C6 pressure excursions and compensation

Finite pressure, mean-budget and compensation obstructions; no global closure.

Part of Cycle winding under Coupling: scope and retained evidence. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

Archived research record. This preserves conditional derivations and source-bound evidence. Its local next-step language is historical and does not schedule work.

26. Pressure-sign sectors and the carried curvature budget

B2.d.28 tests the necessary node-1 sign crossing identified in section 25. It also extends the finite-state drift argument to an entire pressure-sign sector in the local slab. These are statements about the same represented nodal map, with fixed phase, unit capacity, conductance, coefficients and positive step. No restoring term or adjusted pressure is introduced.

A sign sector has a uniform nodal drift bound

The local lattice owner already identifies the monotone scalar pressure function for each node:

p_i(m) = RN(A_i + RN(w_epi * (delta/2) * m)).

Let l_i be the first nonnegative index and u_i the last nonpositive index from section 24. Strictly negative pressure has m<=l_i-1; strictly positive pressure has m>=u_i+1. If D=(upper-lower)/delta, actual local gradients satisfy -2D<=m<=2D. Intersect each sign sector with this range before evaluating its boundary. This avoids treating an unreachable index as a possible local state when the EPI weight is extremely small. An empty sector supplies no drift bound.

derive_c6_pressure_sign_sector uses monotonicity of both nearest-even roundings. In a nonempty negative sector, its largest index gives the least negative pressure bound; in a positive sector, its smallest index gives the least positive bound. Thus s*p_i>=c>0 throughout the selected sector, for s=-1 or s=+1. The index range is an enclosure: no claim that every enclosed integer gradient has a realizable global EPI tuple is needed for this bound.

observe_c6_pressure_sector_exit combines that bound with a validated incoming encoding X=x+r. Let d be its distance to the band endpoint in the direction of the pressure. While all pre-update states remain in the sector and all accepted exact updates remain inside that band, the nodal equation gives

N*h*c <= d,
N <= floor(d/(h*c)).

The next step therefore cannot preserve all these hypotheses. The path may leave the sign sector, leave the declared band, or cease to satisfy the update contract. This is not a guaranteed sign-crossing theorem or a band-exit theorem. At other phase inputs, leaving a strict sign sector can enter a zero-pressure state. The observer reports no initial sector budget if the supplied state is outside that sector. It does not replace the pressure producer or predict live operator admission.

The visible sign index combines exact evolution and carry transfer

The integer Laplacian index is m=-2*L_rw*x/delta. For a retained carried sequence with exact accumulated nodal area A and remainder change dr=r_after-r_before, the existing shared prefix identity gives

x_after-x_before = A-dr,
m_after-m_before = -2*L_rw*A/delta + 2*L_rw*dr/delta.

Both terms are exact rationals. They need not individually be integer; their sum is the observed integer index change. The benchmark uses the existing cycle Laplacian for this readout and checks the identity at every prefix. Neither reconstructed curvature alone nor discarded carry can decide the visible pressure sign. This is the same carry/readout issue as section 21, now expressed at the pressure-sign boundary rather than as an additional field or evolution law.

For node 1 on the inherited closed phase slice, the sign cut remains m_1>=0 for strictly positive pressure and m_1<=-1 for strictly negative pressure. From initial m_1=-1, the two budget terms must sum to at least one before that sign crossing is realized.

One shared bounded cell-exit replay

observe_nodal_remainder_cell_exit now owns the analytic-horizon, shared-prefix replay and unchanged-cell checks used by the section 24, 25 and 26 benchmarks. Its explicit step budget is a resource limit, not a TNFR coefficient. A stationary horizon, an exit beyond that budget or an exact-band departure is rejected before allocating the step schedule. Accepted replays retain their original carry and check every pre-update visible state, each intermediate output and the exact first-exit endpoint. Pressure constancy still requires the caller's fixed visible-state pressure map; the numeric helper does not prove it.

The inherited continuation reaches positive node-1 pressure

c6_winding_pressure_sign.py first replays the B27 report from its retained B26 input and compares the derived payload, including its endpoint and carry. It then continues that endpoint, with a declared ceiling of eight new cell boundaries and 256 new numerical steps. These are work limits only. The benchmark stops at the first positive node-1 pressure, without evolving further under that new pressure. A next boundary exceeding the remaining budget is reported as unresolved before its replay, rather than replaced by a longer run.

The first crossing occurs within these limits, at the third boundary and nineteenth new step. With EPI offsets measured from 0.5 in units delta=2^-54, the finite path is:

Boundary	Additional steps	Visible offsets	m_1	Refreshed pressure sum
B27 endpoint	0	(-1,0,0,-1,2,-2)	-1	-2^-108
First	12	(-1,0,0,-1,4,-2)	-1	3*2^-108
Second	3	(-1,0,0,-2,4,-2)	-1	-2^-107
Third	4	(-1,0,2,-2,4,-3)	1	-5*2^-108

The first two exits alter nodes 4 and 3, which do not enter m_1=n_0+n_2-2*n_1. At the third, node 2 rises by two lattice units; nodes 0 and 1 remain visibly unchanged. This gives Delta m_1=2 and strictly positive pressure. Node 5 changes simultaneously but does not enter this node-1 sign condition. Thus the observed sign reversal is a neighbor-driven change in the existing EPI pressure channel, with fixed phase, capacity and coefficients. It is not a direct write to node 1's pressure or an added feedback law.

The actual node-1 pressure changes from -128295757220873/2^106 to 2786216251278205/2^108. Every pre-update state in these nineteen steps still has m_1=-1. The unchanged-cell checks and fresh boundary observations therefore establish that this is the first positive pressure on this finite continuation, not merely the first positive sampled endpoint.

For the complete nineteen-step prefix, the node-1 index change separates as 86621202941648393/2^58 + 489839549361775095/2^58 = 2. The first term is the reconstructed nodal-curvature change; the second is the visible encoding's carry contribution. The two-unit displayed jump must not be identified with the first term alone. Resetting carry to zero at the B27 endpoint cannot realize this same visible itinerary.

Sign recovery has not yet repaid accumulated evolution

At the stopping point the new positive pressure has not supplied any positive-duration evolution. The exact node-1 accumulated area relative to the B27 endpoint is still

A_1 = -2437619387196587 / 2^110.

Its displayed EPI remains 0.5, while its incoming remainder has changed by this negative amount. The reconstructed mean changes by -11/(3*2^113) over the nineteen steps. The newly generated total pressure sum is negative even though node-1 pressure is positive. Current pressure sign, accumulated nodal area and mean-source sign are three distinct readouts; none may replace the coordinatewise return condition.

Every accepted prefix stays inside the local slab and satisfies the carried nodal and curvature budgets. The complete visible itinerary and its inherited carry are checked by the section 25 inverse-cell owner. No complete carried-state cycle, invariant return class, original full preparation's tail reachability or repeated live-runtime stability follows.

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_pressure_sign.py

The remaining condition at this boundary is the positive sector's admissible duration and signed area, coupled to the other five node budgets. Subsequent transitions would have to compensate prior negative area without violating their carry constraints. A proposed return must still pass the complete vector-area and whole-itinerary tests. The completed sign crossing removes the old four-state separator's hypothesis after that boundary; it does not close the broader trapping problem. Do not restart the sign search or enlarge a run merely to obtain a visual cycle.

27. Finite positive-pressure repayment and exact return obstructions

B2.d.29 continues the first positive-pressure endpoint from section 26. It keeps the B27 endpoint as the reference for accumulated nodal area. Changing the sign of current pressure did not erase the nineteen-step negative area already accumulated relative to that reference.

