Conditional phase locking and circulation geometry

P2 and network locking, capacity adaptation, circulation and cycle periods.

Part of Forced support balance and model boundaries. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

26. Conditional phase locking and form restoration on fixed P2

Declared composition and complete state

Compose the existing U3-gated phase proposal with fresh canonical multichannel pressure on the fixed undirected unit edge 0--1. This explicitly named composition is separate from the optimizer/FFT routes, whose phase updates do not feed back into their pure-EPI pressure. No new oscillator formula is introduced.

The continuous comparison below supplies constant positive capacities nu0,nu1, fixed support, coupling K>0, and effective pressure coefficients e>0,w>=0,v>=0 for EPI, phase and capacity. The topology gradient is zero on P2. The equations nu_dot=0 and support_dot=0 are premises. Capacity as angular speed and the sine interaction are the existing configured model, not deductions from x_dot=nu*p. With a continuous phase lift satisfying |theta1-theta0|<gamma, where 0<gamma<=pi/2 is the effective U3 limit, the complete varying state obeys

theta0_dot=nu0+K*sin(theta1-theta0),
theta1_dot=nu1+K*sin(theta0-theta1),
p0=-e*(x0-x1)+w*(theta1-theta0)/pi+v*(nu1-nu0),  p1=-p0,
x0_dot=nu0*p0,                                      x1_dot=nu1*p1.

Pressure is evaluated from the current state, never evolved independently or reconstructed to fit a desired form. These fixed equations define an autonomous conditional model, without a time-dependent external signal or operator dispatch. They do not explain the origin of their supplied capacities, support or constitutive identification. Coupling is phase to EPI, not bidirectional: EPI does not enter either phase equation.

Locked target, conserved mean and local response

Define sigma=nu0+nu1, d=nu1-nu0, delta=theta1-theta0, q=x0-x1 and b=(theta0+theta1)/2. Direct addition and subtraction give

delta_dot=d-2*K*sin(delta),
q_dot=sigma*(-e*q+w*delta/pi+v*d),
b_dot=sigma/2,
m=(nu1*x0+nu0*x1)/sigma,             m_dot=0.

The weighted mean is generally not the arithmetic EPI mean. Reconstruction is x0=m+nu0*q/sigma, x1=m-nu1*q/sigma. There is exactly one strictly admitted phase lock if and only if

|d|<2*K*sin(gamma),
delta_star=asin(d/(2*K)),
q_star=(w*delta_star/pi+v*d)/e.

At the lock both phases advance at sigma/2, while EPI is stationary at the reconstructed target with its initial m. In particular, maintained contrast does not imply nonzero instantaneous EPI rate or growing pressure. The phase and form contrast linearization is triangular, with eigenvalues -2*K*cos(delta_star)<0 and -e*sigma<0. This proves local attraction of the relative state. The conserved m and common rotation leave neutral directions; no single absolute full-state point attracts all preparations.

Equality in the gate condition places the lock on its excluded boundary. At |delta|=pi/2 the phase linearization also loses strict contraction. For K=0, a nonzero d produces relative drift, while d=0 leaves arbitrary relative phase with no restoring response. Those are not admitted strict locking cases. When capacities agree, the admitted target has delta_star=q_star=0: common phase rotation alone supplies no contrast.

The fresh-pressure chain rule is a further independent consistency check:

p0_dot=-e*sigma*p0+(w/pi)*(d-2*K*sin(delta)),  p1_dot=-p0_dot.

Thus zero pressure away from the phase lock need not remain zero. This identity follows from the same constitutive map; it does not authorize an independently prescribed pressure evolution.

An invariant chart and explicit error bounds

Choose a proof interval [-rho,rho] with 0<rho<gamma, |delta(0)|<=rho and |d|<=2*K*sin(rho). The phase vector field points inward at its endpoints. The interval is therefore forward invariant, with nonzero phasor resultants and strict U3 throughout the ideal evolution. It is an admission region, not a new dynamical coefficient. Put

mu=2*K*cos(rho)>0,       A=e*sigma>0,       C=sigma*w/pi,
D0=|delta(0)-delta_star|, E0=|q(0)-q_star|.

The sine secant on the interval is at least cos(rho). Consequently, for every t>=0,

|delta(t)-delta_star| <= D0*exp(-mu*t),
|q(t)-q_star| <= E0*exp(-A*t)+C*D0*I(A,mu,t),
I(A,mu,t)=(exp(-mu*t)-exp(-A*t))/(A-mu)       if A!=mu,
I(A,A,t)=t*exp(-A*t).

The nonnegative I is the convolution of the two decaying exponentials; the second bound follows by variation of constants. Pressure satisfies |p0|=|p1|<=e*|q-q_star|+(w/pi)*|delta-delta_star|. Errors of the individual EPI coordinates relative to the same-m target are respectively nu0/sigma and nu1/sigma times the contrast error. A perturbation changing m changes the final mean rather than being restored to the old absolute target.

There is also a bounded form interval. Write Q(delta)=(w*delta/pi+v*d)/e and set q_lo=min(q(0),Q(-rho)), q_hi=max(q(0),Q(rho)). The q vector field points inward at these endpoints. Its reconstruction gives bounds for both node forms, which can verify that configured clipping remains inactive. A clipped trajectory is not automatically this smooth model.

For the ideal simultaneous Euler map with a fixed step h, the sufficient conditions h>0, 2*h*K<=1 and h*A<=1 preserve both intervals. Indeed, the phase map is nondecreasing there and maps its endpoints inward, while the form map is a convex combination of q and Q(delta). With alpha=1-h*mu and beta=1-h*A, both in [0,1), the discrete errors obey

D_n <= alpha^n*D0,
E_n <= beta^n*E0+h*C*D0*sum(beta^(n-1-j)*alpha^j, j=0,...,n-1).

The sum is empty at n=0. These are conservative sufficient conditions, not a claim that every larger step is unstable. They refer to exact Euler arithmetic, separately from the continuous theorem and represented execution. Signed numerical defects must be accounted for and chart membership checked before any corresponding finite binary64 claim; no infinite runtime convergence follows merely from these formulas.

Inactive-channel control and implementation boundary

The prospective w=0 control must hold effective e and v fixed. Then phase evolution is unchanged, but q_star=v*d/e and the phase-to-form term vanishes. With v also zero the target contrast vanishes; with v nonzero the old capacity-supported contrast remains. Conversely, v=0 with w>0 and d!=0 isolates the contribution supplied by the phase lock.

The pressure owner normalizes configured weights. Simply deleting the raw phase weight would generally change e and v and invalidate that comparison. On this P2 only, moving the removed phase weight to the inactive topology channel preserves the other effective coefficients; the captured normalized values must still be checked. This uses a zero existing channel, not an additional force or an altered nodal equation.

