Directional maintenance after a complete form-capacity word

The form-capacity analysis showed why a scalar storage sublevel cannot decide retention of the specified local resets. This owner supplies an exact reduction of the retained source and a conditional nonlinear enclosure method. The research program alone selects further investigation and its dependencies. No change of nodal law, radius, source origin or operator gains is part of this comparison.

1. Actual state and the complete word

Use the OZ_completed primitive state and associated execution context from the retained IL/OZ response. Keep the twenty nodes, all 45 bare edges, actual conductances, signed form, positive capacities and primitive phase. The starting structural time is 7. The source has nine exact twin equalities, but nodes 0 and 1 have different forms and need not have the same boundary weights. The generic full-source neighborhood does not share the nominal source's exact symmetry.

The selected bounded extension is contraction,coherence,silence on ordered targets (0,1). NUL uses its default factor; IL and SHA retain the source's configuration, including the supplied SHA_vf_factor=1/2. This is not a pair of reciprocal scale maps. NUL's stored-pressure rescaling, IL's phase reset, SHA's capacity change, live histories and every pressure refresh belong to the full preparation. A result for the isolated primary NUL map cannot be substituted for this word's endpoint.

Read-only sequence admission retains the existing OZ debt of one on each target. The proposed debt progression is 1 -> 1 -> 0 -> 0: NUL supplies no stabilization, IL pays the retained debt, and SHA closes the word. Shadow grammar checks do not execute the word or admit its later physical states; each actual stage must still pass live preconditions without replacement.

Canonical U6 is separately observed from before/after structural potential snapshots, using the shared full-graph distance convention and the declared drift threshold. It is not a token-order rule or a future confinement theorem. Record its baseline, raw/refreshed stored-pressure convention and finite coverage; a passing sequence does not imply a passing U6 observation.

The retained-state reader reconstructs explicitly recorded persistent inputs, including bounded history deques, attribute associations and neighbor order. It neither authenticates a previous execution nor invents omitted cache state. Recorded rebuildable caches remain absent until their owners rebuild them. This is a new declared preparation from retained inputs, not an unrestricted checkpoint facility.

Between events use the same complete conservative law and original clock:

\[ \dot x_i=\frac{\nu_i}{\pi d_i} \sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \dot\theta_i=\frac{\nu_i}{\pi d_i} \sum_{j\sim i}W_{ij}(x_i-x_j). \tag{1} \]

Support, conductances and post-word capacities are held, and forcing and continuous loss are absent. The supplied operator word does not establish an autonomous event-selection law. Future phases are continuous lifts, not modulo-wrapped jumps between integration steps.

2. An exact eleven-cell reduction

Let sigma fix nodes 0 and 1 and exchange each pair (2,3),...,(18,19). Admit the complete bare support, each conductance, held capacities and initial form/phase lifts under this involution. Exact equivariance and smooth uniqueness then preserve its fixed-point family under (1). Event preservation is separate: re-admit membership and symmetry after the complete word.

There are eleven cells: two singletons and nine pairs. Let m_a be their multiplicities, n_ab the number of neighbors in b of one member of a, and w_ab its sum of conductances to b. These are per-member quantities. Internal neighbors still contribute to the original degree d_a, even though their form and phase differences vanish on the fixed family. Zero-conductance edges still contribute to n_ab and d_a. Reciprocity gives

\[ m_a n_{ab}=m_b n_{ba},\qquad m_a w_{ab}=m_b w_{ba}. \tag{2} \]

The reduced field is exactly

\[ \dot X_a=\frac{\nu_a}{\pi d_a} \sum_b n_{ab}\sin(\Theta_b-\Theta_a),\qquad \dot\Theta_a=\frac{\nu_a}{\pi d_a} \sum_b w_{ab}(X_a-X_b). \tag{3} \]

This is an exact 22-coordinate representation of the specified invariant family. It is not an eleven-node simple graph with recomputed degrees, nor an identification of eleven independent collective NFRs. Reconstructing fine observables requires multiplicities and the retained member association. In particular the unchanged identity is

\[ Z^2=\sum_a m_a(X_a-\bar X)^2+ \sum_a m_a(\Theta_a-\Theta_{*,a}-\bar\eta)^2, \quad \bar X=\frac1{20}\sum_a m_aX_a, \quad \bar\eta=\frac1{20}\sum_a m_a(\Theta_a-\Theta_{*,a}). \tag{4} \]

The radius remains r=1/16. An independent uncertainty box around the original fine source need not lie in this family; no robust full-family result follows merely from reducing its nominal member.

