C6 relational regions and exit exclusions

Correlated regions, histories and regional exclusion proofs.

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.

39. B45: the coordinate arithmetic class does not rescue local mean confinement

B2.d.45 strengthens section 38 by imposing a necessary arithmetic restriction inherited from B43's actual starting state. The uniform-carry templates used by B44 need not satisfy this finer restriction. The new construction places both outward witnesses in the correct coordinate arithmetic class while preserving their displayed canonical pressures. It still addresses a proposed region, not reachability of its witnesses along the saved trajectory.

The owner c6_carried_affine_mean.py implements derive_c6_carried_affine_mean_obstruction and observe_c6_carried_affine_mean_escape. The domain is the same conditional fixed-phase, unit-capacity C6 map, with the same represented channel weights, h=1/16, declared band and B37 centered-energy bound. The reference state X0 is B43's actual pre-SHA encoding, not a zero-carry replacement.

Coordinate congruences derived from the allowed nodal areas

The B37 envelope supplies necessary integer-gradient intervals. For each node i, evaluate the existing rounded pressure law at every integer in its declared interval and take the exact rational greatest common divisor of the resulting represented nodal increments h*p_i(m). Denote this coordinate spacing by gamma_i. These are finite arithmetic operations on the fixed numerical pressure law, not an additional physical parameter or a claim of physical quantization. The row intervals can include gradients that are not jointly realizable; including them only weakens the necessary congruence restriction.

Every admitted increment is an integer multiple of its gamma_i. Thus, while the fixed source, timestep and gradient bounds apply, every successive reconstructed coordinate satisfies

X_i in X0_i + gamma_i*Z.

For the declared B43 source the exact spacings are

m = 2^-113,
(gamma_0,...,gamma_5) = (m,m,2*m,4*m,8*m,4*m),
q = m/6 = 1/(3*2^114).

All six actual X0_i have zero residue modulo their respective spacings. The mean of any state in this product of affine lattices belongs to mu0+q*Z. This is a necessary condition, not a converse reachability criterion. It is also coarser than the general encoding grid g=2^-3222: B44's common carry c0 is a half-integer multiple of m and cannot by itself supply the required coordinate congruences.

A mean-preserving lift onto all six coordinate classes

Use the two displayed templates of section 38, with common displayed mean m_x=1/2-delta/3. For a requested arithmetic mean mu=mu0+k*q, set c=mu-m_x. In a general affine class, let a_i be the residue of X0_i-x_i modulo gamma_i, chosen in [0,gamma_i) for the selected displayed row. The present two templates have a_i=0, but the construction retains the affine residues rather than assuming they vanish.

Select coordinate 0, whose spacing is the minimum m, to balance the sum. Define

r_i = a_i + gamma_i*floor((c-a_i)/gamma_i),    i=1,...,5,
r_0 = 6*c - sum_{i=1}^5 r_i.

The last identity makes the mean exactly mu. Moreover, 6*c-sum_i a_i is an integer multiple of m: the requested mean differs from mu0 by k*q, and every coordinate spacing is an integer multiple of m. Each r_i-a_i for i>=1 is already an integer multiple of its own spacing. Consequently r_0-a_0 is a multiple of m as well. All six reconstructed coordinates therefore belong to their initial arithmetic classes. This is a direct exact construction, without searching periods or treating a convex pressure combination as an executable schedule.

The auxiliary common carry c need not itself be dyadic: a requested mean can contain a factor of three in its denominator. It is not asserted to be an intermediate encoded state. Instead, 6*c=sum_i X0_i+k*m-sum_i x_i is dyadic on the shared encoding grid. The affine residues, spacings and five floored carries lie on that grid, so their balanced sixth carry does as well. The final six-coordinate state, rather than the auxiliary uniform vector, undergoes the shared encoding and cell checks.

Write r=c*1+e. The floor operations give

-gamma_i < e_i <= 0,    i=1,...,5,
0 <= e_0 < G,          G=sum_{i=1}^5 gamma_i=19*m,
sum_i e_i = 0.

The largest nonpivot spacing is 8*m. Therefore every constructed carry lies in the shared legal interval [c_min,c_max] of section 38 whenever

c_min + 8*m <= c <= c_max - 19*m,
c_min = -delta/2+g,    c_max = delta/2-g.

The resulting sufficient mean window is

W'_lower = W_lower + 8*m,
W'_upper = W_upper - 19*m.

It still contains the actual mu0 strictly. Its shrinkage comes directly from the coordinate spacings and the balancing construction. It is a sufficient window for this lift, not a maximal domain of possible arithmetic-class states or an optimized physical threshold.

Uniform energy control without enumerating lifted states

Since the lift error has zero mean, the centered state differs from the uniform-carry template by exactly e. Its squared norm obeys

||e||^2 <= G^2 + sum_{i=1}^5 gamma_i^2
        = (19^2+1^2+2^2+4^2+8^2+4^2)*m^2
        = 462*m^2.

The elementary inequality ||v+e||^2<=2*||v||^2+2*||e||^2, applied to the template's centered profile error v, yields

E_lift <= 2*E_template + 924*m^2.

Both right-hand sides are strictly below the rederived B43 envelope by exact rational comparison. In units of delta^2, they are approximately 24.2595163638 for the negative template and 5.77441769843 for the positive template, against E_bar approximately 96.00000000000064. Thus every lift in the sufficient mean window satisfies the centered energy constraint. This analytic estimate covers the complete integer family; no finite residue enumeration is needed to establish it.

The independent mean boundary still has an outward witness

Consider a rational interval satisfying W'_lower<=a<=mu0<=b<=W'_upper. Choose the requested means

mu_upper = mu0 + q*floor((b-mu0)/q),
mu_lower = mu0 + q*ceil((a-mu0)/q).

They remain inside [a,b] and within less than q of its respective boundaries. Apply the exact lift to the positive template at mu_upper and the negative template at mu_lower. Each input now satisfies all four candidate restrictions: legal encoding in the band, the energy envelope, the mean interval and every coordinate congruence inherited from X0. Displayed EPI has not changed, so the freshly recomputed mean increments remain

h*mean(p_negative) = -3*q,
h*mean(p_positive) =  9*q.

They strictly exceed the possible boundary gaps in their outward directions. A valid shared-kernel endpoint therefore leaves the proposed mean interval; a formal candidate outside the physical band already violates admission and is recorded separately. The declared witnesses remain in the band. Their coordinate congruences are also retained after the step, because each canonical nodal increment is a multiple of its derived coordinate spacing.

B45 consequently excludes the local centered-energy/independent-mean candidate even after intersection with this necessary arithmetic class. It strengthens B44's domain statement without turning congruence membership into temporal reachability. Both witness states are hypothetical; neither is claimed to lie on B43's actual future trajectory. Wider mean windows, tighter jointly reachable subsets, and restrictions that couple mean to shape and individual carries remain open. These results supply neither an infinite stability theorem nor whole-band escape.

40. B46: opposite-node relay strips and a signed local budget

B2.d.46 addresses joint carry/shape compatibility at the actual B43 pre-SHA endpoint. The source remains the fixed phase, unit C6 support, unit capacity, canonical pressure weights, band [.375,.625] and h=1/16. All six incoming remainders are retained. This is a conditional numerical branch: B43's historical terminal SHA changed capacity, and replaying its serialized evidence does not recreate a live execution seal.

The owner is c6_carried_relay.py, with a reproducible producer in c6_winding_relay_budget.py. The producer reuses the complete B43 arithmetic replay, shared C6 pressure, closure and carried nodal integrator. No pressure projection, extra oscillator, coefficient fit or carry reset is introduced.

