Native operator dynamics

This owner develops conditional collective descriptions of the general native operator runtime: fresh pressure and diagnostics, configured selection, live grammar, ordered direct glyphs, numerical integration and subsequent policies. The admitted source, complete law, consumed memory and structural clock remain explicit in each result. Restricted closure does not establish autonomous occurrence, formed collective identity or physical identification.

The structural operator contracts, collective action criterion and ordered-language descent own the underlying event and grammar obligations. The research program remains the sole research queue.

The conditional collective calculus combines the five admitted event/flow maps below and states their shared memory and branch obligations. Its finite retention criterion adds explicit selection/admission margins for unequal forms. These are partial mathematical descriptions, not new execution paths. The joint form/capacity extension retains the additional association consumed after unequal-capacity adaptation.

Native selector memory and realized occurrence

The native selector controls use the existing ten-node doubled-C5 support: each adjacent pair of two-member blocks has all four unit links. Every node has signed form x=1/4, capacity one, phase zero and freshly computed pressure zero. Fresh native Si>0.9 makes the actual DefaultGlyphSelector propose Coherence (IL) everywhere. The declared primitive batch path, topology/node order, seed, live grammar and structural clock t=1 are identical in the comparison. No continuous sine trajectory or phase coordinator runs; the pair-state reader supplies coordinates only.

Retain the supplied AL → IL → EN grammar context, including persistent debt and prior Coherence after bounded-trace eviction. All Reception lag counts start at zero. Change only node 0's pre-batch since_AL from zero to the default limit five. The actual native batch increments lag counts before testing the strict limit: one source executes Coherence at every node; the other executes Emission (AL) at node 0 and Coherence elsewhere. Live admission passes both requests. Thus identical instantaneous nodal coordinates, pair coordinates and base proposals do not determine the realized mark when supplied selector memory is discarded.

Let d be the actual represented positive Emission increment at node 0. The default boost keeps its form strictly inside the clamp interval. Coherence leaves this zero-pressure source's physical coordinates unchanged. Emission changes the first pair's form mean by d/2 and its squared internal form by (d/2)^2; other pairs retain their coordinates. These are exact identities for the represented endpoint values, not continuous-rate predictions. The structural clock stays at one. The histories and counters are admitted supplied state, not a proof that the declared continuous law acquired them.

A control changes persistent U2 debt from zero to two while retaining the same physical state and bounded trace. Both sources still execute Coherence: their successor debts are zero and one. Different eligibility information therefore need not cause different realized actions. This distinguishes the new actual selector witness from an admission-only counterexample.

Restricted sufficient policy memory. Suppose every base proposal is Coherence, the proposed Emission, Reception (EN) and Coherence actions pass live admission, no execution fails, and lag limits L,E are fixed nonnegative integers. For pre-batch counts a,e, the implemented priority and successor counters are

mark = AL if a >= L, else EN if e >= E, else IL
AL: (a,e) -> (0, min(e+1,E))
EN: (a,e) -> (a+1, 0)
IL: (a,e) -> (a+1, e+1).

After projecting the successor counts too, (min(a,L), min(e,E)) closes this finite counter/mark system. Indeed, all threshold decisions and capped successors depend only on those capped counts. This does not close the whole NFR state, prove minimal memory, or remove context and target associations. When grammar replaces a proposal, bookkeeping follows the action actually executed; the displayed recurrence applies only on its admitted branch.

One current eligibility bit is insufficient for repeated decisions. With the same explicit Reception limit eight, sources starting with Emission counts zero and one both execute Coherence for four batches, then Coherence and Emission on the fifth. The controls refresh the actual pressure and sense index before each decision and verify the common Coherence proposals. Counts measure selector batches, not elapsed structural time. A schedule linking decisions to time is an additional part of a complete hybrid law.

This result identifies consumed memory in an existing supplied controller. It does not derive the controller, its lag limits or Emission's form increment from the nodal identity. In particular, it does not transfer the closed sine law's conserved form to this hybrid path or establish physical particle formation.

Decision schedule and structural time

When apply_glyphs=True, the native runtime executes one selector batch at each outer invocation's left boundary, after pressure and optional sense-index refresh and before the nodal integrator. Its internal integration substeps do not request more operators. Thus a complete hybrid model includes the decision schedule as well as the state, selector, grammar context and continuous rows. The operator-step index, lag counters, runtime invocation ordinal and structural clock have different meanings; none can be reconstructed from another without additional premises.

The actual runtime controls reuse the uniform doubled-C5 source above, with all lag counters zero, Euler integration and no Gamma. Default phase/capacity policies remain enabled; uniform phase zero and capacity one are fixed by their actual updates. Default limits are five batches for Emission (AL) and three for Reception (EN). The supplied context admits Coherence (IL), Reception and Emission. No auxiliary math engine or scientific callback is supplied; the automatic network REMESH gate's default 25-sample stability window does not activate during this six-call comparison. That gate differs from the node Recursivity advisory.

Before Emission, Coherence preserves zero pressure and Reception preserves uniform form. Every unforced default base choice is therefore Coherence, while the lag policy realizes the six-batch word IL → IL → IL → EN → IL → AL. The final Emission acts on every node, leaving form uniform and stored pressure zero. A fresh evaluation of the default pressure law would also give zero; the runtime does not perform that refresh after the final Emission. Let d>0 be its represented increment, as admitted above. Starting from structural time one gives the following exact finite comparison:

Supplied execution	Realized operator events	Final time	Final uniform form
Three outer calls of duration 1/4	Coherence at 1,5/4,3/2	7/4	1/4
Six outer calls of duration 1/8	Coherence at 1,9/8,5/4; Reception at 11/8; Coherence at 3/2; Emission at 13/8	7/4	1/4+d
Three outer calls of duration 1/4, each internally split into two integration intervals	Same three Coherence events as the first row	7/4	1/4

The represented increments and these dyadic clock sums are checked directly. The discrepancy cannot be assigned to integration error: the form row is zero between events in this source, and the native phase/capacity updates preserve their uniform values. Outer-call refinement has changed the supplied hybrid event schedule. Internal integration refinement retains that schedule here; this control does not prove convergence for arbitrary nonuniform source laws.

Restricting the six finer calls to select operators only at the three original decision times preserves the marks, lag counts and physical endpoint on this source. It does not preserve all runtime state: stability counters and phase telemetry advance three versus six times. Similarly, six calls with dt=0 execute the six-batch word at structural time one and produce the same uniform Emission increment. A zero flow duration is not a request for zero hybrid events.

Consequence. The native batch policy is a supplied sampled controller, not a clock-independent occurrence law derived from the continuous nodal row. A prospective comparison must freeze event times and every per-call policy, or justify how they transform when the clock or decision schedule changes. The existing explicit event-schedule contract and flow-refinement controls provide reusable methods for fixed events. Their executor, pressure-refresh and admission premises remain distinct from the complete native runtime. The runtime ordinal contract records invocation identity, including partial failures; it does not solve occurrence selection.

Disconnected context and the native occurrence boundary

The actual occurrence controls extend the stored-diagnostic comparison to fresh pressure, live grammar and an executed event. They keep the default pressure mixture, diagnostic weights, selector thresholds and operator factors. No replacement selector or componentwise normalization is installed.

Source and law. Take two disconnected unit-weight edges (0,1) and (2,3), with that node order. All four nodes have signed form x=1/2. The first pair has capacities (1,1) and phases (0,3*pi/4); the second has phases (0,0) and capacities (M,M). Compare M=1 and M=4, below the configured capacity ceiling 2*pi. Both preparations retain structural time one, seed 17, supplied Emission → Coherence → Reception context, zero Emission/Reception lag counts, and constant two-sample form evidence at times 3/4 and 1. These histories are admitted preparation, not a reconstruction of earlier executed operators.

Let a denote the effective phase-pressure coefficient and (alpha,beta,gamma) the normalized Sense Index weights. Their default numerical coincidences are configured choices. The mixed pressure law has no EPI, capacity or degree contrast on either edge, so

p_local = (3*a/4, -3*a/4),       p_remote = (0,0),
Vmax = M,                       Pmax = max |p| = 3*a/4,
Si_local(M) = alpha/M + beta/4.                         (M >= 1)

Local phase dispersion is 3/4 and normalized pressure magnitude is one. Thus fresh local pressure and unforced form rates nu*p agree between the preparations. Only the capacity maximum in this diagnostic changes. The represented scalar and vector Si controls give the following separated branches:

Remote capacities	Local Si	Actual local operator	Stored pressure magnitude immediately after the event
(1,1)	approximately 0.8043353561	Coherence (IL)	approximately 0.4315451554
(4,4)	approximately 0.2354251114	Dissonance (OZ)	approximately 0.75

The configured upper/lower Si cuts are 1/2 and 1/4; normalized pressure one exceeds the Dissonance branch's 0.133 cut. Both actions pass the same live grammar, and no lag forces another request. Their direct glyph maps multiply the nonzero pressure by the admitted factors h<1 and f>1, respectively. The opposite pressure signs are retained. This is a difference in executed state changes, not merely in proposed names or a Mutation eligibility test.

Execution boundary. One native Euler call with dt=0, Gamma absent, fresh pressure/Si and strict grammar records each actual glyph before and after its execution. No custom selector, scientific callback or auxiliary math engine runs. Direct glyphs, rather than public operator classes, own these events. Default phase/capacity policies still run afterward: phases do change despite zero elapsed flow time. The table concerns the immediate jump, not a claim that the complete runtime state is frozen. No post-event pressure refresh is inserted. Disabling glyph selection retains equal local physical rows between these two preparations under the remaining one-call policies.

Matched-input control. Compare remote capacities (4,4) and (2,4). The latter creates real remote pressures (+2*b,-2*b), with b the effective capacity-pressure coefficient. Here 0<2*b<3*a/4, so both fresh global maxima, the local rows and local Si agree. The first two executed Dissonance events then agree as well. Remote capacity contrast has not been silently omitted from the declared mixed pressure law.

