One admitted collective Emission--Silence word

This result combines the exact unordered pair state, the primary reset and identity barrier and the partial-transition criterion. It proves one supplied, finite collective intervention on an explicit source domain. Its execution path is the canonical selected-target snapshot stage, not an arbitrary public callback environment or a new coarse-node runtime.

1. A nonempty source domain with automatic geometric and live admission

Use the ten-node doubled C5, with two interchangeable members in each pair k=0,...,4, all four edges between neighboring pairs, and unit undirected support. Keep the conservative normalized-sine complete law e=0,w=beta=1 and the original structural clock. The independently specified rows are

\[ p_i=\frac1{4\pi}\sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad \dot x_i=\nu_i p_i,\qquad \dot\theta_i=\frac{\nu_i}{4\pi}(L_fx)_i. \]

Capacity is held between events, not derived from the observed response. Write theta_*(k,+/-)=2*pi*k/5 on declared lifts. For the arithmetic fine form mean m, phase offset gamma and centered deviations xi,eta, put

\[ x=m\mathbf1+\xi,\qquad \theta=\theta_*+\gamma\mathbf1+\eta,\qquad \mathbf1^T\xi=\mathbf1^T\eta=0,\qquad Z_f^2=\|\xi\|^2+\|\eta\|^2. \]

Fix the following source domain and event parameters:

\[ \boxed{ \frac9{32}<m<\frac{11}{32},\quad Z_f<z=\frac1{512},\quad \frac12<\nu_{k,+}=\nu_{k,-}=v_k<2,\quad b=\frac1{1024},\quad a=\frac12,\quad r=\frac1{16}. } \]

The phase offset may vary in any declared open interval on its circular chart. This is a nonempty positive-width domain relative to the supplied support and equal-member-capacity constraints: the exact twist with any interior m,v_k,gamma belongs, and all sufficiently small independent centered form/phase perturbations belong. Internal pair differences need not vanish. This is a prepared family, not a source-to-identity acquisition claim.

The action mark fixes one whole pair, common boost b, common attenuation a, and the ordered word emission,silence. Select the explicit compatibility_profile="core", now also the default; explicitly selected legacy rejects this pair through its historical adjacency preference. Admit the whole word with the shared validator, then pass its ValidatedSequence.step contexts to execute_network_operator_stage. Set GRAMMAR_REJECTION_MODE="raise" and use the canonical Emission and Silence classes without replacement or subclass hooks.

For this theorem use hard EPI rails [-1,1], EPI_LATENT_MAX=1, VF_BASAL_THRESHOLD=SHA_MIN_VF=1/8, MIN_NETWORK_DEGREE_COUPLING=1, and VALIDATE_OPERATOR_PRECONDITIONS=True. Supply the two stated factors through the shared factor resolver. Set GLYPH_HYSTERESIS_WINDOW=W for a fixed integer W>=2; supply a valid fixed integer execution seed, although neither action draws randomness. Structural time is a fixed finite real during the word. Each stage receives its own fixed, nonempty timestamp as an external event input, common to its two targets. Comparing separate wall-clock calls without matching those inputs is not a deterministic lifecycle comparison.

Every source form satisfies

\[ \frac{143}{512}<x_i<\frac{177}{512},\qquad \frac14<x_i<x_i+b<\frac12. \]

Consequently AL satisfies the strict latent-form threshold and both capacity preconditions have a strict margin. Every fine degree is four. The AL proposal is unclipped and strictly increasing, and SHA leaves the selected capacities in (1/4,1). These inequalities give the actual live-domain intersection; admission is not an additional unverified predicate.

2. Retained execution information and the allowed lifecycle stratum

The physical collective state is the existing (X,Theta,R,U,Q) chart with the five paired capacities. Execution additionally retains the following associated constituent records. This restricted stratum is sufficient for the claimed word and is not asserted minimal for every registered action.

For each selected constituent choose an integer n_i>=0. Its source operator-step index is n_i, trace is the last min(n_i,W) copies of AL, U2 debt is zero and lifetime prior-IL flag is false. If n_i=0, the emission activation, timestamp, origin and lineage fields are absent. If n_i>0, the activation flag is true and the emission timestamp, emission origin and lineage origin are one declared nonempty string o_i; the mutable lineage contains activation_count=n_i, derived_nodes=[] and parent_emission=None. The source has no latency fields. Different members may have different n_i,o_i; these records must remain associated with their physical constituent under a swap. They are supplied provenance, not evidence that a continuous sine trajectory executed those histories. Mutable histories and lineage containers are owned separately by each constituent, without cross-node aliases. Copying their values while retaining a shared mutable container would define a different execution model.