An exact affine budget separates repayment, crossing and overshoot

For an incoming area vector D and constant represented inputs, write a_i=F(h)*F(nu_i)*F(p_i). After an integer number n of those steps,

area_i(n) = D_i + n*a_i.

derive_nodal_area_crossings solves these scalar equations rationally before any replay. A nonzero increment has a continuous zero parameter -D_i/a_i; an exact discrete zero requires this parameter to be a nonnegative integer. For nonzero initial area driven toward the other sign, the first zero-or-opposite integer prefix is its ceiling. An integral root gives exact equality; a nonintegral root gives an overshoot. A zero initial area is recorded separately and is not described as repayment of an earlier deficit.

The observer retains the unrestricted candidates and separately marks those inside the supplied finite step limit. Coordinates that stay identically zero impose no further constraint on a joint return. Every other coordinate must have the same admissible integer root for the complete vector to vanish. Mean-area cancellation is a separate condition and cannot replace this intersection of coordinate conditions.

The inputs are an arithmetic budget, not authenticated pressure, band admission or a future trajectory. The first cell-exit update still uses the old pressure; therefore a candidate through first_exit_step can be bound to the shared cell replay when it remains inside the band. A candidate beyond that step requires fresh pressure and another budget. Holding the source after its actual cell boundary is not a valid way to complete a failed repayment.

The positive episode ends before the proposed repayment prefix

The inherited node-1 area and new per-step input are exactly

D_1 = -9750477548786348 / 2^112,
a_1 =  2786216251278205 / 2^112.

The continuous zero parameter lies strictly between steps three and four. The first nonnegative integer candidate is step four, with positive overshoot 1394387456326472/2^112; no integer prefix of that unchanged source attains exact zero. Step three would still have -1391828794951733/2^112. These are exact conditional calculations, not additional executed states.

The actual next-cell horizon is only two steps. Both use positive node-1 pressure. At the second endpoint node 2 returns from offset +2 to 0 in units 2^-54 relative to EPI 0.5; the other displayed nodes stay fixed. Hence m_1 returns from +1 to -1, and freshly generated node-1 pressure is negative again. This completes the first positive episode on the retained conditional path. No negative-pressure step is taken after that refresh, and the hypothetical four-step budget is not executed.

The new endpoint has offsets (-1,0,0,-2,4,-3). Its two positive steps supply node-1 area 2786216251278205/2^111, leaving

net area_1 = -2089022523114969 / 2^111

relative to the B27 reference. The new local mean area is -5/(3*2^112); including the inherited area gives mean -7/2^113. Every coordinate is checked through X_after-X_B27 = inherited_area + new_area. The node-1 displayed EPI matches its B27 value, but its reconstructed coordinate and remainder do not. Neither a node returning to a visible cell nor the recovery of an earlier pressure sign establishes a complete carried-state return.

Two pressure levels impose a separate integer period constraint

derive_two_level_nodal_return expresses one negative and one positive represented pressure using a common exact denominator: p_minus=-M/Q, p_plus=N/Q, with positive integers M,N. Under a common fixed positive step and capacity, zero accumulated area in that coordinate requires

n_minus*M = n_plus*N.

Writing g=gcd(M,N), every nonempty such return has counts proportional to (N/g,M/g) and length a multiple of (M+N)/g. This necessary condition does not require executing those steps. It supplies neither a realizable itinerary nor a return of any other coordinate.

For the two actually observed node-1 pressure levels,

Q = 2^108,
M = 513183028883492,
N = 2786216251278205,
g = 1,
minimum nonempty zero-area length = 3299399280161697.

Thus a short exact return cannot be obtained merely by rearranging these two levels at the existing fixed step. The bound comes from the represented numerical coefficients; it is not a physical period or a new TNFR constant. Another pressure level, variable step or capacity falls outside this two-level counting argument. It excludes neither bounded motion nor a larger invariant class, and no trajectory of this length is attempted.

Evidence owner and extension conditions

c6_winding_pressure_repayment.py revalidates the retained B28 report through its B27/B26 inputs, preserves the incoming carry, and uses the shared first-cell-exit owner for the two actual steps. Its signed-area reference is explicit. It checks the complete area vector, displayed and reconstructed return separately, and inverse-itinerary compatibility. The single boundary suffices here because its fresh pressure ends the positive episode; otherwise it would only supply a finite duration lower bound. Resource rejection before the complete boundary remains an unresolved observation.

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_pressure_repayment.py

The next structural question is how additional canonical pressure levels or a correlated trapping class can change the signed vector-area budget. Use the pressure-index cuts and incoming-carry constraints to define that test before extending a run. Do not search for a short exact cycle while silently retaining only the two excluded levels. Exact recurrence is one possible certificate, not a requirement to impose on every useful bounded pattern. Original-preparation reachability, future live admission and laboratory correspondence remain distinct open obligations.

28. A third pressure level and a transverse finite-class drift

B2.d.30 starts from the retained section-27 endpoint, with its exact carry, the same fixed phase source, unit capacity and h=1/16. The declared search has at most eight cell boundaries and 256 steps. It stops at the first node-1 pressure outside the two levels from section 27, before applying that new pressure. These are computational limits, not physical parameters.

Finite-level arithmetic reuses the accumulated nodal equation

For finitely many supplied represented pressures, choose a common exact denominator Q and write p_j=z_j/Q. With a common fixed positive step and capacity, a zero-area word has nonnegative integer counts n_j satisfying

sum_j n_j*z_j = 0,    T = sum_j n_j > 0.

Let d=gcd(z_j-z_0). If d>0, the first equation implies d | T*z_0, so every possible length is divisible by

L = d / gcd(d,z_0).

This is a necessary length constraint. For three or more levels it need not be an attainable minimum; L=1 does not establish a one-step return. For example, the levels (-2,3,4) give L=1, but no level is zero. When d=0, identical nonzero levels exclude every nonempty zero-area word, while identical zero levels permit every length. Strictly one-sign inputs also exclude zero area. A selectable zero level or opposite signs permit some scalar zero-area multiset, without proving that its ordering can be realized by canonical pressure refresh and carried transitions.

The shared owner derive_finite_level_nodal_return retains these cases separately and shares exact integerization with the two-level result. Neither arithmetic criterion authenticates its source, changes pressure, executes a long word or proves vector return.

A new level is reached, after actual node-1 overshoot

Four derived boundaries take 1,5,1,4 steps. Their displayed offsets around 0.5, in units 2^-54, are:

start:  (-1,0,0,-2,4,-3)    m_1=-1
step 1: (-1,0,2,-2,4,-3)    m_1=+1
step 6: (-1,0,0,-2,4,-3)    m_1=-1
step 7: (-1,0,2,-2,4,-3)    m_1=+1
step11: (-2,0,0,-2,4,-3)    m_1=-2

The final boundary changes nodes 0 and 2 simultaneously. Its newly generated node-1 pressure is -4325765337928681/2^109. The eleven executed steps use the old negative level twice and the old positive level nine times; none uses the newly captured third level.

The node-1 area relative to the B27 endpoint first becomes positive at new step three, with value 220301106860745/2^110. No observed prefix attains exact zero. At step eleven the local node-1 area is 24049580203736861/2^112, and its B27-reference net area is 19871535157506923/2^112. The local mean is -17/2^113 and the net mean is -3/2^110. Thus this node has overshot its earlier loss while the six-coordinate area and its mean still do not vanish. Every prefix retains net area = inherited area + local area = displayed change + carry change relative to the same reference.