State-sufficiency consequence. Let Q retain the first component's form, capacity, phase, support and stored pressure coordinate at the declared event boundary, together with grammar, temporal evidence, lag memory and decision time. Initial pressure is freshly evaluated; Q reads the coordinate without recomputing it after an event. It omits graph-wide maxima and the graph-normalized Si. This projection identifies the two main preparations, yet their realized marks and raw pressure successors differ. The marked occurrence criterion therefore excludes an autonomous realization of this supplied controller on Q's domain. If a different observation recomputed pressure immediately after these direct glyphs, their unchanged form, capacity and phase would give equal local pressure. The realized marks would still differ, but this raw-pressure successor argument would not apply to that reset-and-refresh observation. The full stored node records do differ in Si after refresh; retaining that externally dependent diagnostic distinguishes them. For valid fresh observations and fixed configuration, Vmax and Pmax are sufficient external values for its default base choice alongside the local readings. Shared admission also validates acceleration; that channel does not affect the default choice, unlike the parametric selector.

This immediate factorization does not give the maxima their own evolution, make remote invalid inputs admissible, or close every operator's execution. The existing native-writer analysis also retains global phase control and Si-dependent capacity policy. An NFR model using the selector must declare these controller/environment inputs or justify their elimination on a restricted family. Replacing global maxima by component maxima would change the controller and needs its own justification. Disconnected transport support alone does not impose locality on a supplied graph-wide policy, and this result identifies no physical remote interaction.

A restricted collective map of one native selector batch

The memory and environment obstructions concern discarded inputs. Retaining those inputs can admit a collective map on a restricted family. The following result concerns the actual DefaultGlyphSelector / _apply_glyphs path, including its sequential direct glyphs and live incremental grammar. It installs no new selector or law.

State and preparation. Let B_i={2i,2i+1}, for i modulo five, be the two-member blocks of the retained doubled-C5 graph. Each adjacent block pair has all four unit-conductance edges; there are no internal block edges or external nodes. The two members of block i have common signed form x_i in [-1,1], phase zero, capacity one and zero stored acceleration. Use the default pressure mixture, diagnostic weights, selector cuts, operator factors, hard clipping and lag limits. Fix a finite structural clock and the seed; a primitive batch advances neither clock time nor continuous flow. The EPI semantic kind (epi_kind) is one common nonempty label, retained unchanged on this family; operator provenance is a separate coordinate. Freshly evaluate default pressure and Sense Index before preparing the selector.

Retain each block's two lag counts (a_i,e_i) and admitted grammar state H_i=(history,debt,prior_coherence,operator_step), equal within the block, with nonempty history and debt at most the configured default capacity two. One sufficient preparation is the supplied Emission → Coherence → Reception context, with zero debt and prior Coherence. Use strict rejection and the default positive history-retention window. These records are supplied context, not reconstructed physical history. The same conclusion holds for subsequent contexts generated by the admitted branches below. Cache objects, source identifiers and diagnostics are not additional physical coordinates; every consumed primitive still passes its shared admission boundary.

Fresh inputs and realized marks. Write k for the effective normalized EPI-pressure coefficient and (alpha,beta,gamma) for the effective Sense Index weights. With P=max_i |p_i|, the ideal real-arithmetic rows are

p_i = k * ((x_(i-1) + x_(i+1))/2 - x_i)
Vmax = 1
Si_i = alpha + beta + gamma * (1 - |p_i|/P).

At P=0, use the implemented unit denominator, so the pressure ratio is zero. The other pressure channels vanish: phases, capacities and degrees agree. Uniform states are included even when the initial preparation is nonuniform; hard-clipped Emission can return a uniform endpoint. The default alpha+beta is approximately 0.9417, above si_hi=0.5. Consequently every base proposal is Coherence, including for nonuniform form. The selector prepares these inputs before any event; it does not recompute them between targets. The existing increment-before-comparison policy gives

g_i = AL if a_i >= 5, else EN if e_i >= 3, else IL.

Thus Emission takes priority when both limits are due. These are supplied controller decisions, not consequences of the unforced nodal differential row. Numerical execution additionally requires all represented input, pressure, ratio and event arithmetic to be admitted; the ideal formulas do not waive overflow or underflow rejection.

Ordered block recurrence. Process the blocks in the fixed order 0..4, with both members of each block consecutive. Initialize y=x. At block i, apply exactly one of the following scalar assignments, retaining updates already made to earlier blocks:

Emission (AL):   y_i <- clip_[−1,1](y_i + b)
Reception (EN):  y_i <- (1-m)*y_i + m*(y_(i-1) + y_(i+1))/2
Coherence (IL):  y_i <- y_i.

Here b, m and h are the actual AL_boost, EN_mix and IL_dnfr_factor settings. Their ideal default expressions are 1/(4*pi), 1/(pi+1) and pi/(pi+1) respectively. The Reception row is a convex blend, so clipping is inactive for its admitted interval. The raw stored pressure after the complete batch is

p_i^raw = h*p_i if g_i=IL, else p_i.

Phase, capacity, support and structural clock remain fixed on this direct glyph path. Advance each block's lag pair by the existing counter recurrence. Append its realized mark once to each member's bounded history, increment each member's operator step once, and retain the corresponding source_glyph. Emission and Reception leave debt and prior Coherence unchanged; Coherence maps debt to max(0,debt-1) and makes prior Coherence true. The default preparation therefore retains zero debt and a nonempty trace.

Proof and admission. Both members have equal prepared inputs and lag counts, hence the same proposal. Before their first event, they also have equal live local grammar and physical coordinates. They are not adjacent: changing the first member cannot change any neighbor read by the second. No other target runs between them. Their actual Reception snapshots therefore have the same form mean and common semantic kind; Emission and Coherence use the same point maps. After the second event, physical rows, retained grammar, provenance and counters agree within the block. Induction over the five blocks gives the displayed recurrence. Both members' U1a checks have a nonempty trace; debt at most two passes U2, and these three marks require no U3/U4 coupling or transformer context. Shared numerical and seed admission remains live at each fine prefix. This establishes a sufficient admitted domain, not unconditional execution for arbitrary histories or a transactional batch.

The pair-constant family can be left after the first member acts and restored after the second. The result is an endpoint map of a complete ordered batch, not descent of every singleton event. In particular, it is not an atomic frozen-neighbor stage, a five-node invocation of the unchanged native engine, or a quotient of arbitrary unequal pairs. The observation retains block labels, the declared multiplicities and target order.

Raw endpoint versus refreshed return. Denote the batch by B and a declared pressure-and-Sense-Index refresh by R. Generally p^raw != p(y): an Emission or Reception changes form while retaining stored pressure; Coherence changes stored pressure while retaining form. Thus B does not return to the freshly prepared pressure submanifold. Its raw output includes p^raw and the unchanged prepared Sense Index. The separately declared composition R o B returns to that submanifold, with pressure given by the same displayed pressure row evaluated at y and newly evaluated Si. It preserves paired form, phase, capacity, kind, grammar and lag association. This gives a closed refreshed discrete map on this restricted family's admitted composition domain, including the refresh arithmetic. It does not turn refresh into an operator or silently erase the raw event observation.

Order counterexample. Set x=(c,c+1/4,c,c,c) with c=1/8, and make Reception due in every block while Emission is not due. For contiguous order, write the first two returned forms as c+z_0,c+z_1. The recurrence predicts

z_0 = m/8,
z_1 = (1-m)/4 + m^2/16.

A frozen-neighbor simultaneous update would omit the positive m^2/16 contribution. Instead interleave the first targets as (0,2,1,3,4,5,6,7,8,9). After node 0, node 2 already reads one changed member of block 0. Node 1 then reads that changed node 2. Direct substitution gives the final first-pair gap

x_1^raw - x_0^raw = m^2*(m-8)/128 < 0    for 0 < m <= 1.

Neither node acts again, so the pair remains split at the batch endpoint. The same initial block coordinates, lag counts and grammar do not determine a pair-constant return when target order is discarded. This scoped failure does not exclude a larger collective state retaining the generated contrast.

The execution controls compare independent block arithmetic with actual default pressure, fresh Si, prepared proposals, realized events and live grammar. Binary64 agreement is checked with an absolute 1e-14 allowance against independent rational evaluation of the supplied coefficients; exact pair equality and discrete bookkeeping use no tolerance. This is not exact matrix evaluation for every represented input. The controls distinguish raw and refreshed pressure, sequential and simultaneous Reception, and a late grammar rejection retaining its executed prefix. No continuous trajectory, new formation mechanism, physical identification or autonomous occurrence law is established here.

One complete native step on the synchronized collective family

The batch result alone does not cover the integrator or later native policies. Here the same doubled-C5 preparation is advanced through one actual runtime.step, with default pressure, fresh Sense Index, the default selector, strict grammar and the default Euler integrator. The block order and consecutive twins are retained. The live Gamma registry retains its built-in zero source; its none label alone is insufficient. No custom callbacks or optional mathematics engine run, and the supplied graph history has no stability-fraction samples admitting automatic REMESH.

Declared step and state. Use structural time t=1, seed 17 and the explicit duration delta=1/16. This equals the default DT_MIN, so preparation selects exactly one Euler substep; the graph's default DT=1/2 remains unchanged. These are declared execution inputs, not a derived physical clock. In addition to the batch state, retain a prior stored form-rate value r_i, a nonnegative stable_count value s_i and any consumed temporal history, equal within each pair. Supplied rates and counts do not certify a past trajectory or stability. For the finite controls the temporal histories start absent, and the runtime starts its tracked invocation epoch at zero. The supplied recent grammar trace contains only Emission, Coherence and Reception; default phase gains and empty phase histories fix the initial global controller state.

