Winding generation and native writer boundaries¶

Acute winding retention, chart escape and native Mutation/capacity writer obstructions.

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

32. Acute-cycle relaxation retains phase winding while form relaxes¶

The supplied joint law and the support it actually uses¶

Fix a simple undirected phase-support cycle with cyclically ordered vertices i=0,...,n-1, n>=3, and positive common capacity kappa. Let every support edge remain U3-admitted. The supplied averaged-sine phase law is

theta_dot_i = kappa + (K/2)*(sin(delta_i)-sin(delta_(i-1))),    K>0,
delta_i = wrap(theta_(i+1)-theta_i).

Indices are cyclic. The phase denominator is two independently of transport conductance. Fix symmetric nonnegative conductances W on that support whose positive part is connected. Write s_i=sum_j W_ij, S=sum_i s_i, B=diag(s)-W and L_rw=diag(s)^(-1)*B. The form law considered here is

x_dot = kappa*(-e*L_rw*x + w*g),             e>0, w>=0,
g_i = Arg(exp(i*delta_i)+exp(-i*delta_(i-1)))/pi.

The phase and EPI coefficients w,e are supplied effective pressure weights. The full configured mixture is admissible: common fixed capacity makes its capacity-gradient channel zero, and every node of the simple cycle has unique support degree two, making its topology-gradient channel zero. Their coefficients need not be disabled or renormalized, even with unequal transport conductances. Gamma, further operator events, support changes and evolving capacities are absent. Any clipping is inactive under the chart condition below. This is a conditional continuous model of the declared channels; the nodal product alone does not derive this phase law or the selection of its support.

Acute gaps form an invariant region¶

Assume all initial oriented gaps lie in [m,M] with -pi/2<m<=M<pi/2. On the corresponding continuous lifts, differentiating the neighboring phase equations gives the exact gap equation

delta_dot_i = (K/2)*(sin(delta_(i+1))-2*sin(delta_i)
                    +sin(delta_(i-1)))
            = -(K/2)*(L_cycle*sin(delta))_i.

At a maximal gap its derivative is nonpositive; at a minimal gap it is nonnegative, since sine is increasing on this interval. Thus [m,M]^n is forward invariant. This also supplies continuation inside the branch-safe acute chart. If the configured U3 gate contains that interval, no edge loses admission along this exact trajectory. A gate narrower than the initial gaps does not meet the premise.

The gap sum has zero derivative. Since it is initially 2*pi*ell for the cycle winding integer ell, let

delta_bar = 2*pi*ell/n,
u = delta-delta_bar*1,
rho = max(abs(m),abs(M)) < pi/2,
lambda_C = lambda_2(L_cycle) = 2-2*cos(2*pi/n).

Then sum u_i=0. Multiplication of the gap equation by u gives

(1/2)*d||u||_2^2/dt
 = -(K/2)*sum_i (u_(i+1)-u_i)
                  *(sin(delta_(i+1))-sin(delta_i))
 <= -(K*cos(rho)/2)*u^T*L_cycle*u
 <= -gamma*||u||_2^2,

gamma = K*cos(rho)*lambda_C/2 > 0.

Consequently ||u(t)||_2<=Q*exp(-gamma*t), where Q=||u(0)||_2. This is convergence to uniform oriented gaps, not to equal phases when ell!=0. The phase-rate error also satisfies ||theta_dot-kappa*1||_2<=K*||u||_2: the cycle incidence has norm at most two and sine is 1-Lipschitz. Its integrability implies that each lifted theta_i(t)-kappa*t has a limit, with a rate-error tail bounded by K*Q*exp(-gamma*t)/gamma. The common rotation is therefore separated from the convergent relative configuration. No global semicircle containing all node phases is required.

The canonical phase source vanishes, but phase geometry remains¶

Both neighbor displacements lie in the center's open half-pi interval. Their resultant is

exp(i*delta_i)+exp(-i*delta_(i-1))
 = 2*cos((delta_i+delta_(i-1))/2)
       *exp(i*(delta_i-delta_(i-1))/2).

Its displayed cosine is positive. The exact phasor direction is therefore the midpoint and

g_i = (delta_i-delta_(i-1))/(2*pi),
sum_i g_i = 0,
||g(t)||_2 <= (Q/pi)*exp(-gamma*t).

This local identity uses the actual neighbor-phasor pressure, not a proposed global replacement of that pressure by a Laplacian. It ceases to be justified outside its chart/resultant premises. At the limiting winding configuration the pressure source is zero, whereas the tetrad's local absolute phase gradient is abs(delta_bar)=2*pi*abs(ell)/n. Thus vanishing phase pressure does not erase nonuniform phase geometry. This maintained relative-phase configuration does not, on its own, identify an autonomous NFR or a particle.

One existing margin controls availability and conditional stiffness¶

The same geometry supplies more information without another fitted parameter. Let r_i=|sum_(j in N(i)) exp(i*theta_j)|/2. The preceding exact identity gives

r_i = cos((delta_i+delta_(i-1))/2) >= cos(rho) > 0.

Thus the preserved acute interval certifies availability of every local phasor direction for this ideal trajectory, even when the global resultant is zero. The existing cosine_lower_bound in CycleRelaxationEnvelope is an outward rational lower bound for this same quantity; a separate tunable availability threshold is unnecessary. The certificate does not assert zero rounding error or that an arbitrary native phase writer preserves the interval.

It also controls the Hessian of the supplied sine law's alignment potential. For an oriented cycle incidence matrix A and V(theta)=sum_edges(1-cos(delta_e)),

Hessian(V) = A diag(cos(delta_e)) A^T >= cos(rho)*L_cycle.

After removing the common rotation, its smallest eigenvalue is at least cos(rho)*lambda_2(L_cycle). Multiplication by the declared phase mobility K/2 gives the existing decay bound. This is a conditional restoring margin, not a new primitive or a phase law derived from the nodal product. The phase-pressure response is a different derivative: within these fixed two-neighbor branches, D_theta g=-L_cycle/(2*pi), independent of the cosine weights. Confusing these two Jacobians would silently substitute the supplied sine evolution for the actual Arg-based pressure. Existing source, response and cycle owners already contain the necessary quantities.

The weighted form response and its nonconserved mean¶

Let m_s=(sum_i s_i*x_i)/S, y=x-m_s*1, and ||z||_s^2=sum_i s_i*z_i^2. Fixed reversible transport is self-adjoint in this metric. Let

lambda = kappa*e*lambda_2(L_rw) > 0,
D0 = ||y(0)||_s^2,
A = Q/pi,
C = kappa*w*sqrt(max_i s_i)*A.

Projection onto the s-centered subspace is an orthogonal contraction. Variation of constants and the preceding source bound yield

||y(t)||_s <= exp(-lambda*t)*sqrt(D0) + C*J(lambda,gamma,t),

J(lambda,gamma,t)
 = integral_0^t exp(-lambda*(t-v))*exp(-gamma*v) dv
 = (exp(-gamma*t)-exp(-lambda*t))/(lambda-gamma), lambda!=gamma,
 = t*exp(-lambda*t),                                  lambda=gamma.

Both terms tend to zero. The equality case is retained instead of dividing by a zero rate difference. Conservative positive lower bounds on either rate remain valid because the defining integral decreases as either rate increases. Squared upper bounds can be formed from these nonnegative terms without treating rounded square roots as exact coefficients.

The s-weighted mean need not be conserved. Diffusion makes no contribution to its derivative, but the actual phase source gives

m_s_dot = (kappa*w/S)*sum_i s_i*g_i
        = (kappa*w/(2*pi*S))*sum_i (s_i-s_(i+1))*u_i.

Define

M = kappa*w*Q*sqrt(sum_i (s_i-s_(i+1))^2)/(2*pi*S).

Then abs(m_s_dot)<=M*exp(-gamma*t). Hence a finite limit m_infinity exists and

abs(m_s(t)-m_s(0)) <= M*(1-exp(-gamma*t))/gamma,
abs(m_infinity-m_s(t)) <= M*exp(-gamma*t)/gamma.

Every EPI coordinate converges to this same limiting mean. The theorem does not assign the limit to the initial mean unless the weighted source actually vanishes. Equal transport strengths make M zero; unequal strengths require the retained source integral or its bound. The mean-tail estimate refers to the exact modeled trajectory at time t, not automatically to a numerically computed endpoint with an unmeasured solver error.

For a sufficient all-time form-chart check, J<=1/lambda. A conservative global disagreement bound is

D_global = C^2/lambda^2,                    D0=0,
           D0,                             C=0,
           2*D0+2*C^2/lambda^2,             otherwise.

Since abs(y_i)<=sqrt(D_global/min_i s_i), the entire exact trajectory lies inside the interval centered at m_s(0) with radius M/gamma+sqrt(D_global/min_i s_i). If this interval lies in the configured hard EPI chart, clipping stays inactive and the preceding linear form equation applies throughout. This sufficient interval can be conservative; failure to fit is not evidence that clipping actually occurs.

Apply the theorem to the actual closing-edge conductance¶

For the ideal arithmetic of Section 31's default Coupling event, put f=1/(pi+1) and retain its five cyclic gaps

delta(0) = (pi/5)*(2-2*f, 2+f, 2, 2, 2+f).

Their mean is 2*pi/5, their winding is one, and Q=f*pi*sqrt(6)/5. Their maximum (2+f)*pi/5 is strictly below pi/2. The theorem therefore predicts convergence toward uniform winding-one gaps under the supplied phase law. The predicted nonzero winding is retained, while the initially nonzero phase pressure decays.

The EPI graph must retain the closing conductance a=1-(2+f)/10, not replace it by one. Its strengths and total are s=(1+a,2,2,2,1+a) and S=8+2*a. In particular,

sum_i s_i*g_i = (a-1)*(delta_0-delta_3)/(2*pi),
sum_i s_i*w*g_i at t=0 = w*f*(1-a)/5,
M = kappa*w*(1-a)*Q/(sqrt(2)*pi*S).

For w=1/2 this recovers Section 31's positive initial compatibility f*(1-a)/10. Uniform initial EPI therefore starts with a positive weighted mean derivative, even though the unweighted phase-source sum is zero. The asymptotic result is uniform EPI at a finite, potentially shifted value, alongside nonuniform winding-one phase geometry. It is not a maintained nonuniform scalar EPI profile.

The ideal expression for f and the captured binary64 event endpoint are separate initial data. The implemented endpoint certificate starts from the actual materialized phases, capacities, conductances and EPI, rather than silently replacing them with the ideal formulas above. Exact real arithmetic then defines the conditional reference model from those inputs.

Reuse and numerical evidence boundaries¶

cycle_relaxation.py bounds this conditional model without advancing a graph. It reuses the shared detached pressure/transport capture, exact rational-pi branch enclosures and reversible spectral-gap owners. An actual materialized oriented gap is represented as r_i+2*k_i*pi, with rational r_i and a verified branch integer k_i. Its winding and deviations from the mean retain mathematical pi; a binary64 gap sum is not rounded to an integer and promoted to a proof.