Two coupled cell coordinates with independent switches

Write delta=2^-54, m=2^-113, X=x+r. From the actual initial visible row (.5 + delta*(-3,0,2,0,8,-5)), select nodes 0 and 3 and their immediate predecessor floats. This gives four displayed rows, in delta offsets:

C00 = (-3,0,2, 0,8,-5),    C10 = (-4,0,2, 0,8,-5),
C01 = (-3,0,2,-1,8,-5),    C11 = (-4,0,2,-1,8,-5).

The two nodes are opposite on C6. Their pressure-response supports are disjoint, but the certificate still recomputes all four canonical pressure rows and verifies the exact binary64 increment identity

A00 = h*p(C00),
d0  = h*p(C10)-A00,    d3 = h*p(C01)-A00,
h*p(C11) = A00+d0+d3.

No idealized -1/2 neighbor coefficient is substituted for these represented differences. Each selected node has strictly inward increments: -a_s in its upper cell and +b_s in its lower cell. They are

Node	a_s/m	b_s/m	(a_s+b_s)/m
0	3039824576180305	3558973984143090	6598798560323395
3	1803052714421816	4795745845901584	6598798560323400

Let t_s be the exact midpoint of its adjacent floats, and u_s=X_s-t_s. Nearest-even rounding assigns the node-0 facet to the lower cell, giving the invariant strip -a0<u0<=b0. The node-3 facet belongs to the upper cell, giving -a3<=u3<b3. The complete strips fit strictly inside their two-cell unions and stay in the declared band. The actual incoming offsets are 2638626784309416*m and 2678732075009744*m, so both membership checks pass without modifying the state.

For the first orientation the exact rotation is

u0(n) = b0 - ((b0-u0(0)+n*a0) mod (a0+b0)),

and for the second it is

u3(n) = -a3 + ((u3(0)+a3-n*a3) mod (a3+b3)).

These identities prove inclusion for every legal point of each strip, conditional on the other four displayed values staying fixed. They do not establish full six-node invariance. They require no enumeration of the very long rational rotation periods.

Removing the oscillatory contribution exposes a signed drift

Define k_s=d_s/(a_s+b_s) and v=A00+k0*a0+k3*a3. Exact pressure independence gives k_s[s]=1, zero cross-relay entries and v0=v3=0. For every step before another displayed coordinate exits, including the step that first exits, the complete vector identity is

X(n) = X(0) + n*v + k0*(u0(n)-u0(0)) + k3*(u3(n)-u3(0)).

The correction columns are exactly

k0 = (1, -3299399280161697/6598798560323395, 0, 0, 0,
         -3299399280161696/6598798560323395),
k3 = (0, 0, -412424910020212/824849820040425, 1,
         -137474970006737/274949940013475, 0).

Thus X_j-k0[j]*X0-k3[j]*X3 has constant increment v_j within this four-cell region. In particular,

v4 = -163545780653844840044770794592 * m / 274949940013475 < 0.

This is a signed recurrent budget derived from the actual nodal increments. It is valid across switches of either relay; a frozen single-cell pressure is not assumed. It is also local: after a non-relay displayed value changes, these four pressure rows no longer describe the next transition.

For each non-relay coordinate bound the oscillatory correction over the two strips by exact rational endpoints c_j^- and c_j^+. If v_j<0, choose the first positive integer N satisfying X_j(0)+N*v_j+c_j^+<RN_lower(x_j). For positive drift use X_j(0)+N*v_j+c_j^->RN_upper(x_j). Strict inequalities keep the bound valid for either rounding-tie orientation. The four individual deadlines for nodes (1,2,4,5) are (205,299,10,2899). Therefore a non-relay cell exit or earlier band failure must occur by step 10 from this input.

The observer first derives that deadline, then evaluates the exact formula and shared nodal transitions only until the first exit. It finds node 4 exiting at step 7, with visible endpoint offsets (-3,0,2,0,6,-5). All seven steps remain inside the physical band. Their complete nodal balance residual is zero in all six coordinates, the local mean area is exactly 1/2^114, and the vector area is nonzero. Adding this conditional area to B43's original budget gives 3296/(3*2^114) relative to .5; it is neither a complete return nor a new live graph observation.

Result and next boundary

This block gives a constructive joint subcell mechanism and its precise limit: the two relay coordinates are conditionally trapped, while a different corrected coordinate forces departure from the local region. The retained report is artifacts/research/c6_winding_relay_budget.json. Its endpoint includes the exact remainder and is the next conditional numerical input; the original B43 record remains unchanged.

Any extension must include node 4's actual new displayed value and recompute the affected pressure rows. Local corrected-coordinate budgets can be combined only after checking exact transition compatibility and the change of correction across their common boundary. Merely concatenating local deadlines, convex pressure balances or projected relay periods does not establish a global signed budget or an invariant region. General trapping, indefinite mean control, future operator admission and physical correspondence remain open.

41. B47: a local relay budget survives the interacting boundary

B2.d.47 resumes the exact conditional B46 endpoint, including all six remainders. Its visible offsets from .5, in delta=2^-54, are (-3,0,2,0,6,-5). The fixed phase, unit capacities, unit C6 support, canonical pressure coefficients, h=1/16 and [.375,.625] band are retained. The complete B43 arithmetic evidence and seven B46 conditional steps are replayed before accepting this input. This binds serialized numerical lineage, without recreating a live graph seal or undoing historical SHA.

The new APIs extend the existing relay owner: derive_c6_carried_local_relay_budget and observe_c6_carried_local_relay_exit. The handoff producer reuses replay_c6_relay_budget_evidence; the original B46 producer shares that reconstruction and record logic. There is no second solver or pressure law.

The third switch cannot be treated as another independent relay

Add node 4's displayed pair 6/8 to B46's node-0 pair -4/-3 and node-3 pair -1/0. Recomputing all eight canonical pressure rows reveals the nonzero mixed increment

h*p(node3 lower, node4 lower) - h*p(node3 lower, node4 upper)
  - h*p(node3 upper, node4 lower) + h*p(node3 upper, node4 upper)
    = -8 * 2^-113 * (e3+e4).

Node 0 is held at its upper value in this expression; the same result holds in its lower branch. Its two pairwise mixed differences and the three-way mixed difference vanish. The nonzero term is an exact consequence of the represented pressure product/assembly. Treating nodes 3 and 4 as independent would omit it. These eight evaluations are a finite control, not the proof of the broader locality statement below.

Local support supplies a common corrected coordinate

On fixed C6 the two required EPI pressure gradients are exactly

g0 = (x5+x1)/2-x0,
g1 = (x0+x2)/2-x1.

The shared indexed pressure reader applies its represented product and fixed phase-source assembly to these gradients. Consequently, holding displayed nodes (1,2,5) fixes both pressure functions of x0, regardless of the admitted displayed values of nodes (3,4). The source/target support union derives this held set; it is not a selected cancellation assumption. The proof allows arbitrary node-3/node-4 values in the declared band, not only the eight boundary cells. Their own pressure rows continue to refresh.

Node 0 therefore keeps its B46 invariant strip -a<u<=b, where u=X0-facet, a=3039824576180305*2^-113 and b=3558973984143090*2^-113. The incoming B46 remainder belongs to it. Recompute the two local pressure rows and set

k = h*(p1_lower-p1_upper)/(a+b)
  = -3299399280161697/6598798560323395,
v = h*p1_upper+k*a
  = -3360475576564941293533924113653 * 2^-113 / 1319759712064679 < 0.

For every transition whose source retains the held neighborhood,

W = X1-k*X0,          W(n)-W(0) = n*v,
X1(n) = X1(0)+n*v+k*(u(n)-u(0)).

The node-0 modular formula remains the same exact rotation as in B46. Only these two coordinates have a closed formula; no independent formula for the other four coordinates is asserted. This common budget crosses the node-4 boundary without an omitted change of correction potential: its coefficient and drift are identical on both sides. It remains valid through the first outgoing step whose endpoint changes a held value.