The three captured node-1 levels, on denominator 2^109, have integers

(-1026366057766984, 5572432502556410, -4325765337928681).

Their difference gcd and length divisor are both 3299399280161697. The third level therefore leaves the earlier length restriction unchanged. This conclusion concerns only words using those scalar levels; it decides neither a larger pressure family nor bounded behavior.

A different node excludes confinement to the three visible states

Only three distinct displayed states occur in this continuation. In all three, node 4 has the same displayed value 0.5+4*2^-54 and the same strictly positive generated pressure

c = 1668868774373469 / 2^105.

Apply the existing observe_finite_nodal_pressure_drift to those three states with functional ell=e_4. Its reconstructed node-4 cell has width W=2^-53. The exact accumulated nodal equation then gives

N*h*c <= W,
N <= floor(2^56 / 1668868774373469) = 43.

Consequently the next, forty-fourth step must leave that finite class or fail a declared update premise. This covers every admissible incoming carry in the class, including carries not visited in the finite replay. It is stronger than a short-period obstruction for this particular class: its permanent confinement is excluded. It does not establish positive-band exit, the next actual transition, or absence of a larger correlated trap. No forty-four-step trajectory is executed to obtain the bound.

The local pressure formula also gives a conditional extension beyond the three-state list. Holding the displayed stencil (x_3,x_4,x_5) at offsets (-2,4,-3) fixes m_4=-13. The row-local EPI reducer therefore fixes its contribution; the fixed phase supplies the same phase contribution, while unit capacities and regular C6 support give zero frequency and topology contributions. Arbitrary admissible changes of remote EPI coordinates x_0,x_1,x_2 cannot change this pressure. The same node-4 cell width and positive increment thus give the same 43-step bound while that entire stencil remains fixed, with the other declared conditions unchanged. This locality deduction is separate from the executable three-state certificate. Its next-step disjunction is a stencil change or premise failure, not a prescribed node transition or full-runtime stability claim.

Evidence and the next structural gate

c6_winding_pressure_levels.py revalidates the complete B29 report through its B28/B27/B26 inputs and binds the four historical input byte hashes. Shared cell-exit and inverse itinerary owners preserve the incoming carry; the signed prefix observer retains the B27 area reference. Resource exhaustion is reported as censoring, not as absence of another pressure level. This remains a conditional numerical path, without live graph or original-preparation reachability.

Extending this retained path requires a compatible change of the neighborhood sustaining the positive node-4 drift, together with the full signed vector-area budget. The node-1 sign search, first positive episode and first third-level hit are complete on this retained path. Do not repeat them or attempt the large arithmetic period. A larger trapping claim must account for the newly identified drift direction and prove forward inclusion; successful compensation of one coordinate does not supply that proof.

29. Frozen-neighborhood bounds and a censored node-4 sign test

B2.d.31 makes the local-stencil deduction from section 28 executable and tests its first observed failure from the retained B30 endpoint. The conditional numerical map keeps the same phase source, unit capacities, support, coefficients, positive EPI band and h=1/16. The incoming carry and B27 accumulated-area reference are retained.

Local pressure constancy does not require a frozen whole graph

observe_c6_frozen_pressure_stencil binds one row of the actual CPU pressure map to its displayed stencil (x_(i-1),x_i,x_(i+1)). Under the declared fixed conditions, the row-local EPI and phase readers give the same pressure while that stencil stays fixed. Remote displayed EPI and carried remainders may change.

For nonzero pressure p_i, the accumulated nodal law is X_i(n)=X_i(0)+n*h*p_i over such a prefix. The existing held-cell owner supplies the selected coordinate's exact directional limit inside its nearest-even cell intersected with the positive band. Its whole-vector first-exit time is not the stencil bound: a remote coordinate can leave its cell while this row stays fixed. The existing finite-drift owner independently supplies the uniform bound floor(W/(h*abs(p_i))) over all admissible incoming center carries. That closed enclosure can be conservative at odd ties; the actual-carry bound retains endpoint parity.

Both bounds are conditional deadlines, not predictions of the first neighborhood change. A neighbor can change earlier, or an update/band premise can fail. Zero pressure supplies no finite center deadline and does not establish invariance of the stencil. The observer changes no state and certifies no graph execution or pressure-sign hit.

At the B30 endpoint, node 4 has

p_4 = 1668868774373469 / 2^105,
r_4 = 4148688737535375 / 2^110,
upper-cell distance = 67908905300392561 / 2^110.

Its exact incoming-carry limit is 20 unchanged steps, with deadline 21. The uniform limit over all carries remains 43, with deadline 44. The difference is entirely determined by the retained numerical state.

The neighbor changes first, without reversing the pressure

The declared test has at most eight new boundaries and 256 steps and stops early if freshly generated node-4 pressure becomes nonpositive. It reaches the eight-boundary ceiling after 1,5,1,4,1,3,2,1 steps, eighteen in total. No nonpositive pressure is observed, so the sign test is censored. No ninth boundary or post-limit step is executed.

The first stencil change occurs at new step 15: node 5 moves from offset -3 to -4, in units 2^-54 around 0.5. Nodes 3 and 4 retain offsets -2 and +4. The node-4 gradient index falls from -13 to -14 and its pressure falls to 1462656319363363/2^105, still positive. Thus the neighbor invalidates the original frozen-stencil premise before the conditional center deadline 21. A stencil exit and a sign exit are different observations.

The endpoint has offsets (-2,0,2,-2,4,-4). The local mean area is -131/(3*2^114) and its net mean relative to B27 is -275/(3*2^114); the complete vector also remains nonzero. Node 4 accumulates strictly positive local area 7355250143423031/2^107. It has not begun to compensate its earlier positive area. Its final carry is 62990689884919623/2^110. Under the newly frozen stencil, the remaining center deadline is four steps, with at most three unchanged center steps. This is an analytic conditional bound, not a continuation after censoring.

The sign threshold constrains the coupled neighborhood budget

The existing exact row thresholds give nonpositive node-4 pressure only when m_4<=-22. Starting at m_4=-13 requires

Delta n_3 + Delta n_5 - 2*Delta n_4 <= -9.

After the observed neighbor change, eight further gradient-index units are still needed. This is a necessary displayed-state condition, not a feasible itinerary or an instruction to modify EPI. In this binade one upward node-4 float move has Delta n_4=2, reducing m_4 by four if the neighbors stay fixed. From the original index, even two such moves would give m_4=-21, still positive. The actual coupled transitions and their carried areas must determine whether the cut can be reached.

The shared observe_nodal_remainder_cycle_gradient now owns the relation Delta m=-2*L_rw*A/delta+2*L_rw*Delta r/delta used by the earlier benchmarks. It first checks the complete identity A=X_after-X_before. Taking only the Laplacian would miss an arbitrary uniform error in the area vector. The previous callers already checked their full prefix areas elsewhere; centralization makes that obligation explicit in the reusable owner and preserves their existing report payloads.

Evidence and remaining requirements

c6_winding_pressure_stencil.py revalidates B30 through the retained B29/B28/B27/B26 chain, uses only the shared nodal integrator, and retains whole-itinerary compatibility and all six signed area budgets. Its ceiling is computational censoring, not an impossibility theorem for reaching negative pressure. Neither a live graph nor original-preparation reachability is promoted by this audit.

Extending this boundary result requires a coupled neighborhood reachability or trapping argument using the remaining gradient cut, exact carry and signed vector budget. Simply adding another fixed number of boundaries would not supply that argument. Pressure reduction alone is insufficient; any candidate bounded class must account for accumulated drift in every coordinate and verify its transitions and forward inclusion. Full-runtime admission and physical correspondence remain separate open obligations.