Rational enclosures bound the initial Q, the acute radius and the phase and transport gaps. For 0<=rho<pi/2, the concavity bound cos(rho)>=1-2*rho/pi supplies a conservative positive phase coefficient. Shared exact exponential enclosures, with outward exponent rounding on a 2^-64 arithmetic grid and the shared exponent work limit 4096, bound the decay, Duhamel response, mean drift/tail and all-time chart. These are exact conditional inequalities for the reference model, not floating-point asymptotic convergence results.

Production execution instead refreshes its represented pressure and uses the shared phase proposal and nodal integrator. Trigonometric reduction, source realization, represented coefficients, Euler truncation, phase writes and possible clipping retain their own numerical discrepancies. Finite production observations can check a declared horizon against the prospective envelopes; they do not remove those discrepancies or certify arbitrary future binary64 execution. The preceding Coupling event remains policy-selected, and this maintenance theorem does not derive autonomous support selection, formation from a uniform preparation, or physical identification.

Frozen finite production continuation¶

The post-event controls reuse the same default UM event through the shared preparation. Before continuing, the evaluator fixes envelopes at t=0,8,16,32 from the actual captured endpoint, with K=1/2. The finite execution uses h=1/8 for 256 steps, with fresh canonical pressure, the shared phase proposal and the existing nodal Euler integrator. There are no further operators, support/capacity changes, controller updates or fitted final targets.

The conservative represented-state reference rates are approximately gamma_lower=0.03573031566296992 and lambda_lower=0.32231115638261676. Its all-time sufficient EPI interval is approximately [-0.09720168,0.34720168], strictly inside the hard chart [-1,1]. These are rounded displays of rational model bounds.

The retained finite observations are:

Read-out	At event endpoint	At t=32
Gap deviation norm from winding-one twist	0.3716106	0.00342818
Canonical phase-source norm	0.1079811	0.000641409
Strength-metric EPI disagreement norm	0	0.00172119
Weighted EPI mean	0.125	0.1280933944

The form disagreement initially rises in response to the source and then falls; its observed peak norm is approximately 0.05042945. All four reserved observations satisfy the prospective envelopes. Each finite step retains winding one, a positive U3 margin and interior EPI, while the exact Euler energy accounting has zero identity residual. The maximum retained state-rounding defect is approximately 1.39e-17; fresh-pressure assembly defect is at most approximately 2.00e-18. These do not bound accumulated phase or Euler truncation error or prove numerical convergence.

Independent 110-digit comparisons check affine-pi initial gaps, source/mean identities, the rational spectral lower bound and the exponential convolution. Reversed orientation, node relabeling, zero source, equal decay rates, unit transport weights and rejected branch/gate/capacity/support domains delimit the implementation. Replacing the closing weight by one makes the ideal mean-drive bound zero, demonstrating why the actual weight cannot be dropped. The raw operator pressure and refreshed pressure produce the same conditional reference but retain different reported stored-pressure residuals.

33. A common phase semicircle obstructs winding generation by the existing maps¶

The question is generation, not maintenance of a prepared sector¶

Sections 31 and 32 distinguish a support event that exposes a prepared phase arrangement as a new cycle period from subsequent conditional maintenance. They do not show that those mechanisms can generate the phase arrangement from an initially common semicircle. The following obstruction addresses that specific missing step. It concerns cycle winding, not every possible form of coherence, nonuniform EPI or NFR identity.

Let a finite set of existing nodes have real phase lifts q_i in a common interval I=[a,b] of width b-a<pi, with physical phases q_i mod 2*pi. Such a closed interval is contained in an open semicircle. The interval can cross the displayed zero-phase cut, and need not have width below the U3 gate pi/2. In particular, this is a global chart hypothesis, not the weaker condition that every existing support edge has an acute gap.

All angles, circular means and arithmetic in the proof initially mean exact real operations. The implemented binary64 maps and detached chart observation have separate boundaries below. Support may change among these same nodes. No assumption about EPI equality, transport conductance or Sense Index is needed for the phase-chart statement.

Actual Coupling proposal and merge formulas preserve that chart¶

The target kernel first selects its U3-compatible existing neighbors. Its phase consensus is the circular mean of either those neighbors alone or, when bidirectional writes are enabled, the target together with those neighbors. An admitted target has a nonempty compatible-neighbor set. Every selected input is in I. After rotation by -(a+b)/2, each input phasor has strictly positive real part. Their sum cannot be zero, and its direction has a unique lift c between the minimum and maximum input lifts. Thus c in I for either bidirectional setting.

For the declared push f in [0,1], each proposed phase write has the form

p_i = q_i + f*wrap(c-q_i) = (1-f)*q_i+f*c in I.

The second equality follows from abs(c-q_i)<=b-a<pi; the shortest arc is the ordinary difference in this common lift. Reduction modulo 2*pi is a representation change, not motion out of the chart. This argument covers the target and every optional bidirectional neighbor write. It does not require the default factor specifically; the kernel's entire admitted interval [0,1] is convex.

The simultaneous stage can give one node several proposals from different targets. All proposals read the same snapshot and use the same original q_i. If they are p_i^(1),...,p_i^(r), the actual stage's idealized shortest-arc displacement merge is

q_i^+ = q_i + (1/r)*sum_alpha wrap(p_i^(alpha)-q_i)
      = (1/r)*sum_alpha p_i^(alpha) in I.

Each difference has magnitude below pi, so the second equality does not identify distinct circle branches. Untouched nodes remain in I. Source-rank ordering is relevant to represented summation but not to this exact convex identity. The post-merge U3 validation remains a separate requirement: the chart result neither promises that every requested stage is admitted nor weakens the configured gate.

Functional-link candidates and their eventual weights are chosen separately from these phase proposals. Adding edges between existing nodes whose final phases stay in I cannot invalidate the common lift. This conclusion holds regardless of whether EPI, Sense Index, candidate sampling or a threshold selects a particular link. It establishes a phase constraint on the existing selection policy, not a derivation or endorsement of that policy as an emergent physical law. The fixed-node premise excludes importing a new node with an independently prescribed phase outside I.

Equal-capacity sine evolution preserves the moving chart¶

Between events consider the supplied averaged-sine law with one common capacity/free angular rate kappa>=0, coupling K>=0, and whatever neighbor subset A_i(t) the U3 gate currently admits:

theta_dot_i = kappa + (K/d_i)*sum_(j in A_i) sin(theta_j-theta_i),
d_i = |A_i|,

with zero coupling contribution when d_i=0. In a frame rotating at kappa, the same formula acts on lifted q_i=theta_i-kappa*t without the free term. While the range is below pi, a maximal q_i has only nonpositive sine contributions, and a minimal q_i has only nonnegative contributions. Consequently the maximum cannot increase and the minimum cannot decrease. The chart I is invariant in the rotating frame, and its width remains strictly below pi.

This extremum argument does not require reciprocity, a fixed neighbor count or a conserved phase mean. It remains valid when the gate changes the admitted subsets, or when the support changes without a phase write. For a discontinuous gate it states invariance for every absolutely continuous trajectory satisfying the supplied equation almost everywhere; it does not assert a new existence or uniqueness theorem for arbitrary switching fields. Grammar and operator events retain their own admission conditions.

The shared helper also has a directly applicable exact-Euler statement. For one simultaneous step with h>0 and h*K<=1, put

a_ij = sin(q_j-q_i)/(q_j-q_i),         q_j!=q_i,
a_ij = 1,                            q_j=q_i.

Within the common chart 0<a_ij<=1. For a nonempty admitted set the ideal update, after removing the common rotation h*kappa, is

q_i^+ = (1-(h*K/d_i)*sum_j a_ij)*q_i
        + (h*K/d_i)*sum_j a_ij*q_j.

All coefficients are nonnegative and sum to one. An empty admitted set leaves q_i unchanged in this rotating frame. Thus the Euler map preserves I for arbitrary simultaneous admitted subsets under this step bound. The bound is sufficient, not a declaration that larger steps always lose the chart. The helper does not enforce h*K<=1 for all callers; a composition invoking this result must establish it separately. Sine of the raw displayed phase difference equals sine of the lifted difference only in this exact-real identity; represented argument reduction remains numerical evidence.

Why neither new edges nor finite compositions can create a winding¶

For any oriented edge between nodes in I, its unique principal gap is q_head-q_tail, since its magnitude is below pi. For every closed support walk these differences telescope:

sum_(edges in a closed walk) wrap(theta_head-theta_tail) = 0.

Every cycle period therefore vanishes, independently of the cycle basis. This includes a cycle created by the next admissible link addition. Existing cycles, new cycles and a different choice of spanning tree have the same zero-period conclusion.

Induction now applies to any finite admitted composition consisting only of the stated exact Coupling stages, common-capacity sine-flow intervals or Euler steps satisfying their step condition, and support changes among the same nodes that leave phases unchanged. There is a common moving phase chart throughout, so none of these compositions generates a nonzero cycle winding from the stipulated preparation. This conditional statement does not require choosing a particular sequence of functional-link decisions. It is not an unrestricted claim about all 13 operators or complete runtime execution.

Section 31's initially prepared five-node path has phases 0,2*pi/5,4*pi/5,6*pi/5,8*pi/5. Its shortest containing circular arc has width 8*pi/5, exceeding pi. It is already outside this invariant chart class, even though each consecutive path gap is acute. Before closure the path has no cycle winding to measure. Closure makes that supplied, non-semicircle phase arrangement observable as a new nonzero period; it does not demonstrate its generation from a common-chart preparation.

Common capacity is an ideal identity with an explicit numerical safeguard¶

The Coupling capacity proposal writes only requested targets. Bidirectional phase writes do not themselves give every affected neighbor a capacity write. Its declared blend, with r in [0,1], is

nu_target^+ = nu_target+r*(mean(nu_compatible_neighbors)-nu_target).

If every node has the same capacity kappa, this formula preserves kappa exactly. Disabling capacity synchronization also preserves that supplied common value. Hence Coupling does not create the heterogeneous free rates that would escape the preceding exact-model premise.

The represented implementation requires an explicit equal-input identity. Unguarded evaluation of fsum(neighbors)/len(neighbors) is not always idempotent: three binary64 values 0.1 have the represented mean 0.10000000000000002. With blend factor one, a target-only write would create a capacity difference from equal represented inputs. Summation can also overflow for several equal large finite capacities even when their mathematical mean is that same finite input.

The shared coupling_capacity_blend therefore returns the target value directly when all compatible-neighbor capacities equal it. This is the declared blend's equal-input identity, not a new capacity-selection law or a change to nonuniform-input arithmetic. Both the direct and staged owners reuse this numerical kernel. A runtime study must still retain the observed capacities: this narrow safeguard does not certify unrelated capacity writes, phase arithmetic or future invocations.