Retain the graph's adaptive phase gains/history, invocation ordinal and REMESH history separately from node coordinates. Fixed support and node labels allow the fine-indexed sample and history records to be reconstructed from paired values. Rebuildable implementation caches are not independent physical state. This is a factorization of the declared retained observations; it is not a claim that five scalar forms encode every Python object owned by the runtime.

Predicted endpoint. Let y_i and q_i=p_i^raw denote the actual ordered batch's form and stored pressure. Let Si_i be the fresh pre-batch diagnostic. The integrator holds the post-glyph pressure and capacity during this substep. In ideal arithmetic its rows are

z_i = clip_[−1,1](y_i + delta*q_i)
stored_dEPI_i = q_i
stored_d2EPI_i = (q_i-r_i)/delta
t_next = t + delta.

The rate and acceleration are integrator telemetry. When clipping is active, the measured form secant (z_i-y_i)/delta need not equal the stored rate. Nor may q_i be replaced by freshly evaluated pressure at y or z: neither the integrator nor the subsequent policies perform that constitutive refresh. The terminal stored pressure remains q_i, and stored Si remains Si_i. The separate continuous and projection contracts retain their own numerical and rate/residual evidence boundaries.

These acceleration rows specifically require one substep. With multiple constant-pressure Euler substeps, each later substep reads the previous rate q_i, so its terminal acceleration is zero. The default duration 1/2 would produce eight substeps here. Neither subdivision nor acceleration history can be silently changed while carrying over this one-step prediction.

Active phase and capacity policies. Every phase is zero after the batch and Euler update. Both global and neighbor phase targets are therefore zero, so the actual coordinator retains theta_i=0. It still evaluates its adaptive controller and records its gains and phase diagnostics once per invocation. For the declared Emission/Reception/Coherence histories, order parameter one and zero disruptor load select its stable branch; each default gain K moves toward its configured lower target by

K_next = K + down*(K_min-K).

The existing gain rails still apply. This is an invoked phase relaxation policy, with no capacity-driven free phase advance and no delta multiplier.

The capacity policy reads the retained Si_i and q_i, not terminal refreshed fields. Let eps=EPS_DNFR_STABLE and tau=VF_ADAPT_TAU (default five). Then

eligible_observation_i = (Si_i >= si_hi and |q_i| <= eps)
s_i_next = s_i+1 if eligible_observation_i, else 0
capacity_update_due_i = (s_i_next >= tau).

All neighbor capacities and the target capacity equal one. The actual uniform-fixed-point guard thus keeps nu_i=1 exactly, even on due nodes. The policy can execute writes and advance counters without changing capacity. Equal per-pair counts yield equal successor counts; physical equality alone does not supply that history condition.

History and support. Runtime records the left flow boundary, runs the batch, then records its right boundary at the same timestamp. A form-changing event restarts that node's temporal history at (t,y_i). For the declared empty-history preparation, the completed record is ((t,y_i),(t+delta,z_i)), including unchanged-form events. It measures the post-event interval and cannot absorb the jump into a physical secant. The terminal graph-wide EPI snapshot is appended once to delayed REMESH history. This does not supply the separate stable_frac history: the actual automatic REMESH gate remains unavailable and performs no network event. The admitted invocation advances the ordinal once, independently of the structural clock and each node's operator step. Support and semantic kinds remain unchanged on this preparation.

Preservation and its boundary. The batch restores paired rows and grammar. The same pointwise Euler/clipping map then acts on both members; the same prior rate gives the same acceleration telemetry. Phase zero and capacity one are fixed points of the invoked policies, and matched counters/history give matched successor records. Consequently the complete call preserves the declared paired physical and local bookkeeping observations, while updating the retained global controller and runtime records. No phase/capacity policy has been disabled to obtain this result.

A countercontrol gives one member of a qualifying pair s=tau-1 and the other s=0. Both still finish with capacity one and equal physical rows, but their successor counts are tau and one. The supplied counter association cannot be discarded from a paired bookkeeping description. This does not claim a physical splitting of the uniform-capacity pair.

The full-step controls compare an independent rational block/Euler prediction with scalar and NumPy execution using absolute allowance 1e-14; exact pair equality, marks and discrete bookkeeping use no tolerance. They retain the actual owner order, due and ordinary capacity gates, form clipping, post-jump temporal histories, phase-controller records, the invocation ordinal and REMESH nonoccurrence. The prior-rate input is varied by block rather than assumed zero.

This establishes one complete native call on a synchronized restricted family. Its constant phase and capacity are model-specific fixed points, not a newly formed collective identity or a demonstration of nontrivial phase/capacity dynamics. Repeated calls with additional retained history, automatic REMESH, arbitrary target orders and autonomous selection remain separate admission questions. The following result admits a nonuniform phase/capacity sector and identifies a default-selector boundary outside it.

Nonuniform collective phase and capacity: closure and a selector boundary

Retain the preceding doubled-C5 support, consecutive twin order, common nonempty epi_kind, strict live grammar, supplied Emission/Coherence/Reception context, default laws and one-substep clock. Keep matched per-pair lag, prior-rate, adaptation and temporal memory, the live built-in zero Gamma source and the insufficient history for automatic REMESH. The source is supplied, not formed by this call. Now let each block carry its own signed form x_i in [-1,1], positive capacity v_i within the configured rails and circular phase theta_i. Both members of a pair still have the same values. Global controller/history state remains separate from these block coordinates.

A sufficient selector sector. Suppose the phases admit one common lift interval of width D<pi. Every neighbor and global phasor resultant is then nonzero. Write V=max_i v_i, and let (alpha,beta,gamma) be the normalized Sense Index weights. On this regular chart, define

ell_i = Arg(exp(i*theta_(i-1)) + exp(i*theta_(i+1)))
d_i = wrap(ell_i-theta_i)
p_i = w_phase*d_i/pi
      + w_epi*((x_(i-1)+x_(i+1))/2-x_i)
      + w_vf*((v_(i-1)+v_(i+1))/2-v_i).

Here w_phase,w_epi,w_vf are the effective normalized pressure weights. These are fresh pressures from the complete default mixture. All fine degrees are four, so the degree-contrast channel is zero. Neither U3 filtering nor the capacity adaptation law replaces the support used by this pressure reading. With P=max_i |p_i|, the fresh diagnostic is

Si_i = alpha*v_i/V + beta*(1-|d_i|/pi)
       + gamma*(1-|p_i|/P),

where the normalized pressure is zero when P=0. The phasor mean belongs to the common lift interval, hence |d_i|<=D. Consequently

alpha*min_i(v_i)/V + beta*(1-D/pi) > si_hi

is sufficient for every default base proposal to be Coherence (IL), with a strict decision margin. Matched lag overrides can still request Emission (AL) or Reception (EN), in their existing priority order. This inequality is sufficient, not necessary; it is not a new selector rule or a stability law. The real-arithmetic proof and finite represented-arithmetic controls are separate; numerical use must retain defined resultants and branch margins.

These three direct glyphs preserve phase and capacity. In particular the native IL glyph contracts stored pressure; it does not execute the public Coherence class's additional phase proposal. Coupling and Resonance are absent because their selection is excluded on this sector. U3 is not disabled: it is inapplicable to these realized marks. Leaving this sector requires checking the actually requested operator and its live grammar/preconditions afresh.

Nonconstant one-call map. The preceding ordered AL/EN/IL batch map still gives y_i and q_i, with q_i=h*p_i for IL and q_i=p_i otherwise. Consecutive nonadjacent twins see the same current neighboring forms before their respective Reception events. The held-capacity Euler rows now read

z_i = clip_[−1,1](y_i + delta*v_i*q_i)
stored_dEPI_i = v_i*q_i
stored_d2EPI_i = (v_i*q_i-r_i)/delta.

Before coordination, runtime centers phase representatives into [-pi,pi); write these values as psi_i. The coordinator uses this immutable snapshot to compute its global target g=Arg(sum_i exp(i*psi_i)) and neighbor targets ell_i. Let K_G,K_L be the gains after the actual adaptive policy has read its order parameter and glyph load. The stored phase proposal is

theta_i_next = psi_i + K_G*wrap(g-psi_i) + K_L*wrap(ell_i-psi_i).

Here wrap is the signed principal circular difference. The coordinator stores the raw proposal; it does not normalize it again. Its circular observation is understood modulo 2*pi. The global and local targets are identical for twins, even when these proposals move. Gains are shared controller state; their adaptive branch and records must be retained. This policy has no free capacity-driven phase advance and no multiplier delta.

Capacity admission still reads pre-batch Si_i and post-event q_i, not fresh terminal diagnostics. Let s_i_next and the due flag be defined by the prior one-call result. The capacity proposals use a second immutable snapshot:

m_i = (v_(i-1)+v_(i+1))/2
v_i_next = (1-mu)*v_i + mu*m_i    if s_i_next >= tau,
           v_i                   otherwise.

The represented implementation preserves exact fixed points and endpoint hulls. A matched due flag and identical snapshot neighbors give equal twin proposals. Thus the batch, pointwise integration, phase coordination, capacity adaptation and declared history updates preserve pairs at the completed-call boundary. This proves one conditional collective map with changing phase and capacity; neither policy needs to be at a fixed point. Stored pressure and Si retain their original observation times after these changes.

A source with both policies active. Take all x_i=1/8, d=e=1/1024, theta=(d,0,0,0,0), v=(1+e,1,1,1,1), all-IL lag preparation and matched counts tau-1. Write a=w_phase*d/pi+w_vf*e. Then

p = (-a, a/2, 0, 0, a/2),       q = h*p,
v_next = (1+(1-mu)*e, 1+mu*e/2, 1, 1, 1+mu*e/2).

Every default stability observation qualifies with a strict margin and all capacity writes are due. With g=Arg(4+exp(i*d)), the stable-branch phase proposals are (1-K_G-K_L)*d+K_G*g in block zero, K_G*g+K_L*d/2 in blocks one/four and K_G*g in blocks two/three. Both phase and capacity change. This is a declared finite-state witness, not evidence of acquired stability.

