Cycle winding under Coupling: scope and retained evidence¶

Conditional cycle transport and winding protection remain reusable. The C6 global-stability campaign is parked: 41 of 56 exit labels are excluded and 15 remain open. Its numerical studies are archived evidence, not the current research queue.

Chapters and scope¶

Chapter	Responsibility
C6 response, contraction and represented rounding	Parked C6 response domains, numerical reserves and rounding-cell restrictions.
C6 carried flow and pressure cells	Exact carry, live-pressure and finite itinerary results under their numerical model.
C6 pressure excursions and compensation	Finite pressure, mean-budget and compensation obstructions; no global closure.
C6 relational regions and exit exclusions	Correlated regions, histories and regional exclusion proofs.
C6 history refinement and retained exclusions	Exact union, safe-memory and refinement evidence with its source-bound budgets.
C6 remaining verification obligations	Retained bounded-search failures and unresolved certification conditions.

Section numbers remain stable across this collection. The execution plan alone assigns research work.

1. The structural state and the actual operator realization¶

Let phi_i be the phase at node i of an oriented cycle C_n, n>=3, and let its oriented shortest-arc gaps be

\[ d_i=\operatorname{wrap}(\phi_{i+1}-\phi_i),\qquad -\pi<d_i<\pi,\qquad \sum_i d_i=2\pi W. \]

Indices are cyclic and W is the integer winding of this declared cycle. The final identity follows from a consistent closed phase loop, not from arbitrarily supplied real gap coordinates. Reversing orientation changes its sign. Retaining a positive wrap-branch margin makes the observation defined; it does not by itself prove that a future operation preserves that margin.

The following theorem uses the existing canonical Coupling implementation in _coupling_stage_kernel.py, with UM_BIDIRECTIONAL=False and UM_FUNCTIONAL_LINKS=False. The first configuration makes each target average only its neighbors and write only its own phase. The second keeps the simple cycle support fixed. Other preconditions and grammar admission remain necessary at execution time.

Let g be the effective UM gate, including any tightening of the hard U3 limit, and suppose every gap lies in one interval

\[ -g<m\leq d_i\leq M<g,\qquad 0<g\leq\pi/2. \]

The interval may be positive, negative or contain zero. Both cycle neighbors are therefore compatible with every target. Their separation in the local lift is d_(i-1)+d_i, whose absolute value is strictly less than pi. The neighbor phasor sum is nonzero, so the circular midpoint is unambiguous. The UM phase factor eta=UM_theta_push is the already configured operator coefficient, with 0<eta<=1. Its repository default is 1/(pi+1) as materialized by UM_THETA_PUSH. The theorem covers the declared factor interval; it does not derive that particular default uniquely from the nodal equation or add an independent phase-evolution law.

The default bidirectional UM configuration has a different effect: it includes the target in the phasor mean and writes neighboring phases too. It is outside this theorem. Even on a uniform twisted cycle, one bidirectional target shrinks the two inner gaps by 1-eta and expands the adjacent outer gaps by 1+eta. The interval invariance proved below cannot simply be transferred to it.

2. The midpoint identity and its canonical pressure connection¶

In the target's local lift, its neighbor phases are phi_i-d_(i-1) and phi_i+d_i. Their phasor sum is

\[ 2\cos\frac{d_{i-1}+d_i}{2}\; \exp\left(i\left[\phi_i+\frac{d_i-d_{i-1}}2\right]\right). \]

The cosine is positive under the stated strict gate. Consequently the actual target-only UM shortest-arc update has displacement

\[ u_i=\frac\eta2(d_i-d_{i-1}). \]

This identity connects directly to the existing canonical phase-pressure channel in dnfr.py. On this same simple cycle, its unweighted neighbor-phasor mean is the midpoint above, hence

\[ \partial\phi_i =-\frac1\pi\operatorname{wrap}(\phi_i-\bar\phi_i) =\frac{d_i-d_{i-1}}{2\pi},\qquad u_i=\eta\pi\,\partial\phi_i. \]

The configured phase-channel weight belongs to the aggregation of DeltaNFR; it is separate from the unweighted channel identity here. The nodal integrator subsequently uses nu_f*DeltaNFR to evolve EPI. Reading the same structural gradient in UM does not identify a phase jump with that EPI evolution or introduce a second continuous-time law. The identity excludes vanishing resultants, arbitrary neighborhood profiles and wrap-crossing configurations. Binary64 phasor evaluation retains its separate residual.