A necessary capacity-contrast budget for leaving the common chart¶

Allow the supplied capacities in the same continuous sine law to vary by node and time. Assume locally integrable finite rates and an absolutely continuous lifted trajectory satisfying that law almost everywhere between finitely many admitted Coupling events. Use unrotated real lifts for this estimate. Up to the first chart boundary, let

D(t) = max_i q_i(t)-min_i q_i(t) < pi,
osc(nu(t)) = max_i nu_i(t)-min_i nu_i(t).

The coupling contribution still points inward at each phase extremum. Consequently the derivative of the maximum is at most max_i nu_i(t) and that of the minimum is at least min_i nu_i(t) almost everywhere. It follows that

D'(t) <= osc(nu(t)),
D(t) <= D(0)+integral_0^t osc(nu(v)) dv.

Each exact Coupling phase event has D(t^+)<=D(t^-), by the convex proof above, so the same bound survives their insertion. Support-only changes among the existing nodes do not change D. These statements concern every trajectory satisfying the premises; they add no existence or uniqueness claim for the discontinuous gate.

If t_* is the first boundary with D(t_*)=pi, a necessary condition is

integral_0^t_* osc(nu(v)) dv >= pi-D(0).

If a uniform bound osc(nu)<=R, R>0, is supplied, this also gives t_* >= (pi-D(0))/R. Zero contrast recovers the invariant common-chart result. A capacity update can change the subsequent rate contrast without an instantaneous phase jump; it contributes through the integral, not as an invented phase displacement.

This budget is necessary, not sufficient. Inward coupling can prevent chart escape even when the bound allows it. Losing the global common chart is itself weaker than creating a nonzero cycle period or a maintained pattern; the first limiting antipodal pair need not be a support edge. The budget does not derive the supplied capacities or their evolution. It supplies a checkable requirement for a proposed escape mechanism, not that mechanism.

Exact chart observation, finite execution and the remaining obligation¶

phase_chart.py observes whether the captured phase set admits a common open semicircle. It interprets represented radians as exact real inputs, retains mathematical pi through the shared rational enclosures, and uses the equivalent criterion that the largest empty circular gap exceed pi. Successful admission provides consistent lifts and zero periods in the shared support-cycle coordinates. Exclusion of the chart and an unresolved enclosure are separate outcomes; neither is a positive winding or a successful formation result.

This detached observer does not establish how its state was produced, seal a trajectory or prove that a future runtime event retains the same chart. Its connected 2-to-32-node, at-most-50-edge implementation budget is narrower than the chart theorem. The actual arithmetic owners are _coupling_stage_kernel.py and phase_evolution.py; the formation controls exercise their finite behavior separately from the exact composition argument. Phasor evaluation, shortest-arc subtraction, displacement averaging, modulo writes and Euler rounding must not be silently identified with exact convex maps. In particular, the equal-capacity safeguard alone gives no binary64 phase-chart theorem for arbitrary repetition.

Generating nonzero winding from the invariant preparation requires leaving at least one of its premises. Heterogeneous free rates can remove the common rotation; a phase-changing map outside the proved convex family can remove the chart invariant; and an introduced or prepared phase outside the chart changes the initial-data hypothesis. None of these possibilities is by itself a sufficient generative mechanism. Topology changes alone among the same common-chart phases cannot supply the missing period. A large numerical step or accumulated rounding discrepancy is also not a derived physical source. Any proposed escape must identify its actual operator/state preconditions or constitutive law and retain the independent evidence for formation, maintenance and selection.

Finite controls for overlapping writes and the excluded preparations¶

The production control fixes a five-node path with displayed signed phases (-0.30,-0.12,0.04,0.16,0.29) represented modulo tau, common capacity one, EPI 1/8, Si 0.8, seed 17 and declared initial AL histories. One actual default simultaneous Coupling stage targets (1,3) with all eligible candidates; no factor or compatibility threshold is tuned. Node 2 receives overlapping phase proposals. The stage adds (0,3), (1,3) and (1,4): cycle rank rises from zero to three while every fundamental period is zero. The exact endpoint observer proves both common charts; their diameters shrink.

Independent 80-digit circular-mean and convex-merge calculations agree with the represented writes within 2e-15 radians while retaining a nonzero arithmetic difference. Reversing target order preserves primary endpoints. Four subsequent shared joint steps at h=1/8, K=1/2 retain the observed common chart, common capacity and zero cycle periods. These finite checks neither seal future execution nor turn the exact model into a binary64 theorem.

A separate three-node path starts with equal zero phase and supplied positive capacities (1,2,3). One phase proposal at h=2, K=1/2 gives (2,4,6) radians, outside every common open semicircle. This control violates the common-capacity premise and is a tree: it demonstrates possible chart escape, not winding birth, an exact continuous trajectory or autonomous contrast. Section 31's prepared wound path and its executed C5 endpoint are also correctly excluded from this obstruction's initial class.

Geometry controls retain repeated phases, node relabeling, a chart crossing zero and transport-weight independence. The represented pair (0,math.pi) is slightly narrower than a mathematical semicircle and is admitted with a small positive exact margin. A valid deliberately coarse pi enclosure instead returns unresolved where required, with unavailable periods rather than zeros. Direct/staged capacity regressions confirm that the repaired shared capacity identity retains 0.1 at a three-neighbor target, plus zero, subnormal and extreme finite kernel inputs.

34. Native runtime admission uses relaxation, not the supplied sine clock¶

Reuse the native owners before transferring the oscillator result¶

The preceding sine-law results do not identify the phase evolution used by runtime.step. That entry point calls coordinate_global_local_phase, a configured relaxation per invocation. It does not call the averaged-sine proposal, add dt*nu_f to phase, or use U3 to filter the coordinator's local neighbors. Capacity contrast therefore is not, by itself, a native angular speed contrast. Section 33's accumulated capacity-contrast budget concerns its supplied oscillator law and cannot be transferred to this native map.

The existing native diameter bound, writer audit and selector reachability result remain the mathematical and policy owners. This section connects those results to the complete step boundary, extends the represented common-capacity identity beyond the earlier retained unit value, and specifies what an actual native formation check must retain. It introduces no substitute phase law.

The actual step order and freshness boundary¶

For the built-in path, runtime.py performs the following operations in order:

Before-step callbacks; mutation-flow boundary and node-sample bookkeeping.
The configured pressure refresh, then Si refresh when use_Si=True.
Selector decisions, lag overrides, grammar processing and sequential primitive glyph execution; another mutation-flow boundary is recorded.
The configured integrator, followed by EPI/capacity clamps and phase re-expression.
Global/local phase coordination, followed by the stored-pressure/Si capacity-adaptation gate and another mutation-flow boundary.
The optional auxiliary math step, delayed-EPI history, automatic REMESH, validators, after-step callbacks and cache telemetry.

There is no intervening pressure/Si refresh after the glyphs, EPI integration or phase coordination. Under ordinary IL execution, the integrator consumes the operator-contracted stored pressure. AL/EN can change form without refreshing that pressure. The later adaptation gate reads the resulting stored pressure together with Si computed before those state changes. Automatic REMESH can introduce a further EPI jump after that gate. These values identify their actual observation times; calling them fresh terminal state pressure and Si would be incorrect.

use_Si=False suppresses the refresh, not consumers of stored Si. apply_glyphs=False suppresses glyph execution, not subsequent coordination and adaptation. The native integrator owns physical EPI time, whereas the coordinator's gains act once per invocation without a dt argument. Changing the number of native steps can therefore change the number of relaxation maps even at a fixed total EPI integration duration.

Default coordination stays in the common chart and consumes its diameter¶

In a common open-semicircle lift, the exact global phasor mean b and every nonempty local mean l_i lie in the same interval; isolates use their own phase. The actual coordinator's ideal formula is

q_i^+ = (1-kG-kL)*q_i + kG*b + kL*l_i.

The existing diameter proof applies when kG,kL>=0 and kG+kL<=1: D^+<=(1-kG)*D. It is stronger than merely retaining the chart. The default enabled adaptive policy clamps both gains after its stable, transition or dissonant branch. Their nominal configured bounds are

1/(8*pi^2) <= kG <= 1/(2*pi),
1/(8*pi)   <= kL <= 1/(2*pi).

For executable inequalities the endpoints are their stored binary64 values, treated as exact coefficients. Their global lower endpoint m is positive and their two upper endpoints sum to less than one. Thus every default adaptive branch supplies the exact coefficient bound D^+<=(1-m)*D for the ideal map. Kuramoto order and recent disruptive-glyph load choose the adaptive branch; neither branch supplies an outward phase source in this domain. The separate fallback defaults for a partially configured policy must not be substituted for the fully injected default configuration.

These facts are already checked by the native diameter controls. They do not guarantee arbitrary configured overrides, disabled adaptation, modified gain bounds, undefined represented resultants or all binary64 trajectories. Legacy native coordination uses ordinary reductions; its optional exact-component evidence version is not silently enabled by runtime.step and does not certify transcendental phase accuracy either.

Which native glyphs and other writers preserve the premises¶

With fresh default Si and equal positive capacities, its capacity-normalized term equals one and the existing selector bound gives a base IL choice. For initialized nodes, meaning nonzero scalar EPI or valid retained nonempty glyph history, IL is admitted by the incremental grammar: U1a no longer requires a generator, and IL supplies stabilization rather than new debt. The default selector has no parametric soft-repetition filter. Lag overrides can request AL or EN; if one is rejected in this initialized class, IL is the first admitted fallback. Thus the relevant reachable primitive set is {IL,AL,EN} under those hypotheses. Outside that class, inspect the actual applied labels and grammar outcome; a label appearing later in the fallback list does not prove either its occurrence or its impossibility.

The native dispatcher does not invoke the public Coherence class or an all-target operator stage. _op_IL changes only stored pressure; _op_AL changes form. For a graph-bound EN call, the adapter captures a ReceptionReadSnapshot and its prepared writer changes form and its kind. These paths leave phase, capacity and support unchanged. Their history and cache writes do not constitute a phase update. A public operator with an additional configured phase action is a different path.

The built-in default integrator advances scalar EPI, its rate bookkeeping and the runtime clock, holding phase and capacity fixed. The ordinary clamps apply common capacity bounds to every node. An equal positive capacity already inside those bounds therefore remains that same value. Exact phase wrapping preserves the circular state; its represented implementation is addressed below. No EPI stability or eventual EPI equilibrium follows just from these phase/capacity facts.

Automatic network REMESH is the protected delayed-EPI map. It does not call topological remeshing or the separate structural-memory phase/capacity interpolation. Its transaction protects phase, capacity and support against its ON_REMESH observers. The standard optional math engine advances a separate state vector and telemetry without an inverse projection onto the nodal phases or capacities. Arbitrary custom pressure hooks, integrators, selectors, mutable-object implementations, validators and before/after callbacks remain additional writers whose effects require their own audit. The theorem excludes such unaccounted changes rather than treating a callback name or a suppressed exception as evidence of a read-only operation.