If only node zero's incoming counter is instead zero, its capacity stays 1+e while its twin becomes 1+(1-mu)*e. The terminal difference v_node1_next-v_node0_next=-mu*e is nonzero, although their initial physical coordinates agree. Discarding counter association now loses a physical prediction, not merely a bookkeeping value. This countercontrol deliberately violates the matched-memory premise.

An actual default-selector obstruction. For all x_i=1/4, theta_i=0 and v=(1/4,1,1,1,1), with ordinary lag/count preparation, fresh pressures are p_0=3*w_vf/4, p_1=p_4=-3*w_vf/8, p_2=p_3=0. The first pair has Si_0=alpha/4+beta, strictly between the default lower and upper cuts, and normalized pressure one. The actual selector therefore requests Transition (NAV) for both twins; the other blocks request Coherence. The supplied grammar admits this request; no U3-dependent operator is substituted.

Default Transition uses NAV_STRICT=False, NAV_RANDOM=True, eta=1/2 and target-associated jitter draws xi_0,xi_1. Its stored pressures are q_j=(p_0+1/4)/2+xi_j for the first two nodes. No form clipping occurs, so after the declared Euler step

x_1_next-x_0_next = (xi_1-xi_0)/64.

Phase remains zero, and the first pair's low Si prevents capacity adaptation; these later policies do not erase the form split. Equal current block state and equal selected operator therefore do not ensure pathwise pair retention. A seed fixes a reproducible generator only together with target association, draw counters and source; it does not impose equal twin innovations. This is an obstruction for the configured native law, not a defect of the nodal identity or evidence that a larger collective description cannot exist. In particular, failure of pair constancy does not prove that the collective form mean lacks a closed law on a separately admitted selector sector.

The nonuniform controls freeze source, node order, seed 17, duration and absolute budget 1e-14 before execution. Independent block arithmetic checks two AL/EN/IL profiles on a broader nonuniform source, the active-policy witness and the asymmetric-memory boundary on scalar/NumPy paths. An independent standard-library reconstruction of the declared jitter stream predicts the NAV form split before execution; it does not reconstruct pressure from the observed response. Exact pair equality, marks and discrete histories are checked without tolerance. These controls do not certify all nonuniform states, repeated calls, a probability law for physical noise, formation or autonomous operator selection.

A next-call collective form-mean prediction after Transition

The preceding Transition counterexample breaks pair constancy. This does not settle whether a declared collective observation can still be predicted. Here the observation is the five signed form means after one further native call, with the same support, default laws, strict grammar and duration delta=1/16. This is a reduction of that output on an admitted sector, not a complete collective NFR state or a new autonomous execution law.

Retained source and discarded information. Continue the preceding source with all initial forms 1/4, zero phases, capacities v=(1/4,1,1,1,1) and seed 17. Its first complete call reaches time 17/16. Retain its actual grammar, lag/counter association, controller state, valid temporal histories, node order and random-generator progress. The next call ends at 9/8. The two first-pair targets have each consumed one jitter sample; their next samples have draw index one. Restarting the streams from the seed or drawing twice from one temporary generator is not the declared continuation.

Write each pair as x_(2i)=M_i-d_i, x_(2i+1)=M_i+d_i. The proposed form forecast retains M, capacities and phase zero, with a bound on |d_i| instead of the exact signed contrasts. Fine histories still admit the actual execution; omitting them from an output formula does not waive that admission. Previous stored pressure and Si are refreshed before selection. Prior stored rates enter new acceleration telemetry, not this default selector's choices.

Let k=w_epi, b=w_vf be the effective normalized pressure coefficients. On the complete bipartite links, the two neighbors from each adjacent block contribute their mean, so fresh pressure satisfies

B_i = k*((M_(i-1)+M_(i+1))/2-M_i)
      + b*((v_(i-1)+v_(i+1))/2-v_i)
p_(2i) = B_i+k*d_i,       p_(2i+1) = B_i-k*d_i
P = max_i (|B_i|+k*|d_i|).

Thus the block pressure mean closes, but the graph-wide normalization P does not generally depend on means alone. It must be bounded rather than silently held fixed. Phase and degree channels vanish on this source; their absence is a declared sector, not a change to the pressure configuration.

Admission from a contrast envelope. Let rho=1/64, |d_i|<=rho, L=B_0-k*rho and U=max_i(|B_i|+k*rho). The first-call means have the independent prospective values

M = (0.25198548108026736, 0.24896352456651732,
     0.25, 0.25, 0.24896352456651732).

These displayed decimals summarize the primitive-source arithmetic; they are not replacement exact inputs. For that source, the envelope gives the strict bounds L>1/25, U<3/64 and L/U>4/5. Both first-pair pressures are therefore positive and their normalized magnitudes exceed the default dnfr_hi=0.133. Writing A=alpha/4+beta, their Si values always lie in [A,A+gamma], strictly inside (si_lo,si_hi)=(1/4,1/2). All other capacities equal the maximum one, so their Si values are at least alpha+beta>si_hi. This admits Transition (NAV) on both first-pair nodes and Coherence (IL) elsewhere throughout the envelope, without assuming an unchanged pressure maximum.

Actual lag counters increment from one to two, below both forcing thresholds. The default selector uses its prepared fresh metric snapshot; the parametric selector's repetition, score and soft-filter policies are not part of this model. Live grammar admits these requests with the retained context, including repeated NAV/IL. No U3-dependent mark is requested or substituted.

Let J be the default jitter amplitude and eta=1/2. For any |xi|<=J, the first-pair Transition changes pressure because eta*(v_0-U)-J>0. The unprojected Euler endpoints throughout this envelope lie strictly between 1/5 and 3/10, so form clipping is inactive. These conditions are separate from grammar and selector admission.

All phases stay zero. Incoming adaptation counts are at most one, hence the next counts are at most two, below tau=5; no capacity write is due. The pressure-based stability flags need not agree across contrasting lifts. The existing missing-history REMESH gate still performs no network event. These facts establish this call's phase, capacity and support boundaries; they do not establish an invariant region for arbitrary later calls.

Reduced mean prediction. Let h be the Coherence pressure factor and let xi_bar=(xi_0+xi_1)/2, xi_delta=(xi_1-xi_0)/2 use the actual next target-associated Transition inputs. The admitted mean/contrast equations are

M_0_next = M_0 + delta*v_0*((1-eta)*B_0 + eta*v_0 + xi_bar)
d_0_next = (1-delta*v_0*(1-eta)*k)*d_0 + delta*v_0*xi_delta

M_i_next = M_i + delta*v_i*h*B_i                  (i>0)
d_i_next = (1-delta*v_i*h*k)*d_i                  (i>0).

Only the first and third rows are needed to forecast the five form means. The contrast equations justify the cancellation and retain the discarded information's evolution; storing all their coordinates would simply recover the original fine forms. The reduced prediction still consumes the declared innovation mean and an admitted contrast envelope. It does not derive a noise distribution, choose a new correlation between targets or authenticate a seed.

An independent reconstruction of draw-index-one samples predicts xi_0=-0.006554716868658765, xi_1=-0.014026349986664306. Before the reserved second-call response, the predicted means are

(0.2541150913718302, 0.24794466913247273,
 0.24999550006091195, 0.24999550006091195, 0.24794466913247273).

The actual generated contrast is contained in the envelope. A scalar and a NumPy continuation compare these prospective predictions against the complete native call using the unchanged absolute budget 1e-14. Binary64 averaging of distinct lifts need not cancel bitwise; exact real-arithmetic descent and finite numerical agreement retain different scopes.

Why the selected word is insufficient outside the sector. Keep the same five means, capacities, phases, supplied admission context and next random inputs, but compare d=0 with d_0=1/2, d_i=0 for i>0. The latter's signed forms remain within [-1,1]. Its first-pair pressures are approximately 0.134748 and -0.048406; both normalized magnitudes still exceed 0.133 and both Si values remain in the middle band. The realized word is therefore the same NAV/NAV followed by eight IL actions in both preparations.

Transition's general pressure-mean row, however, is

q_bar_0 = (1-eta)*B_0
          + eta*v_0*(sigma(p_0)+sigma(p_1))/2 + xi_bar,
sigma(p) = 1 if p >= 0, else -1.

This retains the native convention at zero pressure. Opposite signs cancel the capacity targets. With no clipping or later capacity write, the contrasted preparation's next first-block mean differs from the zero-contrast preparation by exactly

M_0_next(d_0=1/2) - M_0_next(d=0) = -delta*eta*v_0^2 = -1/512.

Thus means and selected marks alone do not determine the next mean globally. Pressure-sign admission matters even when grammar and the operator word agree. This counterexample is a supplied boundary comparison, not a claim that the original first call generated contrast 1/2.

The collective-mean controls retain the genuine two-call histories and generator progress. Separate fresh boundary preparations use common means rounded once to a declared 2^-40 grid, giving exactly equal represented means for different fine contrasts. They supply valid grammar/counter/RNG context without inventing a measured past trajectory. Exact rational envelope checks precede their response; the large-contrast comparison retains the same seed, draw indices and defaults. No pressure is inferred from the observed motion, and no threshold is retuned.

This closes a one-output, one-next-call question. Fine pressure, Si, adaptation counts and temporal histories need not descend to the five means, even inside the admitted envelope. Continuing their prediction, allowing due adaptation, crossing selector/sign/clipping boundaries or changing phase/capacity support requires a new sufficient-state and admission argument. Neither the positive prediction nor the obstruction identifies a physical particle or derives an autonomous collective operator law.

Propagated admission through the first lag-forced Reception

The preceding one-next-call result extends through the first forced Reception (EN), provided the contrast envelope evolves along with the means. This is finite-horizon admission for a reduced output, not a complete collective state or an invariant domain for unlimited execution.