The explicit P2 composition owner exports derive_p2_phase_form_model and propose_p2_phase_form_step through tnfr.physics. The first returns a frozen P2PhaseFormModel: represented capacities and normalized coefficients, an exact rational locking ratio and classification for the ideal real half-pi gate, binary64 target estimates, and the sufficient monotone step ceiling. The displayed phase decay rate is the least binary64 upper enclosure of the exact rational radicand's square root sqrt((2K-d)*(2K+d)); it is not a lower contraction certificate. This calculation avoids intermediate overflow and underflow near the lock boundary. Its default K reuses the existing operational coupling setting, not a newly derived structural constant. The theorem above allows general gamma<=pi/2; this implementation uses the half-pi case and separately checks the represented strict gate.

propose_p2_phase_form_step creates its own detached fixed P2 and returns a frozen P2PhaseFormStep. It admits canonical phases in [0,2*pi), checks the strict represented half-pi gap before and after, and requires initial and proposed EPI in [-1,1] with clipping inactive. The shared phase kernel and DefaultIntegrator use one initial state and one unsplit Euler segment; pressure is freshly evaluated before consumption and again at the endpoint. capture_non_epi_forcing and SupportTransportEuler retain the source and Euler residuals, alongside measured weighted-mean drift. No caller graph or complete runtime invocation is certified. A single proposal is reproducible with the declared recipe:

from tnfr.physics import derive_p2_phase_form_model, propose_p2_phase_form_step
weights = {"phase": 1, "epi": 1, "vf": 0, "topo": 0}
model = derive_p2_phase_form_model((0.95, 1.05), pressure_weights=weights)
step = propose_p2_phase_form_step(model, (0.0, 0.0), (0.25, 0.75), dt=0.25)
print(model.lock_status, model.locked_contrast_estimate)
print(tuple(map(float, step.after_epi)), step.after_phase_gap)

Target estimates are not transcendental interval certificates, and a finite accepted proposal does not prove repeated binary64 stability. The portable controls compare prospective targets, perturbations and the inactive phase channel within this scope.

The result establishes conditional restoration of a differentiated form under a specified source-state law. It does not derive the supporting capacity distribution, full substrate evolution, autonomous NFR creation, a physical clock or a particle interpretation. The single G3 plan owns any subsequent admission or experiment; existing passive and independent- pressure results are not rerun or promoted by this composition.

27. Capacity adaptation moves the conditional P2 target

The existing capacity adaptation owner moves an eligible node toward the immutable-snapshot mean of its neighbors. Its eligibility gate consumes stored pressure, stored Si and a consecutive evaluation counter. It is a configured controller, not a capacity law derived from the nodal equation. The following one-event calculation tests whether this controller preserves the supporting capacities of Section 26.

Exact event calculation and its limits

Retain the positive P2 capacities and definitions of Section 26. Let a0,a1 be the eligibility indicators at one event, and let 0<=mu<=1 be the configured adaptation fraction. In exact arithmetic, with clipping inactive, the simultaneous capacity proposals are

nu0_plus=nu0+mu*a0*d,       nu1_plus=nu1-mu*a1*d,
d_plus=(1-mu*(a0+a1))*d,
sigma_plus=sigma+mu*(a0-a1)*d.

Positive capacities remain positive because the active proposals are convex combinations of the two positive inputs. Moreover, |d_plus|<=|d|, so a strictly admitted fixed-capacity phase lock remains available after this event, at the same K and phase gate. Availability of a new lock does not mean that the previous locked state remains a lock.

When both nodes are eligible, sigma is unchanged and d is multiplied by 1-2*mu. Its magnitude strictly decreases for 0<mu<1 and d!=0. The endpoint mu=1 swaps capacities and reverses d; it is not a strict contraction. If just one node is eligible, d is multiplied by 1-mu, while sigma increases or decreases by mu*d. With neither node eligible, or with mu=0, the capacities do not change. These alternatives matter: partial eligibility also changes the form decay rate e*sigma_plus and the sufficient Euler step ceiling. They must be checked for the next step.

The post-event comparison target follows from the same constitutive law:

delta_star_plus=asin(d_plus/(2*K)),
q_star_plus=(w*delta_star_plus/pi+v*d_plus)/e.

The old target remains the reference for the preservation test. Computing the new target explains its displacement; it does not replace the old reference or fit a trajectory. For w>0, v>=0 and d!=0, reducing |d| reduces the target's magnitude. In the phase-only source witness v=0, eliminating the capacity contrast eliminates the predicted phase-supported form contrast. Indefinitely repeated eligibility would need additional evidence; the one-event identity does not establish it.

Immediate pressure and subsequent form response

The event changes capacity, but leaves EPI and phase unchanged. At the same form and phase, freshly evaluated pressure therefore changes by

p0_plus-p0=v*(d_plus-d),       p1_plus-p1=-(p0_plus-p0).

In particular, with v=0 an immediate pressure check misses the displaced target: pressure is unchanged even though the relative phase derivative changes by d_plus-d. Starting at the ideal old lock and subsequently holding the new capacities fixed gives

delta_dot(0_plus)=d_plus-d,
p0(0_plus)=0,
p0_dot(0_plus)=(w/pi)*(d_plus-d),
q_dot(0_plus)=0,
q_ddot(0_plus)=sigma_plus*(w/pi)*(d_plus-d).

This is the existing phase-to-pressure-to-form path. No independent pressure evolution is introduced. An ideal simultaneous Euler step first changes the phase gap by h*(d_plus-d) while leaving q unchanged; the second changes q by h^2*sigma_plus*(w/pi)*(d_plus-d). Actual binary64 residuals and chart admission remain separate obligations.

The conserved mean of Section 26 is also specific to fixed capacities. Writing its post-event value using unchanged x gives the exact jump

m_plus-m=-mu*d*q*(a1*nu0+a0*nu1)/(sigma*sigma_plus).

Subsequent fixed-capacity segments conserve the new mean. They do not restore the old absolute target simply because each segment has its own weighted-mean conservation law. There is no EPI jump in this capacity event; the mean changes because its weights change.

Finite default-policy witness

The P2 regression controls prepare the prospective Section 26 lock for capacities (0.95,1.05), raw pressure weights phase=epi=1, vf=topo=0, and the existing coupling K=0.1. They compare an active capacity writer with an inactive-writer control, preserving the old target in both branches. Fresh pressure and fresh Si are computed through their shared owners before the gate is evaluated.

For the specified binary64 preparation and current shared phase-pressure kernel, pressure is (0,0) and Si is approximately (0.8972318538363662,0.9694744245977734). Both nodes qualify under the existing defaults: EPS_DNFR_STABLE=0.001, si_hi=0.5, VF_ADAPT_TAU=5 and VF_ADAPT_MU=0.1. Beginning with zero counters, the fifth qualifying evaluation changes capacities to (0.96,1.04). These are five gate evaluations on a prepared state, not five elapsed units of physical time.