30. A coupled profile, disagreement tube and finite numerical-band horizon

B2.d.32 through B2.d.36 replace another short continuation with five dependent mathematical checks. They retain the fixed canonical C6 phase source, unit conductance and capacity, zero Gamma, existing channel weights and h=1/16. Their finite evidence is exactly the eighteen transitions already retained by B31. No new trajectory or additional boundary is used. The model concerns the shared carried numerical map with pressure refreshed from displayed EPI; future live operator admission remains a separate task.

B32: the actual source determines a centered relative profile

Write L=L_rw for the unit-cycle Laplacian, w=w_epi, and A for the represented phase contribution returned by the existing CPU pressure producer. These are inherited coefficients and actual source values, not an externally imposed compensating pressure. Define arithmetic centering by P=I-11^T/6. The existing forced-support owner solves

w*L*z = P*A,    mean(z) = 0.

On unit C6, the reversible metric is H=2I, so this is also its existing metric-centered Poisson profile. The new c6_carried_profile.py adapts that owner instead of introducing another profile solver. It rebuilds the source from primitive phase and weight inputs before using public reference caches.

The source mean is mean(A)=-1/(6*2^109). Thus z is a relative spatial profile with a separate uniform drift; it is not a zero-pressure equilibrium. At the B31 endpoint, node 4 still has displayed index m_4=-14, while nonpositive pressure requires m_4<=-22. With delta=2^-54 and m_i=-2*(L*x)_i/delta, this requires an increase of 4*delta in [L*(P*x-z)]_4. The profile expresses the same necessary neighborhood condition in centered coordinates. It neither constructs a path to the cut nor changes the actual EPI state.

B33: exact carried evolution separates shape, rounding and mean

Let X=x+r, where x is displayed EPI and r is the retained numerical remainder. The actual pressure has the exact decomposition

p = A - w*L*x + eta,
eta = EPI-product rounding error + pressure-assembly rounding error.

Here eta is obtained from the shared pressure observation. It does not include an assumed error of future phase evolution. Replaying the shared nodal step checks X_next=X+h*p; substituting x=X-r gives

T = I-h*w*L,
X_next = T*X + h*A + h*(w*L*r + eta),
y = P*X-z,
y_next = T*y + h*P*(w*L*r + eta).

The independent mean identity is

mean(X_next-X) = h*mean(A) + h*mean(eta),
mean(w*L*r) = 0.

The carried readout feedback can therefore affect shape without supplying a mean correction. The adapter reuses the existing pressure-readout and forced-profile observers, validates raw carried coordinates, and requires the complete supplied step to match the numerical kernel. All eighteen retained steps have zero recurrence and mean residuals. Their signed mean budget, relative to the B30 endpoint, is

phase source:       -3/2^113,
pressure rounding:  -113/(3*2^114),
carry feedback:      0,
total change:       -131/(3*2^114).

This exact finite budget does not establish a bounded infinite signed mean prefix. In particular, a nonuniform admissible carry can coexist with uniform displayed EPI and zero pressure in a synchronized control; the reconstructed centered error then persists. Homogeneous diffusion contraction alone cannot remove this readout feedback.

B34: exact C6 contraction distinguishes norm and energy gains

c6_carried_tube.py uses the shared exact cycle Laplacian. A complete rational orthogonal basis consists of the uniform vector and the following mean-zero vectors:

lambda=1/2: (2,1,-1,-2,-1,1), (0,1,1,0,-1,-1),
lambda=3/2: (2,-1,-1,2,-1,-1), (0,1,-1,0,1,-1),
lambda=2:   (1,-1,1,-1,1,-1).

Their exact Laplacian identities prove, without a numerical eigensolver, that for s=h*w with 0<s<1,

q = max(abs(1-s/2), abs(1-3*s/2), abs(1-2*s)) < 1,
||T*y||_2 <= q*||y||_2          when mean(y)=0.

The sharp squared-energy gain is q^2, whereas the uniform vector has gain one. The retained configuration gives q=573161353023261791/2^59. This is a property of the exact represented coefficients and fixed support, not a measured decay fit. The statement concerns the homogeneous map; the actual carried recurrence still has the additive term from B33.

B35: uniform numerical bounds give a conditional disagreement tube

All bounds are derived from the declared numerical band and coefficients, without extrapolating observed error maxima. Let its width be D=upper-lower, let u=2^-53 and t=2^-1075, and write Amax=max_i abs(A_i). The local slab theorem gives the exact gradient g=-L*x with abs(g_i)<=D. Under nearest-even binary64 arithmetic, the product and assembly errors therefore satisfy

e1 = u*w*D + t,
e2 = u*(Amax+w*D+e1) + t,
abs(eta_i) <= epsilon = e1+e2.

The implementation checks finite-operation envelopes before applying these bounds. The half-subnormal term retains underflow cases. With R=ulp(upper)/2, every admissible carry obeys abs(r_i)<=R; on the retained band [3/8,5/8], R=2^-54. Consequently

B = 2*w*R + epsilon,
abs((w*L*r+eta)_i) <= B,
||P*(w*L*r+eta)||_2^2 <= 6*B^2.

For E=||P*X-z||_2^2, Young's inequality supplies

E_next <= q*E + 6*h^2*B^2/(1-q),
E_floor = 6*h^2*B^2/(1-q)^2,
E_bar = max(E_initial,E_floor).

The affine bound uses q, not the homogeneous squared gain q^2. It gives a forward-invariant disagreement envelope while the numerical band and fixed-source premises hold. A persistent nonzero floor is not an asymptotic-convergence claim. Separately, the exact zero carry mean gives the per-step bound

abs(mean(X_next-X)) <= b = h*(abs(mean(A))+epsilon).

For the retained prefix, E_initial is approximately 8.781440801771191e-32, E_final is approximately 6.230400442940049e-32, and E_floor is approximately 6.656013887802292e-31. The mean-increment bound is approximately 6.354411872153111e-19. The code retains exact fractions; these decimals are summaries. The observed energy decrease neither removes the uniform floor nor proves future mean cancellation.

B36: close a finite band premise and assess the remaining cut

The disagreement bound also supplies abs(y_i)^2<=5*E_bar/6. Let mu_0=mean(X_0) and define the minimum initial margin using the carried state's own band, including when it is narrower than the pressure slab:

a = min_i(mu_0+z_i-lower, upper-mu_0-z_i).

An initially admitted tube remains inside that band through every integer prefix n satisfying

a-n*b >= 0,    (a-n*b)^2 >= 5*E_bar/6.

This closes the premise by induction rather than assuming the conclusion. An admitted current state supplies the pressure and carry bounds; these bounds first place the exact next candidate inside the band. Monotonic nearest-even rounding keeps its displayed coordinate inside the binary64 endpoints, and the shared dyadic remainder representation remains valid. For b>0, an exact monotone binary search finds the largest integer satisfying the sufficient inequalities. An admitted tube with b=0 would instead have an unbounded conditional prefix; a rejected initial tube supplies neither conclusion.

The retained inputs give the maximal sufficient integer horizon N=196713720348826219. That horizon is derived, not executed. The next integer fails the sufficient inequality; this is not a prediction of actual band exit. The result certifies the fixed carried numerical map through a finite horizon, not live grammar, future operators, changing phases or support, original-preparation reachability, or infinite stability.

For the node-4 sign question, put m_i_star=-2*(L*z)_i/delta and let k be its nonpositive-pressure cut. Since a C6 Laplacian row has squared norm 3/2, reaching m_i<=k requires

delta*(m_i_star-k)/2 <= sqrt(3*E_bar/2) + 2*R.