Analytic deadline and first held-neighborhood exit

Let c_plus=max(k*(-a-u(0)), k*(b-u(0))). Since v<0, the least integer

N = floor((X1(0)+c_plus-RN_lower(x1))/(-v)) + 1

puts even the largest possible corrected coordinate strictly below its original rounding cell. For the actual incoming state this gives 198. Until that deadline, either a held displayed value changes, or numerical band admission fails. The existing independent band-horizon certificate covers 196713720348826205 steps from this input, which exceeds 198. That sufficient bound is not a simulated duration or a claim of indefinite stability. Here it eliminates band failure as the earlier alternative.

The observer derives the deadline before taking any new step, refreshes all six canonical pressure rows at every step and stops at the first held change. This occurs at 197, node 1, with endpoint offsets (-3,-1,2,0,8,-5). All 197 shared-kernel steps and all 198 formula points have zero nodal and corrected-budget residuals. The complete vector area is retained and nonzero. The local mean area is 349/(3*2^114); together with B43 and B46 it is 3645/(3*2^114)=1215/2^114 relative to the original uniform .5 preparation. There are 204 conditional steps after B43, and no additional live graph-owned word.

This observed branch stays within the eight boundary cells until the same step 197. The broader theorem's benefit is its independence from any restriction on the free coordinates, not a claim that this particular trajectory visited additional cells. Membership in the local theorem alone does not prove that an arbitrary supplied state descends from B46; the report establishes that numerical lineage separately by replay.

The retained result is artifacts/research/c6_winding_relay_handoff.json. Its B47_conditional_first_exit.endpoint is the next conditional input, with the full remainder. Node 1's changed value now modifies node 0's pressure as well as the corrected-coordinate budget. Continuing from this dependency boundary requires rederiving the affected relay to account for any change in correction potential before composing another budget. Repeating B47's formula beyond this endpoint is invalid. The result does not establish a global signed budget, invariant trapping, future operator admission or empirical correspondence.

42. B48: exact correlated-set viability and the remaining global gap

The target is indefinite boundedness of the current fixed C6 numerical branch, with the unchanged B47 endpoint and complete carry. No theorem about arbitrary TNFR networks or future operator words is being inferred. The canonical pressure, phase tuple, unit capacity, h=1/16 and positive band remain fixed. No post-B47 trajectory steps are taken in this audit.

Universal preimages, rather than an observed return

The shared carried nodal equation is a translation in each displayed cell:

X = x+r,   F(X) = X+h*p(x),   x_next = RN(F(X)).

For a finite declared visible family, derive each coordinate spacing as gamma_i=gcd({h*p_i(x)}). These are arithmetic consequences of the source table, not fitted physical parameters. If a coordinate has identically zero increment, the implementation uses the shared 2^-3222 encoding grid. The affine cosets through the unchanged origin are preserved while pressure comes from that family. On them, a subcell is a closed integer box inside the exact band-clipped nearest-even cell. Different pieces must be disjoint on these cosets; their geometric hulls may overlap only off that grid.

Starting from the candidate K_0, compute

K_(n+1) = K_n intersect F^-1(K_n).

Canonical pressure is freshly read for every source cell. Exact translation and intersection with target boxes retain the incoming-carry relation. Adjacent boxes are merged only when their other coordinate bounds agree. Because the sets are finite and nested, equality of exact integer point counts proves equality of sets. It is not a floating volume comparison.

The reusable owner derive_c6_carried_viability has three distinct outcomes:

• fixed_point: a complete nonempty fixed set contains the supplied origin. Induction proves arbitrary repetition remains in the retained boxes and band. This is conditional numerical boundedness, without convergence or live-graph provenance.
• origin_excluded: the origin leaves the initial candidate or its band within the recorded descent depth. This does not prove whole-band escape.
• resource_limit: only the last fully completed descent is returned. No infinite-time or origin-exit conclusion follows from unfinished work.

The positive proof contract is tested on a stationary C6 case and a nonstationary alternating C6 two-cycle, including retained carry. Neither control replaces the pressure source of the research branch.

The actual profile candidate

Use the existing exact forced profile z and the actual B47 mean mu. The adjacent represented values bracketing mu+z_i give the lower and upper offsets, in units delta=2^-54 from .5:

lower = (-4,-1,2,-1,6,-6)
upper = (-3, 0,4, 0,8,-5).

Their 64 combinations include the unchanged B47 endpoint at mask 57, where bit i selects the upper value of coordinate i. The candidate-pressure gcds give gamma=2^-113*(1,1,4,8,8,8), which are specific to this family. Three complete descents retain 64, 416, and 4,285 boxes respectively, with strictly decreasing exact point counts and the origin still present. The fourth descent exhausts the 500,000-work-item guard. The result is therefore undecided. The guard controls computational work, not physical time. No new trajectory, invariant set or escape is asserted by these counts.

Exact scope of the quadratic controls

Write the cube's nodal increments in units m=2^-113 as a_sigma=b-M*sigma+e_sigma, using the six one-bit differences to define M. With K=3299399280161697, the positive diagonal symmetrizer is

D = diag(1,1,2K/(K+3),K/(K+3),2K/(K-1),K/(K-1)).
v = (1,1,1/2,1,1/2,1).
M*v = (1,0,-8,-8,0,0).
v^T*D*M*v = (3-15K)/(K+3) < 0.

Thus the proposed inverse-affine positive quadratic cannot supply the desired energy. This is a failure of that surrogate, not a dynamical instability proof. The exact mixed remainders range over (0..1,0,-8..0, -8..0,-8..0,0) in these units; they cannot be dropped in a mean argument.

A second control uses the actual nonlinear table. Let Q_i=e_i-e_5. Every symmetric matrix can be written H=Q*S*Q^T+Q*a*1^T+1*a^T*Q^T+K*beta*1*1^T. A retained 21-term positive rational dual, recomputed and checked against all its exact coefficients, gives

sum_j y_j * sign(sigma_j[i_j]-1/2) * (H*a_sigma_j)[i_j]/K
    = lambda * trace(S),
y_j > 0, sum_j y_j=1, lambda > 0.

Positive definite H implies positive definite S by congruence, so at least one selected oriented component must be outward. This excludes the all-orthant inward common-quadratic sufficient criterion. It does not exclude every quadratic invariant of a finite region, shifted or piecewise potentials, or a reachability-trimmed subset. No numerical LP tolerance is used to admit either exact obstruction.

Reproduction and unresolved target

Run benchmarks/c6_winding_invariant_region.py. It reuses the centralized B47 evidence replay, verifies the complete B43/B46/B47 numerical lineage, and retains the source digest, input hashes, canonical pressure table, exact quadratic witnesses and all last-completed subcell bounds in artifacts/research/c6_winding_invariant_region.json. The original B47 endpoint remains the next dynamical input; the proof search does not create a later state or reverse historical SHA.

Global boundedness of this actual C6 branch remains open. The required next mathematical object is a closed carry-correlated region containing that state, or an exact temporally compatible signed budget proving escape. Independent pressure balances, pairwise existential edges and longer finite prefixes cannot substitute for this universal inclusion or budget argument.

43. B49: protected pair contrasts and relational past envelopes

This block keeps the B47 input, canonical pressure source, unit capacities, h=1/16, positive band and complete carry. It changes the representation of a proof candidate, not the nodal dynamics. Both search directions now share source, state, pressure and exact RN-cell preparation in c6_carried_viability.py.

Conditional contrast strips from the nodal map

Let g be the rational gcd of all declared nodal increments, with the shared encoding grid used if all increments vanish. Write X_i=X_initial_i+g*k_i on the initial state's preserved arithmetic class. All k_i are integers; this common grid may contain more states than the separate coordinate gcds. It is a sound relaxation, not a change of carry. Each exact RN cell gives integer coordinate bounds, including nearest-even endpoint ownership, and one fixed translation a_c.