Use ordinary graph/node/edge mappings with valid canonical scalar channels. Any retained stored pressure is transported with its node. It is not used as a substitute for the complete law's evaluated pressure. Unselected nodes and any inert retained data are unchanged. There is no graph monitor, pressure callback, operator metrics, optional held-step nodal comparison, user callback, mutable external resource, or optional jump certificate in this execution claim. Initially the recognized-pattern sink is absent or an empty list. Support, node inventory and the declared ordering convention are fixed. This specifies a supported runtime configuration rather than disabling the hard grammar or live preconditions.

Retain the actual graph stage diagnostic, _last_operator_applied and the source-glyph alias written by each stage. Rebuildable maximum caches, weak node wrappers, transaction snapshots and temporary proposal objects are implementation state, not additional primitive NFR coordinates. The claimed observations are nodal state, associated lifecycle/history, fixed support, stage status and reset work. A separately exposed cache witness or order-sensitive event stream requires its own transported mark or quotient. In particular, a tied maximum does not select a distinguished physical constituent. Cache correctness and availability remain implementation obligations of the shared setters. This execution domain admits absent caches with canonical reconstruction, or warm maximum caches maintained under the declared shared-setter mutation scope, and only correctly graph/node-bound adapters. Arbitrarily stale warm caches or incorrectly rebound adapters are unsupported execution context. They do not shrink the mathematical source domain, but cannot certify a complete runtime observation without repair and re-admission.

3. Explicit prefix maps and descent of the complete declared word

Let j=0,1,2 denote the source, post-AL and post-AL/SHA strata. The execution description is the simultaneous pair-swap orbit of physical state and associated records, together with j, configuration, action mark and event timestamps. It is not the product of a physical multiset and a separate history multiset. Equivalently, supplement the physical pair coordinates with the orbit of the records attached to their two reconstructed members. At a synchronized physical tip the records can still differ.

The AL stage reads one immutable source and proposes both targets before committing either. On each selected member it writes x_i+b, leaves phase/capacity/stored pressure fixed, and performs precisely these lifecycle and history updates:

• If n_i=0, initialize the emission timestamps and origin to the supplied AL timestamp and lineage count to one. Otherwise preserve those origins and increase the lineage count from n_i to n_i+1.
• Append AL with the declared window, set the operator-step index to n_i+1, preserve debt zero and prior-IL false, and set source glyph AL. No source latency requires clearing and no latency warning is emitted.

The next SHA stage reads the committed AL state. It writes capacity v_k/2 on both selected members and leaves form, phase and stored pressure fixed. It sets latent=True, latency_start_time to the supplied SHA timestamp, preserved_epi=x_i+b, silence_duration=0, and was_initial_on_silence=False; the last value follows from x_i+b>1/4. It appends SHA, advances the operator-step index to n_i+2, retains the emission lineage, debt zero and prior-IL false, and sets source glyph SHA.

At every prefix, the last two recorded glyphs are AL/AL or AL/SHA when two exist. Neither is an IL pattern or antipattern in the shared recognizer, so there is no recognized-pattern append or related warning. U2 remains zero because AL and SHA are neutral in the registered debt accounting. AL is an admitted initiation and SHA an admitted closure; neither requires the U3 coupling gate, U4 transformation context, a recursion-depth witness, or a U6 monitor reference on this path. The whole-word core validation is therefore consistent with the independently checked live prefixes.

In exact real arithmetic the collective primary maps are

\[ \begin{array}{c|ccccc|c} &X_k&\Theta_k&R_k&U_k&Q_k&v_k\\\hline \mathrm{AL}&X_k+b&\Theta_k&R_k&U_k&Q_k&v_k\\ \mathrm{SHA}\ \text{after AL}&X_k+b&\Theta_k&R_k&U_k&Q_k&v_k/2. \end{array} \]

All other physical pairs remain unchanged. For each member the metadata map uses only its own associated source record, the shared parameters and the shared stage timestamp. Swapping a pair therefore swaps its proposals, its admission inequalities and its complete selected nodal output. The targets are the whole pair, so their transported set is unchanged. Disjoint target writes do not depend on the other target's committed value. The stage status is the same two-phase schedule with two processed targets. Thus both admission and projected successor are constant on each joint orbit at j=0, and the same argument holds on the reached j=1 stratum. The ordered-language criterion now gives an exact collective partial action for the prefixes empty, AL, AL,SHA, with their declared lifecycle outputs. This verifies its hypotheses by the actual maps and inequalities; it does not merely assume equivariance or infer it from a sampled swap.

This is a typed, two-step domain. It does not establish closure under arbitrary repetitions, other histories, every operator, optional monitors or opaque extensions. In particular, it proves no arbitrary graph-wide bytewise relabeling of cached identifiers. If a represented source fails admission, report that prefix and its state; a successful AL followed by a rejected SHA is not a completed word. The stage transactions do not supply whole-word rollback or rollback of external effects.