The implementation uses rational comparisons: a positive remaining distance d=delta*(m_i_star-k)/2-2*R with d^2>3*E_bar/2 excludes the cut under the tube premises. The actual node-4 assessment is inconclusive. Failure to exclude the cut does not prove a feasible transition, compensation or trapping class; the current index remains -14 against the threshold -22.

c6_winding_coupled_budget.py revalidates the six retained B31-through-B26 reports, checks the complete eighteen-step pressure and carry history, and binds every observed defect, energy and signed mean contribution to these shared owners. It retains the B30 endpoint as the envelope origin and the B31 endpoint as the current state. Reproduction uses

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_coupled_budget.py

The next work is to tighten the signed coupled mean/shape budget and its correlations with the reachable integer-gradient states. The absolute mean bound grows linearly and the present tube leaves the sign cut undecided. Another arbitrary short continuation would not close either obligation. A stronger result must derive cancellation, a tighter admissible set or a controlled exit from the canonical pressure law and retained carry, without fitting a new coefficient or inferring an infinite claim from finite data.

31. B37: closing the spatial and rounding bounds on each other

The band-wide B35 estimate controls an entire interval of EPI values, while the carried trajectory is restricted by its centered profile. B2.d.37 uses that relation to close a sharper bound analytically. The existing source, support, capacity, timestep and pressure producer remain unchanged. Write E=||y||_2^2, y=P*(x+r)-z, G=max_i abs((L*z)_i), and retain R=ulp(upper)/2, u=2^-53, t=2^-1075.

A row of the unit C6 Laplacian has squared Euclidean norm 3/2. Consequently, the actual displayed gradient satisfies

abs((L*x)_i) <= G + sqrt(3*E/2) + 2*R.

Applying the same product and assembly rounding bounds as B35 gives

a = u*(2+u)*w*(G+2*R) + u*max_i abs(A_i) + (2+u)*t,
b = u*(2+u)*w,
abs(eta_i) <= epsilon(E) = a + b*sqrt(3*E/2).

The centered projector is nonexpansive and ||L*r||_2<=2*sqrt(6)*R. The exact recurrence from B33 and the contraction from B34 therefore imply

sqrt(E_next) <= Q*sqrt(E) + h*sqrt(6)*(2*w*R+a),
Q = q + 3*h*b.

The factor 3 is the exact identity sqrt(6)*sqrt(3/2)=3. For Q<1, the closed invariant energy envelope is wholly rational:

E_floor_closed = 6*h^2*(2*w*R+a)^2/(1-Q)^2,
E_bar_closed = max(E_initial,E_floor_closed).

This is an algebraic closure, without repeated numerical fitting of an error estimate. The condition Q<1 is checked separately from q<1: a homogeneous map arbitrarily close to its noncontracting boundary need not pass this sufficient rounding-feedback test. The new owner c6_carried_closure.py rebuilds the profile and initial carried state through existing owners, checks finite-operation envelopes, and retains large initial disagreement instead of replacing it by the smaller floor.

At the retained B31 endpoint, the closed energy envelope is approximately 2.958228394578814e-31, compared with the earlier 6.656013887802292e-31. Its combined per-coordinate rounding bound is approximately 6.731922543446726e-32. These are summaries of exact rational results. In particular, the sharper error bound is derived from the structural profile, carry and binary64 format rather than from the largest error observed in a finite trajectory.

For each node, put m_i_star=-(L*z)_i/(delta/2). The necessary gradient condition is

abs((delta/2)*(m_i-m_i_star)) <= sqrt(3*E_bar_closed/2)+2*R.

Exact rational square comparisons and monotone integer searches produce the following integer hulls at the B31 origin:

Node	Necessary integer gradient interval
0	[-26,29]
1	[-28,27]
2	[-33,22]
3	[-17,38]
4	[-49,6]
5	[-13,42]

The tests verify both admitted endpoints and rejected adjacent integers. The intervals are coordinate conditions; their Cartesian product is not a set of jointly reachable states. Node 4's cut m_4<=-22 remains inside its interval, so the closure alone neither excludes nor establishes it.

The signed mean interval remains separate:

h*(mean(A)-epsilon_bound) <= mean(X_next-X)
                         <= h*(mean(A)+epsilon_bound).

It still straddles zero. The carry term contributes exactly zero mean; sharper spatial control cannot be relabeled as mean cancellation. This envelope remains conditional on the fixed numerical source and admitted band. The finite band bootstrap in section 30 remains available, but neither result alone proves infinite containment or live operator admission.

32. B38: static compensation refutes a class-wide linear drift argument

B2.d.38 asks whether a fixed linear functional could separate all canonical pressure vectors in the closed envelope from zero. The answer is negative for the admitted class, including one slice with exactly the same reconstructed mean as the B31 origin.

The new observer c6_carried_balance.py admits detached numerical states only after revalidating their carried encoding, common declared band, centered energy and freshly recomputed canonical pressure. The seven displayed EPI vectors in the witness are .5+delta*n, with delta=2^-54 and the following exact offsets:

(-6,-1,2, 2,8,-7),
(-5, 2,4,-2,6,-7),
(-4,-2,0, 2,6,-4),
(-4,-1,2,-2,8,-5),
(-3, 0,2,-2,8,-7),
(-2,-1,4,-2,6,-7),
(-2, 2,0,-2,6,-6).

Every row has offset sum -2. Assigning each coordinate the same admissible carry 96076792050570541/2^113 gives all seven states the same exact reconstructed mean as the retained B31 origin. This constructs static comparison states; it does not reset or change the carry on the actual trajectory. All seven centered energies lie below 21*delta^2 and satisfy the rebuilt B37 envelope.

Let their freshly computed pressure vectors be p_1,...,p_7. The shared exact linear-algebra owner inverts the rational matrix whose columns are (p_j,1) and derives unique weights satisfying

lambda_j > 0 for every j,
sum_j lambda_j = 1,
sum_j lambda_j*p_j = (0,0,0,0,0,0).

The coefficients are an algebraic certificate, not a new TNFR parameter, probability distribution or dynamical mixing rule. If one fixed linear functional c had a strict common sign on every admitted pressure vector, the same sign would hold for their positive weighted sum, contradicting the exact zero vector above. Hence a strict class-wide linear drift separator cannot prove escape on this envelope or its demonstrated constant-mean slice.

This result supplies no temporal ordering of the points, no transition compatibility of their carries, and no reachability from the B31 state. A smaller reachable subset can still have a different drift constraint. In particular, a static convex cancellation is not an accumulated zero vector area or a periodic orbit. The obstruction redirects the finite sign question toward the actual nodal recurrence rather than treating the source's negative mean as a universal pressure-mean sign law.

33. B39: a finite first-passage theorem for the pressure cut

The coupled spatial envelope does more than constrain static gradients. Together with the nodal equation, it excludes indefinitely positive pressure at node 4 while its mean drift remains much smaller than the least possible positive nodal pressure. B2.d.39 proves this before executing a new continuation.

The owner c6_carried_passage.py rebuilds the earlier B35 tube and its independently proven B36 band horizon. It uses the sharper profile identity

abs((L*x)_i) <= abs((L*z)_i)+sqrt(3*E_bar/2)+2*R

to derive row-specific product and assembly rounding envelopes. Their average gives an upper bound M on mean(p); zero-mean carry feedback does not contribute to M. The use of the already certified tube and band horizon makes the finite premise independent of the desired sign change. The new B37 closure is useful additional information but is not required to promote this particular first-passage certificate.