The old target estimates are delta_star=0.5235987755982994 and q_star=0.16666666666666682. Reapplying the unchanged model to the observed new capacities gives delta_star_plus=0.4115168460674884 and q_star_plus=0.13098988043445475. The new target is displaced, although strict locking remains available. With only node 0 or only node 1 eligible, the respective ideal proposals are (0.96,1.05) or (0.95,1.04); both give a phase target near 0.46676533904729683 and a form contrast near 0.1485760219466835, with different total capacities. The unchanged high-threshold regression declares si_hi=0.875 before execution. Both nodes now qualify. The earlier pressure arithmetic produced (-2.7755575615628914e-17,2.7755575615628914e-17) and Si approximately (0.8389323,0.91117487), admitting only the higher-capacity node. The current shared principal-phase difference and division by represented pi instead cancel the prepared form gradient exactly in this witness. We retain the threshold and record the changed outcome rather than retuning it to restore partial eligibility. The single-node proposals above remain conditional blend identities, not the outcome of this current runtime control. This is a separate configured policy control, not another canonical threshold.

After the writer window is closed, two existing P2 proposals with h=0.25 expose the delayed form response. In the active branch the first phase-gap change is approximately -0.005; the second contrast change is approximately -0.00039788735772973427. The inactive-writer branch retains the old contrast within the existing 1e-14 regression criterion in this finite comparison. These decimal values are represented observations or target estimates, not exact transcendental certificates or infinite-time runtime bounds.

The adaptation owner itself refreshes neither pressure nor Si. Si normalizes pressure by the current maximum magnitude, so an arbitrarily small nonzero antisymmetric pressure can lose the full pressure contribution to Si compared with exact zero. The audit retains the actual fresh values instead of silently substituting their ideal equilibrium limits. Counters count qualifying calls, reset on a failed gate and are not reset by an adaptation event; continued eligibility can therefore cause a write on the next call.

This detached, finite policy audit rejects preservation of this nonzero fixed-capacity target by the existing default writer. It does not establish instability of the new lock, a complete default runtime trajectory, permanent eligibility, or an emergent law for Si, capacity or substrate formation.

28. Geometry-limited regional phase locking on the unit barbell

Declared model and the bridge constraint

Use triangles (0,1,2) and (3,4,5), joined only by edge (2,3), with unit symmetric conductance and degrees d=(2,2,3,3,2,2). Capacities are constant within each region: nu_L=kappa-epsilon, nu_R=kappa+epsilon, with kappa>|epsilon|. These positive supplied capacities are also the free angular rates in the existing averaged-sine phase law. Set the effective pressure coefficients to e>0,w>0 for form and phase, with capacity and topology coefficients zero. The topology channel is disabled, not identically zero on this irregular graph. The ideal equations are

theta_dot_i = nu_i + (K/d_i) sum_(j~i) sin(theta_j-theta_i),  K>0,
x_dot = diag(nu) [-e L_rw x + w g(theta)],
g_i(theta) = Arg(sum_(j~i) exp(i(theta_j-theta_i))) / pi.

All edges must remain U3-admitted. This composition excludes Gamma, events, clipping, capacity adaptation and other controllers. The scalar form chart is signed real; zero capacity is outside this result. These assumptions specify an additional phase/capacity model, not a consequence of the nodal product alone.

Multiplication of the phase row by d_i cancels the sine interaction on each undirected edge. Any common locked angular rate must therefore be omega=sum(d_i nu_i)/14=kappa. Summing only the left region gives the necessary cut balance K sin(theta_3-theta_2)=7 epsilon: its volume is seven, and its sole outward edge carries all of its phase imbalance. Consequently no lock under this model exists when |7 epsilon|>K.

For |7 epsilon|<K, let the principal angles and lock offsets be

alpha = asin(2 epsilon/K),       beta = asin(7 epsilon/K),
psi = (-alpha-beta/2, -alpha-beta/2, -beta/2,
        beta/2,        alpha+beta/2, alpha+beta/2).

Substitution in all six rows gives theta(t)=psi+kappa*t*1. The interior and bridge loads are different: sin(alpha)=2 epsilon/K and sin(beta)=7 epsilon/K. A two-region average erases that distinction. With the half-pi U3 gate the strict threshold is |epsilon|<K/7; a narrower configured gate additionally requires |beta| to lie strictly inside that gate. At equality |7 epsilon|=K, the bridge cosine vanishes and strict relative restoring stability is lost. The overloaded result uses the cut equation and does not depend on imposing equal interior phases.

Full six-node local stability

The phase Jacobian at the lock is -K D^(-1) B_cos, where B_cos is the weighted graph Laplacian with edge weights cos(psi_j-psi_i). On the strict branch every weight is positive. Similarity by D^(1/2) gives one neutral common-rotation mode and five strictly negative modes. This includes perturbations that split either interior pair; it is not merely stability within the four-fiber reduction.

A finite neighborhood follows without linearizing. For a fixed common chart offset c0, write y=theta-psi-(c0+kappa*t)*1 in a continuous lift. For any edge its nonlinear error coupling equals a_ij(y)(y_j-y_i), where

a_ij(y) = integral_0^1 cos(psi_j-psi_i+s*(y_j-y_i)) ds.

If |beta|+osc(y(0))<pi/2 and is strictly inside the configured gate, the coefficients are symmetric and positive. At a largest error the derivative is nonpositive; at a smallest error it is nonnegative. Thus the error range is invariant and the degree-weighted error mean is conserved. With that mean set to zero,

||y(t)||_D <= exp(-K*cos(|beta|+osc(y(0)))*lambda_2*t) ||y(0)||_D,
lambda_2 = (11-sqrt(73))/12.

The same range argument holds for ideal simultaneous Euler when h*K<=1. For the norm bound it suffices to require h*K<=1/2; the symmetric error map then has nonnegative eigenvalues and contracts by at most 1-h*K*cos(|beta|+osc(y(0)))*lambda_2 on the centered subspace.

The actual phasor source and its compatible form

The phase source is not the sine interaction or a global Laplacian substitute. At the lock define

gamma = atan2(3 epsilon/K, 2*cos(alpha)+cos(beta)),
g_star = (alpha/2, alpha/2, gamma, -gamma, -alpha/2, -alpha/2) / pi.

The two-neighbor rows are phasor midpoints; a bridge row instead has the resultant 2 exp(-i alpha)+exp(i beta). Its imaginary part is 3 epsilon/K and its real part is positive on the strict branch. Reflection makes sum(d_i*g_star_i)=0 exactly, so the source is compatible with a stationary form. With s=w/(e*pi), an independent solution of L_rw*z=(w/e)*g_star is

F = 2*alpha+3*gamma,
z = s*(F/2+alpha, F/2+alpha, F/2, -F/2, -F/2-alpha, -F/2-alpha).

The complete stationary family is x_star=z+c*1. Its physically relevant mean for this declared nodal model uses H=diag(d_i/nu_i), not uniform weights and not D alone. To specify initial weighted mean m, set

m_H(z) = s*(epsilon/kappa)*((11/7)*alpha+(3/2)*gamma),
x_star = z + (m-m_H(z))*1.