4. Work and all-time identity of the postword continuous law

Let H=x^T L_f x/2+V(theta) and let H_* be its winding-one target value. The largest fine Laplacian eigenvalue is 5+sqrt(5)<8: the pair-constant subspace has spectrum twice that of C5 and the antisymmetric subspace has eigenvalue four. The target is a phase critical point. Along the straight segment of declared phase deviations, the potential Hessian is B diag(cos(edge difference)) B^T <= L_f. Taylor's integral remainder therefore gives the uniform upper bound

\[ H-H_*\le\frac{\lambda_{\max}(L_f)}2 Z_f^2<4z^2. \]

No unmeasured source energy ceiling is needed in addition to the stated norm domain. For the whole-pair increment h=b*1_pair, the existing exact work identity is

\[ \Delta H=(L_fx)^T h+\tfrac12h^TL_fh =4b\sum_{\ell\sim k}(X_k-X_\ell)+4b^2. \]

It is invariant under the pair swaps and is thus a well-defined collective event cost in this structural storage. It can have either sign on the source domain. No external reservoir or thermodynamic meaning is inferred. The graph constants ||P_f h||=sqrt(8/5)*b, ||L_f h||=4*sqrt(3)*b give

\[ Z_f^+<\frac1{512}+\frac2{1024}=\frac1{256}<r, \qquad H^+-H_*<4z^2+7bz+4b^2=\frac{17}{524288}. \]

For r=1/16 the barrier coefficient

\[ \kappa_f=\frac{5-\sqrt5}{2} \cos\left(\frac{2\pi}5+\frac{\sqrt2}{16}\right) \]

satisfies kappa_f>1/8. One entirely rational lower estimate uses 3<pi<22/7, sqrt(2)<3/2, sqrt(5)<9/4. The complementary angle pi/10-sqrt(2)/16 exceeds 1/5; concavity gives sin(t)>=2*t/pi on [0,pi/2], so the cosine exceeds 7/55 and kappa_f>7/40>1/8. In particular

\[ H^+-H_*<\frac{17}{524288} <\frac1{16384}<\frac1{2048}<\kappa_f r^2. \]

SHA changes neither H nor Z_f and keeps positive equal capacities within each pair. The existing capacity-independent first-exit argument therefore traps the entire postword fine state in the acute winding-one tube for all subsequent times under the declared held-capacity conservative law. All admitted pair charts remain nonantipodal. This is geometric identity retention, not waveform recovery, an activity lower bound or autonomous acquisition.

The postword weighted form mean uses rho_i=4/nu_i^+. At the word's endpoint every form lies in (1/4,1/2), so its positive weighted mean also lies in that open slab. This supplies the mean condition of the existing invariant family with excess ceiling 1/16384; the arithmetic source mean is not claimed conserved after heterogeneous capacity changes. Continue both original rows with the new capacities and the original clock, without clipping or additional events. The initial binade restriction need not hold throughout that continuous continuation. Here pressure depends only on phase, so these two resets do not change its exact instantaneous law value, while AL changes the phase rates through L_f x. Subsequent pressure must still be evaluated from the evolving phase. Stored pressure or a one-row continuation cannot replace that complete law.

5. Exact real domain, represented execution and remaining scope

The open-domain theorem concerns exact real state. Binary64 values are a discrete set; sampling them does not prove openness. Conversely, a source already materialized in binary64 and independently certified inside the stated domain has particularly simple primary arithmetic:

1. Every source form and its boosted result lie in the same binade [1/4,1/2), whose spacing is 2^-54. The boost 2^-10 is an integer multiple of this spacing. The exact sum is representable within the same binade, so round-to-nearest addition introduces no AL defect.
2. Halving a normal capacity in (1/2,2) is exactly representable and remains normal in (1/4,1). SHA cannot underflow on this domain.

Thus this represented subdomain realizes the same primary resets, without pretending every requested boost or attenuation is exact elsewhere. Admission concerns the materialized source; converting an arbitrary rational or decimal input may change it. Trigonometric pair coordinates, source norm relative to the irrational twist, storage and barrier verdicts retain independently checked enclosures. Equal floating summaries do not authenticate a joint physical/history orbit. Outside the binade domain use actual represented increments and the general reset identity instead.

The static primary controls and selected-stage controls exercise separate existing boundaries; their earlier finite fixtures do not by themselves prove this source-family theorem. Further consumers must re-admit the complete state and configuration described here. Optional pure-EPI gain certificates use another storage and all-node target premises; their selected-pair abstention does not refute this sine identity barrier.

The word-domain controls separately certify a fresh-context represented member of this source domain and exercise the whole runtime word. Their broader prespecified family also probes other means, factors and associated histories. Those additional finite cases do not enlarge this theorem's exact-real or binade domain.

The result supplies an admitted ordered TNFR intervention on an existing mathematical object, including associated history, reset work and continued identity. It neither derives the two event times, target, amplitude or occurrence rule nor selects the grammar policies as fundamental natural laws. The positively dissipative acquired C9 objects, delayed-memory maps and unit-capacity global polynomial retain their own model premises. The execution plan alone determines what investigation follows.