For a pair (i,j), project that cell onto d=k_i-k_j, obtaining [L_c,U_c] and signed increment b_c=a_c[i]-a_c[j]. Start a candidate strip [l,u] at the actual value zero. Whenever it intersects a projected cell with b_c<0, extend l to include L_c+b_c; for b_c>0, extend u to include U_c+b_c. Only the finite set of cell endpoints can change either bound. At termination, verify directly, for every nonempty intersection,

max(l,L_c)+b_c >= l,   min(u,U_c)+b_c <= u.

Thus every pair strip is preserved by a step whose source stays in the declared cube. Their simultaneous intersection contains the unchanged initial state. This is a conditional result: it does not itself keep the next visible state in the cube. All 15 pair contrasts tighten the current candidate. For example, the contrast between coordinates 4 and 2, after subtracting their central-facet difference, is confined to roughly [-0.0125*delta, 2.022*delta], compared with the original [-4*delta,4*delta], where delta=2^-54. Exact bounds and their source premises are retained.

A compact envelope of states with a compatible past

One relational zone per visible cell stores the integer inequalities k_i-k_j<=B_ij, including a seventh fixed coordinate k_6=0 for absolute bounds. Shortest-path closure gives exact implied bounds. Translating a zone adds a_i-a_j to each entry. Intersections are closed exactly; the entrywise maximum of closed nonempty matrices is their least zone hull. The hull can include extra states, so membership alone never proves a compatible past.

Let D be the RN cube intersected with the independently protected pair strips, and let alpha_D form this hull separately inside each cell of D. The successive outer envelopes are

R_0=D,   R_(n+1)=alpha_D(F(R_n) intersect D).

Monotonicity implies R_(n+1) subset R_n. If a complete image F(R_n) stays inside the original cube, the conditional pair-strip theorem also keeps it in D. Consequently

F(R_n) subset alpha_D(F(R_n)) = R_(n+1) subset R_n.

This proves an invariant core without requiring equality between successive envelopes. The complete Cartesian RN cover is checked before an outer-box test is used; holes in an arbitrary family cannot be silently filled. An empty core, a stationary abstraction with outgoing states and an unfinished layer do not establish boundedness of the supplied origin.

Actual initial membership is a separate gate. If a certified core contains the original carried state, induction applies immediately. Otherwise only a replay of the proof-derived finite entry prefix can bind that state to the core. Its entire prefix must stay admitted and end inside the retained core. Union with those prefix points then gives an invariant set containing the original state. No trajectory is advanced merely to look for recurrence.

Current result and reproduction

Run benchmarks/c6_winding_invariant_region.py --method forward. The same campaign and B47 lineage replay serve both B48 and B49, with distinct output files. The current result is retained in artifacts/research/c6_winding_forward_envelope.json.

The relational representation keeps only 64 zones. With the pair strips, the retained bounded search completes 259 image layers and reduces possible outgoing coordinate facets from 72 to 56, while retaining the original B47 point. Its 250000-intersection guard is computational. Since outgoing facets remain, the result is inconclusive, not a trapping theorem or a trajectory escape. The report makes no new live-graph or asymptotic claim.

An independent exact replay recomputes all 259 layers and verifies 15 pair barriers, 128 initial/final closed matrices and all 56 outward facets. Ten of those outward slabs contain explicit hypothetical states with exactly the original B47 mean and its separate coordinate cosets g*(1,1,4,8,8,8). Six leave through node 0's lower boundary (masks 8,12,16,20,24,28); four through node 5's upper boundary (masks 49,51,57,59). Every witness satisfies all retained zone inequalities and the exact RN cell. Consequently, intersecting this particular retained envelope with any independent mean interval containing the original mean cannot make it invariant. This does not exclude a mean/shape/carry-correlated subset, and none of these hypothetical points is claimed reachable. The independent replay and witnesses are retained in artifacts/research/c6_winding_forward_envelope.validation.json and artifacts/research/validate_b49_envelope.py.

A separate static refinement restores those coordinate cosets by snapping difference bounds to their exact gcds and reclosing the matrices. It tightens 1,588 entries, but all 56 outward facets still have exact coset-admissible witnesses. One complete refined image also retains 56 such facets. These witnesses are checked with the shared nodal kernel; they are hypothetical one-step states, not a continuation of B47. Thus restoring coordinate arithmetic alone does not close this retained envelope. Evidence: artifacts/research/c6_b49_coordinate_coset_probe.json.

Validation includes 636 targeted passing tests. A separate canonical zero-phase control has 729 exactly enumerated states and forward layers 729 -> 45 -> 9 -> 3; the implementation verifies entry into that invariant core from a nonstationary initial state. Other controls prevent an escaping origin, a partial image or a stationary outer envelope with outgoing states from being certified. These are verifier checks, not parameter changes to the current fixed-phase h=1/16 research branch. The retained B49 source delta and validation record bind the report to its scientific source digest.

A separate bounded falsification probe took 10000 shared-kernel steps from the same B47 state and found no exit from the 64-cell cube. Its explicit computational budget and full finite trace are retained in artifacts/research/c6_cube_counterexample_probe.json. This does not prove stability and does not replace B47 as the primary proof input. Global boundedness remains open: the surviving outward states may require stronger correlations than pair differences, a smaller reachable region, or a different candidate. Their presence in an outer envelope does not establish their reachability from B47.

44. B50: exact point predecessors and the temporal-correlation gap

B49's outward states are hypothetical members of an outer envelope. B50 asks whether selected points can have an exact carried past, using the same C6 pressure, phase, unit capacity, timestep, RN rule and full remainder. It adds no physical law or parameter. The public predecessor observer shares source and RN-cell preparation with both existing viability methods in c6_carried_viability.py.

Exact predecessor sets, not a graph of visible labels

Fix a complete target state X, a declared source domain D and the original affine coordinate cosets. In visible cell c, the only possible predecessor is

Y_c = X - h*p(c).

It is admitted only if it has that exact nearest-even visible row, stays in the band and domain, and belongs to the preserved coordinate cosets. Every accepted edge is checked with the shared nodal integrator, including its full carry. Each complete predecessor layer retains exact states and successor indices, so adjacent edges cannot silently change their intermediate state. Pressure is reconstructed from the canonical source rather than trusted from a cached or serialized table.

Starting from P_0={X}, compute P_(n+1)=D intersect F^-1(P_n). These sets are not generally nested, and repeated cardinalities do not imply equality. If a complete P_n is empty, no admitted depth-n history ends at X. This excludes occurrence after n transitions during a path wholly inside D; it does not exclude earlier transient visits. A nonempty layer certifies finite compatible pasts, without proving connection to the actual B47 origin. Resource interruption discards the partial next layer.

Origin matching is a separate exact test across all retained layers. A match plus its linked steps proves a finite origin-to-target path under the declared map. If the complete predecessor tree exhausts without any origin match, every origin-to-target path wholly inside D is excluded, including transient ones. Neither result applies to paths that leave D and later return. A point certificate says nothing universal about its containing facet or other points with the same displayed EPI.

Current branch and pointwise first-exit exclusions

The campaign benchmarks/c6_winding_invariant_region.py --method predecessors first reconstructs all B49 payload fields from B47's unchanged lineage. It checks the selected witness bytes, exact original mean, RN cell and actual outgoing nodal step. Each target is then audited in two distinct domains: the full 64-cell RN cube and the retained relational envelope R259.

The campaign also verifies the exact clipped identity F(R259) intersect D subset R259, where D is the cube intersected with B49's protected pair strips. Since the original state belongs to R259, every prefix before its first cube exit remains in R259. Complete point-predecessor exclusion from that origin therefore rules out that point as the cause of the first cube exit. It does not exclude the entire facet, prove confinement or control a later trajectory that has already left the cube.