Source and horizon. Retain that doubled-C5 source, consecutive pair order, default mixed pressure and selector, strict grammar, seed 17, zero phase, v=(c,1,1,1,1) with c=1/4, and live built-in zero Gamma. Start with forms 1/4, time 1, zero lag/adaptation counters and the same supplied grammar context. Each complete native call has duration delta=1/16 and one Euler substep. The horizon is four calls, ending at 5/4. Histories, controller records, target-associated random progress and prior rates continue on the actual graph; none is reconstructed from its mean output.

The strict inequalities below admit Transition (NAV) on the first pair and Coherence (IL) elsewhere for calls one through three. Before the fourth selection, the actual selector increments since_EN to four, exceeding EN_MAX_LAG=3. Emission's higher-priority override remains inactive because since_AL=4<=AL_MAX_LAG=5. All ten targets therefore execute Reception. Live grammar admits these requests with the retained context; no repetition policy from the parametric selector or public-class-specific Reception ceiling is silently added to the direct-glyph path.

A box that propagates its own admission. Use the preceding M_i,d_i,B_i, and write D_i=b*((v_(i-1)+v_(i+1))/2-v_i) for the fixed capacity contribution to pressure. Suppose l_i<=M_i<=u_i and |d_i|<=rho_i. Exact interval bounds are

B_lo_i = k*((l_(i-1)+l_(i+1))/2-u_i)+D_i
B_hi_i = k*((u_(i-1)+u_(i+1))/2-l_i)+D_i
L = B_lo_0-k*rho_0
U = max_i(max(abs(B_lo_i),abs(B_hi_i))+k*rho_i).

Let h=IL_dnfr_factor, eta=NAV_eta=1/2, J=NAV_jitter, a_0=delta*c*(1-eta), a_i=delta*h for i>0, and A_i=1-a_i*k. All A_i are positive. The NAV/IL mean map is consequently monotone in each incoming mean, giving the following update for every permitted |xi_j|<=J:

l_i_next = A_i*l_i + a_i*k*(l_(i-1)+l_(i+1))/2 + a_i*D_i
           + indicator(i=0)*delta*c*(eta*c-J)
u_i_next = A_i*u_i + a_i*k*(u_(i-1)+u_(i+1))/2 + a_i*D_i
           + indicator(i=0)*delta*c*(eta*c+J)
rho_i_next = A_i*rho_i + indicator(i=0)*delta*c*J.

The two bounds |xi_bar|<=J and |xi_delta|<=J overapproximate their joint possibilities. Their use is conservative; the proof does not assume that both extrema occur together. Exact means still use the declared innovation mean, while the radius update omits the exact signed contrasts.

Starting either at l=u=(1/4,...,1/4), rho=0, or at the preceding first-call means with every rho_i=1/64, propagate these bounds to the fourth call. The latter comparison is a supplied postfirst family, not a claim that all its contrasting lifts were generated by the initial call. At every entry, exact rational arithmetic over the represented defaults gives

L > 3/80,       U < 3/64,       L/U > 4/5.

Thus both first-pair pressures remain positive, and their normalized magnitudes exceed dnfr_hi=0.133. Their Si remains in [alpha*c+beta, alpha*c+beta+gamma], strictly inside (1/4,1/2); the other blocks have Si>=alpha+beta>1/2. These inequalities justify the first three base choices, without assuming a fixed pressure maximum. The Transition change margin eta*(c-U)-J is strictly positive; forms at entry, Reception intermediate states and unprojected flow endpoints remain strictly inside (1/5,3/10). No clipping or zero-action replacement is needed.

A constant radius would be incorrect. From d_0=rho_0=1/64, allowed inputs xi_0=-J, xi_1=J give

d_0_next-rho_0 = delta*c*(J-(1-eta)*k*rho_0) > 0.

The supplied family's first-block radii after calls two and three are about 0.01622434 and 0.01682283, both above 1/64. The evolving box, rather than the previous fixed box, establishes the continued admission.

The actual Reception boundary. Set m=EN_mix and initialize Y=M. In block order i=0,1,2,3,4, perform

Y_i <- (1-m)*Y_i + m*(current Y_(i-1)+current Y_(i+1))/2.

Consecutive twins are nonadjacent and see identical current neighbor means at their own events. Reception therefore multiplies their contrast by 1-m. It retains the fresh pressure computed before the batch. The subsequent held-pressure Euler map is exactly

M_i_next = Y_i + delta*v_i*B_i(M)
d_i_next = (1-m-delta*v_i*k)*d_i.

This differs both from simultaneous neighbor mixing and from refreshing pressure after Reception. In particular, multiplying (1-m) by (1-delta*v_i*k) gives the wrong contrast coefficient. Apply the same ordered convex sweep to l and u, obtaining Y_lo,Y_hi; a conservative successor enclosure is

l_i_next = Y_lo_i + delta*v_i*B_lo_i
u_i_next = Y_hi_i + delta*v_i*B_hi_i
rho_i_next = abs(1-m-delta*v_i*k)*rho_i.

All contrast factors have magnitude below one on this configuration. The supplied family's first-block radius falls to about 0.01271276; this one contraction does not establish global synchronization or permanent retention.

Retained execution boundaries. Phase remains zero under the actual policy. Initially zero adaptation counts stay at most four, below tau=5; supplied postfirst counts at most one obey the same bound. Fine stability flags and counts may differ, but no capacity write is due. The automatic REMESH gate still abstains on absent stable_frac history. These facts preserve capacity and support without disabling either policy. They do not authorize discarding their memory once a write becomes possible.

The three Transition calls consume draw indices zero, one and two for each first-pair target. Reception consumes no random input and resets its lag to zero, while since_AL becomes four. At a Reception form jump, the temporal history restarts at its right-hand form at time 19/16; the terminal sample is at 5/4. The resulting flow secant excludes the jump. Stored rate is v_i*p_i using pre-batch pressure, and acceleration retains the preceding stored rate. The projected mean alone supplies none of that historical evidence.

Prospective finite controls. The Reception-envelope controls freeze every continuation's predictions before evaluating that preparation's response: four calls for the source, three for each supplied boundary. Independent target-addressed BLAKE2b/standard-library draws give index-two inputs (0.022444905166719355,-0.018236360145937148). The predicted mean at the Reception endpoint is

(0.2547985213089609, 0.2468856193270596, 0.2497511113424707,
 0.24957551867428465, 0.24683809383481498).

These decimals summarize the primitive-source calculation, not exact new inputs. Its ordered Reception splits previously equal block-two/three means by more than 1/8192; simultaneous mixing incorrectly preserves that equality. The held-pressure prediction also differs from a post-event refresh. Scalar and NumPy controls compare the complete continued execution using the unchanged absolute budget 1e-14, with exact discrete checks. Separate boundary preparations round common means once to 2^-40 and use zero or opposite alternating 1/64 contrasts. Their identical predicted mean continuations coexist with different fine histories and adaptation counts. Supplied metadata does not certify an acquired temporal past.

The interval proof covers ideal real arithmetic over the represented constants and all bounded innovations on this finite horizon. It is not a validated enclosure of every binary64 execution; the numerical controls separately check the declared finite preparations and paths against their fixed allowance. Neither derives physical noise, a laboratory clock, a collective identity, autonomous operator selection or a complete quotient of the native state. Capacity feedback and later threshold crossings require new admission evidence.

Active capacity adaptation exposes a discarded internal association

The preceding Reception result excludes due capacity writes on its horizon. When adaptation actually changes capacity, a form-mean prediction must retain capacity-pressure correlations. The following two-call comparison establishes that requirement on the actual native path, with unchanged policies and live grammar. It also identifies a restricted sector where active adaptation still permits a mean reduction.

Projected flow and its missing information. Retain the doubled-C5 support, unit conductances, common epi_kind, contiguous twin order, zero phase and built-in zero Gamma. Admit Coherence (IL) everywhere, one unclipped Euler substep of duration delta=1/16, and positive capacities within their rails. Write the signed form and capacity of each pair as

x_(2i)=M_i-d_i,          x_(2i+1)=M_i+d_i
v_(2i)=V_i-e_i,          v_(2i+1)=V_i+e_i
W_i=(V_(i-1)+V_(i+1))/2
B_i=k*((M_(i-1)+M_(i+1))/2-M_i)+b*(W_i-V_i)
H_i=k*d_i+b*e_i.

Here k=w_epi, b=w_vf are the effective normalized pressure coefficients and h=IL_dnfr_factor. The fresh fine pressures are B_i+H_i,B_i-H_i. Direct Coherence multiplies them by h without changing form or capacity. Integrating with these held pressures gives the exact ideal-real map

M_i_star = M_i + delta*h*(V_i*B_i-k*e_i*d_i-b*e_i^2)
d_i_star = d_i + delta*h*(e_i*B_i-V_i*(k*d_i+b*e_i)).

The population covariance of capacity and form is e_i*d_i; capacity variance is e_i^2. Thus replacing the mean nodal product by the product of its means loses both terms. This is a consequence of the supplied pressure law and nodal row, not an additional pressure law. With [k]=[p]/[x] and [b]=[p]/[v], all three terms inside the mean-rate parentheses have units [v][p].

Capacity and threshold memory. For each fine node, the actual adaptation gate reads pre-batch Si and post-Coherence stored pressure. A qualifying gate increments its retained count; a failing gate resets it to zero. A write is due when the new count is at least tau=5, and a successful write does not reset that count. Let a_minus,a_plus be the two resulting due flags, and set a_bar=(a_minus+a_plus)/2, a_delta=(a_plus-a_minus)/2. The immutable capacity snapshot gives

V_i_next = V_i + mu*(a_bar*(W_i-V_i)-a_delta*e_i)
e_i_next = e_i + mu*(a_delta*(W_i-V_i)-a_bar*e_i),
mu = VF_ADAPT_MU.