The shared integer sign-sector owner gives p_i>=p_min>0 whenever the selected pressure is positive. Its integer sector is relaxed with respect to joint coordinate realizability, which makes the pressure lower bound valid without fabricating a graph state. At node 4, the smallest positive sector index is -21, with

p_min = 38338268585241/2^106,
M approximately 6.242107045064891e-32,
v = h*(p_min-M) > 0.

The carried nodal equation gives exactly

y_i(next)-y_i = h*(p_i-mean(p)).

If every pressure readout in the first N transitions were positive, then y_i(N)>=y_i(0)+N*v. But the tube imposes abs(y_i(N))^2<=5*E_bar/6. The least integer N for which

y_i(0)+N*v > 0,
(y_i(0)+N*v)^2 > 5*E_bar/6

therefore yields a contradiction, provided the independent band horizon covers all N transitions. Exact monotone search gives N=30255, well inside the previously proved finite band horizon. Thus some pressure readout with index at most 30254 must be nonpositive; index zero is the B31 endpoint. This resolves the finite conditional reachability question for the numerical map. It does not assert the first actual hit, infinite trapping, original-preparation reachability or future live graph admission.

34. B40: the realized first passage and a finite mean-budget reversal

The finite theorem supplies a justified stopping rule for a new continuation. B2.d.40 resumes the exact B31 endpoint, including all six retained carries. It keeps the closed phase source, unit C6 support and capacity, existing channel weights and h=1/16. Every step uses the shared carried nodal kernel; pressure is refreshed whenever the displayed state changes. There is no pressure projection, carry reset or fitted coefficient.

c6_winding_coupled_passage.py revalidates the existing source chain and binds the B37 closure, B38 static witness, B39 passage theorem and B40 realized continuation to their shared owners. The continuation stops at the first nonpositive node-4 pressure, with the derived B39 bound as its maximum stopping budget. This replaces the earlier arbitrary boundary ceiling by a decisive test.

The first hit occurs at pressure-readout index 118, after 118 shared transitions and 59 displayed-state boundaries. Readout zero is the inherited B31 endpoint; the source readouts at indices 0,...,117 all have positive node-4 pressure. The fresh endpoint readout gives

displayed offsets from .5 in delta=2^-54:
    (-3,0,2,-1,8,-5),
integer gradients:
    (1,-1,-5,12,-22,15),
p_4 = -187043320717485/2^105 < 0.

Thus the pending m_4<=-22 cut is reached by the actual finite carried map, rather than merely admitted by a coordinate bound. The hit at index 118 lies within the proven maximum index 30254. This resolves the previously censored finite sign question on its declared numerical domain. The certificate still concerns the supplied endpoint and source; it does not reconstruct the endpoint from the original full preparation or certify future live operator admission.

The continuation also shows that its signed mean pressure need not retain the earlier negative sign. Relative to the explicit B27 endpoint reference, the accumulated mean area immediately before transition 59 is -1/(3*2^114). Immediately afterward it is 1/2^113. This is the first crossing in the new continuation: the prior negative mean budget has been overcompensated. It is not an exact zero-vector return or an assertion that some sampled mean prefix equals zero.

At the pressure-cut endpoint, the exact local mean budget relative to the B31 origin is

phase-source contribution:       -59/(3*2^113),
pressure-rounding contribution:  305/(3*2^113),
carry-feedback contribution:       0,
total local mean area:             41/2^112.

Including the inherited B27-referenced budget gives the final accumulated mean area 217/(3*2^114). The full coordinate budget is retained and all six B27-referenced net areas remain nonzero. The finite mean reversal therefore establishes actual temporal compensation of the scalar mean budget while leaving complete vector recurrence unresolved. In particular, the negative fixed source mean is overcome here by signed pressure-rounding contributions; a zero-mean carry feedback is not credited with that change.

Reproduction uses the consolidated benchmark:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_coupled_passage.py

The next unresolved question is whether the reachable carried states admit a bounded signed vector budget and forward inclusion for arbitrarily long evolution. B38 rules out one class-wide linear separation strategy, while B39-B40 resolve one finite pressure cut and a finite scalar compensation. These results do not prove infinite trapping, a periodic orbit, convergence, complete-runtime stability or physical correspondence. Any broader claim must retain the actual transition order, pressure refresh, all coordinates and carry compatibility rather than replacing them by static convex weights or another unmotivated extension of the execution horizon.

35. B41: complete carry cells cannot provide invariant trapping

B2.d.41 tests a candidate class rather than extending the saved B40 trajectory. A displayed six-tuple fixes the canonical pressure when phase, support, capacity and channel weights are held as declared. Its complete carry fiber consists of every legal exact encoding X=x+r that rounds to that tuple and remains in the positive band. A finite family of displayed tuples can have arbitrary spatial correlations while still assigning the complete independent carry fiber to each tuple.

The new owner c6_carried_cell_escape.py proves that no nonempty finite family of this kind can be forward invariant when its displayed integer gradients satisfy B37's bounds. This strengthens the earlier Cartesian obstruction: the visible tuples need not form a Cartesian product. It does not exclude a family that restricts carry jointly with visible shape or reconstructed energy.

The proof uses the existing nodal update, pressure lattice and inverse rounding-cell itinerary. Let g=2^-3222 be the shared exact encoding grid and delta=2^-54 the local EPI quantum. Band endpoints are represented; every band-clipped nearest-even cell has width at least delta/2. Removing open tie endpoints reduces the distance between its first and last legal grid points by at most 2*g. The explicit remainder magnitude bound is inactive on these cells. Monotonicity of the source-plus-rounded-product pressure law supplies exact pressure bounds from the endpoints of B37's integer-gradient intervals. The largest possible negative increment obeys

max_i max(-h*p_i,0)/delta = 5757725321284133/2^55
                         < 0.16 < 1/2,
max_i max(-h*p_i,0) < delta/2-2*g.

These are bounds over the declared gradient class, not maxima fitted to a trajectory. The pressure lattice also proves that every displayed tuple in the slab has at least one positive pressure component: an all-nonpositive vector would require integer-gradient sum at most -4, whereas the cycle identity makes that sum exactly zero.

For an arbitrary nonempty finite candidate family, choose a displayed tuple maximizing sum_i x_i. In each coordinate, select the greatest admissible grid point of its band-clipped rounding cell. This simultaneous choice belongs to the complete carry fiber. Every negative increment is smaller than the cell's available grid width, so its displayed coordinate stays fixed. Every positive increment lies on the same exact grid and therefore moves beyond the greatest admissible grid point. Its displayed coordinate must increase, or its exact candidate must leave the declared band. Consequently the witness either has strictly larger displayed sum than every tuple in the family or fails band admission. Both outcomes contradict forward invariance of that full-cell union.

The executable witness uses B38's seven displayed rows together with the B40 displayed endpoint. The maximum-sum row is the eighth entry, index 7. The hypothetical greatest-carry state produces

source offsets:   (-3,0,2,-1,8,-5),
endpoint offsets: (-3,0,4, 0,8,-4),
increased nodes:  (2,3,5),
increase in displayed sum: 4*delta = 2^-52.

The source pressure is freshly recomputed and the admissible endpoint is verified by the shared carried nodal kernel. The selected carry is an existence witness from a complete fiber; it is not B40's retained carry, is not written into a live graph, and is not claimed reachable from B40. Its reconstructed energy need not remain in the smaller B37 energy set. Thus B41 proves that some legal carry escapes each candidate full-cell union. It does not prove that every carry escapes, that the saved trajectory leaves the band, or that all correlated invariant regions fail.

