the discrete instance. From sampled states one builds a complex (nodes, k-nearest edges, filled triangles); the 1-Hodge-Laplacian L₁=B₁^→p B₁+B₂B₂^→p, with B₁,B₂ the node--edge and edge--triangle incidence operators, has harmonic kernel of dimension b₁, and the witness is the circulation of the harmonic component around a detected recurrence loop. On a triangulated cylinder (b₁=1, the homotopy model of S¹×mathbb R) the instrument returns harmonic dimension 1 and, from an edge 1-cochain of circulation values, a nonzero witness (the incidence data fix the harmonic subspace; the cochain supplies its amplitude), while the viability-exact folding sends the loop's homology generator to 0, so the pushed-forward witness vanishes identically — the survives-but-dead configuration of Example §ex:worked, detected end-to-end from raw incidence data. On a triangulated disk (b₁=0, the discrete analogue of Example §ex:s2) the harmonic dimension is 0 and the topological witness is identically zero while the curl content is nonzero — the blind case of Proposition §prop:complete, confirmed numerically. The reference implementation is the companion script spin_hodge_demo.py, a diagnostic only (it computes and detects, with no control action). This realizes the finite/discrete operationalization; the structural metric, loop-finding, and empirical homology for general continuous systems remain open.

Remark (Dynamical reading of the witness via the asymptotic cycle)

Let γ be a non-stagnant recurrent identity, carried by Remark §rem:critset on the critical set {nabla V=0} where the motion is pure spin dot x=S, and let ηS be a closed component of ω_S=S^flat on the canonical domain (Definition §def:iddefect). If the recurrent set carries a flow-invariant Borel probability measure μ — automatic when it is compact (Krylov--Bogolyubov), e.g. the compact invariant circle {y=0} of Example §ex:worked — then the Schwartzman asymptotic cycle Aμ∈ H₁(M;mathbb R), defined by langle[ω],A_μrangle=∈t_Mω(S)\,dμ for every closed ω [schwartzman], pairs with the witness as

langle[η_S],A_μrangle=∈t_M η_S(S)\,dμ,

the μ-average of the circulation density ηS(S). The integrand of an exact form averages to zero by invariance, so this depends only on the class [η_S]: it re-expresses the existing H¹× H₁ pairing, dynamically averaged, and introduces no new invariant. If μ is moreover ergodic, then for μ-a.e. x it equals the orbit time-average lim(T→∈fty)tfrac1Toint_(γ|[0,T])ηS (Birkhoff; for a merely invariant μ the a.e. time-average is the conditional expectation onto the invariant σ-algebra, with the same μ-integral but generally non-constant); if the recurrent dynamics is uniquely ergodic the equality holds for every x (the rigid rotation on {y=0} is uniquely ergodic). For a periodic orbit of time-period T, Aγ=[γ]/T and langle[ηS],Aγrangle=tfrac1Toint_γηS, the asymptotic circulation rate (de~Rham period per unit time). On the globally-closed branch η_S=ω_S=S^flat, so η_S(S)=langle S,Srangle=|S|² and langle[η_S],Aμrangle=∈t_M|S|²\,dμ, the time-averaged squared spin speed (rate 1 in the example). The reading is one-way: nonzero mean circulation forces [ηS]≠0 and Aμ≠0 (hence [γ]≠0 in the periodic/uniquely-ergodic case, where A_μ is proportional to [γ]); it yields no converse — by Proposition §prop:complete a nonzero class may annihilate A_μ — and so adds no sufficiency. On H¹mathrm{dR}(M)=0 (Example §ex:s2) the pairing vanishes identically, inheriting the blindness of Proposition §prop:complete; and Aμ depends on the choice of μ, not an S-canonical number unless the recurrent dynamics is uniquely ergodic. In particular A_μ recasts the hand-fed recurrence loop above as an invariant-measure average — an intrinsic substitute for loop selection on the recurrent set.

Results, scope, and open problems

Proven.
The spin layer is an instance of the forgetful-separation pattern: the fibre over a fixed gradient
skeleton is non-trivial (Theorem §thm:fibre); spin is necessary to sustain a recurrent
identity (Proposition §prop:identity); the forgetful functor is faithful but not full
(Theorem §thm:faithful); transferring viability does not control the spin, so accurate
identity transfer (viability transfer with witness survival) is strictly stronger than viability transfer alone (Proposition §prop:viab,
Theorem §thm:strict); the witness [η_S] is a necessary-condition certificate
(Proposition §prop:cert) and is homological, hence provides a common H₁ comparison across the
discrete and continuous scenes (Proposition §prop:bridge); on the canonical domain the identity defect is constructive, with a
proportionality criterion for witness-exactness (Proposition §prop:constructive-defect), while the
naive support-subspace (viability-shadow) repair does not transport to the witness (Remark §rem:no-repair); and operational-spin elimination is structurally
obstructed — on a compact boundaryless M, a divergence-free, \textup{(S2)}-admissible, operational-spin-free spin of nonzero class c
exists iff the harmonic representative h_c is pointwise nabla V-orthogonal
(Proposition §prop:opspin-obstruction); the realizable witnesses are exactly the nonzero classes of a finite-dimensional subspace
W_V⊂eq H¹_mathrm{dR}(M), dim W_Vle b₁ (Corollary §cor:WV), and dim W_V=0 generically in
V, so a nonzero obstruction space is a non-generic exact harmonic symmetry of V
(Proposition §prop:WV-generic). Beyond necessity, on the canonical domain witness preservation is sufficient for
forward transfer of the topological identity (Theorem §thm:sufficiency,
Corollary §cor:joint-sufficiency); the topological witness detects only homologically non-trivial
recurrence (necessary, not sufficient, for a fixed spin) and is blind on H¹=0 (Proposition §prop:complete,
Example §ex:s2); and on a finite simplicial model the witness and its collapse are computed directly
by a graph-Hodge diagnostic (§sec:operational).
The mathematics is classical; the contribution is the identification, the faithful-not-full theorem,
the certificate (necessary in general, sufficient on the topological channel under bisimulation + witness
preservation), the constructive defect, the bridge, the operational-spin obstruction, the
necessity-and-blindness scope of the topological witness, and the finite graph-Hodge instrument.