Preserve the declared adaptation fixed point in represented arithmetic¶

The adaptation owner uses a held neighbor-capacity snapshot and eligibility from stored pressure, stored Si and a consecutive-call counter. In exact arithmetic, a common capacity kappa is unchanged by

nu_i^+ = (1-mu)*nu_i + mu*mean(nu_neighbors),      0<=mu<=1,

regardless of which subset qualifies. This does not require the stored gate inputs to describe the latest phase/form state: arbitrary eligibility still leaves that equal-capacity fixed point unchanged.

A represented implementation must preserve this identity explicitly. For example, with target and neighbor capacities 0.104 and default mu=0.1, the two-product expression can return 0.10400000000000001. If only that target is eligible, it creates a numerical contrast between previously equal capacities. This is not evidence of a structurally generated capacity mechanism. The shared adaptation proposal retains the current value when its represented neighbor mean equals that value, avoiding the redundant multiply/add operations. Section 38 additionally enforces the neighbor and convex-proposal intervals for nonuniform inputs. Interior arithmetic and the eligibility policy remain unchanged; neither guard changes a threshold or event-selection law.

Phase clamps have a corresponding representation boundary. The native contract re-expresses phases in [-pi,pi), not necessarily [0,2*pi). Recomputing (theta+pi) % (2*pi)-pi for an already valid centered value can lose small increments unnecessarily. The clamp therefore retains that valid value directly and uses the existing re-expression for values outside the centered range. The coordinator can subsequently store a raw unwrapped proposal. This guard preserves the existing range contract where it applies; it is neither a phase evolution law nor a guarantee of exact re-expression at other branches.

The detached common-chart observer currently accepts canonical represented inputs in [0,math.tau). A centered or raw native endpoint outside that input domain is not admitted merely because a periodic interpretation exists. Finite controls that use this observer must state their actual input range; they may not normalize the live graph just to manufacture an admissible observation. A separately retained compatible lift and its representation error are different evidence.

Conditional whole-step conclusion and its limits¶

Start with initialized scalar nodes, fixed existing support, equal positive capacity inside the configured rails and a common open-semicircle phase chart. Retain fresh default Si, default selection and adaptive gains, the built-in integration path and no unaccounted phase/capacity/support writer. For each admitted exact-model native step, the primitive glyphs preserve these phase/capacity conditions, adaptation preserves common capacity and the coordinator contracts the chart diameter. For any finite number N of such steps,

D_N <= (1-m)^N*D_0 < pi.

Every support-cycle period remains zero. Indefinite continuation satisfying these same ideal premises would exhaust phase contrast, rather than renew it. This conditional conclusion is the existing native theorem with its actual writer/admission requirements; it is not a new autonomous formation law or an unrestricted claim about every grammar history and runtime option.

The numerical fixed-point and clamp repairs remove identifiable arithmetic defects from this path. They do not make phasor evaluation, shortest-arc arithmetic, normalization or complete runtime repetition exact. A finite native experiment must retain actual glyphs, effective gains, primitive phase/capacity endpoints, pressure/Si freshness and every intervening writer. The existing diameter ledger then keeps phase realization and intervening changes as explicit defects; it is not permission to assign unobserved changes a zero value. Whether a different admitted native mechanism can supply and maintain phase contrast remains a separate formation question.

Finite ordinary-step control¶

Native step controls execute two independently initialized unit-conductance C5 graphs with seed 17, actual injected defaults, common capacity 0.3, phases (0.10,0.13,0.11,0.17,0.14) and EPI (0.125,0.128,0.121,0.127,0.123). The cases differ only in supplied stability counters: all zero versus VF_ADAPT_TAU-1. Those initial counters do not certify preceding stability. The default dt=0.5, selector, pressure reader, Si weights, coordinator, adaptation and REMESH gate are retained. Transparent wrappers observe the original functions without replacing their decisions.

Both executions refresh pressure and Si, apply native IL at every node and advance EPI nontrivially through the nodal integrator. The coordinator takes its adaptive stable branch with represented gains kG=0.04461342911494398, kL=0.140157221158957. The observed chart diameter falls from approximately 0.07 to 0.05566448226923737; the exact endpoint observer separately admits both charts and finds zero cycle period. All observed phases remain in the observer's input range without re-expression. Support and capacity remain unchanged. Only node 4 passes the pressure/Si gate; only the counter-ready case reaches the capacity writer, which retains 0.3 exactly. Its counter advances rather than being reset.

The trace also retains the timing limitations: adaptation reads the IL-scaled stored pressure and pre-glyph Si. The detached terminal pressure comparison has a nonzero freshness residual. Mutation-flow timestamps and delayed-EPI history contain the actual integrated endpoints. Automatic REMESH abstains because the required stable-fraction history is absent; its gate was executed, but this control does not claim an applied REMESH map. These are two finite whole-step observations, not a proof of arbitrary binary64 repetition, unobserved prior history or autonomous pattern formation.

35. Heterogeneous capacity opens a conditional Mutation admission gate¶

Section 34 closes a uniform-capacity class; it does not establish that fresh default selection always excludes phase writers. The question addressed here is whether heterogeneous supplied capacities can simultaneously satisfy the default selector, live Mutation evidence and incremental grammar. The owners remain native selection, Mutation evidence, its runtime adapter and incremental grammar. Admission of an endpoint and production of its preceding history are separate results.

The ideal fresh-state compatibility inequality¶

Let V=max_i(nu_i)>0, P=max_i(abs(p_i)), r_i=nu_i/V, and z_i=abs(p_i)/P when P>0. Let d_i be the absolute wrapped displacement from the target phase to its neighbor-phasor mean, divided by pi. This is the ideal phase dispersion used in the default Si definition; an isolate has zero dispersion. Write a,b,c for the normalized default Si weights, approximately (0.75854699,0.18315345,0.05829955). Then

Si_i = a*r_i + b*(1-d_i) + c*(1-z_i).

With no lag override, the default selector proposes ZHIR exactly when Si_i<=lo and z_i<=h, where the injected thresholds are lo=0.25 and h=0.133. Thus its ideal low-Si condition is equivalent to

r_i <= [lo-b*(1-d_i)-c*(1-z_i)]/a,       z_i<=h.

If P=0, both normalization owners use their zero-pressure fallback and z_i=0; no division by zero or positive-pressure premise is required. Mutation still requires strictly positive target capacity. Its additional configured floor ZHIR_MIN_VF defaults to zero; 0.1 is the signed growth threshold ZHIR_THRESHOLD_XI, not a capacity floor.

In an exact common open-semicircle chart of diameter D<pi, every defined neighbor mean lies in that chart, so d_i<=D/pi. A necessary bound is

r_i <= R(D) = [lo-b*(1-D/pi)-c*(1-h)]/a.

For the nominal defaults, R(0) is approximately 0.02148955, R(pi/4) approximately 0.08185280, and its limiting value as D tends to pi approximately 0.26294256. These are restrictions on relative capacity, not a proof of selection: current pressure, history, grammar and represented numerical margins still have to be checked. An additional capacity floor larger than V*R(D) would exclude this chart's gate. Executable comparisons use captured represented coefficients and thresholds; the rounded displays above are not directed interval certificates.

For the pressure bound, use the separate effective pressure weights A,B,C, whose nominal defaults coincide with a,b,c. On nonnegative conductance and support with defined phase resultants, the ideal default pressure is

p_i = A*g_i + B*(weighted_neighbor_mean(x)-x_i)
              + C*(neighbor_mean(nu)-nu_i),
abs(g_i) <= D/pi,
P <= A*D/pi + B*(x_max-x_min) + C*(V-nu_min).

The capacity difference is raw capacity, not the normalized Si quantity r_i. The EPI mean uses transport conductance; the phase and capacity means use support neighbors. The default topology coefficient is zero. The bound therefore uses the actual EPI and capacity ranges, including their units; replacing the last term by C would generally be incorrect. With the default rails it is at most A*D/pi+2*B+C*VF_MAX, where VF_MAX is the stored binary64 value near 2*pi. This bounds fresh current pressure only.

In particular, a small current product nu_i*p_i does not invalidate a larger previously observed signed EPI secant. Mutation validates two retained samples ending at the live EPI, with timestamped evidence authoritative. The preceding interval may have had different pressure, phase or capacity. Conversely, supplying a valid sample pair does not establish that native execution generated it. EPI rails alone do not exclude the history gate: two samples in [-1,1] can exceed the 0.1 threshold over an appropriately short positive interval. No measured derivative is reconstructed from current pressure to manufacture this evidence.

Which additional reachable primitives can supply a phase write¶

For initialized nodes and fresh default selection, the base choices are {IL,OZ,ZHIR,NAV,RA}. Lag overrides add AL and EN. An incremental-grammar rejection has IL as its first admitted fallback in this class; it does not make every later fallback label reachable. Successful native primitive writes have the following scope:

Reachable glyph	Direct state writes relevant here	Support change
IL, OZ, NAV	Stored pressure	None
AL, EN	EPI, with EN's form-kind contract	None
RA	EPI, optional capacity amplification and phase alignment	None
ZHIR	Phase, after live evidence and grammar admission	None

This is the primitive dispatcher audited in section 34, not the public operator classes, arbitrary hooks or an all-target stage. In particular, the native selector does not directly choose UM or a child-creating THOL operation. Optional network REMESH retains its separate protected EPI scope.

The RA primitive uses only its actual U3-compatible neighbor subset. In a common chart, its ideal phase write has the form q_i^+=(1-eta)*q_i+eta*b_i, where b_i is that subset's phasor mean and default eta=1/(4*pi) lies in [0,1]. It therefore cannot expand the common lift hull. This remains true for successive admitted primitive writes; it does not remove represented mean/wrap errors. Fresh default RA selection also requires Si_i<0.5. Nonnegativity of the other Si terms gives nu_i/V<0.5/a. Its optional default multiplier 1+1/(8*pi) consequently leaves that target below approximately 0.68538189*V, with V from the selection snapshot. This limited bound does not supply a general capacity-convergence theorem.

ZHIR supplies a different write. Its shared proposal uses the current pressure sign and the default configured shift factor, giving a shift near +0.25 or -0.25 radians. The trigger is the observed signed EPI growth, not the sign or magnitude of this current pressure. Positive growth, phase-shift direction and subsequent useful source work must not be identified with one another.

Declared-history endpoint: the admission intersection is nonempty¶

Consider connected unit-conductance P4 with nodes (0,1,2,3), injected defaults and the following supplied state:

capacity = (0.6, 1.2, 6.0, 6.0)
phase    = (2.5, 1.0, 0.2, 2.0)
EPI      = (-0.85, 0.95, -0.9, 0.9).