This writer runs after form integration and phase coordination. It changes the capacity consumed by the next call; substituting its output into the current call's flow is incorrect. The next call refreshes pressure from its own form and capacity before selection. Neither the former pressure nor a terminal product of newly updated capacity with old pressure is its fresh rate. Zero phase remains a fixed point while the real phase-controller records continue to evolve. This configured adaptation policy is not derived from the nodal identity.

If a_delta=0 and e_i=0 in every block, capacity remains equal within each pair even when the writes change block capacities. On an admitted selector, stability and clipping sector, the mean rows then close using M,V, matched due flags and retained execution context. Nonzero internal form contrasts do not enter those mean rows. Their bounds still matter for admitting the sector. Asymmetric due flags can instead create e_i_next from initially zero e_i.

A prospective two-call obstruction. Specialize the earlier active-policy witness to zero phase, with epsilon=rho=1/1024, all M_i=1/8, V=(1+epsilon,1,1,1,1) and initial capacity contrasts zero. Compare d_0=+rho and d_0=-rho, with other form contrasts zero. Both preparations retain exactly the same associated grammar, lags, prior rates and counters: node zero starts at count zero, every other node at tau-1. This metadata is supplied, not a certified past trajectory. Use seed 17, time 1, the unchanged default laws and two complete calls ending at 9/8. Neither call requests or consumes random jitter.

Both sources have identical means, contrast bounds, unordered within-pair form values and current covariance/variance e_i*d_i=e_i^2=0. A common sufficient Si bound is alpha/(1+epsilon)>1/2, so both calls select Coherence before either lag override is due. A conservative form-radius bound about 1/8 is

r_0=epsilon
r_(n+1)=r_n+delta*h*(1+epsilon)*b*epsilon.

Indeed, the EPI part of the Euler map is convex when 0<delta*h*v*k<1; capacity adaptation preserves [1,1+epsilon]. At both call entries (n=0,1), h*(2*k*r_n+b*epsilon)<EPS_DNFR_STABLE=0.001. All adaptation gates therefore qualify with a strict margin; endpoints stay inside (0.124,0.126) without clipping. These source-specific guards admit the actual branches, rather than imposing selected marks or disabling a policy.

Let D=(1-delta*h*(1+epsilon)*k)*rho>0. The first flow leaves identical mean forms in the two preparations and contrasts +D,-D in block zero. Its later capacity update gives the same fine capacities in both preparations:

v_0_next=1+epsilon,       v_1_next=1+(1-mu)*epsilon
e_0_next=-mu*epsilon/2.

Other pairs retain equal capacities within each pair; neighboring blocks also undergo nonzero adaptation. Counts, phase, grammar and lag records agree between the two preparations. At the second flow, however, the first mean gap is exactly

M_0_second(+rho)-M_0_second(-rho)
    = delta*h*mu*epsilon*k*D
    = 8.208928540582729e-10  (displayed approximation).

All other block mean gaps are zero at this boundary. This is a strictly nonzero prospective prediction from primitive state and defaults, not a pressure reconstructed from measured motion. Counter association is already retained: the missing information is its relation to the signed form contrast that becomes correlated with capacity after the write.

Adding only the current covariance and capacity variance does not repair multistep closure. Initially both are zero in the two sources, yet their next covariances e_0_next*(+D) and e_0_next*(-D) differ. The term mu*a_delta*(W_i-V_i)*d_i_star in e_i_next*d_i_star explicitly retains that association. Reconstructing every fine form and capacity from a coordinate change would be sufficient, but is not a demonstrated reduction.

Controls and a bounded alternative to exact means. Two matched-counter controls start every count at tau-1 and use the same opposite form contrasts. Their capacity writes are active, but preserve e_i=0; their second mean forecasts agree. A third asymmetric-counter source starts with d_i=0. Its first flow preserves zero form contrast, so the second first-block mean has the isolated correction

M_0_second - (M_0_first+delta*h*V_0_first*B_0_first)
    = -delta*h*b*(mu*epsilon/2)^2
    = -6.589733336656356e-12  (displayed approximation).

These distinguish covariance from capacity variance. More generally, with |d_i|<=rho_x and |e_i|<=rho_v, the one-call mean-only error obeys

|M_i_star-(M_i+delta*h*V_i*B_i)|
    <= delta*h*(k*rho_x*rho_v+b*rho_v^2).

A bound can therefore remain useful when a unique mean prediction fails. Propagating it requires updating both radii and the mean domain, then re-admitting the actual selector and adaptation branches. It does not by itself certify subsequent admission or an autonomous coarse execution law.

The capacity-feedback controls freeze both calls for all five preparations before response evaluation, with absolute binary64 budget 1e-14. Independent rational fine predictions verify the actual scalar and NumPy paths, fresh/held pressure, old-capacity flow, snapshot adaptation, rate/acceleration telemetry, temporal evidence and live grammar. Wrong predictions using adaptation too early, stale capacity at the next pressure refresh or a product of means are discriminated separately. The finite runs retain phase-controller history, actual capacity writes and the missing-history REMESH abstention. The fine oracle verifies execution; it is not presented as a reduced engine or a complete checkpoint.

The constituent-memory study already establishes hidden-correlation feedback under its own organized sine law. The result here resolves the separate native grammar/adaptation question: counter-dependent writes can make previously invisible form contrasts affect the next collective output. Coefficients, law and admission evidence do not transfer between those models. Ideal-real identities and finite numerical agreement establish neither a formed collective identity nor physical matter.

Joint enclosures distinguish certified decisions from unresolved branches

The preceding covariance obstruction does not prevent bounded prediction. An interval description can preserve the actual fine trajectory while stating when its retained information determines a native decision and when it does not. Here the question is adaptation admission across an actual threshold, under the same doubled-C5 support, zero phase, default laws, strict grammar, live built-in zero Gamma and one-substep clock delta=1/16. No new selector, adaptation law or generic predictor is installed.

Retained description. For each block retain mean intervals M_i in X_i, V_i in N_i, and bounds |d_i|<=r_x_i, |e_i|<=r_v_i, using the preceding pair coordinates. Retain associated sets S_n of possible counters for each fine node, plus the consumed grammar, lag, clock, support and controller state. The exact signed contrasts are omitted from the enclosure calculation. Its output is a set containing allowed states; it need not identify one state or certify that every combination inside the set is realizable.

Let interval arithmetic act on the full incoming boxes. Define

W_i = (N_(i-1)+N_(i+1))/2
B_i = k*((X_(i-1)+X_(i+1))/2-X_i)+b*(W_i-N_i)
R_i = k*r_x_i+b*r_v_i
J_i = h*(B_i+[-R_i,R_i]).

Both fine post-Coherence pressures belong to J_i. Their possible signs are not reconstructed from a mean. A positive capacity hull [v_min,v_max] and zero phase give Si>=alpha*v_min/v_max+beta; the stated sources satisfy the default high cut with a strict margin. This admits Coherence (IL) everywhere before any lag override, independently of normalized pressure. Live grammar and scalar domains are still required. Without this Si certificate, the following pressure-only classification does not admit the complete gate.

For T=EPS_DNFR_STABLE, classify the retained pressure interval as

qualifies:     max(abs(J_i)) <= T
fails:         min(abs(J_i)) > T
undetermined:  otherwise.

The minimum is zero when the interval contains zero. Equality qualifies, matching the existing native contract. An undetermined enclosure is not a failed gate, missing observation or execution rejection. It can arise from conservative bounds; only actual admissible lifts with different decisions establish intrinsic ambiguity of the proposed collective description.

Update each node's counter set by {s+1: s in S_n} for qualification, {0} for failure, and their union when undetermined. Its due-flag set consists of s_next>=tau for those successor counts. Qualification can be undetermined while a write is certainly not due: from count zero, the next count is in {0,1}, both below tau=5. From count four, the same uncertainty yields {0,5} and a genuinely unresolved due flag. Keep these sets associated with their targets; replacing them by a block-average counter loses this decision.

Joint propagation. Let maxabs denote an interval's largest absolute endpoint. The preceding exact pair identities imply the sufficient bounds

X_i_next = X_i + delta*h*N_i*B_i
           + delta*h*[-k*r_x_i*r_v_i-b*r_v_i^2, k*r_x_i*r_v_i]
r_x_i_next = maxabs(1-delta*h*k*N_i)*r_x_i
             + delta*h*maxabs(B_i-b*N_i)*r_v_i.

The asymmetric correction interval retains the nonnegative capacity variance: e_i^2 cannot supply a positive contribution to the mean flow. Dependency between interval factors may widen the result but cannot exclude a valid lift. For each possible associated pair of due flags, use the previous a_bar,a_delta definitions and compute

N_i_candidate = N_i + mu*a_bar*(W_i-N_i)
                + [-mu*abs(a_delta)*r_v_i, mu*abs(a_delta)*r_v_i]
r_v_i_candidate = abs(1-mu*a_bar)*r_v_i
                   + mu*abs(a_delta)*maxabs(W_i-N_i).

Take the hull of candidate mean intervals and the maximum candidate radius. The dependency in N_i+mu*a_bar*(W_i-N_i) may widen its interval; the controls re-admit those computed bounds without requiring a sharper expression. Cartesian combinations of possible flags can include incompatible branches, so this is an enclosure rather than an assertion that every branch occurs. Flow uses the incoming capacity; the capacity writer follows it. Rebuild B_i from the successor boxes at the next call. Admission, positive capacity and unclipped forms must be rechecked at every prefix; no all-time conclusion follows from the recurrence alone.

A certified threshold crossing with active feedback. Use initial means M_0=1/8+57/8192, M_i=1/8 otherwise, capacity means V=(1+epsilon,1,1,1,1), epsilon=1/1024, zero capacity contrasts and r_x_i=rho=2^-20 in all blocks. Initial counter sets are singletons: node two has count three, every other node count four. Supply the same grammar context as the prior native controls, lags zero, seed 17, time 1 and two calls. The source declares possible fine forms; it does not assert their formation.