The target evaluator derive_regional_phase_lock reuses derive_forced_support_balance for the represented source and mean constraint. The analytic formula above provides an independent reference. Exact rational cut loads, binary64 inverse-trigonometric estimates, the exactly odd represented source and the actual production phasor source are distinct evidence. In particular, enforcing reflection in a comparison source is not permission to replace or silently correct the production pressure.

Mean conservation is conditional on the phase source

In general the full form mean obeys

d(m_H(x))/dt = w*sum(d_i*g_i(theta)) / sum(d_i/nu_i).

Canonical phasor directions do not generally cancel in that sum. Thus an arbitrary phase perturbation can move the absolute form target even with fixed capacities. The phase degree mean, whose sine terms do cancel, does not supply a form conservation law.

There is a useful invariant restricted preparation. Let reversal exchange 0<->5, 1<->4, 2<->3, and require the rotating phase offsets to be odd under that reversal. The supplied detuning and graph share this symmetry, so both the ideal continuous phase law and its Euler map preserve it. The fresh phase source remains odd, its degree sum vanishes, and m_H(x) is conserved for any initial form. This symmetry permits a fixed full form reference during recovery without repeatedly recentering the observed trajectory. General preparations retain the measured drift.

Prospective finite recovery protocol

The following preparation and criteria precede runtime observations: kappa=1, K=1/2, epsilon=1/32, e=w=1/2, h=1/8, and 256 simultaneous phase/form Euler steps. Fix m=0, use x_star above, and initialize

theta(0) = psi + a common canonical chart offset
           + (2,-1,1,-1,1,-2)/128,
x(0) = x_star + (31,31,31,-33,-33,-33)/256.

The phase perturbation splits both interior pairs and preserves reflection. It has degree mean zero and Y0^2=||y(0)||_D^2=13/8192. The form perturbation is H-centered with X0^2=||x(0)-x_star||_H^2=7/32. Neither input nor target is fitted to an observed endpoint.

Elementary bounds give alpha<1/7, beta<1/2, total phase width below one radian, edge separation at most 17/32, and lambda_2>1/5. The following rational lower rates suffice:

lambda_p = 1759/20480,       lambda_x = 31/320,
a = 1-h*lambda_p,            b = 1-h*lambda_x.

For the form rate, e*x^T B*x >= e*nu_min*lambda_2*||x||_H^2 on the H-centered subspace follows by comparing the minimizing constant in the D and H variances. The ideal Euler form map has nonnegative spectrum under h*e*nu_max<=1/2, satisfied here.

To bound the changing source, on a common lift of width below one the phasor-direction derivative is (R-I)/pi, with nonnegative entries R_ij<=2/d_i on support. Hence ||R||_D<=2 and the phase-source Lipschitz bound is strictly below one, since pi>3. This yields

Y_n <= a^n*Y0,
X_n <= b^n*X0
       + w*sqrt(nu_max)*Y0*(a^n-b^n)/(lambda_x-lambda_p).

Here w*sqrt(nu_max)*Y0/X0=sqrt(429/229376)<1/23. Exact rational evaluation at n=256 therefore verifies, before execution,

a^256 < 1/10,
b^256 + (a^256-b^256)/(23*(lambda_x-lambda_p)) < 1/5.

The selected acceptance is a phase error below one tenth and a full fixed mean form error below one fifth of their initial norms. These are scoped ideal-Euler predictions, not a posteriori choices of a favorable time. The finite execution controls use the existing phase proposal, fresh canonical pressure and shared nodal integrator. They separately retain source and Euler defects, non-symmetric mean drift, overload and configured-gate controls. Common rotation crosses the canonical phase wrap during this experiment; relative circular errors must be reconstructed against the fixed rotating reference, rather than mistaking a wrap jump for physical phase recovery or instability.

No ideal estimate above certifies infinite binary64 stability or a complete runtime containing other writers. This mechanism conditionally maintains regional differentiation while its supplied capacity contrast and phase law persist. It neither selects those inputs autonomously nor establishes a physical NFR, particle or endogenous substrate.

Finite outcome with the frozen target

The production control passes both reserved recovery criteria after 256 steps (t=32 in the supplied clock). The target estimates are alpha=0.1253278311680654, beta=0.4528165947449256 and

x_star = (0.10786151777372711, 0.10786151777372711, 0.06796843009895889,
         -0.07382420301501220, -0.11371729068978044, -0.11371729068978044).

Error norm	Initial	Final	Observed ratio	Precomputed ideal upper ratio
Phase, D norm	0.0398361	0.000883519	0.0221789	0.0630834
Full form, H norm	0.467707	0.0160276	0.0342685	0.118917

No target is recentered or fitted after execution. The reflected preparation's largest observed weighted-mean drift is below 2.77e-16; its signed Euler defects are below 1.42e-17. A separate unperturbed target control keeps form unchanged at its represented coordinates and has final phase error below 5.48e-15. Numerical comparisons permit a separately declared 1e-11 allowance; neither that allowance nor these measured residuals certifies asymptotic binary64 accuracy.

The controls also retain a genuinely nonconserving mean response to an asymmetric phase perturbation, failure to preserve the old form when phase pressure is disabled, exact overload classification and tighter-gate rejection of an otherwise strict ideal lock. The target is independently checked against 80-digit phasor and closed Poisson formulas. These results support conditional regional maintenance and recovery, while the source of the supplied capacity contrast and phase law remains open.

29. Graph-independent locking and phase-source constraints

Exact model: phase support is not transport conductance

Let the phase support be a fixed connected, simple undirected graph with at least two nodes, unique neighbor sets N_i and counts d_i=|N_i|. Capacities nu_i>0 and the shared coupling K>0 are fixed. Every support edge is admitted by the U3 gate. The supplied phase law is the same averaged-sine law as in Sections 26 and 28:

v_i = theta_dot_i = nu_i + (K/d_i) sum_(j in N_i) sin(theta_j-theta_i),
Z_i = sum_(j in N_i) exp(i*(theta_j-theta_i)),
g_i = Arg(Z_i)/pi, when Z_i != 0.

Here g is the ideal canonical neighbor-phasor pressure channel, not the sine coupling itself. The formulas use circular edge displacements; they do not require all phases to lie in one global semicircle. The identification of capacity with free angular rate and the value of K remain constitutive inputs.

Transport may separately use symmetric nonnegative conductances W_ij with strengths s_i=sum_j W_ij and L_W=I-diag(s)^(-1)W. Phase counts d_i include every support neighbor, even one with zero transport conductance. Replacing d_i by s_i in the phase law changes the model. Statements about stationary form below require the positive-conductance transport graph to be connected, so every s_i>0.

Locked rate, cut load and the sign of the phase source

At a common-rate lock theta_i(t)=psi_i+omega*t, multiply each phase row by d_i. Contributions from the two orientations of an edge cancel exactly. Therefore

omega = sum_i d_i*nu_i / sum_i d_i,
Im(Z_i) = d_i*(omega-nu_i)/K.