Supply node 0's glyph history IL,OZ and timestamped EPI pairs (-0.5,-0.95),(0.0,-0.85), with zero lag counters before selection. These histories are declared initial data, not a claimed native prefix. An independent binary64 endpoint refresh gives approximately

pressure = (0.00247643659, -0.12731743356, 0.50822502874, -0.76429162695)
Si       = (0.22966947381,  0.36304588940, 0.88544357994,  0.83676124714)
base     = (ZHIR, NAV, IL, IL).

At target 0, the endpoint pressure can also be checked directly as -A*1.5/pi+B*1.8+C*0.6. The shared history gate reports an observed rate approximately 0.2, above 0.1, while the current nodal product is only approximately 0.00148586195. The incremental grammar admits ZHIR for the declared history. All supplied EPI and capacity values are inside the default rails. The exact common-chart observer admits diameter 2.3<pi. Thus the selector, live history, capacity and grammar conditions are jointly compatible; the equal-capacity obstruction does not extend to this endpoint.

The admitted default positive shift would move target 0 from 2.5 to 2.75, outside the previous lift hull [0.2,2.5]. The new diameter is still 2.55<pi: outward phase motion is not a chart-crossing claim. P4 is a tree, so its zero support-cycle winding is vacuous. The maintained native execution controls below use a separate declared graph; endpoint arithmetic and read-only admission do not certify an applied P4 batch or a complete native P4 step.

A finite native prefix produces its own Mutation evidence¶

The heterogeneous Mutation controls retain one prospective six-node case: a complete graph on nodes (0,1,2,3,4) with pendant node 9 joined only to node 1. All conductances are one. Inject defaults, seed 17, initial runtime time zero, empty glyph histories and no supplied Si. Initialize

node 0:    phase=0.0, EPI= 0.6, capacity=0.8
nodes 1-4: phase=2.8, EPI=-1.0, capacity=0.1
node 9:    phase=4.7, EPI= 1.0, capacity=6.25.

Three ordinary native steps, each with the default physical interval 0.5, produce the target-0 sequence IL,OZ,ZHIR. The pressure reader, fresh Si, selector, grammar, nodal integrator, coordinator and adaptation remain the original owners; observation wrappers do not replace their decisions. Before the three target decisions the retained fresh values are approximately

Native decision	Target pressure	Target Si	Global maximum absolute pressure
IL	0.34221320754	0.15845231195	1.18360987888
OZ	0.18135858879	0.19478282220	0.70716117563
ZHIR	0.04528547527	0.23430789739	0.63220290369

The first two low-Si base choices request OZ; the initial lack of a recent handler makes grammar supply IL first. The next OZ is then admitted. Before the third decision, normalized target pressure is below 0.133, and the actual timestamped history supplies a positive secant approximately 0.19126945644>0.1. Its current nodal product is only approximately 0.03622838022; that prediction is not substituted for the observed rate. The actual preceding IL,OZ supplies the grammar context. This is a native history-producing trace, unlike the P4 endpoint's supplied histories.

The paired control changes only all initial capacities to 0.8; it yields target IL,IL,IL. This comparison identifies a finite consequence of supplied capacity heterogeneity under the same configured policy; it does not establish how that initial heterogeneity arose.

The initial phases (0,2.8,4.7) do not admit a common open-semicircle chart. Therefore this case is not an escape from section 34's common-chart class. Before the third decision, however, an independent exact-pi lift check already admits a common chart. The target phase is 1.063472935200046; its neighbors lie between 2.751877932766905 and 2.8637878070029825. The actual ZHIR jump is +0.25, toward every one of those neighbors. The pendant's raw phase -2.2705373138792972 requires adding true 2*pi, enclosed by the shared rational pi bounds. This check does not pass an unsupported negative value to the detached common-chart observer or replace true 2*pi by represented math.tau.

The retained lift's diameter is approximately 2.949175<pi before ZHIR and decreases by 0.25 at that phase-only event. This causal occurrence therefore supplies observed phase alignment, not outward hull renewal. Capacity remains heterogeneous, so the event does not contradict section 34's equal-capacity selection restriction. A new cycle period, coherent identity or continued source renewal does not follow. The ensuing coordinator's contribution and the corresponding source-work balance are measured in section 36.

From admission to the retained source budget¶

The native trace closes finite occurrence with genuine temporal evidence; section 36 measures whether that actual write renews source work and how the ensuing relaxation changes it. One local distinction is already exact. Hold a neighbor-phase center fixed, choose a consistent branch, and write u=center-phase and delta for the admitted Mutation shift. Provided the before/after separations remain on that branch,

(u-delta)^2-u^2 = delta^2-2*u*delta.

For the default direction sign(delta)=sign(p), opposite signs of p and u increase this squared separation. Matching signs decrease it unless the shift overshoots sufficiently. This is a fixed-center local identity, not a global hull, winding, pressure-work or field-energy theorem. The P4 endpoint has u=-1.5 and positive pressure, but a positive Mutation shift can instead align a target whose neighbor center lies ahead. Triggering Mutation therefore does not imply contrast renewal.

The completed budget in section 36 reuses this actual third-step case and its uniform-capacity control, retaining phase endpoints around the glyph batch and coordinator. It introduces no additional case or longer horizon. The forcing capture reports a pressure vector F, not diag(nu)*F. At fixed EPI, capacity and support, the prospective refreshed Dirichlet-rate increment is (B*x)^T*diag(nu)*Delta_p; a unit-capacity formula cannot be copied directly to this heterogeneous case. The instantaneous EPI energy remains unchanged. Moreover, ZHIR retains stored pressure: the ensuing held-pressure interval does not automatically use that refreshed-law increment. Keep actual flow provenance separate from the detached fresh-source comparison. Do not replace the phase source by its linear comparison outside an admitted chart or attribute earlier glyphs' changes to ZHIR. The observed secant, current nodal product, operator jump and subsequent relaxation remain separate evidence. This bounded continuation does not authorize an open-ended search for a favorable trace.

A reachable configured mechanism does not derive the Si policy or prove repeated renewal of a coherent pattern. The formation objective still requires useful retained source work against relaxation and its maintenance under the same admitted execution. This section adds no autonomous selection law, substrate-creation result, particle identification or new active research branch.

36. Native Mutation and coordination in the Dirichlet source budget¶

The native occurrence in section 35 is now examined through the existing pressure and energy owners, using the same three-step heterogeneous case and its uniform-capacity control. No additional initial condition, longer run, substituted glyph or revised selector is introduced. The distinction is between a phase event's effect on a prospective fresh nodal rate and the work of the pressure actually retained by the following integrator.

One detached channel ledger, with heterogeneous mobility retained¶

Forcing realization owns ForcingDirichletBalance and observe_forcing_dirichlet_balance. The observer consumes an existing NonEpiForcingObservation, reuses its shared channel decomposition and rebuilds the transport snapshot. It performs no graph evolution, pressure recapture or new phase-kernel call. The primitive full-pressure vector remains detached data; cached gradient, energy and pressure-defect fields are recomputed rather than trusted as evidence.

For fixed symmetric nonnegative conductance, write B=D-W, x=EPI, E_D=x^T*B*x/2, nu for the capacity vector and e for the EPI pressure weight. The existing EPI gradient is g_epi=-D^(-1)*B*x, with zero rows at zero strength. Let F be the sum of the captured phase, capacity and topology pressure channels. Define

p_model  = e*g_epi + F
eps_k    = p_fresh - p_model
eps_s    = p_stored - p_fresh
score    = diag(nu)*B*x
R_epi    = score dot (e*g_epi)
R_source = score dot F
R_model  = R_epi + R_source
R_fresh  = R_model + score dot eps_k
R_stored = R_fresh + score dot eps_s.

The three source-channel works are reported separately and retain their signs. The final rate agrees exactly with the rebuilt snapshot's stored nodal energy rate. These are rational identities for represented input values, not claims of exact transcendental phase evaluation, authenticated history or universal runtime stability. The captured kernel discrepancy is relative to the represented channel reference, not an error enclosure against an exact-real phasor model. Zero capacity and zero-strength rows retain their existing degenerate-mobility convention.

The EPI term satisfies

R_epi = -D_loss,
D_loss = e*sum_{d_i>0}(nu_i/d_i)*(B*x)_i^2 >= 0.

Thus instantaneous nondecrease for the prospective refreshed nodal row requires R_source + score dot eps_k >= D_loss. A positive change in phase-source work alone is insufficient; capacity-source work, diffusion and the declared numerical reference matter too. This criterion concerns Dirichlet form contrast. It does not identify that contrast with coherent identity, tetrad energy, a particle or a maintained NFR.

At fixed EPI, capacity and support, a phase-only ZHIR event has zero instantaneous EPI-energy jump. It can nevertheless change the fresh phase pressure at its target and at neighbors whose resultants include that target. The channel ledger retains those nonlocal contributions instead of inferring source work from the target's phase displacement alone. Stored pressure remains unchanged by ZHIR, so its held nodal rate does not automatically become the newly observed fresh rate.

Reuse the finite Euler endpoint identity¶

The existing observe_support_transport_euler owns the finite energy accounting. With fixed conductance and capacity, held r=diag(nu)*p_stored, interval h, exact reference y=x+h*r and observed endpoint defect delta=x_after-y, it reports

E_D(x_after)-E_D(x)
  = h*R_stored + h^2*r^T*B*r/2
    + (B*y) dot delta + delta^T*B*delta/2.

The linear drift, nonnegative Euler quadratic contribution and signed endpoint-defect contribution remain separate. The last term can contain rounding, clipping or other deviations from the declared reference. The algebra alone does not identify their origin. A distinct clipping budget would require an observed raw pre-clip endpoint; no such intermediate endpoint is inferred from the final value. Likewise, an instantaneous rate sampled at an endpoint is not an integral of refreshed rates along the actual held-pressure interval.

The quadratic contribution is also present under exact continuation with constant held rate; it is not automatically a numerical error. A nonpositive initial energy rate therefore does not exclude positive finite energy change. At consensus, for example, the initial rate is zero while a nonuniform held forcing can generate positive energy quadratically. The instantaneous source criterion above must not replace the integrated or finite endpoint budget when assessing maintenance.

What the retained third step shows¶

The enhanced native admission controls retain the actual target event, integrator and coordinator boundaries from section 35. The third heterogeneous batch applies (ZHIR,OZ,IL,IL,IL,IL) in node order (0,1,2,3,4,9). Thus the pressure at flow start also includes the later nodes' pressure writes; it is not attributed solely to the target Mutation. Approximate global rates are:

Boundary	EPI term	Phase work	Capacity work	Stored-residual work	Fresh rate
Before target ZHIR	`-1.939783470`	`+2.456864795`	`-0.207571077`	`0`	`+0.309510247`
After target ZHIR	`-1.939783470`	`+2.107507269`	`-0.207571077`	`+0.349357525`	`-0.039847278`
Subsequent flow start	`-1.939783470`	`+2.107507269`	`-0.207571077`	`+0.326643961`	`-0.039847278`
Before coordinator	`-1.974221035`	`+2.138489438`	`-0.192984463`	`+0.340073217`	`-0.028716059`
After coordinator	`-1.974221035`	`+1.660508618`	`-0.192984463`	`+0.818054037`	`-0.506696879`

Topology work is zero at these boundaries. The retained kernel-rate defects have magnitude below 6e-16; the table's decimals are rounded displays, while each underlying rational ledger has zero identity residual.