The joint enclosure places both first-pair held pressures strictly outside the adaptation band on call one, near -0.001010. Their counters reset to zero. On call two, the entire interval lies inside the band, near -0.0009951, so both counters become one. Every other target qualifies on both calls. Node two consequently changes from not due to due as its count crosses five; node three is due on both. Active neighboring writes produce internal capacity contrast, which the enclosure propagates along with form contrast and means. Thus both the pressure threshold and the counter threshold are crossed under unchanged defaults, without supplying a branch selector.

Two prospective fine controls use d_i=+(-1)^i*rho and its negative. Their first mean outputs agree. After the first capacity update, e_1=mu*epsilon/4; their second block-one mean gap is

M_1_second(positive)-M_1_second(negative)
    = delta*h*k*mu*epsilon/2*(1-delta*h*k)*rho
    = 4.008300175634783e-13  (displayed approximation).

The common propagated interval contains both means. It therefore remains predictive on this horizon where an exact mean-only forecast would fail. Capacity and form radii are both active; no discarded signed contrast is fed back into the coarse calculation to obtain the enclosure.

Actual ambiguity at the same collective input. Return to all mean forms 1/8 and the same capacity pulse, with r_x_0=59/8192 and other radii zero. Keep node zero's incoming count zero and every other count four. Compare the two admitted fine lifts d_0=+59/8192 and its negative. They have the same coarse state, associated context and unordered forms. Both native words remain all-Coherence. For positive contrast, the first-pair held pressures are approximately +0.0009574116113 and -0.0010437845641; negative contrast swaps them. Both distances from the absolute threshold exceed 1/25000.

Positive contrast qualifies node zero, whose count becomes one without a due write, and rejects node one's adaptation gate. Negative contrast rejects node zero's gate and qualifies node one, whose count reaches five and writes. Their first flow means agree, but their terminal first-block capacity means differ by mu*epsilon/2, approximately 4.8828125e-5. The interval descriptor must leave node one's write unresolved while still certifying that node zero does not write. Treating all unresolved gates as false loses a real successor; treating them all as true invents a common decision that these lifts refute. The enclosing union retains both outcomes without assigning probabilities. Abstention concerns the detached collective verdict; the actual native motor uses its fine state and executes its own definite decision.

Verification boundary. The joint-admission controls reuse the shared outward interval kernel for the coarse calculation and independent rational fine arithmetic for its controls. Each preparation, horizon, complete forecast and fixed absolute binary64 allowance 1e-14 precedes the reserved response. Scalar and NumPy paths retain actual pressure/Si timing, strict grammar, controller records, counter resets, due writes, temporal history and REMESH abstention. The shared threshold controls retain inclusive boundaries and nearby rejected values.

Outward arithmetic encloses the stated ideal-real maps, including its own dyadic rounding. It is not an enclosure of all binary64 runtime executions; the declared finite controls check those responses separately. This result supplies conditional prediction and an explicit limitation of retained information, not autonomous coarse dynamics, a new NFR identity or a physical identification.

Native Resonance with primitive phase dispersion

Resonance (RA) admits a bounded collective prediction under the actual native selector, with primitive circular phases retained independently of form. This extends the preceding joint enclosure beyond zero phase. It reuses the collective RA geometry and the native phase budget, but admits the default pressure, sequential direct glyphs and subsequent native policies anew. The public atomic word and its normalized-sine continuation are different models; their coefficients are not imported.

Frozen domain. Keep the doubled C5, unit support, consecutive twins (2i,2i+1), order 0..9, common epi_kind="seed", strict grammar and live built-in zero Gamma of the preceding native controls. Supply their AL -> IL -> EN grammar context, zero AL/EN lags, zero adaptation counters, prior rates and zero acceleration; no temporal samples are invented. Use seed 17, structural time 1, one call with delta=1/16, and unchanged default pressure, selector, glyph, integrator, phase and adaptation policies. The source family is

M = (3/8, 1/8, 1/4, 1/8, 1/8),   V = (1/4, 1, 1, 1, 1),
x_(i,even) = M_i-d_i,             x_(i,odd) = M_i+d_i,
|d_i| <= rho = 2^-20,             v_(i,even) = v_(i,odd) = V_i,
theta_n in I = [1, 1+D],          D = 1/1024.

Phases use this declared lift, far from a wrapping boundary. Its pair means need not agree, and the coarse calculation receives neither signed form contrasts nor fine phases. The support and initial organization are supplied, not formed by this calculation. All claims below concern the ideal-real maps on represented configuration constants; actual binary64 responses have a separate finite error check. Write P for the represented math.pi divisor used by pressure, with P>3; the circular geometry uses ordinary phasors.

Selection and live U3. Let w,k,b be normalized phase, form and capacity pressure weights, and alpha,beta,gamma the normalized Si weights. Define

B_i = k*((M_(i-1)+M_(i+1))/2-M_i)
      + b*((V_(i-1)+V_(i+1))/2-V_i),
E = k*rho + w*D/3.

Every neighbor resultant lies in the nonzero cone I, so fresh fine pressure belongs to [B_i-E,B_i+E]. In particular B_0=(3*b-k)/4 and B_2=-k/8. The normalized pressure at either block-zero target is at most (|B_0|+E)/(k/8-E)<0.104<dnfr_hi=0.133. Its Si lies between alpha/4+beta*(1-D/3) and alpha/4+beta+gamma, strictly inside (0.25,0.5). Every other node has Si>=alpha+beta*(1-D/3)>0.5. With the retained lag and grammar context, the default batch therefore selects RA on nodes zero and one, and Coherence (IL) on the other eight nodes throughout this family.

This is a subset result. With these default Si weights, a positive maximum-capacity node has Si>=alpha>0.5 and selects IL before lag overrides. Equal positive capacities consequently cannot produce an all-RA batch under this ordinary selector. Changing that conclusion would require a different admitted policy or context, not an inference from the name Resonance.

All four neighbors of each RA target satisfy U3 because D<pi/2. RA phase mixing is convex inside I; the twins are not adjacent and share the same four neighbors. Neither RA changes its partner's propagation inputs, and IL changes no phase or form. These facts certify the live, ordered U3 checks, not just the initial snapshot. The positive neighbor form mean is 1/8, above the RA amplification trigger; scalar sign/kind and clipping gates have strict margins. No public-class optional readiness policy is substituted for the direct-glyph contract used by this runtime.

Joint successor. Let r,c,a denote the default RA form-mixing, phase-coupling and capacity-amplification factors, and h the IL pressure retention. Set

Y_0 = (1-r)*M_0+r*(M_4+M_1)/2,    Y_i = M_i otherwise,
A_0 = 1-r,                       A_i = 1 otherwise,
u_0 = (1+a)/4,                    u_i = 1 otherwise,
h_0 = 1,                         h_i = h otherwise.

Both RA targets acquire capacity u_0. RA leaves their stored pressure unchanged, whereas IL multiplies its targets' stored pressure by h. Integration then uses these post-event capacities and the held pre-event pressure, without refreshing the pressure after the phase jump. The terminal form mean and contrast satisfy

M_i_next in Y_i + delta*u_i*h_i*B_i
             + [-delta*u_i*h_i*w*D/3, delta*u_i*h_i*w*D/3],
|d_i_next| <= |A_i-delta*u_i*h_i*k|*rho
             + delta*u_i*h_i*w*D/6.

The second bound retains a useful correlation: twins have the same neighbor resultant, so their phase-pressure difference is exactly -2*w*t_i/P, where t_i=(theta_(i,odd)-theta_(i,even))/2. Treating the two pressure intervals as independent would lose this cancellation. All proposed forms stay strictly inside the positive scalar rails. Capacity adaptation follows integration; the initial zero counters cannot reach five. Moreover an RA target fails its Si qualification because the writer retains the pre-batch Si<0.5. Its counter resets and its new capacity remains u_0 on this call. The pressure bounds also determine all terminal counts: block four qualifies, the other IL blocks fail, giving (0,0,0,0,0,0,0,0,1,1) from the supplied zeros.

The RA batch stays in I, with order parameter at least 1-D^2/8 and no disruptor history, hence the default phase controller takes its stable branch. Retain its effective gains K_G,K_L, computed from the supplied initial gains and stable-branch update. They obey K_G>0, K_L>=0, K_G+K_L<1. The subsequent coordinator preserves I and gives global diameter at most (1-K_G)*D. Shared local and global targets cancel within each twin pair:

t_i_next = (1-K_G-K_L)*s_i*t_i,    s_0=1-c, s_i=1 otherwise.

Thus the phase contrast radius is at most (1-K_G-K_L)*s_i*D/2. Phase pair means remain in I; a diameter bound alone is not an absolute phase-location bound. Controller records advance even though the sufficient enclosure need not retain every fine phase.

Prospective controls and limits. Freeze two lifts with identical forms d_i=(-1)^i*rho, phase center theta_c=1+D/2 and phase contrasts t_i=+(-1)^i*D/2 or their negatives. Every neighbor resultant initially has angle theta_c; the RA batch retains this conjugate-pair geometry. The independent fine oracle therefore uses

M_i_next = Y_i+delta*u_i*h_i*B_i,
d_i_next = (A_i-delta*u_i*h_i*k)*d_i-delta*u_i*h_i*w*t_i/P.

Opposite phase assignments give a nonzero contrast difference -2*delta*u_i*h_i*w*t_i/P despite identical initial forms, capacities and unordered phase marginals. Their mean outputs agree and both successors fit the common enclosure. With L_i=A_i-delta*u_i*h_i*k>0 and H_i=delta*u_i*h_i*w/P>0, their squared contrast difference is (d_i_positive)^2-(d_i_negative)^2=-2*L_i*H_i*rho*D<0. This label-invariant dispersion separates the two form/phase associations; it is not a failure of mean prediction in this symmetric family. It isolates primitive phase participation in the nodal form response, rather than identifying phase with a form observation.