More generally, summing only over a vertex set S gives the necessary cut balance

sum_(i in S) d_i*(omega-nu_i)
    = K*sum_(i in S, j outside S, i~j) sin(psi_j-psi_i).

For strictly acute edge gaps, its absolute value is less than K*|boundary(S)| for every nonempty proper S. If the gap magnitudes are at most rho<pi/2, the sharper upper bound is K*|boundary(S)|*sin(rho). The one-bridge load in Section 28 is one instance. Cut inequalities are necessary, not sufficient: compatible phase differences on cycles and the actual gate must still be satisfied. No enumeration of graph cuts is required to use the identity for a specified region.

Now assume every circular edge gap has magnitude strictly below pi/2. Every cosine is positive, so define

r_i = Re(Z_i)/d_i > 0,
g_i = atan((omega-nu_i)/(K*r_i))/pi.

The resultant lies in the right half-plane, removing both the zero-resultant and argument-branch ambiguities. Consequently

sign(g_i) = sign(omega-nu_i),
g_i = 0  if and only if  nu_i = omega.

For an edge bound rho<pi/2, cos(rho)<=r_i<=1, hence

atan(|omega-nu_i|/K)/pi
    <= |g_i|
    <= atan(|omega-nu_i|/(K*cos(rho)))/pi
    <= |omega-nu_i|/(pi*K*cos(rho)).

The nodal capacity imbalance fixes the source's sign and constrains its magnitude. It does not reconstruct the source without the retained resultant geometry r_i, nor prove that a lock exists for the proposed capacities.

What equal capacities obstruct, and what they do not

With common capacity nu_i=kappa>0, the locked rate is omega=kappa and the strictly acute, fully admitted lock has g=0. For the unforced form row

x_dot = diag(nu) [-e*L_W*x + w*g],   e>0, w>=0,

all stationary forms therefore satisfy L_W*x=0. Connected nonnegative reciprocal transport makes x constant. This conclusion requires the capacity and topology pressure contributions to be absent or zero, Gamma to be zero, and no other event, history, clipping or controller term to alter the row. Equal capacity itself makes the capacity-gradient channel zero; it does not make topology pressure vanish on an irregular graph.

This is an obstruction to differentiated stationary scalar EPI through this phase-source mechanism. It is not a prohibition of nonuniform phase, a nontrivial triad geometry, moving patterns, or every possible NFR identity. In particular, on a unit cycle C_n with n>=5, set

psi_j = 2*pi*j/n  modulo 2*pi.

Each node has two relative neighbor angles +2*pi/n and -2*pi/n. Their sine sum is zero and Z_i=2*cos(2*pi/n)>0. Common capacity therefore gives an exact rotating phase lock with g=0, despite winding once around the circle and lacking a narrow global phase chart. The local absolute phase gradient is 2*pi/n, not zero; for C6 it is pi/3. The pattern retains nonuniform phase while its phase channel supplies no differentiated stationary form. This identity alone asserts neither perturbation stability nor autonomous formation of that winding state. The existing winding owner already records this pressure/phase distinction; its additional UM/IL preservation and recovery theorems retain their own execution hypotheses.

Zero capacity, disconnected positive-conductance transport and other source channels are different mechanisms. Zero capacity can freeze nonzero pressure; disconnected transport permits componentwise constants. They are not counterexamples to the positive-capacity connected-transport statement.

A residual bound away from exact locking

Use the same degree-weighted capacity mean omega at an arbitrary admitted strictly acute state, and define the exact phase-rate residual epsilon_i=v_i-omega. Edge cancellation still gives sum_i d_i*epsilon_i=0 and

Im(Z_i) = d_i*(omega-nu_i+epsilon_i)/K,
g_i = atan((omega-nu_i+epsilon_i)/(K*r_i))/pi.

For common capacity and the edge bound rho this yields

|g_i| <= atan(|epsilon_i|/(K*cos(rho)))/pi
       <= |epsilon_i|/(pi*K*cos(rho)).

Thus a nonzero observed phase-rate residual can support a nonzero transient source even when all capacities agree. A small residual is not an exact lock, and this algebra does not prove the residual will decay. Computed sine sums, resultants, production pressure and binary64 phase-step residuals retain their separate numerical errors; a numerical value near zero is not a symbolic proof.

Capacity contrast is necessary here, not sufficient for stationary form

For any fixed positive capacities, multiplication of the full form row by H_i=s_i/nu_i gives the mean balance

d(m_H(x))/dt = w*sum_i s_i*g_i / sum_i s_i/nu_i.

A stationary form with w>0 therefore requires sum_i s_i*g_i=0. For this fixed connected transport, this is also the Poisson compatibility condition for the declared source. It is not implied by the phase identity sum_i d_i*(omega-nu_i)=0: taking phasor arguments is nonlinear, and transport strengths need not equal phase-support counts. The existing derive_forced_support_balance owns the compatible-profile and drift calculations.

An exact countercontrol uses the unit star K1,3 with center first. Declare K=1 and rational capacities

nu = (5/6, 3/2, 3/2, 1/2).

The common rate is one. The principal leaf equations give the strictly acute lock psi=(0,pi/6,pi/6,-pi/6). Set gamma=atan(1/(3*sqrt(3))). Its canonical source and compatibility sum are

g = (gamma/pi, -1/6, -1/6, 1/6),
sum_i d_i*g_i = 3*gamma/pi - 1/6 > 0.

For the strict inequality, 0<gamma<pi/6 and tan(3*gamma)=10/(9*sqrt(3))>tan(pi/6); both compared angles lie in (0,pi/2). Thus the lock's nonzero source has nonzero mean drive. No stationary EPI solves this phase/form model, although the forced-support owner can still describe its relative profile and common drift. The capacities precede the phase and form calculation; no desired EPI target defines them retrospectively. Section 28's reflection symmetry supplies compatibility for its barbell; heterogeneity by itself does not.

Admission boundaries and represented observations

• Missing gated edges: phase motion averages only admitted neighbors, while canonical pressure reads all support neighbors. On P2, a gap 1/2 with an effective gate 1/4 removes both phase interactions. Equal capacities then give common free advance while full-support pressure has g=(1/(2*pi),-1/(2*pi)). This is outside full-support admission even though the gap is acute. The full-support identities cannot silently use the smaller admitted-neighbor denominator.
• Half-pi boundary and zero resultants: ideal C4 quarter-turn winding has opposite neighbor phasors and Z_i=0; its direction is undefined. Binary64 trigonometric evaluation can leave a tiny nonzero resultant, which must not be mistaken for an exact nonzero ideal direction or exact degeneracy.
• Nonacute branches: an ungated antipodal P2 has zero sine sum and a negative real resultant, so equal capacity does not imply zero phasor pressure. This is a mathematical boundary control, not an admitted production U3 branch: the engine's hard gate cannot exceed pi/2, and drops those interactions.
• Nonreciprocal support: a directed C6 with one outgoing neighbor per node and winding one has common rate kappa+K*sin(pi/3) and g=1/3. The reciprocal cancellation and degree-weighted rate formula do not apply.
• Other dynamics: capacity evolution, changes of support, different phase laws, additional pressure channels and Gamma require their actual balances. The theorem does not authenticate an executor or admit all configured graph writers merely because one snapshot satisfies the geometric assumptions.