The retained point audit is artifacts/research/c6_winding_temporal_predecessors.json. It introduces no new actual trajectory steps or live execution; shared-kernel checks on hypothetical edges are explicitly distinct from the B47 research trajectory.

Of the ten selected original-mean outward points, nine have exhausted exact predecessor trees in R259, with no B47 origin in any complete layer. Their first empty depths, indexed by visible mask, are 8:12, 12:9, 16:3, 20:1, 24:3, 49:1, 51:1, 57:1, 59:4. Those nine particular points cannot cause the first cube exit of the retained branch under the fixed map. Mask 28 remains undecided after 21 complete predecessor layers and the 32,768-row-check guard; a partial deeper tree is not promoted. The broader complete RN cube gives five exhausted trees: 20:42, 49:91, 51:120, 57:91, 59:40; its other five searches are resource limited. These two domains and depth bounds must not be conflated.

The implementation passes 690 targeted tests, including 54 new point/past and report cases. A separate validator independently reconstructs all 20 serialized certificates and replays 4,014 exact predecessor edges plus ten hypothetical outgoing steps. It verifies 893 nonempty clipped image pieces for the R259 inclusion. Artifacts: artifacts/research/c6_winding_temporal_predecessors.validation.json, artifacts/research/validate_b50_predecessors.py and artifacts/research/b50_final_validation.json. These tests and mathematical checks do not supply empirical physical evidence or a whole-region theorem.

Why retaining temporal labels helps, and what remains open

A separate bounded prototype keeps the preceding visible cell as part of each abstract mode before taking any matrix hull. One preceding cell gives 870 modes after seven complete refinements, reducing distinct outward facets from 56 to 48 and retaining six of the ten selected point witnesses. Two preceding cells give 7,387 modes after one complete refinement, 46 outward facets and four surviving selected witnesses. Each lifted clipped-image inclusion is verified; no actual entry or invariant core is claimed. The mode counts are computational state representations, not extra TNFR variables. Evidence: artifacts/research/c6_temporal_mode_envelope_probe.json.

For comparison, exact full unsafe-set preimages without any hull retain 56, 280, 861, 2,553 and 7,963 zones through depths zero to four before a 20,000-zone guard interrupts the next complete layer. This is again a representation limit, not an escape or stability theorem; see artifacts/research/c6_b50_unsafe_past_probe.json. These probes show precisely where temporal information is lost and where retaining every disjunction becomes expensive. The remaining task is a compact, universally checked temporal refinement or a source-derived invariant region. More finite trajectory steps cannot replace that inclusion proof.

45. B51: whole outgoing regions excluded by their complete pasts

B51 replaces selected point targets with the entire outward slabs of the retained R259 domain. The same nodal source, unit capacity, h=1/16, band and complete carried B47 state are unchanged. Both point and region campaigns now use one B49 reconstruction and clipped-inclusion helper, and the physics owner reuses the existing source, RN-cell and relational-domain validation.

Region targets and the complete-past argument

Write X_i=X_initial_i+g*k_i, with B49's common increment grid g=2^-113. The full 64-cell RN cube has exact integer endpoints l_i,u_i, including nearest-even ownership. These endpoints are reconstructed from complete RN cells: the protected pair strips can tighten coordinate projections and must not redefine the original cube's exit boundary. In source cell c with integer increment a_c, the lower and upper outward slabs are respectively

T_(c,i,lower) = R259_c intersect {k_i <= l_i-a_c[i]-1},
T_(c,i,upper) = R259_c intersect {k_i >= u_i-a_c[i]+1}.

The nonempty slabs cover every possible first-cube-exit source in R259. They can overlap; their number is not a probability or a progress percentage. The common grid relaxes the finer coordinate cosets, so exclusion is sound for the actual state even though an admitted hypothetical point need not have its full arithmetic provenance.

For each complete slab T separately, begin with P_0=T. In each source cell i, translate every target zone backward by the actual increment a_i, intersect with R259_i and take the least closed difference-bound hull. Thus

P_(n+1) = alpha_R259(R259 intersect F^-1(P_n)).

Every exact n-step predecessor is included. The layers need not be nested; cardinality equality is insufficient. If a complete layer is empty, there are no predecessors at that or any later depth. If consecutive complete abstract layers are exactly equal, the deterministic abstract recurrence stays equal thereafter. Either case proves no origin-to-T path within R259 when the origin is absent from every earlier complete layer as well. An origin admitted by a hull is only inconclusive, not an actual path.

The established identity F(R259) intersect D subset R259, with D the cube and its protected pair strips, combines with the conditional pair-strip preservation proved in section 43 and initial membership to keep every actual pre-cube-exit prefix in R259. Therefore an entire excluded target slab cannot cause the first cube exit. This includes early transients because every complete preceding layer is checked for the origin. It does not assert that the slab is empty, that arbitrary hypothetical starts are safe, or that later exit-and-return paths are excluded.

Current C6 result

The production batch owner derive_c6_carried_region_exclusions advances all 56 labeled slab queries in round-robin order under one computational intersection guard. It records each completed layer and discards an interrupted partial layer. The campaign is benchmarks/c6_winding_invariant_region.py --method regions; its report is artifacts/research/c6_winding_region_exclusions.json.

The retained 250,000-intersection search excludes 24 whole slabs: all 16 node-1 upper slabs and all eight node-5 upper slabs. Their complete predecessor layers become empty within depths two through eight for node 1, and at depth six for node 5, with the actual origin absent throughout. These two node/direction classes cannot produce the first cube exit under the fixed map. The remaining 32 slabs are eight each for node-0 lower, node-2 lower, node-3 upper and node-4 upper. Their bounded searches remain inconclusive. The engine does not certify indefinite trapping or actual escape, and no new actual trajectory or live graph execution is run.

Complementary probes and the next boundary

An independent increasing backward-reachability hull gives nine complete closed-set exclusions, all within the node-1 upper class. The hull of the union of all unsafe targets instead admits the origin immediately; that is loss of information, not an observed exit. This supports keeping target labels separate. Evidence: artifacts/research/c6_unsafe_backward_hull_probe.json.

A separate point-tree lifting calculation translates the same perturbation through every ancestor of an exhausted B50 tree. Preserving one violated domain inequality for each rejected row prevents new branches; preserving one signed coordinate of every admitted ancestor excludes the origin. Intersecting these exact difference constraints gives nontrivial excluded regions around the nine former point witnesses. Five lie in node-0 lower slabs (masks 8, 12, 16, 20 and 24), outside the two fully excluded exit classes. These partial-region exclusions remain complementary evidence; they do not exclude their whole slabs. See artifacts/research/c6_b51_point_region_probe.json.

Coarsening temporal labels to only the prior bits of node 1 and its two neighbors reduces memory cost, but leaves 51 outward facets, versus 48/46 with one/two complete preceding cells. The local pressure dependency alone does not preserve all useful cross-node temporal constraints.

The regional exclusions also support a sound domain refinement: remove each proven unreachable slab by its exact complementary bound in that source cell. Every actual prefix before the first cube exit remains in the refined domain, although universal forward invariance of that domain is not claimed. The retained probe rechecks all 24 exclusions, tightens 104 matrix entries across 22 cells and preserves the actual origin. With the same 250,000 intersection guard, all 32 remaining queries are still inconclusive after 33 or 34 complete layers; no further slab is excluded. Evidence: artifacts/research/c6_b51_iterated_certified_cuts.json.

The unresolved obligation at this stage is a compact proof over the four remaining exit classes, preserving more of their coupled pressure, carry and temporal relations. Merely repeating the certified cuts, increasing a resource guard or testing additional isolated points cannot supply the missing universal proof.