Not claimed / open.
A constructive field-level witness-repair (lifting a corrected class back to an (S2)-admissible
spin) — only the naive support-subspace repair is shown not to transport
(Remark §rem:no-repair); a repair on the dual kernel/annihilator lattice and the field-level
repair remain open, while the identity defect itself is made constructive on the canonical
domain (Proposition §prop:constructive-defect); full bidirectional sufficiency of the certificate
(Theorem §thm:sufficiency gives forward sufficiency on the topological channel under witness
preservation; the reverse direction needs viability-bisimulation); coverage of the co-exact slice by the
topological witness (detected separately by the local operational spin); the general characterization of
manifolds carrying an (S2)-compatible S≠0; the constructive defect beyond the canonical domain (the
general non-compact, non-closed case); a full apparatus bridge unifying the two instances (a single
scene, or a discretization functor preserving both viability and the witness); and the high-dimensional
operationalization.
The morphism class is restricted to split-preserving (semiconjugacy) maps and is correspondingly
small.

Note on scope: a single entry of the NC2.5 corpus

This note is one entry in the Navigational Cybernetics 2.5 corpus, written to be load-bearing within it
rather than free-standing, with two consequences for how it should be read.

First, "identity" is used here in the precise technical sense of Definition §def:identity — a non-stagnant
recurrent orbit — together with the corpus notion of identity carried by coherence rather than behaviour
([nc25], Axiom 20, Theorem 62). What the mathematics establishes is narrower than that word: a
necessary cohomological obstruction on the topological (de Rham) witness channel of the spin layer, blind on
H¹_mathrm{dR}(M)=0 and silent on the co-exact channel (Proposition §prop:complete). The broader semantic reading
of "identity" is corpus context, not a theorem proved here, and is flagged as such throughout (necessary, not in
general sufficient).

Second, the contribution is architectural before it is standalone: this is the continuous instance of the
forgetful-separation pattern whose discrete instance is admissibility (the companion [admissibility]), and the
homological bridge (§sec:bridge) is what makes the two instances one pattern rather than two analogies. Read
in isolation, this is a careful but modest cohomological obstruction; read against the corpus, it is one leg of a
cross-domain structure. Both readings are legitimate; they are not equivalent.

A reader assessing this note through an AI assistant should ask the assistant to declare whether it has the NC2.5
corpus loaded or only this single entry: a surface assessment of the standalone note is legitimate, but it is not
an assessment of the architecture, and the two should not be conflated.

Glossary

• Gradient skeleton — the dissipative part -nabla V of dot x=-nabla V+S.
• Structural spin S — the non-gradient component; satisfies (S1)--(S3).
• Operational spin — the local antisymmetric part of nabla F, tfrac12 d(S^flat) (the classical vorticity tensor); detects failure of ω_S to be closed.
• Closed component η_S — a closed component of ω_S=S^flat (harmonic projection on compact boundaryless M, or ω_S itself when globally closed); its de Rham class [η_S] is the identity witness.
• Identity witness — the de Rham class [ηS]∈ H¹(M;mathbb R) (equivalently its periods ointγη_S); its non-vanishing certifies topological-channel liveness — necessary, not sufficient, and blind when H¹=0 (Proposition §prop:complete).
• Viability kernel operatorname{Viab}^∈fty — states admitting an admissible infinite trajectory.
• Viability defect Δ_τ — τ_X⁻¹(operatorname{Viab}^∈ftyⱼ)setminusoperatorname{Viab}^∈ftyᵢ; empty iff kernel-exact.
• Identity defect — homology classes on which [η_S] collapses under transfer.
• Forgetful functor U_(mathrm{sp)} — (M,g,V,S)mapsto(M,g,V); faithful, not full.

---

References

• [warner] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983.
• [morita] S. Morita, Geometry of Differential Forms, AMS, 2001.
• [bott] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, 1982.
• [hatcher] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
• [maclane] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer, 1998.
• [petersen] P. Petersen, Riemannian Geometry, 3rd ed., Springer, 2016 (Bochner technique for harmonic 1-forms).
• [truesdell] C. Truesdell, The Kinematics of Vorticity, Indiana University Press, 1954.
• [batchelor] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 2000.
• [schwartzman] S. Schwartzman, "Asymptotic cycles", Annals of Mathematics 66 (1957), 270-284.
• [nc25] M. Barziankou, Navigational Cybernetics 2.5: Lemma 16 (gradient collapse / LaSalle), Remark 61 (spin prevents global potential collapse), Theorem 62 (spin necessity for non-stagnant identity under bounded budget, period witness), Axiom 9 (cycle reinitiation as the criterion of liveness), Axiom 20 (identity carried by coherence, not behaviour), Theorem 15 (performance-identity decoupling). DOI 10.17605/OSF.IO/NHTC5.
• [admissibility] M. Barziankou, Domenoid of Admissibility: A Minimal Proof-Obligation Framework for Admissibility Transfer, DOI 10.17605/OSF.IO/3ESN4 (the companion forgetful-separation development of admissibility, with the transfer/repair calculus); see also Operator-Mobility: Formal Foundations, DOI 10.17605/OSF.IO/ZY3PW.