36. B42: every carry leaves the seven static compensation cells

B2.d.42 addresses the actual temporal compatibility of B38's seven displayed compensation rows. The owner c6_carried_cell_graph.py constructs their complete finite transition graph using the fixed canonical pressure source, unit capacity, h=1/16 and the declared band [.375,.625]. Each of the 7*7=49 directed edge tests asks whether any legal incoming carry can execute that one transition. The shared inverse itinerary intersects translated nearest-even cells exactly, including tie parity, band clipping and the 2^-3222 encoding grid.

All seven rows have a self-loop. The only nonself edge is from row 6 to row 5, using the one-based order in section 32; in zero-based indices it is 5 -> 4. Removing self-loops therefore leaves a directed acyclic graph. An edge is existential: consecutive edge witnesses need not carry the same incoming residual. The graph is an overapproximation of actual carried trajectories, which is sufficient for an upper escape deadline. It must not be interpreted as a realizable schedule of arbitrary edges.

For a fixed displayed row, pressure remains constant until that row changes. To maximize residence, choose each positive-pressure coordinate's first legal grid point and each negative-pressure coordinate's last. These independent choices simultaneously maximize every directional residence limit. If a coordinate's legal grid interval is [a_i,b_i] and its exact increment is g*k_i, its maximal unchanged prefix is floor((b_i-a_i)/abs(k_i)) when k_i is nonzero; zero increments impose no finite limit. The shared cell-horizon owner verifies these bounds and their nearest-even endpoint convention. Taking the minimum across coordinates gives the sharp maximal unchanged residence for that row.

Reverse topological induction then bounds residence in the entire family:

first_exit_i = maximal_unchanged_steps_i + 1,
D_i = first_exit_i + max(D_j : i -> j, j != i),

with the maximum over an empty successor set equal to zero. The first exit step includes the transition into a successor, if one occurs; the successor's bound starts from its resulting carried state. Enlarging the set of possible paths by ignoring carry compatibility between edges can only make this upper bound more conservative.

B38 row, one-based	Maximum unchanged steps	First exit from its cell	Upper deadline to leave the family or fail band admission
1	76	77	77
2	50	51	51
3	38	39	39
4	50	51	51
5	39	40	40
6	29	30	70
7	49	50	50

Every legal carried trajectory starting in these seven complete cells must therefore leave this family or fail band admission within 77 steps, conditional on the fixed numerical map. This excludes a periodic carried orbit confined to those seven displayed rows. A larger band may retain the trajectory after it leaves the family; no whole-band exit is proved. Nor does this result prove that B40's saved state ever enters the seven cells. Unlike B41's existence of an outward carry, B42's deadline covers every legal incoming carry in this particular seven-cell family.

The positive static convex pressure balance of section 32 remains correct. It cannot supply a closed temporal class using its seven displayed points: exact carry compatibility and residence expose the missing dynamical condition. A cyclic transition graph or a stationary-pressure row would make this sufficient deadline test abstain, not certify recurrence.

The consolidated producer c6_winding_temporal_compatibility.py replays the retained source chain, rebuilds the B41 bounds and witness, and checks the B42 graph without advancing the saved B40 endpoint. Reproduction:

.venv313\Scripts\python.exe -X utf8 benchmarks/c6_winding_temporal_compatibility.py

B41 already rules out every finite full-cell union in its bounded-gradient class, so searching a larger family of complete carry cells is not the next invariance strategy. The remaining candidate must restrict translated carry subcells jointly with visible shape and signed accumulated vector area. It needs forward inclusion over its entire declared domain, or a signed recurrent-budget obstruction. Neither a static convex cancellation nor another longer sampled trajectory supplies that proof.

37. B43: a finite live bridge from the original winding preparation

The phase projection in section 23 and the carried pressure studies address different provenance obligations. The former reaches an exact numerical phase tail using detached proposal inputs; the latter evolves explicitly supplied EPI and carry under that fixed source. Neither fact alone supplies the EPI/carry reached when the original graph preparation executes the complete operator word. B2.d.43 addresses that finite bridge through the existing carried event executor.

The declared preparation is the original null winding on ordered unit C6: EPI_i=.5, nu_i=1, stored phases i*pi/3, seed 17, and canonical default channel weights, with Gamma disabled as in the existing study. The declared numerical band is [.375,.625], inside the configured graph bounds. The finite word and its flow partition are

(UM IL)^89 SHA,
after each IL: one interval of duration 1/4,
each positive interval: four refreshed carried steps of duration 1/16,
no flow between UM and IL or after terminal SHA.

The budget therefore specifies 179 operator events and 356 numerical steps, ending at represented time 22.25. It targets entry into the phase tail already identified by the detached map, rather than adding an arbitrary trajectory length. Complete-word grammar admission is checked before execution. The whole invocation uses one graph-owned transaction; failure rolls back the graph and its carried encoding.

Two implementation boundaries matter. A fresh graph receives a zero carry encoding from its actual initial EPI. An existing nonzero encoding can be continued only through its intact graph-owned binding. Consequently B43 does not attach a detached B40 carry to a new graph or bypass the executor seal. Moreover, canonical Silence attenuates capacity. The final SHA is placed after all unit-capacity measurements; unit capacity is not claimed for a subsequent invocation on the closed graph.

Actual phase evidence accompanies each finite nodal record

The shared nodal_remainder_runtime.py already retains actual pressure, EPI, capacity, carry, support and clock evidence. Its nodal-flow snapshot intentionally has no phase channel. Since pressure is not an injective phase readout, reconstructing a phase path from its pressure values would leave a provenance gap.

Flow and event records now additionally capture the ordered stored phase_before and phase_after tuples. These raw binary64 values, including signed zero, enter the existing complete execution seal. No phase formula, operator behavior, pressure law or integration step changes. Tests verify actual two-cycle UM/IL proposals against the existing phase owner, unchanged phases during flow and SHA, and loss of certification if any retained phase field is tampered with.

The finite adapter c6_winding_phase_tail_runtime.py compares every actual UM/IL capture with the shared phase projection and checks every refreshed pressure against its captured phase and displayed EPI. It separately verifies unit capacity for all measured flows and exact preservation of EPI/carry across each named event. It retains both the state immediately after the last IL, before its four flow segments, and the state immediately before SHA. The former is the appropriate input for a new fixed-source carried profile and self-consistent closure.

Completed finite capture and remaining scope

The complete 89-cycle invocation succeeds with 179 admitted operator events and 356 refreshed carried steps. Its executor seal is valid at capture. Every observed UM/IL transition matches the shared phase owner, and every measured flow has unit capacity and the recorded phase-bound canonical pressure. The final phase tuple repeats exactly under the detached composite UM/IL projection. This establishes entry into that phase tail on an actual finite graph-owned word.

The two relevant captured states are distinct. Their displayed offsets from .5, in units delta=2^-54, are

Live boundary	Time	Displayed offsets
Immediately after the last IL, before its flow	22	(-3,0,2,-1,6,-5)
After its four carried segments, before SHA	22.25	(-3,0,2,0,8,-5)

Both states retain their actual nonzero carry vectors in the result. The first supplies the fixed-phase profile and B37 closure directly, without rewriting its numerical band or substituting a detached carry. Its closed energy envelope is approximately 2.958228394578814e-31. The complete invocation's mean nodal area from the original preparation is 3293/(3*2^114). All six exact nodal balance residuals vanish. An independent replay of all 356 recorded steps, their signed areas and the extracted closure agrees with the retained result.