3. One target: conserved circulation and interval protection¶

Updating target i changes just its incident gaps. Put a=d_(i-1), b=d_i. Then

\[ \begin{pmatrix}a'\\b'\end{pmatrix} =\begin{pmatrix}1-\eta/2&\eta/2\\\eta/2&1-\eta/2\end{pmatrix} \begin{pmatrix}a\\b\end{pmatrix}. \]

Each new gap is a convex combination of the old pair, and their sum is unchanged. Every gap remains in [m,M], strictly inside U3 and away from the wrap branch. The same is true along the affine shortest-arc completion of the declared phase jump. Thus the closed cycle retains W. This completion is a mathematical path witness; the engine event itself is a discrete update.

For the gap mean d_bar=sum(d)/n, define the diagnostic squared spread V(d)=sum((d_i-d_bar)^2)/2. Direct expansion gives

\[ V(d)-V(d')=\frac{\eta(2-\eta)}4(a-b)^2\geq0. \]

Every finite sequence of these admitted target-only updates preserves the same interval and winding, even if its target order varies. Arbitrary target selection does not imply convergence to uniform gaps: a sequence can keep updating an already equal pair while leaving other differences untouched.

4. A simultaneous stage is the canonical cycle diffusion map¶

An immutable all-target UM stage proposes the displacement in section 2 for every node. Since target-only UM has no overlapping neighbor writes, the new edge gaps are

\[ d_i'=d_i+u_{i+1}-u_i =(1-\eta)d_i+\frac\eta2(d_{i-1}+d_{i+1}). \]

Equivalently,

\[ d'=(I-\eta L_{\rm rw,C_n})d. \]

This is precisely the normalized cycle Laplacian already used by the canonical pure-EPI diffusion channel. Here it acts on an oriented gap observation under a UM stage; the index counts operator stages, not elapsed physical time. Its nonnegative, doubly stochastic matrix preserves the gap sum, the initial gap interval, U3 compatibility and winding in the same exact model.

The exact nonnegative spread drop is

\[ V(d)-V(d')= \frac{\eta(1-\eta)}2\sum_i(d_{i+1}-d_i)^2 +\frac{\eta^2}{8}\sum_i(d_{i+1}-d_{i-1})^2. \]

The Fourier eigenvalues are 1-eta+eta*cos(2*pi*k/n). For fixed 0<eta<1, their absolute values are strictly below one for all k!=0, so repeated exact all-target stages converge to d_bar=2*pi*W/n. At eta=1, an even cycle has an alternating gap mode with eigenvalue -1; its spread is preserved, although winding and interval protection still hold. An odd cycle has no such nonconstant mode. These are exact-model statements, not asymptotic binary64 execution certificates.

For same-sign gaps and nonzero total S=sum(d)=2*pi*W, circulation concentration can be read without another dynamical parameter:

\[ I(d)=\sum_i\left(\frac{d_i}{S}\right)^2 =\frac1n+\frac{2V(d)}{S^2}. \]

It cannot increase under the protected maps. Under the strictly contracting all-target regime it tends to 1/n. The preserved winding therefore becomes an evenly distributed twist; this class supplies no mechanism for sustained localization of circulation. It neither constructs a localized physical entity nor supplies a restoring law for localized EPI or capacity support.

5. Nonzero winding can coexist with zero canonical pressure¶

A uniform twist phi_i=2*pi*W*i/n with |2*pi*W/n|<g is fixed by target-only UM. Its phase-pressure channel vanishes because its two neighbor phasors have their mean at the target phase. With uniform EPI, uniform positive capacity and the uniform degree of the simple cycle, the other three canonical gradient channels also vanish. Therefore the complete canonical DeltaNFR is zero in this exact restricted state, despite nonzero W.