The detached observe_phase_lock_source read-out keeps the support, capacity, actual phase-pressure capture and numerically evaluated lock/source relations together. It reports numerical residuals, not an is_lock decision or an exact trigonometric certificate. The independent controls exercise the production owners and the symbolic identities and boundaries above. Exact theorem hypotheses, represented coefficients, finite numerical evidence and autonomous-law selection remain distinct.

30. Acute phase locks are circulation states with integral cycle periods

Fixed support, common capacity and the declared phase law

Retain Section 29's connected simple undirected phase support, full U3 admission, common fixed capacity nu_i=kappa>0 and supplied coupling K>0. The phase law is

theta_dot_i = kappa + (K/d_i)*sum_(j in N_i) sin(theta_j-theta_i).

Every oriented shortest-arc edge gap is strictly inside (-pi/2,pi/2). This is a local condition; no common real-valued phase chart is assumed. The theorem concerns exact common-rate solutions of this supplied phase law. It does not identify continuous sine evolution with UM events, establish stability, or derive a phase or support law from the nodal EPI equation.

Choose an arbitrary orientation of each of the M support edges. Let B be the N by M incidence matrix with -1 at the tail and +1 at the head. Write

delta_e = wrap(psi_head(e)-psi_tail(e)),
f_e = sin(delta_e),
-pi/2 < delta_e < pi/2.

The neighbor sine sum at node i is -(B*f)_i. Section 29 fixes the locked rate to kappa, so the equal-capacity locking equations are exactly

B*f = 0.

Thus f is an edge circulation with no nodal sources or sinks. This is an algebraic observation derived from the existing phase coordinates, not a new primitive or an independently prescribed physical current. Transport conductances do not enter this unweighted phase balance.

Integral periods and reconstruction

Choose a spanning tree T. For each edge outside T, orient the corresponding fundamental cycle so that this chord has coefficient +1; complete the cycle with its unique return path in T. The resulting matrix C has one signed cycle column per chord. It has b=M-N+1 columns, B*C=0, and its chord rows form the identity. These columns span all real circulations: subtracting the combination given by a circulation's chord entries leaves a circulation supported on a tree, which must vanish successively at leaves. The same argument for integer entries shows that C is also an integer lattice basis.

For a circle-valued phase assignment, each closed cycle returns to its starting phase, hence

C^T*delta = 2*pi*w,       w in Z^b.

Conversely, these fundamental-cycle period conditions suffice. Fix a root phase and integrate delta along the tree to assign real lifts to every vertex. The period condition for each chord says that its supplied gap differs from the difference of those lifts by an integer multiple of 2*pi. Reducing all lifts modulo 2*pi therefore reconstructs the required edge gaps. Strict acuity makes each supplied gap the unique shortest-arc value. Connectedness makes the reconstruction unique modulo one common rotation.

The integer-lattice qualification matters. An arbitrary real cycle-space basis cannot be used with an unmodified integer-period test. Even an integer basis need not generate the whole integer lattice: on C5, replacing its oriented cycle c by 2*c would incorrectly admit constant gaps pi/5, since (2*c)^T*delta=2*pi while c^T*delta=pi. Those gaps cannot come from closed circle-valued node phases. Fundamental cycles prevent this false admission. Another integer lattice basis changes winding coordinates by an invertible integer matrix; the underlying phase configuration does not change.

Necessary and sufficient locking equations, and sector uniqueness

Since sine is one-to-one in the acute interval, every lock has a unique circulation coordinate z and satisfies

f = C*z,
abs((C*z)_e) < 1                         for every edge,
C^T*asin(C*z) = 2*pi*w,                   w in Z^b.

Conversely, a solution of these equations supplies delta=asin(C*z). The period conditions reconstruct phases, the circulation condition supplies zero neighbor sine sums, and common rotation at rate kappa gives the required exact lock. Actual configured U3 limits must also admit every reconstructed gap; the nonlinear equations do not override a tighter gate. The criterion is an existence equivalence, not an assertion that every integer vector w has a solution.

There is at most one strictly acute lock per winding sector w, up to common rotation. To prove this, suppose f=C*z and f'=C*z' solve the same sector. Then

sum_e (f_e-f'_e)*(asin(f_e)-asin(f'_e))
    = (z-z')^T*C^T*(asin(f)-asin(f'))
    = 0.

Each summand is nonnegative and is strictly positive when f_e!=f'_e. Consequently f=f', then delta=delta', and the reconstruction differs only by common rotation. In particular, the zero-winding sector contains only phase consensus, since zero gaps supply one solution in that sector. The result gives uniqueness of a locked geometry, not convergence toward it or invariance of the acute domain under future evolution.

An equivalent variational formulation makes the same distinction. On the open convex domain abs(C*z)<1, define

F_w(z) = sum_e [f_e*asin(f_e) + sqrt(1-f_e^2) - 1]
         - 2*pi*w^T*z,       f=C*z.

Its gradient is C^T*asin(C*z)-2*pi*w, and its Hessian is C^T*diag(1/sqrt(1-f_e^2))*C, positive definite when b>0. This provides the same at-most-one interior critical point. It does not guarantee an interior critical point or equate this auxiliary circulation functional with the engine's EPI pressure or dynamical Lyapunov function.

What graph structure permits or excludes

• Trees: b=0, so every circulation is zero. Every acute equal-capacity lock is phase consensus.
• Bridges: sum B*f=0 over either side of a bridge. Its single cut term forces f_e=0, hence delta_e=0. Distinct cycle-bearing regions can carry their own allowed circulation, but the bridge endpoints have equal phase in a lock. A bridge does not generate a capacity-free phase source.
• Short cycles: a simple cycle of length l has abs(sum_cycle delta)<l*pi/2. For l<=4, its integral winding must be zero. If cycles of length at most four span the real cycle space, all acute locks are consensus. Indeed, delta is then orthogonal to every circulation, so delta^T*f=0; every term delta_e*sin(delta_e) is positive unless delta_e=0. Here real spanning is sufficient because the periods already vanish, unlike the integer reconstruction problem above.
• Long cycles are necessary somewhere, not sufficient by themselves: a nonzero acute lock must have nonzero winding on some simple cycle, which must have at least five edges. The mere presence of a long cycle does not imply a nonzero lock. K5 contains a five-cycle, but its cycle space is generated by triangles, excluding any nonconsensus acute lock.
• A single cycle with attached trees: bridges have zero gap and circulation is constant around the unique n-cycle. The complete acute lock family is delta_cycle=2*pi*w/n, with integer abs(w)<n/4, additionally restricted by the actual U3 gate. Tree vertices share the phase of their attachment vertex. Thus the unit windings of C5 and C6 are the first nonzero members, while C4's unit winding lies on the excluded half-pi boundary.

