Joint structural support and nodal evolution on a cycle¶

Status: Exact finite reset/flow budgets, a default-factor retention obstruction and finite canonical runtime checks. Autonomous, self-restoring localized structures remain unproved. Research links: B2.d.3/O3.a, S3, S8, S9 and S16.

Use a fixed simple unit-conductance cycle with normalized Laplacian L=I-W/2 and conductance Laplacian B=2L. Let x denote scalar EPI and nu>=0 capacity. In a consistent phase lift, write

\[ \phi_i=2\pi Wi/n+p_i,\qquad a_i=p_i/\pi, \]

where p is periodic. Require every oriented lifted gap to have absolute value strictly below both the effective U3 gate and pi/2. Each neighbor phasor sum then has a nonzero resultant on its unambiguous midpoint arc. The canonical phase-pressure channel is exactly g_phi=-L*a; this is the midpoint identity of Coupling winding persistence, not an independently imposed phase differential equation.

The regular unit cycle has zero topology gradient. Its EPI, capacity and phase walks coincide, so with effective normalized channel coefficients e=w_epi>0, f=w_vf>=0, w=w_phase>=0, the canonical pressure becomes

\[ \Delta\mathrm{NFR}=-L(e x+f\nu+w a)=-eLy,\qquad y=x+\frac f e\nu+\frac w e a. \]

The four-channel normalization remains intact even though the topology gradient vanishes. A constant change in phase lift adds only a constant to y and changes neither its pressure nor its Dirichlet energy. The detached observer accepts a as a declared dimensionless coordinate; rational inputs are not asserted to reconstruct a circle, a winding or mathematical pi.

This scope excludes weighted/support walk mismatches, changing topology, clipping and ambiguous phase branches. The counterexamples in capacity-conditioned balance continue to apply. No confinement potential or additional physical coefficient is added.

2. Canonical support resets have a signed energy budget¶

Here "support reset" means a capacity/phase reset on the fixed cycle, not an edge-set event. The latter has a separate relational storage budget.

Use target-only Coupling with UM_BIDIRECTIONAL=False and UM_FUNCTIONAL_LINKS=False. Its immutable all-target stage, followed by uniform Silence, gives the exact reference reset

\[ \nu^+=q(I-sL)\nu^-,\qquad a^+=(I-\eta L)a^-,\qquad x^+=x^-. \]

Here s=UM_vf_sync, eta=UM_theta_push and q=SHA_vf_factor are the existing operator coefficients. For factors in [0,1], capacity remains nonnegative. The phase gap interval is preserved under the preceding strict chart hypotheses. Runtime operator-local pressure writes do not replace the subsequently refreshed three-channel pressure.

Define the support change and a fixed, capacity-independent diagnostic

\[ d_b=\frac f e(\nu^+-\nu^-)+\frac w e(a^+-a^-),\qquad F(y)=\tfrac12y^\top By=\tfrac12\sum_i(y_{i+1}-y_i)^2. \]

The same matrix B before and after the event gives the exact identity

\[ J:=F(y^+)-F(y^-) =(y^-)^\top B d_b+\tfrac12d_b^\top B d_b. \]

The cross term is signed: a support reset can increase or decrease F. If the initial pressure is zero, y^- is constant and the cross term vanishes. The new pressure is then

\[ \Delta\mathrm{NFR}^+ =f(1-q)L\nu^-+fqsL^2\nu^-+w\eta L^2a^-. \]

This is not a universal one-step release theorem. Correlated capacity and phase contributions can cancel. If Lv=lambda*v, take nu=nu_bar+b*v and a=A*v, with

\[ A=-\frac{f[(1-q)+qs\lambda]}{w\eta\lambda}\,b \quad (w\eta\lambda>0). \]

Preparing x=c-(f/e)*nu-(w/e)*a then gives zero pressure both before and after that reset. For the eigenvalue-one cycle mode, e=1/2, f=w=1/4, q=s=eta=1/2, b=1/8, the exact control is A=-3/16. This algebraic cancellation does not infer repeated invariance or a live phase chart; smaller amplitudes retain the same cancellation when needed to meet a prescribed strict chart margin.

3. Held support supplies the flow part of the budget¶

Between resets, hold nu,a fixed and evolve only EPI through the nodal law:

\[ \dot y=\dot x=-e\operatorname{diag}(\nu)Ly,\qquad \dot F=-e(By)^\top\operatorname{diag}(\nu/2)(By)\leq0. \]

No inverse capacity occurs. The identity remains valid when capacity vanishes, although zero rate then does not imply zero pressure. In particular, q=0 freezes subsequent EPI evolution while phase support may still change under later declared Coupling events.

Any finite compatible sequence of these exact resets and held-support continuous flows obeys

\[ F_{\rm final}-F_{\rm initial} =\sum_k J_k- e\sum_k\int_{\text{flow }k} (By)^\top\operatorname{diag}(\nu/2)(By)\,dt. \]

An exact refreshed Euler segment has a different, explicit identity. With r=diag(nu)*DeltaNFR, x^+=x+h*r and the same support,

\[ F(y+h r)-F(y)=h\dot F(y)+h^2F(r). \]

The quadratic remainder is nonnegative. Since L<=2I, the condition h*e*max(nu)<=1 is sufficient for both a row-convex update of y and Dirichlet nonincrease. Larger steps can increase energy even though the continuous derivative is negative. A finite discrete telescope therefore retains reset changes, negative linear flow terms and positive Euler remainders separately. Runtime pressure-realization and arithmetic defects are further measured contributions, not part of this exact equality.