Independent validation and retained source

artifacts/research/validate_b51_regions.py independently reconstructs the 64 canonical pressure rows, complete RN cells, all 56 unsafe slabs, 893 clipped forward-image pieces and all 1,176 completed backward layers. Its exact 250,000-intersection replay confirms the 24 whole-region exclusions and 32 resource-limited queries. The validation is bound to the four input hashes, report bytes and implementation source in artifacts/research/c6_winding_region_exclusions.validation.json. The source archive chain ends at b51_scientific_source_delta.json; the combined software and documentation checks are retained in artifacts/research/b51_final_validation.json. The targeted suite passes 729 tests in 163.37 seconds, including 39 new regional-owner and report cases. Tests cover independently enumerated finite models, exact stationary frontiers, nonnested layers, false origin membership caused by hulls, interrupted budgets and evidence integrity.

46. B52: a nodal excursion budget excludes the node-2 lower boundary

The remaining regional queries need information that survives their hulls. A useful exact relation is now available for the half of the original 64-cell cube in which node 2 has its lower represented value. Keep B47 as the primary origin and use the same integer carried coordinates X=X_B47+g*k, g=2^-113. Define the proof coordinate

W(k) = -2*k_0 + 2*k_2 + k_3 - k_5.

This is a linear read-out of the existing nodal state. It adds no dynamics, pressure coefficient, capacity, timestep or state reset.

Its coefficients are derived from the unit-C6 nodal diffusion geometry: they are the primitive integer zero-mean Poisson contrast satisfying L_rw*w=(3/2)*(e_2-e_0). The campaign solves that rational system rather than selecting new physical weights. This geometric identity motivates the read-out; the complete binary64 pressure table separately proves its drift, including all nonlinear rounding terms:

W(k_next)-W(k) >= d = 1418638309884336 > 0.

The complete node-2 lower unsafe slabs inside R259 have W <= M = 237475340900157211. Every admitted transition from the other half of R259 into this half has W >= L = 449419196811482809 > M. Each extremum is certified by an integer transport dual and a matching admissible primal point; every crossing intersection is enumerated exactly. These bounds already hold in R259, without the optional B51 cuts.

The infinite claim reduces to one derived finite gate

After any reentry, W starts above M and increases throughout the active visit, so that visit cannot reach a node-2 lower unsafe slab. The original B47 point starts in the active half at W=0. During its initial visit, a target could therefore be reached only at source ordinals n <= floor(M/d) = 167. This is a mathematical bound, not a selected simulation horizon.

The shared carried integrator checks the initial visit with freshly rebuilt canonical pressure and complete incoming remainders. It can stop as soon as the visit ends or W exceeds M. On the actual B47 input this occurs at step 56, with W = 247713108641769776 > M. Every checked source avoids the targets, all 56 steps remain in R259 and the full RN cube, and every nodal balance residual is exactly zero. The earlier scratch check of all 168 steps is redundant for this certificate; its endpoint never replaces B47.

Consequently, no path from the unchanged B47 origin that remains in R259 can reach any of the eight node-2 lower unsafe slabs. Section 43's conditional pair-strip preservation and the rebuilt clipped forward inclusion bind this to every actual prefix before a first cube exit. Thus the eight complete slabs cannot cause that first exit at any later time. This is neither invariance for arbitrary starts nor a claim about paths that have already left the proof domain.

Shared implementation and present coverage

src/tnfr/physics/c6_carried_excursion.py owns the excursion theorem, integer extremum certificates, complete ingress checks and derived-prefix observer. It shares canonical source, RN-cell and relational-domain preparation with c6_carried_viability.py, and uses the shared nodal integrator for the finite gate. A nonpositive drift, nonstrict ingress gap or unfinished prefix remains inconclusive. Domain departure only settles domain-confined path exclusion; the owner never certifies global trapping.

The existing campaign's --method excursion reconstructs B49, rechecks all 24 positive B51 queries, and applies the new certificate to the full node-2 lower target family. Full RN geometry and unsafe-slab construction are now shared by the regional and excursion campaigns. The combined result excludes 32 of 56 complete first-exit slabs. The remaining 24 are eight each for node-0 lower, node-3 upper and node-4 upper. Indefinite boundedness of this C6 case remains open; no live graph invocation occurs. Evidence: artifacts/research/c6_winding_excursion_exclusion.json and its independent validation; source/check retention uses the B52 archive and artifacts/research/b52_final_validation.json. The combined targeted suite passes 772 tests in 215.06 seconds, including 43 new owner/report cases. Independent validation checks all 199 ingress intersections, 271 attaining primal/dual extrema, the 56 shared prefix steps, and the 24 prior exclusions through 112 complete predecessor layers.

Complementary controls and next proof target

A second exact coordinate, with coefficients (1,-1,-3,-5,5,3), has positive drift in the node-3 upper half, but its ingress lower bound is below the unsafe ceiling. That coefficient vector supplies no corresponding reentry exclusion. A bounded separation search further produces an exact obstruction to this entire single-linear-coordinate template on the full common-grid R259 domain: seven strictly positive rational coefficients balance five active pressure vectors and two admissible ingress-minus-target point differences to zero. No linear functional can be strictly positive on all those vectors. Thus changing the weights alone cannot simultaneously give positive drift on every node-3 upper row and strict separation of every ingress from every unsafe target in this domain. The obstruction does not cover smaller reachable domains, mode-dependent coordinates or nonlinear barriers. Evidence: artifacts/research/c6_b52_node3_excursion_separator_probe.json. The next attack is a coupled ingress/exit budget or finer certified domain for the three remaining classes, retaining the distinctions that made node 2 decidable.

One concrete next candidate is a cell-dependent proof coordinate V_c(k)=w*k+b_c. Every admitted active transition c -> d would need a strictly positive verified increment w*a_c+b_d-b_c, with target and ingress extrema using their own cell offsets. The initial budget must start at V_initial, not silently at zero. The current seven-vector obstruction does not include such cell offsets. Their feasibility and finite-prefix gate remain unproved; the offsets would belong only to the proof, never to the nodal update.

Three bounded refinements explain why more geometric detail alone has not closed the argument. Successor-cell modes preserve immediate temporal correlation but leave all former 32 queries undecided at depths 13/14 under 250,000 intersections. Removing the five lifted node-0 excluded regions creates 102 disjoint domain pieces and reaches depths 23/24 without another exclusion; joining those pieces per original cell fills all five holes. Two affine read-outs in a lifted DBM reach depths 34/35 but likewise yield no additional exclusion at the same guard. These are conditional outer approximations, not actual exits. Retained probes are c6_b52_successor_modes_probe.json, c6_b52_lifted_domain_probe.json and c6_b52_affine_lift_probe.json under artifacts/research.

47. B53: separate cell-offset budgets exclude four further regions

The original B47 state and canonical carried map remain unchanged. B53 first removes the 32 whole first-exit slabs already excluded by B51/B52, using their exact complementary inequalities in the original RN cube. Call this restricted domain D32. Every actual prefix before the first cube exit remains in D32; this is not a forward-invariance assertion for every point of D32.

A shared origin-containing forward envelope

The new derive_c6_carried_reachable_envelope in src/tnfr/physics/c6_carried_viability.py computes

R_0 = D32
R_(n+1) = hull_per_RN_cell({B47 origin} union (F(R_n) intersect D32)).

The complete layers descend and contain the unchanged origin. Induction therefore puts every domain-confined origin path inside every retained layer, including a complete layer retained after a resource limit. Each layer satisfies F(R_n) intersect D32 subset R_n. In this instance, 264 strict descents are followed by exact matrix equality, after 232,802 intersection checks. The resulting R* still has 24 outward labels. Its clipped fixed point is not a trapping certificate.