For the third and fourth statements, if every simple-cycle period were zero, the fundamental-cycle periods would be zero and sector uniqueness would give consensus. No enumeration of every graph or a new numerical recovery sweep is needed to obtain these obstructions.

At each admitted acute lock, B*f=0 and positive neighbor cosines imply zero canonical phase pressure, as proved in Section 29. Nonzero winding still gives nonzero local absolute phase gradients on its nonzero-gap edges. A unique phase geometry in a sector can therefore retain information that a zero-pressure label alone does not describe. This classifies candidate phase identity; it does not prove differentiated stationary EPI, autonomous NFR formation, physical identity or a complete state description by winding alone.

Persistence of a sector is not its formation

For a continuous circle-valued phase trajectory on fixed support, retain the same oriented fundamental cycles. If no edge reaches antipodal separation, its wrapped gap varies continuously inside (-pi,pi). Every cycle period is both continuous and an integer multiple of 2*pi; its winding is therefore constant on that time interval. This argument does not require the sine law, equal capacities, locking, or even acute gaps.

The U3 boundary and the winding boundary are different. Crossing a configured gate at or below pi/2 can change which neighbors the phase law uses, without changing a support-cycle winding. A winding change on fixed support requires an antipodal branch encounter at abs(delta)=pi in any continuous phase history. Loss of U3 admission is therefore not evidence of a winding slip. Conversely, gate admission at finitely sampled endpoints does not certify the unobserved path or exclude cancelling slips. Discrete operator events need their own path witness or endpoint interpretation.

Changing support can create or remove the cycle whose winding is being classified. For example, prepare a five-vertex path with consecutive phase gaps 2*pi/5, then add the edge joining its endpoints. All five resulting cycle gaps are acute and the new cycle has winding one, with no phase change or branch encounter. The prepared phase arrangement and edge addition are inputs in this example; it is not a derivation of their endogenous selection. Comparing sectors across a support change therefore requires identifying which old cycles survive and which cycle classes are new.

This separates two research obligations: maintaining a phase configuration within an existing sector, and explaining how the retained configuration and its support sector are produced. The protected UM results in the winding owner address specified preservation and recovery maps. They neither replace the present continuous sine law nor supply an autonomous mechanism that selects nonzero winding from a zero-winding, fixed-support, branch-safe preparation.

Exact implementation boundary

phase_cycle_geometry.py centralizes oriented support and the fundamental integer cycle basis. Its exact rational-turn reconstruction uses a_e=delta_e/(2*pi), requires abs(a_e)<1/4, and tests that C^T*a has integer entries. This certifies declared edge-phase compatibility without converting rounded trigonometric residuals into an exact lock claim. The represented live phases and actual U3 execution still require their separate observations and admission.

The implemented symbolic odd-sine cancellation test groups equal absolute turns with opposite incidence signs at each node. A successful exact cancellation supplies a sufficient locking witness; failure to cancel this way is inconclusive, since more general trigonometric identities may also balance a node. The implementation does not solve the full nonlinear C^T*asin(C*z)=2*pi*w existence problem. The independent controls retain these topology, reconstruction and sufficient-witness boundaries separately from finite checks against the production phase and pressure owners.

One control makes this incompleteness explicit: three internally disjoint paths between the same endpoints have lengths (10,6,10) and constant oriented turns (1/20,1/12,-3/20). Their endpoint differences agree modulo one turn. Internal nodes cancel pairwise, while the endpoints use sin(pi/10)+sin(pi/6)=sin(3*pi/10), verified by the exact coefficients of 1 and sqrt(5). This is an acute lock under the supplied equal-capacity law, but the oddness-only checker correctly reports unresolved. A failed sufficient symbolic check must therefore not be presented as nonexistence.

31. One added chord extends the cycle lattice, not a phase-generation law

Retained support and the exact lattice identity

Let G=(V,E) be a connected simple undirected phase support. Add exactly one missing edge e_*=(u,v) between its existing vertices, retaining every old edge and vertex. Orient the new edge from u to v and retain the orientations of the old edges. No conductance, capacity, phase law or locking assumption is needed for the following topology identity. Connecting distinct components, deleting edges, changing node identity, loops and parallel edges are different operations outside this one-chord result.

Retain an old spanning tree T. Write R for the old fundamental cycle basis in rows, so B*R^T=0 for Section 30's oriented incidence B. Each row has coefficient +1 on its own non-tree edge. Embed R in the new edge space by inserting zero in the added-edge column. Let r be the signed old-edge vector of the unique tree path from v back to u. Appending the row

c_* = (r, 1)

gives the retained-tree basis R_ext of the new graph. Here the displayed last coordinate denotes the added edge; implementations may retain their lexicographic edge order instead. The edge e_* followed by this tree path is a closed oriented cycle, so B_+*c_*^T=0.

The new cycle rank is b_+=b+1. More strongly, these rows generate the full integer cycle lattice. For any integer circulation h on the new support, subtract its added-edge coefficient times c_*. The remainder has zero on the added edge and is an integer circulation on G, hence an integer combination of R's rows. Independence follows first from the added-edge coordinate and then from independence of R. Thus

Z_1(G_+; Z) = embedded Z_1(G; Z) + Z*c_*

is a direct sum for this chosen tree, and the quotient by the embedded old lattice has rank one. This identifies the surviving old cycles before any comparison of winding coordinates.

A fresh deterministic spanning-tree calculation can produce a different post-event basis R_+. Both are integer lattice bases, so

R_ext = U*R_+,            U and U^(-1) have integer entries,
det(U) = +1 or -1.

One can compute U without a real-valued inverse: its columns are the entries of R_ext on the non-tree edge columns of R_+, whose corresponding submatrix is the identity. The reciprocal construction supplies the integer inverse. Basis changes therefore cannot create or erase a winding; they change its coordinates. A comparison that simply subtracts two independently chosen basis-coordinate lists need not compare the same cycles.

The new period comes from the actual endpoint gap

Express a circle-valued phase field in turns, with oriented edge gaps a_e=wrap_turn(phi_head-phi_tail) in (-1/2,1/2). This statement excludes antipodal edge pairs so that the shortest-arc gap is unambiguous. It does not require strict acuity or U3 admission. For the final field on the new support, let a_old^+ be its gaps restricted to old edges and a_*^+ its new-edge gap. Then

w_old^+ = R*a_old^+,
w_*^+ = r*a_old^+ + a_*^+,
w_+ = U^(-1) * (w_old^+, w_*^+).

Every displayed cycle period is an integer. The return-path sum differs from phi_u-phi_v by an integer, while the new-edge gap differs from phi_v-phi_u by an integer. Their sum is therefore integral. No desired period supplies the gap: the existing path and actual endpoint phases determine it. Conversely, appending an arbitrary prescribed edge gap without checking this relation can fail circular reconstruction.