4. Default attenuation can retain EPI as capacity tends to zero¶

The default factors obey 0<q<1 and 0<s<1/2. If each UM/SHA event precedes a fixed-duration flow of length h, the capacity at flow k>=1 is

\[ \nu_k=q^k(I-sL)^k\nu_0,\qquad \max\nu_k\leq q^k\max\nu_0. \]

Consequently the accumulated capacity exposure is finite. For variable durations this conclusion requires sum_k h_k*q^k<infinity; arbitrarily increasing physical durations do not satisfy it automatically. With spatially uniform initial capacity and a regular twist, an initial EPI mode epsilon*v, Lv=lambda*v, retains the positive limiting amplitude

\[ \epsilon\exp\!\left[-\frac{e\lambda h\nu_0q}{1-q}\right]. \]

The leading q records the event-before-flow ordering. This is inherited contrast retained through a finite reorganization budget, not restoration against perturbations.

A stronger witness begins at exact zero pressure and uses the same default factor ranges. On an even cycle let v_i=(-1)^i, so Lv=2v. Choose nu_0=A_0+D_0*v>0, D_0<0, regular twist a=0, and x_0=m_0+b_0*v, where b_0=-(f/e)*D_0>0. Initially the capacity and EPI gradients cancel. At flow k, write

\[ A_k=A_0q^k,\qquad D_k=D_0[q(1-2s)]^k,\qquad T_k=-\frac f eD_k>0. \]

This two-mode subspace is preserved by the nodal equation. During that held-support segment,

\[ \dot b=-2A_k(e b+fD_k),\qquad \dot m=-2D_k(e b+fD_k). \]

The exact endpoint recurrence is

\[ b_k=T_k+(b_{k-1}-T_k)e^{-2e A_kh}. \]

Starting from b_0=T_0, induction gives T_k<=b_k<=b_{k-1} and

\[ b_\infty\geq b_0 \exp\!\left[-\frac{2eA_0hq}{1-q}\right]>0. \]

Moreover m_k-m_{k-1}=(D_k/A_k)(b_k-b_{k-1})>=0; its total increase is at most (|D_0|/A_0)*(b_0-b_infinity). Both EPI amplitudes converge, while capacity and its spatial contrast vanish. The remaining nonuniform EPI has nonzero limiting EPI pressure and vanishing nodal rate. Thus even an initially balanced structural profile need not remain an active balance when its supporting capacity is synchronized and attenuated. The selected checkerboard preparation is an exact counterexample, not spontaneous localization or a prediction of a particle-like object.

5. Positive mobility and genuine mixing give a different boundary¶

As a formal control, omit attenuation by taking q=1. This is not an admitted Silence factor. Suppose initial capacity lies in [m,M], m>0, both 0<s,eta<1, and every flow lasts a fixed h>0. The cycle maps I-sL and I-eta*L then mix their nonconstant modes. For g_k=f*nu_k+w*a_k, its gradient has the bound ||L*g_k|| <= f*rho_s^k*||L*nu_0|| + w*rho_eta^k*||L*a_0||, where rho_b=max_{lambda>0}|1-b*lambda|<1 over the cycle spectrum. This is a geometric envelope, not a claim that the combined gradient norm decreases at every event. The ordinary EPI Dirichlet energy E_x=x^T*L*x, unchanged by these capacity/phase resets on fixed edges, satisfies

\[ \dot E_x\leq-e m\|Lx\|^2+ \frac{M^2}{em}\|Lg_k\|^2 \leq-em\lambda_2 E_x+\frac{M^2}{em}\|Lg_k\|^2. \]

The resulting forced scalar bound gives E_x->0 and an integrable nodal rate, hence a uniform limiting EPI. The positive-capacity, strictly mixing control cannot sustain a nonuniform EPI profile.

The mixing hypothesis matters. On an even cycle, eta=1 retains the checkerboard phase mode as a_k=epsilon*(-1)^k*v; its amplitude can be chosen within the strict gap margin. With uniform capacity, write rho=exp(-2*e*nu*h), c=w*epsilon/e and tau=(1-rho)/(1+rho). Its forced EPI amplitude has the exact solution

\[ b_k=\rho^k(b_0+c\tau)-c\tau(-1)^k. \]

For w*epsilon!=0 this approaches a nonzero period-two response. Preserving winding and a strict gap interval therefore does not itself prove loss of supporting phase structure or EPI consensus.

6. Executable scope and reusable evidence¶

cycle_support_dynamics.py provides detached exact balance, UM/SHA reset and refreshed-Euler observers. It shares the ordered rational reader and unit-cycle algebra with capacity_localization.py and coupling_winding.py. Rationals remain exact; other supported real values become exact representations of their materialized binary64 values. Default coefficients come from the full canonical normalization. General explicit positive coefficients are algebraic inputs and need separate runtime identification. Public cached fields are recomputed before transitions, not trusted as provenance. No live graph, chart, continuous trajectory or future execution is certified by the detached records.

Exact API tests cover channel cancellation, signed reset budgets, zero-capacity boundaries, Euler remainders, finite telescoping and altered cached records. Pressure bridge tests bind the three-channel identity and UM/SHA release to actual canonical pressure on strict twisted cycles with both pressure backends. Limit controls independently check the default checkerboard reduction and the nonmixing phase boundary. Runtime tests and the benchmark use existing grammar-admitted events and executor-owned physical partitions. Their normalized phase reconstruction, endpoint pressure defects and arithmetic remainders remain measured finite evidence. They establish neither generic solver convergence nor future binary64 stability.

The supported conclusion is a structural distinction: fixed capacity and phase profiles can condition a nonuniform zero-pressure EPI balance; canonical evolution of those supporting fields must enter the energy budget; loss of support with finite accumulated mobility can leave passive EPI contrast. Demonstrating a self-generated, localized profile that restores itself while sustaining reorganization requires further evidence.