Preserve each target's own proof coordinate

For a visit to the 32 cells with node 3 at its upper displayed value, use V_c(k)=w dot k+b_c. Its exact change on an admitted active transition c -> d is w dot a_c+b_d-b_c. The added constants are proof read-outs, not new nodal parameters, pressure terms or state writes.

Four separate candidates succeed for the whole node-3 upper unsafe slabs with masks 29, 31, 61 and 63. Each has strict positive change on every one of the 242 admitted active transitions. Each of the 189 ingress pieces has potential strictly above that candidate's complete target ceiling. The original B47 point is already above the ceiling as well, so the derived initial gate is zero: these four results require no additional trajectory steps. The independently checked 760 attaining primal/dual extrema cover the four target maxima and all their ingress minima.

The candidates are stored as exact rational proof proposals in benchmarks/c6_winding_mode_excursion_candidates.json. A shared positive normalization converts each complete weight/offset family to integers. derive_c6_carried_mode_excursion_exclusion shares source, RN, grid, extremum and prefix machinery with the B52 owner, and checks every active edge explicitly. The initial value and all ingress/target bounds include their own cell offsets. A family with no internal edge has visits of at most one point; it receives no fictitious positive drift certificate.

The existing campaign's --method modes reconstructs the prior evidence and validates these proposals against one common pre-candidate domain. No candidate assumes its own exclusion. It then removes the four newly proved slabs and closes the remaining domain, D36. This second closure is already fixed at its first complete image, with 830 intersection checks. The report binds the B47, B46, B43, B49, B51 and B52 input bytes and the candidate bytes. Replaying B52's 56-step gate is prior-proof verification, separate from the zero new steps required by B53.

Remaining gap and useful negative controls

Coverage is now 36 of 56 complete first-exit slabs:

Pending class	Masks	Count
Node 0 lower	0, 4, 8, 12, 16, 20, 24, 28	8
Node 3 upper	28, 30, 60, 62	4
Node 4 upper	56, 57, 58, 59, 60, 61, 62, 63	8

Twenty slabs remain unresolved. Neither these counts nor a successful restricted proof establish global C6 stability, convergence or a live post-SHA runtime theorem. All statements still concern the fixed-phase, unit-capacity carried continuation from B47, with h=1/16.

The successful single-target functions do not combine into a proof for all eight original node-3 targets. Bounded common-gradient/offset searches retain exact template obstructions or explicit inconclusive outcomes for the residual targets. An apparently nonincreasing mean proposal also fails exactly: an admissible coordinate-coset transition 50 -> 0 increases sum(k)/2^59 by 13/2^59. This hypothetical edge is checked with the shared nodal kernel, but is not claimed reachable from B47. Ignoring the small residual would create a false theorem. Reparameterized mean-corrector searches keep this integer defect visible and have not produced a validated barrier. These failures restrict those proof templates; they do not prove instability of the actual path.

The exact next structural reduction is to preserve complete short return relations, including every intermediate exit gate, instead of joining away the coupled ingress and pressure information. Retained evidence and validation for the integrated four-region result are in artifacts/research/c6_winding_mode_excursions.json and its independent validation. The B53 source archive records the exact scientific source.

Validated return reduction and stopped refinements

Inside D36, no admitted transition connects two cells whose node-2 bit is high. The original B47 point has that bit low. Retain each direct low-to-low transition and each low-to-high-to-low path as a separate translated guard; there are 1,265 such pieces. Origin-containing closure on the low modes reaches equality after three iterations and 3,795 intersections. Rebuilding every high intermediate from every retained low source tightens 26 of the 64 cells. An intermediate is retained even if it has no subsequent low return, so an intermediate outward transition cannot disappear from the audit. All 20 remaining labels still occur.

The coverage argument is induction over successive low visits, followed by one-step reconstruction of high visits. It is not a claim of one-step invariance for the projected per-cell hulls. Independent validation checks all 830 ordinary edges, every compressed guard and all three complete layers. The retained prototype is artifacts/research/c6_b53_compressed_return_probe.json, with its domain loader and validation. Sixty strict and sixty nondecreasing coupled-return separator queries produce no positive certificate; 728 independently checked constraints and overlap witnesses certify their stated negative results, with other queries explicitly inconclusive.

The exact coordinate cosets also admit a finite temporal check. Write k=8*z+r, with 128 allowed residues: r_0,r_1 range from 0 through 7, r_2 is 0 or 4, and r_3=r_4=r_5=0. Integer closure of z_i-z_j <= floor((B_ij-r_i+r_j)/8) decides every guard/coset intersection. All 106,240 intersections are nonempty; abstract reachability from B47 reaches all 8,192 cell/residue vertices. Each of the 20 unsafe slabs remains nonempty in all 128 residues. This rules out exclusion by this particular necessary graph test, not by exact trajectory analysis. Evidence and full independent enumeration are retained in artifacts/research/c6_b53_mod8_temporal_graph_probe.json and its validation.

A lifted total-coordinate envelope also retains all 20 labels after 301 complete layers and its 250,000-intersection guard. Increasing these guards or retaining the same residue labels supplies no new established mechanism. The next proof needs a stronger coupled temporal relation or a barrier verified on complete return guards, preserving their common source state and all intermediate exits. An abstract path assembled from separate edge witnesses does not provide such a state.

The final production campaign passes independent validation of 64 pressure rows, 1,016 primal/dual extrema, 968 active transitions and 756 ingress pieces. The targeted suite passes 845 tests in 267.02 seconds, including 73 new reachable-owner, mode-owner and campaign cases. The unchanged B47 origin, seven input hashes and scientific source archive are recorded in artifacts/research/b53_final_validation.json. This validation closes the four additional regional claims, while the requested remaining twenty-region and indefinite-boundedness proof remains open.

48. B54: complete return guards exclude two node-0 lower regions

The two remaining layers of information in section 47 have different roles. The return envelope is a forward outer bound on possible origin paths. A backward query asks whether a complete unsafe slab can have an origin-compatible past. Combining them now excludes node-0 lower masks 4 and 12, without advancing the actual trajectory.

Preserve the intermediate state in one shared relation

src/tnfr/physics/c6_carried_return.py owns both the exact short-return envelope and its whole-region predecessor queries. It rebuilds canonical pressures, RN cells, the full carried origin and the common integer grid. The transient partition selects node 2's upper displayed value; the complete one-step relation must have no transient-to-transient edge, and the original state must be in the complementary base set. These are verified proof premises, not changes to the dynamics.

For a direct return a -> c, retain its exact source guard and increment. For a two-step return a -> b -> c, intersect the incoming guard in cell b with the outgoing guard at that same intermediate state. Translate this intersection back to a and forward to c, retaining the complete six-coordinate increment. Separate first-step records retain every base-to-transient visit, including visits with no subsequent return. The current D36 instance has 830 ordinary edges and 1,265 return records.

The origin-injected base closure and reconstruction of all transient visits reproduce the independently validated envelope from section 47. Resource accounting includes ordinary candidate pairs, return construction, complete closure layers and transient reconstruction. Incomplete relation construction retains the supplied domain and exposes no usable partial relation. An interrupted closure layer is discarded. A return fixed point still does not assert one-step invariance of joined intermediate hulls.

Backward exclusion covers every possible visit length

Let Q0 contain unsafe base states together with every base predecessor of an unsafe transient state. The latter is computed from the complete first-step records, even where no later return exists. For each exact return guard G with displacement a, the next backward piece is

G intersect (Q_n - a).

Only after applying each complete guard are these pieces joined per base RN cell. Each resulting complete layer is checked against the unchanged B47 origin. For both newly excluded slabs, layer 7 is empty, and the origin is absent from every preceding layer. Earlier lengths are therefore excluded directly; every later layer remains empty. Seven counts returns, not a selected physical-time simulation horizon. These layers need not be nested, and an abstract origin collision would be inconclusive.