Pressure equilibrium thus need not mean phase consensus. A positive-winding example with the strict canonical gate requires n>4*W. The spectral convergence above concerns uniformity of gaps, not uniformity of phases. It supplies neither attraction for arbitrary multichannel trajectories nor a dynamic preparation of nonzero winding from a zero-winding state. The protected UM class cannot perform that preparation because it conserves W.

6. Canonical loss, branch crossings and observation limits¶

Outside the protected class, canonical events can change winding. For the regular unit-winding eight-cycle, change only the phase at node zero from its initial value zero to theta. The edge from node seven to node zero reaches the wrap branch when theta=3*pi/4. Immediately below this value the winding is one; immediately above it, before the next incident branch crossing, it is zero. The cycle support can remain unchanged throughout.

The production Transition (NAV) word supplies a finite instance. On the prepared regular eight-cycle with repository default regime policy and seed 17, repeated admitted single-Transition words give theta_0=2.35 after 13 steps and theta_0=2.5500000000000003 after 14. Their winding observations are respectively one and zero. The reported minimum branch margins are approximately 0.006194490192344748 and 0.19380550980765499. The actual EPI, capacity and pressure effects of Transition remain part of those events; the example does not impose an auxiliary phase trajectory. Its endpoint change requires a branch crossing in any continuous fixed-cycle completion, but does not certify an unobserved continuous path inside a discrete event.

U3 admission applies to the operator's required compatible relations; it is not a universal promise that every cycle edge stays admissible under every canonical operator. Equal endpoint winding likewise cannot establish a protected intervening path: opposite edge slips can cancel. Neither a phase-independent EPI operation nor a static snapshot is a proof about all future phase writes. Cycle deletion, changed support, absent phase values, branch hits and nonzero signed branch slips must remain distinguishable.

7. Exact companion, runtime checks and remaining work¶

coupling_winding.py implements the single-target and all-target exact gap maps with conserved sum, interval and spread identities. It reuses the shared exact-or-represented real reader: integer and rational inputs remain exact, including NumPy integers promoted to Python integers; other real values retain their binary64 rational value. The observer accepts signed gap coordinates and does not round their sum into a purported winding integer. A separate declared-cycle phase observation must establish closure and winding. Its factor, state and gate hypotheses also do not admit a live operator word by themselves.

Exact tests cover convex transport, Jensen dissipation, finite compositions, the even-cycle boundary and the shared pure-EPI pressure map. Pressure tests compare the actual scalar and vectorized canonical phase channel with the midpoint identity, then use the shared nodal integrator for EPI evolution. Their regular twisted controls bound observed binary64 pressure residuals; a selected finite tolerance is not a rigorous bound on every libm evaluation.

Runtime tests and the benchmark compare actual canonical UM histories with the exact gap companion and record endpoint residuals, branch/U3 margins, winding and actual operator histories. They use eight UM/SHA pairs on perturbed C8 and C16 with W=0 and W=1. SHA leaves phase unchanged and attenuates capacity by its existing default factor; the word uses no phase-factor override. Both sequence validators and live operator requirements remain active. Separate DERIVED and MEASURED manifests record the working-source digest and the corresponding evidence boundary. The word observer now uses live primary-EPI admission, shared adjacency validation, runtime U3 limits and node/edge identity comparisons; its integrity tests cover these boundaries without changing committed-prefix behavior after later failure. The production controls also exercise canonical winding loss through Transition. These finite checks are separate from the exact repeated-map theorem and do not prove future binary64 protection, arbitrary grammar stability or a physical memory experiment.

The capacity-localization study now tests physical nodal flow in addition to the discrete phase words above. It finds EPI diffusion under uniform capacity and a conditional nonuniform equilibrium when a retained capacity profile balances that diffusion. The UM/SHA word above has no physical EPI-flow interval, so retention of an EPI shape under that word alone could not establish resistance to diffusion. The remaining question is joint maintenance when canonical operators evolve the supporting phase and capacity fields. That requires their complete effects and cannot be inferred from integer winding alone.

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