The tree is a coordinate choice. Replacing its return path by another old return path changes c_* by an old integer cycle, so

w_*' = w_* + z^T*w_old              for some integer vector z.

The quotient lattice has a single generator up to orientation, but a winding functional descends to that quotient only when it vanishes on all old cycles. In general, a particular numeric "new winding" is therefore a retained-path coordinate, not an intrinsic new scalar charge. The complete embedded old periods together with the chosen new period are unambiguous data.

Separate a support extension from phase writes at the same event

For branch-safe initial and final phase fields, the old cycles have endpoint change

Delta_w_old = R*(a_old^+ - a_old^-).

If the support-only operation leaves the phase gaps unchanged, this is zero. The new cycle can nevertheless have nonzero winding. For example, prepare a five-vertex path with ascending edge turns 1/5 and add the endpoint edge. With the implementation's orientation 0 -> 4, its gap is -1/5 and the tree return path has sum -4/5, giving w_*=-1. Traversing the same cycle as 0 -> 1 -> 2 -> 3 -> 4 -> 0 gives +1. All gaps are strictly acute; there is no phase change or pre-existing cycle whose winding slipped. The initial nonuniform path field and the added edge remain supplied inputs.

When an actual event also writes phase, compute the final new period from the final path gaps and endpoint gap. A useful counterfactual is to add the same edge to the old phase field. With S^-=r*a_old^-, it has

a_*^0 = wrap_turn(-S^-),
w_*^0 = S^- + a_*^0,

provided the old endpoints are not antipodal. Its gap can be nonacute even when both actual endpoint states satisfy their strict-acute domains. If the counterfactual endpoints are antipodal, retain its unavailability instead of choosing a winding by a rounding convention. When defined, w_*^+-w_*^0 records an endpoint difference relative to that old-phase extension.

This decomposition does not claim that the counterfactual stage actually executed, or impose an order on simultaneous phase and support writes. An executor's event provenance and before/after fields are separate evidence. Neither equal endpoint periods nor a changed endpoint period describes an unobserved continuous phase path. On fixed support, an actual continuous winding change requires an antipodal branch encounter; a discrete operator jump instead supplies an endpoint change unless a path is also specified. Crossing the U3 gate alone is still not a winding slip.

A larger cycle lattice does not guarantee a maintained lock

The rank increase is topological. It neither supplies sine balance nor preserves the set of acute locks. Adding a diagonal to C5, for example, produces independent triangle and square cycles spanning its two-dimensional cycle space. Section 30 then excludes every nonconsensus acute equal-capacity lock under the fully admitted supplied sine law. The original C5 unit winding has a nonacute gap across either such diagonal, consistently failing the new full-support acute premise. Adding a connection can thus remove a previously available lock class while increasing the number of cycle coordinates.

A topology-only path closure, an operator event that closes the path while writing phase, and subsequent relaxation to a lock are three distinct claims. This support-reset identity establishes the first comparison and accounts for the second's observed endpoint phases. It does not establish the third, derive candidate selection, or show autonomous generation of a coherence pattern or a physical entity.

Shared implementation and production boundary

phase_cycle_geometry.py owns the one-chord support extension and exact endpoint reset alongside its existing fundamental-cycle basis. The implementation requires the same ordered nodes and exactly one added edge, rederives supplied geometry/state data, and retains both integer basis-coordinate maps. Its exact phase reset uses the existing rational-turn PhaseCycleState domain: both actual endpoint fields are strictly acute and reconstructible. The topology and branch-safe mathematical identities above have broader scope; this finite implementation does not silently certify that larger domain.

The counterfactual old-phase extension is reported separately, including antipodal unavailability. Exact turns are declared angles divided by mathematical 2*pi; dividing binary64 runtime radians by represented tau does not turn an observed event into an exact-angle certificate. Actual operator endpoints instead retain the shared branch-aware winding observer and their numerical residuals. The exact reset controls and Coupling event control exercise those different evidence paths. Neither the graph-theoretic result nor a finite observed sector birth derives the Coupling writer's configured candidate, threshold or Sense Index policy.

Prospective default-Coupling event and policy control

The finite control initializes the five-node path with theta_j=2*pi*j/5, EPI=1/8, nu=1, Si =0.8, unit old conductances and refreshed pressure. Its four path gaps span 4/5 of a turn; there is no cycle or initial winding. It uses the existing atomic two-phase Coupling executor on target 0 with declared history AL, seed 17, all eligible candidates, and the default UM factors, bidirectional phase writes, capacity synchronization, pressure stabilization and functional-link threshold. No factor or threshold is tuned. The pressure observation uses effective phase/EPI weights (1/2,1/2) and zero capacity/topology weights, without Gamma.

Write f=1/(pi+1) for the default phase push. Before execution, the kernel predicts node 0's displacement +f*pi/5 and node 1's -f*pi/5; other phases are unchanged. Only nonneighbor 4 is initially phase-compatible. The final closing gap and maximum old-edge gap are (2+f)*pi/5<pi/2. The resulting compatibility score for equal form and Si is

a = 1-(2+f)/10 = 0.7758546992994776,
threshold = pi/(pi+1) = 0.7585469929947761.

The executed stage adds exactly edge (0,4) with represented weight a. Its observed winding is +1 around (0,1,2,3,4), or -1 in the exact reset owner's added-chord orientation. The finite U3 margin is approximately 0.16244986676002404 radians. Actual phase writes agree with the prediction; EPI and capacity stay unchanged. The retained stage result and histories identify this finite invocation, without asserting an unobserved future.

The final state is not a sine lock. On its acute cycle, the ideal phasor source is

g = (f/10)*(-3,3,-1,0,1).

The runtime source matches this within the test's separate binary64 allowance. Coupling's target-local pressure reduction leaves a positive stored pressure at node 0, while a canonical refresh gives negative pressure there. A direct operator write is therefore not identified with freshly computed pressure. The read-out retains both fields and their difference.

The closing edge is not unit conductance: transport strengths are s=(1+a,2,2,2,1+a), whereas phase-neighbor counts are all two. With F=g/2, the initial source compatibility sum is

sum_i s_i*F_i = f*(1-a)/10 > 0
               approximately 0.005412055686023124.

The shared forced-support observer confirms its positive represented value. Thus the subsequent weighted EPI mean cannot simply be assumed conserved, even though the unweighted phase-source sum is zero. Replacing the new edge by a unit weight would change this actual event's form dynamics.

Two controls delimit the result. Changing only candidate 4's Si from 0.8 to 0.0 lowers compatibility by 0.2 and suppresses edge creation. Disabling functional links also prevents closure. Both preserve the same primary phase, EPI, capacity, stored-pressure and history writes; the differing support changes the subsequent pressure refresh. This is evidence of the existing writer's policy dependency, not a proposal to use Si as a fundamental support law. The event is admitted as finite, policy-selected sector birth; autonomous selection remains open. The event alone does not establish subsequent maintenance; Section 32 supplies its conditional phase/form result.