Maintained-parent SHA: grammatical closure and the changed complete law

The committed UM successor provides a concrete VAL,THOL,UM prefix with genuine parent history and a connected child. This study admits one parent-target SHA closure, accounting for its capacity reset, latency metadata and complete subsequent law. The research program owns task status.

SHA is a deliberately simple control for isolating capacity effects. The canonical grammar admits other closure roles, including OZ, NAV and REMESH. A shared role does not make their effects equivalent or allow automatic substitution at this prefix: the whole word, live preconditions, reset, history and successor law require their own admission. No unique, preferred or universal closing operator is derived.

1. Source, identity and execution boundary

The source reader verifies the retained UM transaction, its original 25/27 assessment and the separate 27/27 review. It re-admits the actual THOL source and rebuilds the UM prediction and endpoint checks from primitive records. Archive receipts associate bytes; they do not independently authenticate acquisition history. A mutable live endpoint must still match freshly re-admitted primitives. The reader executes no event, pressure refresh or trajectory.

Retain all 21 nodes, the 53 undirected positive-conductance edges, node and neighbor order, signed form x, primitive circular phase with its declared real lifts, positive capacities nu and the structural clock. Parent 2 has actual prefix VAL,THOL,UM, zero U2 debt and its acquired two-sample history. The child 2_sub_0 retains its identity, parent association and missing temporal baseline. SHA targets only parent 2. Membership in the new support does not make the child an additional word target or give it a copied history.

The word continues existing form under the core initial_epi_nonzero=True context; this is not an initialization-from-empty claim. All strict live gates remain enabled, with grammar rejection by raising. No IL intervention or alternative closure is inserted into the reserved composition.

The instantaneous storage is

\[ H(x,\theta;W)=\frac12\sum_{\{i,j\}}W_{ij}(x_i-x_j)^2 +\sum_{\{i,j\}}[1-\cos(\theta_j-\theta_i)]. \]

Before and after the event, the chosen complete segment law is

\[ p_i=\frac{S_i(\theta)}{\pi d_i},\quad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\quad \dot x=MS(\theta),\quad \dot\theta=ML_Wx,\quad M=\operatorname{diag}\frac{\nu_i}{\pi d_i}. \]

Each segment holds its own capacities, support and conductances fixed, with zero forcing and loss. This detached weighted two-row model differs from native relational execution and the general EPI-only event runtime. The shared atomic stage executes SHA; detached field admission supplies the complete continuation contract, without advancing either continuous row.

2. Default reset and exact arithmetic

The prediction owner resolves the actual runtime factor through the shared factor registry. The source has no SHA override. Its represented default is

\[ f=\frac{518151819521135}{562949953421312} \approx0.9204225284540524. \]

This is the implementation's represented default associated with 1-1/(4*pi), not an assertion of exact equality to that transcendental expression. The factor 1/2 used in an earlier contact study does not transfer. For parent capacity nu_2^-, distinguish the ideal exact product and the binary64 product admitted by the shared scalar kernel:

\[ \widehat\nu_2^+=f\nu_2^-,\qquad \nu_2^+=\operatorname{fl}(f\nu_2^-),\qquad r=\frac{\nu_2^+}{\nu_2^-}. \]

On this source the represented product changes approximately 117.87376674577816 to 108.49367042655233. Its absolute error relative to the exact product is approximately 1.2901876873056387e-15. All other capacities are unchanged. The represented ratio r, not an unrounded f, governs identities for the represented successor state.

SHA leaves form, phase, support and conductances untouched. Consequently H^+-H^-=0 exactly from identical storage arguments. Subtracting separately rounded storage enclosures would not establish this identity. Stored pressure also remains unchanged during SHA. In this particular sine law, the independently specified fresh pressure depends on phase and bare support, so its constitutive value is unchanged as well. General pressure mixtures that consume capacity do not inherit that last conclusion.

The public SHA lifecycle writes its declared latency snapshot, including preserved form and a wall-clock timestamp. This timestamp does not advance structural time; latent=True installs no continuous stopping rule.

3. Changed mobility, means and continuation

With positive represented capacity, only the parent coefficient changes:

\[ M_2^+=rM_2^-,\qquad \rho_2^+=\rho_2^-/r, \qquad \rho_i=\frac{d_i}{\nu_i}. \]

At the event's common x and theta, both parent evolution rows multiply by r. All other instantaneous rows remain unchanged. On this source the predicted parent form rate changes approximately 0.01647756447 to 0.01516632156, and phase rate 0.3963025696 to 0.3647658132. Both remain nonzero. The same instantaneous neighbor rates do not imply unchanged future motion, because later phase and form evolve through the altered coupled field.