The original full supplied targets remain in the certificate. Their intersection with the origin-containing return envelope is justified by its path-coverage theorem, not by an assumption that every point in that envelope is reachable. Independent reconstruction of the two positive queries checks 1,375 intersections, six transient target preimages, the complete 830-edge ordinary relation and all 1,265 return records. The production envelope accounts for 10,303 intersections: 5,578 during construction and 4,725 during closure and intermediate reconstruction.

The existing campaign adds --method returns. It reconstructs B53 and checks its complete retained payload against eight frozen input byte streams before using D36. The shared unsafe-slab helper uses original RN cube facets throughout. The result is 38 of 56 complete first-exit slabs excluded, with 18 unresolved:

Pending class	Masks	Count
Node 0 lower	0, 8, 16, 20, 24, 28	6
Node 3 upper	28, 30, 60, 62	4
Node 4 upper	56, 57, 58, 59, 60, 61, 62, 63	8

The original fixed-phase, unit-capacity h=1/16 map and B47 carry are unchanged. B52's finite gate is replayed only as a prior proof premise; B54 requires zero new trajectory steps and no live graph invocation. Global C6 boundedness, convergence and post-SHA runtime promotion remain open. Report: artifacts/research/c6_winding_return_regions.json.

A tested stronger temporal refinement

A separate exact prototype composes pairs of returns before joining their middle state. Of 85,272 compatible-label pairs, 9,590 have a nonempty common carried-source guard. Twenty-six complete even-return closure layers use 249,340 intersections before the computational guard. Coverage then includes the one-return image for odd visits and every transient intermediate, avoiding an even-endpoint-only claim. This tightens 53 cells relative to the single-return envelope, but excludes no additional label by itself. Its independent validator reconstructs every pair, every complete layer and the odd/intermediate coverage.

Evidence: artifacts/research/c6_b54_double_return_probe.json and its validation. A backward search on this stronger outer domain finds the same two positive slabs, with the other 18 still resource-limited. The useful gain in B54 comes from complete guarded backward exclusion; tighter forward hulls alone have not supplied the missing full confinement proof.

Removing the two newly excluded slabs and repeating the return audit does not add an exclusion at the same 500,000-query-intersection guard. A further backward prototype keeps direct and transient first-return components separate and intersects each exact guard before joining. All 18 queries still reach their guard at complete depths 19 or 20, without an empty or stationary layer or an origin collision. These retained controls are c6_b54_post_exclusion_probe.json and c6_b54_partitioned_return_predecessors.json under artifacts/research. Their unfinished searches do not prove that their targets are reachable.

The final targeted suite passes 925 tests in 346.90 seconds, including 80 new return-owner, regional-query and campaign cases. Independent validation binds all eight input hashes, the archived scientific source, all complete return guards, every original unsafe target and both positive proofs. Software, documentation and provenance checks are retained in artifacts/research/b54_final_validation.json; source reconstruction uses b54_scientific_source_delta.json. The remaining 18-region proof is open.

49. B55: exact unions exclude another complete first-exit region

This block preserves the B47 origin, full carry, source pressure, phases, unit capacity and h=1/16. It changes only the representation of possible predecessors. A convex hull can fill a gap between two regions and thereby introduce a spurious path. The new query retains a finite union of closed integer difference-bound regions for each base RN cell.

For a return edge e with exact source guard G_e, displacement a_e and target cell d, each individual target piece Z contributes

Pre_e(Z) = G_e intersect (Z - a_e).

The union is formed without a hull. A piece may be removed only when a retained piece in the same cell contains it. This pairwise subsumption preserves the exact union. Initial targets remain the full original unsafe slabs, clipped only by the previously justified path envelope; transient targets contribute every unjoined first-step preimage. Thus possible last visits to transient cells are not lost.

Every complete layer is checked against the unchanged origin. A complete empty layer, or identical consecutive normalized unions with all earlier layers excluding the origin, excludes all domain-confined visit lengths. Equal counts alone are insufficient. These layers need not be nested. The common integer grid and the inherited forward envelope remain outer approximations, so origin membership would not prove actual reachability.

For mask 0, node 0 lower, the complete zone counts are 1 -> 5 -> 9 -> 14 -> 0. The origin is absent at every depth. Production accounting verifies 1,237 intersections (one initial clipping plus 1,236 predecessor pairs) and 90 subsumption comparisons. Layer 4 is empty; all later layers are therefore empty as well. This excludes the entire original target slab as a first-cube-exit source, not merely a chosen outward point. It requires no new trajectory step.

The implementation lives in the existing c6_carried_return.py owner as derive_c6_carried_return_union_exclusions. Shared preparation for both query representations rebuilds pressure, RN cells, carried origin and all intermediate guards. The previous B54 envelope and all twenty B54 query payloads remain identical. Query limits bound intersections, pairwise comparisons and the number of pieces; interrupted initialization publishes no absence claim, and later interruptions retain the last complete layer.

The existing benchmark adds --method unions. It reconstructs the whole B54 result before comparing its retained bytes and binds nine input files. The final campaign spends 500,000 query intersections and 894,744 subsumption comparisons. Its positive proof raises coverage to 39/56:

Pending first-exit class	Masks	Count
Node 0 lower	8, 16, 20, 24, 28	5
Node 3 upper	28, 30, 60, 62	4
Node 4 upper	56, 57, 58, 59, 60, 61, 62, 63	8

The other seventeen queries remain resource-limited. Report: artifacts/research/c6_winding_return_unions.json. Its independent validator binds the source and nine-input chain, reconstructs original targets and replays the positive finite-union proof using separate matrix arithmetic. Global C6 boundedness, convergence and post-SHA runtime promotion remain open.

Bounded complementary controls and reuse

Independent exact-union probes with 250,000 intersections per query find no additional positive among the other seventeen targets. Coordinate- coset tightening adds none; 61 verified adjacent integer-cut union merges reduce fragmentation but add none. Rebuilding every return guard after removing node-0 masks 0, 4 and 12 leaves all completed residual backward layer hashes unchanged.

A signed embedding (k0,-k0,...,k5,-k5,0) additionally preserves pair sums. Its coherence, integer unary tightening and difference closure preserve all embedded integer points, as checked by independent finite-set oracles. The 100,000-intersection forward probe and eighteen backward queries with 50,000 intersections each produce no further exclusion. A separate fixed partition at zero nodal carry preserves each sign combination before joining. Four representative targets reach their 100,000-intersection limits at depths 52, 51, 30 and 21 without a proof. These are inconclusive bounded searches, not impossibility results for richer domains.

The earlier complete spatial increment lattice is reverified against all 64 B54 pressure rows: its exact basis is diag(1,1,4,8,8,8), of index 2048. It supplies no additional time-free spatial congruence. The existing necessary full-return period divisor remains 2721508902482880257845796439528; it proves neither a periodic orbit nor confinement. Reuse that arithmetic result rather than searching the period.

A cumulative backward worklist additionally drops a newly encountered region if an earlier visited region in the same cell contains it. Across all seventeen residual targets, its 134 complete frontiers (depths 5--12) show no cross-depth subsumption and match the ordinary exact-union layers. Sixteen queries stop at comparison limits and one at an intersection limit. This bookkeeping alone supplies no further proof; the retained control is c6_b55_cumulative_union_return_probe.summary.json.

Source reconstruction and final checks are retained in artifacts/research/b55_scientific_source_delta.json and artifacts/research/b55_final_validation.json. The remaining proof needs inductive nonconvex coverage or a stronger coupled temporal barrier, with the original full-target and pre-first-exit quantifiers preserved. The focused shared-return and campaign suite passes 146 tests in 168.17 seconds, including 66 new cases; documentation integrity and lint checks also pass. These checks concern the implementation and its scoped proof, not empirical confirmation of the paradigm.