Terminal SHA preserves phase, EPI and carry but changes every capacity from 1 to the represented value 0.9204225284540524. The final live graph therefore does not retain the unit-capacity premise of the preceding measurements. A later invocation must use its actual capacity; the historic pre-SHA snapshot is not a mutable live continuation checkpoint.

The completed artifact retains its scientific source digest. A subsequent input-hardening correction makes scalar captures resolve exact-string aliases through the runtime's nonvirtual mapping reader. It rejects string-subclass core aliases rather than silently substituting a default, and prevents mapping read hooks from making uncertified auxiliary writes during phase observation. The campaign uses ordinary canonical dictionaries; its preserved source archive and independent verification retain the precise provenance of the completed run. Serialized records do not recreate the original executor seal or acquire a later source identity.

B43 closes original-preparation reachability of these finite causal tail states under the declared carried, refreshed solver. This solver is distinct from older held-pressure visible-EPI experiments. The displayed states and their carries also differ from B40's conditional trajectory, so they cannot retroactively authenticate its endpoint or replace that historical branch. Subsequent use of the extracted fixed-phase closure remains conditional on its declared source and capacity. Indefinite trapping, future complete-word admission and empirical correspondence remain open.

38. B44: a bounded mean interval does not close the centered-energy tube

B2.d.44 tests a specific candidate for joint trapping from B43's actual pre-SHA state. It adds a bounded reconstructed-mean interval to the existing centered-energy envelope. This trims the complete carry fibers excluded by B41 and retains the exact numerical encoding. The new result nevertheless refutes forward invariance for every such interval inside one derived local window containing the actual initial mean. It supplies counterexamples to that proposed region, not an escape claim about the actual starting carry.

The owner c6_carried_mean_cylinder.py reuses the canonical pressure, forced profile, self-consistent closure, inverse nearest-even cells and carried nodal kernel. No pressure source, coefficient, capacity or numerical step is adjusted. Its model remains the conditional fixed-phase, unit-capacity map with h=1/16; the terminal SHA in B43 does not supply this capacity for a future live invocation.

Candidate region and actual starting-state membership

Let X=x+r, P*X=X-mean(X)*1, and let z solve the existing centered Poisson identity. Rebuild the B37 bound E_bar at B43's actual pre-SHA endpoint and write

E(X) = ||P*X-z||^2,
mu0 = 1/2 + 3293/(3*2^114),
delta = 2^-54,
g = 2^-3222.

The candidate K[a,b] consists of every legal carried encoding in the declared band [.375,.625] satisfying both E(X)<=E_bar and a<=mean(X)<=b. Its mean endpoints are bounds on a proposed certificate domain, not new dynamical parameters. The actual B43 endpoint satisfies the energy bound and belongs whenever a<=mu0<=b. Its energy is about 2.82386869748*delta^2, while E_bar is about 96.00000000000064*delta^2.

Two existing displayed rows from B38 provide independent static templates: the first row, with negative mean pressure, and the fourth, with positive mean pressure. Their offsets from .5 are

negative template: (-6,-1,2, 2,8,-7),
positive template: (-4,-1,2,-2,8,-5).

Both have displayed mean m_x=1/2-delta/3. Giving each coordinate the same carry

c0 = mu0-m_x = 384307168202283423/2^114

places both templates at mu0, with a legal dyadic encoding. Their exact centered energies satisfy the rederived envelope; the ratios to delta^2 are approximately 12.1297581819 and 2.88720884922, respectively. These are hypothetical comparison states. This construction neither changes B43's retained carry nor claims that its trajectory reaches a template.

Exact common translation window

The shared inverse-cell owner derives every legal grid endpoint, including nearest-even parity and the physical band. For these two displayed tuples, the largest common interval of equal coordinate carries is exactly

-delta/2 + g <= c <= delta/2 - g,     c on the g-grid.

The subtraction of one grid quantum at each end follows from open odd-tie boundaries in the common intersection; it is not a fitted tolerance. Thus the common mean window for these uniform translations is

W_lower = 1/2 - 5*delta/6 + g,
W_upper = 1/2 + delta/6 - g.

The actual mu0 lies strictly inside it. Relative to mu0, the window is

W_lower-mu0 = -5*delta/6 - 3293/(3*2^114) + g,
W_upper-mu0 =    delta/6 - 3293/(3*2^114) - g.

This is the maximal shared uniform-translation window for the chosen templates, not a maximal domain for all possible pressure states. Translation by any admissible k*g*1 keeps each displayed row, and hence its refreshed canonical pressure, unchanged. It also leaves centered energy unchanged exactly, because P*(X+k*g*1)=P*X.

Outward witnesses for arbitrary rational mean endpoints

For the selected negative and positive templates, fresh canonical pressure gives the exact mean increments

mean(p_negative) = -1/2^110,    h*mean(p_negative) = -1/2^114,
mean(p_positive) =  3/2^110,    h*mean(p_positive) =  3/2^114.

Both signed magnitudes exceed g. Consider any rational interval satisfying W_lower<=a<=mu0<=b<=W_upper. Its endpoints need not lie on an encoding grid. Choose the uniform template translations

k_upper = floor((b-mu0)/g),
k_lower = ceil((a-mu0)/g).

Since the interval contains mu0, the translated positive template has mean between mu0 and b, and the translated negative template has mean between a and mu0. The common translation window makes both carried states legal. Their centered energies still satisfy E_bar, so both belong to K[a,b]. Their distances to the respective mean boundaries obey

0 <= b-(mu0+k_upper*g) < g,
0 <= (mu0+k_lower*g)-a < g.

Applying the actual nodal increments therefore sends the positive witness strictly above b and the negative witness strictly below a. Each valid endpoint is verified through the shared carried kernel. If an exact candidate leaves the physical band, it already violates the candidate domain and is recorded as band-admission failure without fabricating a valid endpoint. For the two declared templates and their common window, the coordinate bounds keep these candidate endpoints in the band; their mean inequalities provide the escape.

This proof uses a uniform-translation sublattice of admissible means. It does not assume that every legal reconstructed mean is attainable by uniform dyadic shifts: the full mean grid is finer. A witness within less than g of each boundary suffices because its signed increment is larger than that gap. No grid enumeration, statistical mean argument or arbitrary long trajectory is needed.

Scope of the obstruction and the next dependency

derive_c6_carried_mean_cylinder_obstruction rebuilds the closure and both templates, checks their common initial mean and energies, recomputes their pressure and derives the common legal translation window. The paired observe_c6_carried_mean_cylinder_escape constructs and verifies the two outward witnesses for a supplied admissible rational interval. Derived caches, static convex weights and endpoint equality are not substituted for these checks.

B44 excludes the natural centered-energy tube closed by any independent mean interval in this local window containing B43's actual mean. Its counterexamples remain hypothetical states inside that proposed region; the theorem does not establish their reachability from B43 or mean escape of the B43 trajectory. The candidate does not impose additional coordinatewise congruences inherited from the initial state and its allowable nodal increments. Legal encoding alone does not establish membership in that finer reachable arithmetic class. Wider mean windows, shape-dependent mean limits, and joint restrictions on individual carries remain unresolved. It proves neither whole-band exit nor an infinite stable regime.

Within the tested local window, a successful invariant certificate must therefore distinguish the pressure signs through additional correlations between shape, carry and mean, or demonstrate a smaller reachable domain that excludes the outward templates. Translation-invariant centered energy plus an independent mean bound is insufficient. Any subsequent piecewise domain still needs exact initial membership and universal forward inclusion of its translated pieces, with the full signed vector budget retained.