The weighted symmetry owner re-admits primitive source and law before constructing SineSupportInvolution. It distinguishes invariant family, source membership and trajectory reduction. Cached verdicts cannot replace these premises. Its interval/Taylor field reuses the shared sine rate evaluator, retaining the actual normalization.

3. A metric that retains directional error

Let K be the symmetric reduced form Hessian: off-diagonal entries are K_ab=-m_a w_ab, with row sums zero. Let C_0 be the reduced phase comparison Laplacian whose fine old edges have coefficient 309/1000 and whose five new edges have coefficient one. The true reference coefficient on old edges is cos(2*pi/5), strictly between 309/1000 and 310/1000. C_0 is a rational comparison metric, not a replacement of the sine field in (3).

Define

\[ \rho_a=\frac{m_a d_a}{\nu_a},\quad R=\sum_a\rho_a, \quad B=\operatorname{diag}\frac1{\pi\rho_a},\qquad Q=\operatorname{diag}\left(K+\frac{\rho\rho^T}{R}, C_0+\frac{\rho\rho^T}{R}\right). \tag{5} \]

Connected positive support makes each Laplacian positive on relative coordinates; the added rank-one term removes the common null direction. Thus Q is positive definite, including common origins. The two rho-weighted means are constant along the continuing law, and rho^T B=1^T/pi, so the added terms contribute zero to the field's metric-Jacobian balance. They make the error metric invertible without changing the actual dynamics.

For the true phase Hessian C(Theta), the Jacobian is

\[ J=\begin{pmatrix}0&-B C(\Theta)\\BK&0\end{pmatrix},\qquad QJ+J^TQ= \begin{pmatrix}0&KB(C_0-C(\Theta))\\ (C_0-C(\Theta))BK&0\end{pmatrix}. \tag{6} \]

The large conservative tangent terms cancel. Only the change of the phase Hessian relative to C_0 enters the error-growth estimate. This is why a directional enclosure can remain useful after the scalar energy-sublevel certificate has failed.

On the larger validation domain Z<1/8, every fine edge perturbation from its reference gap has magnitude below sqrt(2)/8<5/28. The cosine Lipschitz bound and 0<cos(2*pi/5)-309/1000<1/1000 give the relative old-edge bound

\[ \frac{5/28+1/1000}{309/1000}<\frac35. \]

For new zero-reference-gap edges, 1-cos(delta)<=delta^2/2<1/64. Consequently

\[ -\kappa C_0\preceq C(\Theta)-C_0\preceq\kappa C_0, \qquad \kappa=3/5. \tag{7} \]

Actual conductances and capacities must not exceed one for the following comparison. The normalized-Laplacian upper bounds imply ||K^(1/2) B C_0^(1/2)||<=2/pi. Combining (6)--(7) and the quadratic inequality 2ab<=a^2+b^2 yields the logarithmic norm bound

\[ \mu_Q(J)\leq\kappa/\pi<1/5. \tag{8} \]

This bound applies to the true nonlinear Jacobian throughout the admitted convex domain, not just the stationary tangent. It does not assert that Q is conserved or that the complete flow contracts. It permits the existing metric Taylor validator to carry one full uncertainty ball with growth rate 1/5 rather than replacing the carried uncertainty by independent coordinate boxes at each step.

4. Bounded nonlinear evaluation and its possible conclusions

The numerical comparison uses the exact reduced nonlinear field (3), a Picard tube for each whole interval, order-12 interval Taylor bounds and the shared outward 128-bit arithmetic. The fixed mesh has duration 1/4 and 160 steps, covering elapsed structural time [0,40] after the word at time 7. This is [7,47] in the retained absolute clock; operator counts are not elapsed seconds. Source, laws, observation, horizon, precision and failure policy must be frozen before the response is evaluated.

The horizon has an independent scale justification. An untouched internal twin-antisymmetric tangent direction has form and phase Hessian eigenvalues 4 and 4*cos(2*pi/5), and mobility 1/(4*pi). Its angular frequency is sqrt(cos(2*pi/5))/pi, with period 2*pi^2/sqrt(cos(2*pi/5))<40. This calibrates a nontrivial structural horizon without looking at a response; it does not make the actual nonlinear trajectory periodic or transfer that mode's dynamics to the excited directions.