The target ZHIR write preserves EPI, capacity, conductance and stored pressure. Its observed phase alignment therefore changes the prospective fresh rate in the dissipative direction while leaving instantaneous energy and the stored rate unchanged. The following coordinator also preserves EPI energy at its phase event and makes its own prospective fresh rate more negative. These comparisons use each event's actual EPI endpoint; the intervening integration changes B*x, so their work increments must not be added as if they used one unchanged score.

The heterogeneous run's actual EPI-energy increase is accounted for by its positive retained-pressure drift, Euler quadratic contribution and endpoint defect. At flow start, R_stored is approximately +0.286796684. The finite third-flow accounting is:

Contribution	Heterogeneous case	Uniform-capacity control
Held drift `h*R_stored`	`+0.14339834185`	`-0.12701644047`
Held-flow quadratic term	`+1.13459958299`	`+0.01902551467`
Combined endpoint-defect term	`-1.16456266791`	approximately `-1.38e-15`
Observed energy change	`+0.11343525693`	`-0.10799092580`

Both exact endpoint identities have zero residual. The heterogeneous endpoint-defect contribution is substantial; without an actual raw pre-clip capture it is not assigned entirely to clipping or to rounding. The observed increase is not evidence that the aligning Mutation or coordinator supplied a positive fresh-rate increment. The uniform control, which applies IL at every node in all three steps, instead loses EPI energy over its corresponding interval. This is a finite comparison of the supplied capacity configurations and their resulting trajectories, not a universal claim that heterogeneous capacities increase energy or that homogeneous capacities always decrease it.

Detached ledger controls independently check heterogeneous mobility, signed channel contributions, fresh-versus-stored pressure and reconstruction of altered caches. They also retain the distinction between exact arithmetic over detached data and evidence of a live kernel or causal execution.

Interpretation boundary¶

This retained case demonstrates native Mutation occurrence, observed phase alignment and a finite EPI increase under held pressure. The fresh-law and executed-flow ledgers explain why these observations coexist. Neither the energy increase nor the operator label establishes sustained renewal of a coherent identity. The unresolved modeling distinction concerns how pressure refresh and the interval law contribute to retained source work; changing that law would be a declared model change, not an explanation retroactively applied to this trace. The sole task queue remains the execution plan.

37. Separate the frozen hard-rail reference from its represented endpoint¶

Section 36 leaves a substantial combined endpoint defect in the same heterogeneous third interval. A detached exact reference can separate the energy effect of the configured hard rails from the remaining endpoint difference without inventing an internal clipping trace. This section uses that interval and its existing uniform-capacity control only.

Frozen constant-velocity hard clipping has an exact semigroup¶

Let the common scalar rail be [L,U], with L<=U, and define C(a)=min(U,max(L,a)) componentwise. Hold the velocity vector v fixed, start with every x_i inside the rail, and put

T_t(x) = C(x+t*v),       t>=0.

For a coordinate with v_i>=0, the lower rail is inactive and

T_s(T_t(x))_i
  = min(U, min(U,x_i+t*v_i)+s*v_i)
  = min(U, x_i+(s+t)*v_i)
  = T_(s+t)(x)_i.

If the inner value has reached U, nonnegative s*v_i leaves it there; otherwise both expressions apply the same upper cap to the same sum. For v_i<=0, the corresponding lower-cap calculation is

T_s(T_t(x))_i
  = max(L, max(L,x_i+t*v_i)+s*v_i)
  = max(L, x_i+(s+t)*v_i).

Zero velocity satisfies both cases. Thus T_0 is the identity on the rail and T_s o T_t=T_(s+t) for nonnegative times. By induction, exact repeated hard clipping of this fixed velocity over any nonnegative partition gives the same final reference C(x+h*v) for the same total duration h.

The in-rail initial state and fixed velocity are essential premises. Refreshed pressure, changing capacity, forcing, operator jumps, different rails or a different clipping map require separate laws. The identity is not binary64 partition invariance: multiplication, addition and time accumulation can round differently under different partitions, even when the exact reference agrees. Native phase coordination and adaptation also act per invocation and are outside this frozen reference.

The hard-rail reference is a constrained map with declared rails. At a saturated outward-moving coordinate it suppresses further displacement; it is not an unconstrained solution of dx/dt=nu*p there. Its algebra does not derive the rails or remove their role from the execution provenance.

Common hard rails cannot increase Dirichlet energy relative to raw input¶

The common scalar projection is 1-Lipschitz, so for any two coordinates abs(C(y_i)-C(y_j))<=abs(y_i-y_j). On symmetric nonnegative conductance, each edge's squared separation therefore decreases or stays fixed. Summing the weighted terms gives

E_D(C(y)) <= E_D(y).

This compares the projected endpoint with its raw input y, not with the initial state x. It does not imply E_D(C(x+h*v))<=E_D(x): a nonuniform held velocity can create contrast before projection. The common hard-rail assumption must not be replaced by different per-node rails or a configured soft clipping function.

An exact reference split without an internal clipping claim¶

For the retained held-pressure interval, set

v = diag(nu)*p_stored
y = x+h*v
z = C(y).

Treat the captured scalar state, capacity, pressure, duration and rail values as exact rationals. With fixed conductance and the actual captured endpoint x_actual, the energy telescope is

E_D(x_actual)-E_D(x)
  = [E_D(y)-E_D(x)]
    + [E_D(z)-E_D(y)]
    + [E_D(x_actual)-E_D(z)].

The first bracket is the existing held-flow drift plus quadratic term. The second is a nonpositive exact hard-rail reference correction. The last is the signed remaining energy difference between the represented actual endpoint and that projected reference. The corresponding state difference x_actual-z remains available; a small energy difference alone would not establish a small state difference.

The support transport owner exposes SupportTransportClippedFlow through observe_support_transport_clipped_flow(before, after, dt, lower=..., upper=...). It reuses the existing Euler budget for this decomposition rather than introducing another integrator or an inferred pressure law. The reference y is not asserted to be a captured raw runtime array, and z is not asserted to be an observed internal clipping output. Agreement with this reference does not authenticate how many substeps or clipping calls occurred. Without their trace, the last bracket is an endpoint difference, not an exclusively rounding or exclusively clipping measurement.

The same retained interval is close to its projected exact reference¶

The retained cases use the actual third interval h=0.5 and captured common hard rails [-1,1]. No new trajectory or pressure refresh is performed. Approximate energies and signed contributions are:

Quantity	Heterogeneous case	Uniform-capacity control
Initial energy `E_D(x)`	`6.427820565521181`	`7.184031670285270`
Raw reference energy `E_D(y)`	`7.705818490365846`	`7.076040744486429`
Projected reference energy `E_D(z)`	`6.541255822452809`	`7.076040744486429`
Actual endpoint energy	`6.541255822452811`	`7.076040744486428`
Raw reference change	`+1.277997924844665`	`-0.107990925798841`
Hard-rail reference correction	`-1.164562667913037`	`0`
Actual-minus-projected-reference energy	approximately `+3.16e-15`	approximately `-1.38e-15`
Observed total energy change	`+0.113435256931631`	`-0.107990925798842`

The exact rational telescopes have zero residual; these decimal displays are rounded separately. In the heterogeneous reference only pendant node 9 crosses a rail: its raw value is approximately -2.498611286112409 and the projected value is -1, which also equals its observed endpoint. Every other reference coordinate remains unprojected. The maximum absolute actual-minus-projected-reference state difference is approximately 4.16e-16. In the uniform control no reference coordinate is projected, and the maximum absolute state difference is approximately 3.51e-16.

Thus the heterogeneous case's large combined Euler defect from section 36 is resolved, at the endpoint level, into a substantial nonpositive hard-rail reference correction and a small remaining difference from that reference. This does not turn the constructed y or z into observations of internal runtime clipping. It also does not establish binary64 partition invariance. The reference controls and enhanced retained native controls exercise the same shared observer; the existing runtime partition boundary retains the distinction between exact composition and represented execution.

Consequence for the maintenance claim¶

The heterogeneous projected exact reference still gains Dirichlet energy. The positive observed change is therefore not explained away as the small remaining endpoint discrepancy. It belongs to this configured bounded held-pressure evolution, whose hard projection removes part of a larger raw increase. Its source still includes supplied heterogeneous capacity, retained operator pressure and the declared clock and boundary policy.

Sections 35 and 36 separately show that the actual Mutation and coordinator align phases and reduce their prospective fresh-rate work in this case. The surviving finite form-energy increase does not contradict those results and does not establish a restoring law, persistent differentiated identity or autonomous NFR maintenance. No internal clipping history, new feedback law, physical interpretation or additional research queue follows from this endpoint decomposition.

38. Default capacity writers cannot restore an inward-contracted profile¶

The source and interval audits above establish finite native behavior; they do not explain regeneration of the capacity contrast that supports a differentiated profile. This section examines that missing restoration condition directly. It uses the admitted initialized default writer class from sections 34 and 35, without extending the prior native prefix or introducing a new capacity law.

Writer and timing premises¶

At a native decision, require freshly computed default Si and the ordinary default selector, with its injected thresholds and operator factors. Use initialized scalar nodes, so IL is admitted as the first grammar fallback. Retain the complete node set and exclude custom selectors, operator hooks, callbacks or other unaccounted capacity writers. Capacities are nonnegative and initially inside the fixed common configured rails. The maximum M is positive. These are execution premises, not properties inferred from a label saying that a run is canonical.

The selection snapshot precedes the sequential primitive batch. Its base choices are {IL,OZ,ZHIR,NAV,RA}; lag overrides add AL and EN. In this initialized class a rejected candidate does not reach a later capacity-changing fallback because IL is already admitted. The actual IL, OZ, ZHIR, NAV, AL and EN primitives preserve capacity. RA writes only its own target's capacity. Consequently, before a target's own RA write, its capacity is still the value used for its selection, even if earlier targets have already changed their own states. Public operator classes and deliberately supplied words are different execution paths.

The built-in nodal integrator and phase coordinator preserve capacity. The capacity adaptation owner uses a held neighbor snapshot, updating only its eligible subset. Its pressure, Si and counter gates need not be fresh for the interval argument below: any eligible subset has the same convex-range property. The protected optional network REMESH operation does not supply a hidden capacity update. The relevant source owners are selection, primitive operators, runtime ordering and adaptation.