For either retained coordinate vector y, let m^-=sum(rho_i^- y_i)/R, R=sum(rho_i^-) and a=rho_2^-(1/r-1). Then the reset changes the corresponding mean by

\[ m^+=m^-+\frac{a}{R+a}(y_2-m^-). \]

This is reweighting with unchanged coordinates, not an unrecorded form or phase jump. The phase mean uses the declared real lifts, including the unchanged negative lifts at nodes 10 and 11. Each mean is conserved on its own positive-capacity held-law segment; a common value across the event is not generally conserved. H remains the same storage functional, with the usual cancellation between both complete rows on each segment.

The general scalar SHA contract allows intentional zero attenuation or zero source capacity, while rejecting a positive product lost to represented zero. That boundary is outside this selected positive-capacity continuation. If the stated reciprocal two-row law is explicitly extended to nu_2=0, both parent rows vanish because M_2=0, and the H cancellation still holds. However, rho_2=d_2/nu_2 and the positive-weight mean theorem are unavailable. The nodal identity alone would freeze only the unforced form row, not every coordinate or boundary flux of the rest of the network.

The new parent capacity lies strictly outside the narrow parameter family of the UM half-unit certificate. Parent-only attenuation is not a common clock rescaling. Equal instantaneous geometry, unchanged H and grammatical closure therefore do not transfer that certificate or its turning time to the post-SHA law. Fresh law-specific retention needs its own proof; an additional trajectory is not automatically required by admission alone.

4. Reserved closure result

The protocol and primitive source with prediction were frozen before one attempt, with 823 Python/configuration/instrument source members and receipts for all 702 prior artifacts. The full source archive includes uncommitted implementation. The target is parent 2, with actual seed 17, unchanged defaults and structural time 7+2^-19. The latency timestamp 2030-01-02T03:04:08+00:00 is declared event metadata. The budget is one event, one attempt and zero trajectories.

The primitive response and assessment retain one completed canonical atomic SHA with 19/19 checks. The raw record is durable before pressure refresh, and the response precedes assessment. The source used by the checker is freshly re-admitted UM evidence, not the already-mutated execution graph. No response, factor or threshold is retuned.

All 21 forms and phase lifts, the 53 edge payloads, node/neighbor order and hierarchy remain unchanged. Only the parent's capacity takes the independently predicted represented product, with error below the frozen 2^-40 allowance. Raw pressure remains unchanged, and the separate shared refresh satisfies its pressure-only boundary and independent 2^-46 error allowance. The actual full rows and weighted means match the rebuilt continuation.

The actual parent word is VAL,THOL,UM,SHA, with zero U2 debt, preserved temporal samples and the exact five-field SHA latency overwrite. The child receives neither an executed glyph nor invented temporal evidence. Structural time is unchanged. Alias maxima, graph context and final atomic-stage metadata satisfy their declared contracts. The actual storage jump is exactly zero from unchanged primitive storage arguments.

This closes one selected grammatical composition through an admitted sequence of event/complete-law interfaces. It does not select SHA over other closures or imply that every grammatical word preserves one object identity. The post-SHA instantaneous geometry is admitted, while the earlier UM half-unit certificate remains outside its changed-capacity domain. No post-SHA motion, new U5 observation or cross-birth U6 certificate is inferred from this event assessment. The separate continuation below supplies its own finite proof.

5. Verification boundary

The source controls reject altered primitive histories, endpoint state and archive associations. Scalar and law controls independently check the exact product, representation boundaries, both rows, reciprocal-capacity means and exclusion from the old certificate family. Executor controls use an unrelated graph to check actual atomic-stage semantics, declared latency and history, support preservation and pressure-only handoff. They do not evaluate the reserved source. The retained-response reader independently reconstructs the actual product, complete rows, means and zero-work arguments while forbidding operator, refresh and trajectory replay.

The distinction is structural: grammar admits a composition; the event changes its declared coordinates and context; a complete law defines the subsequent motion; an additional certificate establishes retention over a specified interval. None of those steps alone establishes autonomous occurrence, universal organization, a particle interpretation or a physical measurement bridge.