The validation radius 1/8 and observation radius 1/16 have separate roles. The former supplies a domain for (8); it does not relax the identity target. Whole-step bounds Z^2<1/256 prove finite maintenance on the covered intervals. A certified endpoint lower bound Z^2>1/256 proves that this particular identity has been left by that time. A straddling interval, failed Picard tube or exhausted fixed budget remains unresolved. None may be turned into a dynamical failure or a passing flag. Finite maintenance is not all-time stability, formation or autonomous invocation of the word.

The existing scalar-sublevel obstruction remains valid. A successful directional enclosure would answer a different question using more information: the complete initial direction, actual nonlinear evolution and certified accumulated error, rather than all states sharing a scalar storage bound and two means.

5. Frozen complete-word result: finite departure

The protocol, source freeze and archived implementation preceded the single evaluation. Its response and verdict retain all three executed stages and 57 certified continuation steps. The prospective stop rule ended the calculation at elapsed time 57/4, before the maximum forty-unit horizon, because the endpoint was provably outside the unchanged identity radius. No numerical retry, alternate word, radius change or source/gain adjustment was performed.

Grammar and events. The actual ValidatedSequence and all six live target-stage checks passed. Each atomic stage executed its requested glyph with strict preconditions and no replacement. Both targets retained the debt progression 1 -> 1 -> 0 -> 0; IL compensated the inherited OZ debt. All eight canonical U6 comparisons passed on their declared snapshots. The complete-word mean absolute potential drift was approximately 1.137030752337261e-5, below the represented pi/2 threshold. These are finite stored-state observations, not U6 bounds on the continuing trajectory.

Future identity. Whole-interval bounds certify Z<1/16 through elapsed time 113/8. At time 57/4 the outward endpoint bounds imply

\[ 0.003923728663717655 < Z^2 < 0.003923728663717657, \qquad Z^2>1/256=0.00390625. \]

Equivalently 0.06263967324082760 < Z < 0.06263967324082762, whereas the frozen radius was 0.0625. Continuity therefore places the first boundary hit in elapsed (14.125,14.25), or absolute structural time (21.125,21.25). The final metric error radius is below 1.54e-22; the departure margin is not an unresolved enclosure overlap.

This rejects the specified finite-retention hypothesis for this complete word and nominal preparation. It does not establish destruction of the organization, loss of constituent winding, eventual escape from every neighborhood or failure of the TNFR operator contracts. A valid grammar word and passing finite U6 observations do not imply preservation of this finer geometric identity under the chosen subsequent law. Conversely, an isolated NUL reset or a different complete word cannot inherit this trajectory verdict.

The read-only evidence tests verify the frozen source associations, reconstruct primitive event states, recheck live grammar/debt and U6, rebuild the exact reduction/metric and recompute endpoint identity bounds from retained centers and error radii. They execute neither the reserved word nor the continuing trajectory.

6. Retained organization through the certified departure

The same evidence supports a separate interpretation without a new forecast or a changed identity criterion. Every certified Picard tube remains in Z<1/8 through elapsed 57/4. Therefore each fine phase-edge perturbation from the original reference has magnitude below sqrt(2)/8. Old reference gaps have principal magnitude 2*pi/5, and new reference gaps are zero; 2*pi/5+sqrt(2)/8<pi/2. All 45 edges consequently stay acute throughout that interval. No original cycle edge crosses a principal cut, so each constituent's originally oriented C5 winding remains +1. Held support, conductances and positive capacities are premises of this continuation; the exact eleven-cell symmetry is preserved by its law.

Let A and B denote the original ten-node constituents, and let eta=theta-theta_star use the same fixed lifts as (4). Define Z_A and Z_B by centering form and eta separately within each constituent. The exact variance decomposition is

\[ Z^2=Z_A^2+Z_B^2+ 5(\bar x_A-\bar x_B)^2+5(\bar\eta_A-\bar\eta_B)^2. \]

Reconstructing the final metric ball gives the following approximate contributions; the read-only evidence controls retain outward bounds.

Contribution at elapsed 57/4	Squared distance
Internal constituent A	0.002749906304
Internal constituent B	0.001020360841
Relative form means	0.000126069321
Relative phase-error means	0.000027392199

Each internal contribution is below 1/256 at this endpoint, while their sum with relative registration exceeds that threshold. The failed joint-tube claim thus coexists with retained acute geometry and winding. Neither a winding label nor these endpoint component bounds define a closed collective state or prove indefinite constituent maintenance. The separate identity-transition contract applies the existing operator and grammar descent criteria to this marked contact, with explicit source and successor domains. Its ideal event-sector and finite U6 results preserve this negative original observation and do not extend the certified future interval.