Exact interval invariance under those writers¶

Let m=min_i(nu_i) and M=max_i(nu_i) at the selection snapshot. Write a for the default normalized Si capacity weight and hi=0.5 for its upper selection threshold. The other Si contributions are nonnegative, so the ideal fresh definition gives Si_i>=a*nu_i/M. An ordinary RA choice requires Si_i<hi; hence

nu_i < (hi/a)*M.

The optional default RA amplification multiplies the target by gamma=1+b, where the effective default boost b has nominal recipe 1/(8*pi). The effective coefficients satisfy

gamma*hi/a < 1       (approximately 0.68538189),

Thus even an amplified RA target remains below the selection snapshot's M. The multiplier is at least one, so it cannot cross below m. An unamplified target is unchanged. All other reachable primitives preserve capacity; therefore the entire admitted sequential batch remains in [m,M]. This argument uses the selection snapshot bound, not the false claim that RA is itself a convex neighbor average.

For an eligible adaptation target, the ideal update is

nu_i^+ = (1-eta)*nu_i + eta*mean(nu_neighbors),       0<=eta<=1.

Every neighbor value and the target lie in the current interval, so both the mean and the convex update do too. An isolate uses its own value; an ineligible target is unchanged. The common rails are inactive on that already in-rail interval. Thus one admitted exact-model native step obeys

min(nu_after) >= min(nu_before),
max(nu_after) <= max(nu_before).

Induction gives nested capacity intervals for every finite continuation that satisfies these premises. No common semicircle, phase lock, fixed pressure, fixed eligible subset or convergence to consensus is required. In particular, this argument must not import the phase-chart observer's phase representation, connected-graph or resource restrictions.

An arbitrarily small inward perturbation prevents profile recovery¶

Declare a nonuniform target capacity profile nu_star, with extrema 0<=m<M, and arithmetic mean mu strictly between them. For any 0<epsilon<=1, consider the nearby inward perturbation

nu_epsilon = (1-epsilon)*nu_star + epsilon*mu*1,
l = (1-epsilon)*m + epsilon*mu,
u = (1-epsilon)*M + epsilon*mu.

Then m<l<=u<M. Every future capacity profile w admitted by the nested interval result lies in [l,u]^n. At target nodes attaining m and M, comparison with the original profile immediately gives

norm_inf(w-nu_star) >= max(l-m, M-u) > 0.

Removing a common positive capacity scale does not remove the obstruction. For any c>0, the same two coordinates give

norm_inf(w-c*nu_star) >= max(l-c*m, c*M-u)
                      >= (M*l-m*u)/(M+m)
                       = epsilon*mu*(M-m)/(M+m) > 0.

The second inequality follows by averaging the two affine lower bounds with weights M/(M+m) and m/(M+m), which cancels c. The bound is sharp over this full interval box: use c=(l+u)/(M+m) and project c*nu_star onto [l,u] coordinatewise. This mathematical construction proves the bound's sharpness; it is not a proposed engine operation.

For positive m, the equivalent ratio statement is u/l<M/m; the quantitative bound above also handles m equal to zero without dividing by it. The argument uses extrema and is unchanged by a permitted relabeling of the target nodes. It does not assume that the original target was a fixed point: every future state in the interval box remains a positive distance from its original positive-scaling orbit.

Because epsilon can be arbitrarily small, this rules out local asymptotic restoration of a nonuniform capacity profile, including recovery modulo common positive scale, within this admitted exact writer class whenever such capacity perturbations belong to the restoration claim. It does not rule out transient organization, a family with different capacity profiles, or restoration of an identity definition that does not retain this profile.

The adaptation implementation must preserve its declared convex range¶

Exact convexity cannot simply be attributed to rounded arithmetic. With default eta=0.1, target capacity 0.104 and a singleton neighbor equal to the immediately preceding binary64 value, the former two-product blend can return the immediately following value above 0.104. It therefore exceeds both inputs. The equal-input guard from section 34 does not cover this nonuniform case.

The shared adaptation proposal now bounds its represented stable mean by the actual neighbor minimum and maximum, and bounds the represented blend by the target/mean interval. These comparison guards retain the existing arithmetic result when it is already inside the declared interval. They change neither eligibility, counters, factors nor the intended exact convex map. They enforce the adaptation range in represented arithmetic without treating numerical overshoot as a structural source.

That implementation guarantee concerns adaptation. The full interval theorem still has its stated exact-model selector and RA hypotheses; binary64 Si normalization, threshold comparisons and multiplication require their own evidence before upgrading it to a represented runtime theorem for this class. No assumed future rounding margin supplies that evidence.

Reuse and scope of the obstruction¶

The conditional algebra belongs with capacity feedback, using the shared exact scalar/vector readers. derive_capacity_interval_recovery_bound returns a CapacityIntervalRecoveryBound containing the original and perturbed intervals, their ratios, both separation bounds and a minimizing common scale. Its executable domain is nonempty strictly positive capacity vectors and contraction in (0,1]; a uniform vector returns zero separation, not a false nonrecovery claim. The zero-minimum extension of the proof above is not an additional admitted evaluator domain. Its existing P2 Coupling envelope and separate binary64 lattice theorem concern different supplied maps; neither authenticates the present selector, grammar or runtime class. The new interval separation likewise remains conditional on the writer admission established above, rather than a seal on arbitrary executions.

Recovery-bound controls exercise the exact algebra and its supported domain; native writer controls check the declared selector and primitive paths. The adaptation regressions retain the adjacent-value boundary independently of the conditional recovery theorem. These checks supply finite implementation evidence, not admission of unobserved future choices.

The native control uses one default step on unit path4, seed 17, constant EPI 0.125, zero phases, capacities (0.3,0.3,0.3,1) and empty initial histories. Its actual word is (RA,RA,NAV,IL); the first two capacities become approximately 0.31193662073189216, while the global interval stays [0.3,1]. All four decisions precede the first write. Nodes 1 and 2, initially equal in capacity, become unequal: the interval theorem does not assert that every local pairwise contrast decreases.

This result identifies a concrete missing regeneration property for the default capacity-supported route. It is not a theorem about all thirteen operators: explicitly chosen capacity-expanding operators, changed selection laws, new nodes or other capacity writers lie outside its premises. It does not identify capacity with every possible scalar-form, phase-sector or support identity, prove physical emergence, or prescribe a replacement feedback law. The investigation's task queue remains owned by the execution plan.

39. The same stationary reduced identity constrains capacity differences¶

Section 38 obstructs recovery of an original capacity profile, but an identity definition may deliberately discard capacity. The narrower question here is whether altered capacities can still realize the same EPI shape and phase geometry with zero fresh pressure. This is an algebraic stationary-realizability gate, not a theorem about arbitrary convergence, form derivatives or complete-runtime persistence.

Fix the reduced identity and the pressure realization¶

Compare two states on the same connected reciprocal unique-neighbor support, with the same transport conductances and effective pressure coefficients. Let their EPI values differ by one common additive constant and their exact phase configurations by one common rotation. Require a well-defined phase source under the declared circular conventions. The weighted EPI gradient and ideal phase source then agree; the fixed topology channel agrees as well. Let b>0 be the active capacity-pressure coefficient.

Write the unweighted capacity-support operator as

(G*q)_i = mean(q_neighbors)-q_i.

This G uses unique support neighbors, including edges with zero transport conductance. It is generally different from the weighted EPI operator -L_rw. With delta_nu=nu_after-nu_before, the ideal fresh-pressure difference between the two states is exactly

delta_p = b*G*delta_nu.

All capacities must be positive to identify zero unforced nodal velocity with zero pressure. Gamma is absent or zero, and neither clipping nor an operator jump is being used to keep EPI at an endpoint. Under those premises, if both states realize zero fresh pressure, G*delta_nu=0.

On connected reciprocal support, the kernel of G consists precisely of constant vectors. At a maximum of a harmonic vector, its neighbor average can equal that maximum only if every neighbor has the same value; connectivity propagates this equality. Conversely, a constant vector has zero neighbor difference. Therefore the necessary and sufficient capacity condition for preserving the original zero-pressure realization, with the other declared identity data fixed, is

nu_after = nu_before + c*1,

for a common additive capacity offset c that preserves positive capacities. This is not common capacity scaling: adding a constant preserves capacity neighbor differences, whereas scaling a nonuniform profile generally does not. Unequal transport weights do not alter this conclusion because G belongs to the capacity channel's support convention.

The capacity interval obstruction survives this reduced identity¶

Let a nonuniform original capacity profile have extrema m and M, and contract it toward its arithmetic mean as in section 38. The perturbed capacity difference is epsilon*(mu*1-nu_star). Since G annihilates the constant mean, its prospective ideal pressure change at the original form/phase identity is

delta_p = -epsilon*b*G*nu_star = -epsilon*F_vf_star.

The perturbed interval [l,u] has width (1-epsilon)*(M-m). Conditional on subsequent capacities w remaining in that interval, for every real additive offset c,

norm_inf(w-nu_star-c*1)
  >= max(l-m-c, M+c-u)
  >= [M-m-(u-l)]/2
   = epsilon*(M-m)/2 > 0.

The second inequality averages the two extreme-coordinate bounds. It therefore holds uniformly over all future profiles in the interval box, not merely for the initial contracted profile. The unconstrained bound is sharp at c=(l+u-m-M)/2 with w obtained by projecting nu_star+c*1 into [l,u]. That minimizing offset can make some shifted target capacities negative; no sharpness claim is made after imposing positive capacity on the shifted target. The positive lower bound remains valid for every model-admitted positive shift because that is a smaller set of comparisons.

The existing CapacityIntervalRecoveryBound now also reports additive_separation and minimizing_offset. These quantities reuse the same declared perturbation and interval geometry; they introduce no evolution law or additional run.

Combining the connected-support identity with this interval separation excludes an exact zero-pressure stationary realization of the original EPI shape and phase geometry at every subsequent state satisfying the interval and fixed-model premises. It does not establish a positive lower bound on a trajectory's pressure, prove nonconvergence of its EPI or phase, or rule out a different identity. Such conclusions would need additional compactness, continuity, admission and dynamical hypotheses.

Retain phase realization and kernel differences in represented captures¶

Forcing realization supplies ForcingCapacityDifference through observe_forcing_capacity_difference. The detached comparison reuses the validated channel decomposition and rebuilt transport snapshots. It requires matching support, conductance and coefficients, positive capacities, an exact common EPI offset and an exact common raw phase offset in the captured coordinates.

The mathematical theorem allows a common circular rotation, whereas this observer's raw-offset admission is a narrower represented-coordinate condition. Independently rewrapping or rounding phases must not be treated as proof of that condition. Even an admitted represented common offset can change the materialized phase-kernel coefficients slightly; their observed difference remains explicit rather than being declared zero.