6. Complete-law retention after the actual closure

The continuation question fixes the actual represented endpoint, all 42 coordinates and 53 original branches, with elapsed structural horizon T=1/2. The committed-SHA reader rebuilds all nineteen event checks through the actual UM source. The motion adapter then rebuilds the full held field and static bounds from those freshly admitted primitives. Cached predictions, reports or mutable live graphs cannot supply this admission. The source is a point in represented coordinates and parameters; it does not inherit the earlier hypothetical UM family's independent 2^-40 allowances.

For each original oriented edge e=(i,j), retain its initial integer k_e and the real-lift difference delta_e=theta_j-theta_i-2*pi*k_e. The observation is strict |delta_e(t)|<pi throughout the window. Endpoints are never wrapped into a newly chosen branch. This preserves the declared principal geometry; it does not reinstate the old twenty-node radius identity displaced by UM.

The shared energy-bound owner recomputes H, mobility and per-edge speed bounds under the post-SHA law. Writing sigma_i=sum_j W_ij, its Cauchy--Schwarz coefficient is

\[ a_e=M_i^2\sigma_i+M_j^2\sigma_j+2M_iM_jW_{ij},\qquad |\dot\delta_e|\le\sqrt{2H a_e}. \]

Conserved H and nonnegative phase storage justify this bound on the whole held-law flow. The initial margin minus T*sqrt(2H*a_e) proves 41/53 branches, with minimum certified margin above 0.593. Exactly the twelve parent-incident edges remain unresolved. For the child-parent edge the energy speed bound is about 298.366, while its actual initial speed is about -5.34632; discarding dynamical correlations makes the global bound too wide. A negative remaining margin proves no crossing. Fresh initial speed and global acceleration bounds independently retain all branches through 1/25, but this shorter window does not replace the fixed target.

The shared retention loop therefore admits one fresh full-state Picard/Taylor calculation. Both this consumer and the older UM adapter use that loop; the older adapter retains its separate velocity-reversal target and historical record schema. No old derivative jet, remainder, tube or endpoint is reused for the new law. The shared held-field and validated Taylor owners remain unchanged.

Each recorded step combines a whole-time tube margin with the independent fixed-window energy margin for the same edge by taking their maximum. Both bound the same actual branch throughout that step. Strict Picard inclusion establishes a valid whole-step enclosure; Taylor arithmetic provides the carried endpoint. The field is smooth beyond principal cuts, so a wide tube does not invalidate the global energy alternative. The source retains actual node and neighbor order, positive capacities, conductances, zero forcing/loss and the original clock.

The numerical budget is fixed at 256 steps of 1/512, order 12 and outward dyadic 128-bit arithmetic, with 900 wall seconds checked between steps. Preparation is included in that resource budget. Each record is saved before the next step. An unavailable enclosure, unresolved branch margin or resource stop retains its completed prefix and forbids a retry or parameter change. No velocity reversal, turning time, new event or IL intervention is a target.

6.1 Reserved retention result

The protocol and actual source with rebuilt prediction were frozen before one attempt. The archive retains 826 source/configuration/instrument members and receipts for 711 unchanged earlier artifacts. No reserved producer was replayed.

The result accepts 256/256 steps and certifies all 53 original principal branches throughout elapsed structural [0,1/2]. The minimum certified margin is the exact dyadic lower bound

\[ \frac{538123979266280542631226827700816151763} {340282366920938463463374607431768211456}>1.58. \]

The reported wall time, including preparation, is about 588.68 seconds, within the fixed budget. All 256 step archives retain their initial box, whole-time tube, Taylor series and remainder, carried endpoint and branch bounds. No mesh, threshold, capacity or target was changed after evaluation. This result closes finite retention under the actual capacity-changed law; it is not an application of the older UM certificate to new parameters.

The retained-evidence audit re-admits the actual source, checks every strict Picard inclusion under its field, reconstructs Taylor endpoint arithmetic and every branch margin, and rebuilds the accepted prefix and summary. Altered primitives, endpoint carry, chronology and cached verdicts reject. Recorded derivative jets and remainder bounds retain the frozen producer's provenance; the read-only audit does not independently regenerate those coefficients or evaluate another trajectory. Hashes establish byte association, not independent acquisition authentication.

Thus an actually executed selected word has a separately certified finite complete-law continuation preserving its declared principal geometry. This does not establish an open preparation family, indefinite confinement, periodicity, autonomous birth or physical identification. It also installs no runtime time history for the child: mathematical enclosures are not observed samples, and the parent's history does not become the child's. The separate child temporal analysis rebuilds a genuine short-window capability from this source while retaining its independent strict-form, measured-history and U4b boundaries.