Consequently, the represented pressure ledger separates

fresh_pressure_change
  = capacity_pressure_change
    + phase_realization_change
    + kernel_assembly_defect_change.

The stored-pressure difference additionally retains the change in stored versus fresh residual. Neither a supplied capture nor an exact arithmetic identity authenticates a kernel call, a stationary baseline or a causal trajectory. The observer does not infer pressure from an observed EPI derivative or declare a numerical near-zero value to be exact stationarity.

In particular, the general nodal-rate difference is

delta_rate = diag(nu_after)*delta_p
             + diag(delta_nu)*p_before.

The second term vanishes only for a zero-pressure baseline. Actual captured baseline residuals must remain in that identity; a model's exact stationary construction does not erase them.

A prospective stationary construction and its capacity controls¶

The stationary identity controls use a declared three-node path with transport weights 1 and 2, injected default pressure coefficients, seed 17 and the default NumPy kernel. Before any kernel observation, prescribe

phase = (0, pi/8, 0),
capacity = (1/2, 1, 1/2),
x_i = 3/4 - (w_phase/w_epi)*(phase_i/pi)
            - (w_vf/w_epi)*capacity_i.

At this particular target the two endpoints have equal phase and capacity, so the weighted EPI and unweighted capacity rows share the relevant three-coordinate pattern. This special coincidence does not make the capacity channel conductance-weighted in general. The phase source is (1/8,-1/8,1/8) and the transport strengths (1,3,2) give the required zero weighted sum. The prescribed exact-real target therefore has zero pressure; no measured pressure is fitted to define its EPI.

Its materialized EPI is approximately (0.5908450569081046,-0.08600896788251489,0.5908450569081046). The shared signed phase-pressure realization captures approximately (0.12500000000000003,-0.125,0.12500000000000003): singleton phasor rows and the certified two-neighbor midpoint round separately. Their exact represented degree-weighted sum is 3/2^55, rather than the ideal zero. Baseline fresh pressure is approximately (+2.43e-17,+3.47e-18,+2.43e-17). The separately retained phase-realization, modeled-pressure and kernel residuals account for this nonzero result. Supplied stored zeros are initialization, not evidence of an exact runtime stationary state or mean conservation.

Half contraction toward the capacity mean gives (7/12,5/6,7/12) while holding materialized EPI and phase fixed. The shared observer finds

capacity_pressure_change = w_vf*(-1/4,+1/4,-1/4),
fresh_pressure_change = capacity_pressure_change,

with zero phase-realization and kernel-defect changes in this comparison. Each nonzero component has magnitude approximately 0.014574888647966173. The final captured pressure also retains the small baseline residual; it is not replaced by that difference vector. Stored pressures remain their supplied zeros, exposing the fresh-versus-stored discrepancy rather than concealing the loss of stationary realizability.

The complementary controls retain the theorem's boundaries:

Declared change	Observed pressure-difference consequence
Add `1/4` to every capacity on the connected path	Exact fresh-pressure change is zero
Raise only node 0 capacity by `1/4`	Capacity contribution is `w_vf*(-1/4,1/8,0)`; its middle entry is not the transport-weighted `w_vf/12`
Shift capacities independently on two disconnected edges and an isolate	Component offsets `1/4`, `1/2`, `3/4` leave fresh pressure unchanged
Set the capacity-channel weight explicitly to zero, then contract capacities	Fresh pressure is unchanged

The asymmetric control retains its small nonzero kernel-defect change. The rate comparison also keeps diag(delta_nu)*p_before rather than treating the numerical baseline as exactly zero. These are detached production-kernel observations at prospectively specified inputs, not an operator schedule, a trajectory, a perturbation-recovery experiment or an infinite-time stability claim.

Difference-observer controls check the shared arithmetic, admission and represented phase/assembly contributions; recovery-bound controls check the additive separation without assuming that its minimizing shifted target stays positive.

Limits of this stationary gate¶

Disconnected support allows an independent constant capacity shift on each connected component. A zero capacity-pressure coefficient makes this channel insensitive to capacity differences. Changed phase geometry, transport, topology or coefficients can compensate a changed capacity source and are outside the fixed-identity comparison.

Positive capacities, no Gamma and no clipping are also substantive conditions. Zero capacity can suppress a nonzero pressure; forcing can cancel it; a rail can hold an endpoint under outward pressure. None is a zero-pressure stationary realization of the declared unforced model. Likewise, a common EPI drift may preserve differences while using nonzero pressure, and moving or periodic patterns require their own treatment. The present result addresses stationary pressure realizability only and does not introduce a second task queue or a replacement constitutive law.

40. Static phase compensation and the native direction obstruction¶

Section 39 fixes both form and phase geometry. Here the phase is allowed to change while the original EPI shape is retained. The same prospectively declared weighted P3 supplies the comparison; no new oscillator, selected target after observation or extended trajectory is introduced. Static pressure compatibility, the actual native writer and complete-runtime restoration remain different questions.

A different phase geometry can compensate the lost capacity source¶

Write a=w_phase>0, e=w_epi>0 and b=w_vf>0 for the effective pressure coefficients. In the P3 family, both endpoints have the same EPI X, phase and capacity; the middle has EPI Y, a relative phase gap delta and a capacity excess d. Transport edge weights remain 1 and 2. Because the endpoint values agree, the weighted EPI row and the unweighted support rows yield

p = [e*(Y-X)+a*delta/pi+b*d]*(1,-1,1).

The topology contribution is zero because the injected topology weight is zero, not because this irregular support has zero topology gradient. The original construction has delta_0=pi/8, d=1/2 and zero ideal pressure. Inward capacity contraction reduces d to (1-epsilon)*d. Keeping the same EPI shape therefore requires

delta_required = delta_0 + pi*(b/a)*epsilon*d.

This is a positive phase-gap increase. The effective default ratio b/a<1/12 keeps delta_required<pi/6 for 0<epsilon<=1, so the required state remains in a regular acute chart. Static feasibility is not blocked by a phase-wrap singularity. For the retained half contraction, delta_required is approximately 0.45306233345003016, compared with the original gap 0.39269908169872414.

This construction preserves EPI shape through a different phase geometry; it does not preserve the full form-and-phase identity of section 39. Prescribing that compensating phase is initialization of a compatible state, not evidence that the engine generates its restoring change.

The native phase update moves in the opposite direction¶

In this common chart, let g be the shared global phasor target. Each endpoint's local target is the middle phase; the middle's local target is the common endpoint phase. The ideal formula implemented by the native coordinator therefore gives

delta_after = (1-kG-2*kL)*delta_before.

The global target cancels from this difference. Unequal transport conductances do not change the phase-neighbor convention. The injected default adaptive gain bounds from section 34 imply 0<1-kG-2*kL<1, independently of which admitted adaptive branch supplies the gains. Thus this writer decreases the positive gap, whereas lost capacity support requires an increase.

At fixed EPI and capacity, a gap contraction by factor rho contributes

delta_p_phase = -a*(1-rho)*delta_before/pi*(1,-1,1).

Starting from the uncompensated capacity contraction, this adds to the existing residual -b*epsilon*d*(1,-1,1). Starting instead from the statically compensated ideal zero-pressure state, the same native phase write produces a nonzero residual. The compatible stationary surface is therefore not invariant under this phase map.

The default selector does not supply a compensating Mutation in either retained segment. After half contraction, min(nu)/max(nu)=0.7. With alpha denoting the Si capacity coefficient, fresh default Si is bounded below by 0.7*alpha>0.5; every base choice is IL. Initial lag counters remain below their forcing thresholds, and the nonzero EPI values admit IL in the incremental grammar. This argument concerns the actual default selection path, not a deliberately supplied operator word.

Two isolated native segments confirm the direction¶

The finite controls reuse the materialized EPI and capacities from section 39. One starts with the original phase gap; the other initializes the analytically prescribed compensating gap before any observation. Each segment refreshes pressure and Si, executes the actual default glyph batch and invokes the native coordinator. No nodal integration, capacity adaptation, full runtime step or temporal-growth history is performed.

Both batches request and apply IL at all three nodes, pass grammar and leave the lag counters at one. The coordinator selects its stable adaptive branch with represented gains kG=0.04461342911494398, kL=0.140157221158957, giving rho=0.675072128567142. The retained observations are:

Initial phase choice	Gap before coordination	Gap after coordination	Maximum fresh pressure magnitude before	Maximum fresh pressure magnitude after
Original, uncompensated gap	`0.39269908169872414`	`0.2651002049687197`	approximately `0.014574888647966177`	approximately `0.04538402112491463`
Prescribed static compensation	`0.45306233345003016`	`0.3058497538157081`	approximately `3.99e-17`	approximately `0.03554492002170308`

Pressure magnitudes are displayed; after coordination the endpoint pressures are negative and the middle pressure positive. The statically compensated capture retains its small nonzero baseline rather than claiming an exact represented equilibrium. Its pressure increase is not explained by a large initialization residual.

EPI, capacity and support remain unchanged in both segments. IL contracts stored pressure, and the coordinator leaves that held pressure unchanged. The fresh pressures in the table are detached read-outs; they are not substituted into an executed physical interval. Neither segment is evidence of complete-runtime recovery or failure of every future trajectory.

An exact invariant source class closes this native P3 route¶

No further run is needed to combine the earlier conditional results for the uncompensated half-contraction case. Retain the initialized, fresh-Si, default writer class of sections 34 and 38, this fixed support and these coefficients, with no additional phase/capacity writers. The capacity interval stays within [l,u]=[7/12,5/6]. Its minimum-to-maximum ratio cannot fall below 0.7, so every future fresh base selection remains IL. Lag AL/EN and admitted fallback IL also preserve phase and capacity. Adaptation may change any eligible subset without leaving the interval.

The equal endpoint phases remain equal under the coordinator, and its positive gap factor keeps 0<=delta_t<=delta_0. These phase facts do not require intermediate EPI to remain at its original value. At any subsequent state whose EPI does equal the original shape modulo a common offset, capacity differences satisfy

nu_middle-nu_endpoint <= u-l = (1-epsilon)*d,
mean(nu_endpoints)-nu_middle >= -(u-l).

Consequently its ideal fresh pressure necessarily obeys

p_endpoint <= -b*epsilon*d < 0,
p_middle   >= +b*epsilon*d > 0.

This is a uniform exclusion of zero-pressure stationary realizability of that original EPI shape within the declared exact source class, including unequal future endpoint capacities. It is conditional on the stated writer, chart, source and capacity-envelope premises. It is not an unrestricted binary64 continuation theorem, a statement about every actual held pressure, or a proof that hybrid EPI jumps cannot maintain another kind of pattern.

The separately initialized compensated gap is larger than delta_0 and lies outside this source-box hypothesis. Its one-invocation failure above is retained separately; it is not retroactively included in that invariant class. Together the results distinguish an available static compensation from its absence as a restoring native phase action in the audited route. They supply no replacement phase law or additional execution queue.