The native Resonance controls freeze all four source/backend forecasts before the four native calls. They use the shared outward rational interval kernel for coarse bounds and exact rational fine formulas on represented constants, with a separate 1e-14 absolute binary64 allowance. Live U3 subsets, actual glyph effects, held pressure, post-RA capacity, coordinator gains, counters, histories and REMESH abstention remain observable. This proves a conditional one-call enclosure; it does not admit a later call, a partial U3 neighbor subset, arbitrary target order, a formed collective identity, autonomous occurrence or physical matter.

Fresh Resonance selection does not imply live U3 admission

The preceding small phase cone admits every neighbor. At the U3 boundary, fresh native selection and executable Resonance (RA) are different predicates. The following supplied family proves this distinction with unchanged defaults; it is not a solver defect or a reason to weaken the phase gate.

Keep that doubled-C5 support, ordered targets, strict grammar, supplied context, zero counters, live zero Gamma, seed 17, time 1 and one attempted call with delta=1/16. The new pair-constant preparation is

M = (-7/8, 1, -1, -1, 1),    V = (1/4, 1, 1, 1, 1),
theta_(0,even) = theta_(0,odd) = 1+g,
theta_n = 1 for all other nodes,
g in [L-epsilon,L+epsilon],  L=P/2,  epsilon=1/1024.

Here P is the represented pressure normalizer and L the default represented U3 limit. Keep the common nonempty kind seed; negative form is admitted by the signed-scalar chart. No pressure, Si or selected glyph is supplied. The endpoints have exactly representable phase preparations. The ideal-real formulas below use the represented coefficients; finite numerical checks at the endpoints retain their separate allowance, rather than claiming an exact floating-point verdict at the cutoff itself.

Uniform fresh selection. Let w,k,b be the pressure weights and alpha,beta,gamma the separately owned Si weights. Since 0<g<pi, every neighbor resultant is defined. Its angular displacement from its target is (-g,g/2,0,0,g/2) across the five blocks. The fresh pressures simplify to

z(g) = 15*k/8 + 3*b/4 - w*g/P,
p(g) = (z, -k-z/2, k, k, -k-z/2),
max_n |p_n| = k+z/2.

Exact rational bounds on the represented defaults give 7/1000<z<9/1000 and 0<z/(k+z/2)<1/20<dnfr_hi=0.133 throughout the interval. The first pair has

Si_0 = alpha/4 + beta*(1-g/P) + gamma*(1-z/(k+z/2)),
1/3 < Si_0 < 17/50.

Every other Si is at least alpha+beta*(1-(L+epsilon)/(2*P))>7/8. Thus the actual default selector requests RA on nodes zero and one, and Coherence (IL) elsewhere, on both sides of U3. The small total pressure at the RA targets results from cancellation between phase, form and capacity channels. It does not bound each edge's phase mismatch.

Admission separates the same requests. Every neighbor of either RA target has phase 1. Its wrapped phase distance is therefore g. The ideal gate admits all four neighbors when g<=L and none when g>L. The two frozen endpoint controls have strict opposite margins epsilon. For the outside control, enabled strict grammar rejects the first target with StructuralGrammarError(rule="U3", candidate="RA"), before direct RA executes. No fallback glyph, integrator, phase coordinator, capacity writer or REMESH step is reached. This is an actual failed admission, not an indeterminate coarse bound or an unavailable phase observation.

For the inside control, write r,c,a for the default RA form, phase and capacity factors. Each selected target's immediate proposal is

x_RA = -7/8 + 15*r/8 < 0,
theta_RA = 1+(1-c)*g,
v_RA = (1+a)/4.

The compatible-neighbor form mean is 1, which activates amplification; the proposal preserves the target's negative sign and established kind. Twins are nonadjacent, so the first event leaves the second target's four propagation inputs unchanged. The native call completes. Its Euler row uses these new forms/capacities and held pressures (z,h*p_1,h*p_2,h*p_3,h*p_4), where h is IL retention. All forms then lie inside the scalar rails, every adaptation counter is zero, and no adaptation write is due. This positive control distinguishes U3 rejection from an independent scalar identity failure.

Failure state and evidence. First-target rejection is not whole-call rollback. Pressure and Si have already been refreshed, the invocation ordinal consumed, AL/EN lags incremented and a real left-boundary form sample recorded. Physical time, form, phase, capacity, adaptation counts and glyph occurrence history remain unchanged for this source because no glyph has yet executed. The prepared selector binding is cleared. A later-target rejection has the separate accepted-prefix contract; neither result promises callback rollback.

The boundary controls freeze both endpoint preparations and their pressure, Si, request, admission and accepted-event predictions before four attempted native calls on scalar and NumPy paths. Numerical readouts use a fixed absolute 1e-14 allowance; exception, subset and occurrence checks are discrete. Existing shared U3 and runtime rejection tests retain their broader invalid-input and prefix scope. No production policy changes are required: the live grammar already rejects the inadmissible request. Selection diagnostics cannot replace its geometric premises. A strict nonempty partial neighbor subset, autonomous occurrence and physical identification remain outside this result.

Partial compatibility retains the association between form and phase

Resonance (RA) propagates from its live U3-compatible neighbors, whereas the fresh default pressure and Si use their own full-support inputs. A full-neighbor form mean or separate form and phase inventories therefore need not determine the action. This is an information boundary within the implemented native law, not a proposed pressure replacement or an autonomous occurrence principle.

Keep the preceding doubled-C5 support, node order, supplied grammar context, common kind seed, zero counters, strict rejection, seed 17, clock t=1, delta=1/16, live zero Gamma and unchanged defaults. Set

A=15/16, epsilon=rho=1/1024, P=Q(math.pi), L=P/2,
M=A*(-7/8,1,-1,-1,1), V=(1/4,1,1,1,1).

Here Q denotes the exact rational value of a represented source constant. Both nodes of block zero have phase 1+L. Blocks two and three have phase 1. In blocks one and four, associate the even/odd labels sigma=-1,+1 with phase 1+sigma*epsilon and form A+s*sigma*rho, where s=-1,+1 denotes the two preparations. All other forms equal their block means. These are supplied initial conditions, not a formation theorem. Both preparations have the same support, capacities, block means, node phases, unordered forms in each block and full-neighbor form means. They differ in the association between neighboring form and phase.

Fresh selection. With pressure weights w,k,b and separately resolved Si weights alpha,beta,gamma, the ideal-real fresh pressure is

z = 15*k*A/8 + 3*b/4 - w/2,
p_0 = z,
p_(1 or 4,sigma) = -k*A-z/2-sigma*(k*s*rho+w*epsilon/P),
p_2 = p_3 = k*A,                   max_n |p_n| = k*A.

The phase resultants used here have angles 1, 1+L/2, 1, 1, 1+L/2 across the five blocks; they are all nonzero. Exact rational inequalities on the represented defaults give -14/1000<z<-13/1000, a competing-peak margin above 6/1000, and abs(z)/(k*A)<8/100. The RA targets have

Si_0 = alpha/4 + beta/2 + gamma*(1-abs(z)/(k*A)),
1/3 < Si_0 < 17/50.

All other Si values exceed alpha+beta*(1-(L/2+epsilon)/P)>89/100. These inequalities certify native RA requests on nodes zero and one and Coherence (IL) elsewhere, with positive selector margins in both preparations. The target pressure, target Si and global pressure maximum coincide exactly in this algebra. Individual other-node pressures and Si values need not coincide.

Actual selected mean and complete form response. Each RA target has two neighbors at distance L-epsilon and two at L+epsilon. Its live subset is exactly nodes (3,9), with strict margin epsilon; the compatible form mean is m_s=A+s*rho. These twins are nonadjacent, so the first RA does not alter the second target's propagation inputs. With default RA factors r,c,a:

X_s = (1-r)*(-7*A/8)+r*(A+s*rho),
g_star = (1-c)*L+c*epsilon,
theta_RA = 1+g_star,               u_0=(1+a)/4.

Both X_s are negative and retain the common established kind. Both selected means exceed the amplification threshold, so the capacities agree. Stored target pressure remains z; IL multiplies the other held pressures by its factor h. With Y_n=X_s on the first pair and the unchanged form elsewhere, u_n=u_0 on that pair and one elsewhere, and q_n=z or h*p_n respectively, the unclipped complete form row is

rate_n=u_n*q_n,                    x_n_next=Y_n+delta*rate_n.

All endpoint forms are strictly inside the scalar rails. All held pressure magnitudes exceed the stability threshold, so adaptation counters reset to zero and no adaptation write occurs. The two target block means differ by exactly 2*r*rho after RA and at the complete-call endpoint, approximately 0.000310849498226. Replacing the selected mean by the all-neighbor mean A would lose this response. Full-neighbor means and separate form/phase marginals are therefore insufficient even for the target block mean, not only for an internal dispersion statistic.

Successor geometry without another invocation. After the two RA actions, all phases lie in I=[1-epsilon,1+g_star]. Its width D=g_star+epsilon is strictly below L, since the represented defaults satisfy c*L>(1+c)*epsilon. Thus even the previously excluded edges are compatible in this post-event state. This does not change the subset that the already executed actions consumed. The global resultant's real projection relative to angle one obeys

R >= (4+4*cos(epsilon)+2*cos(g_star))/10
  > 4/5-epsilon^2/5 > R_hi.

Here 0<g_star<3/2<pi/2, and the prior/realized context contains no disruptor. The coordinator takes its stable branch. Its updated global/local gains are nonnegative with sum below one. On this short common lift, every neighbor and global circular mean lies in I; the coordinator's convex row retains that arc and gives diameter at most (1-kG)*D<L. Consequently every existing edge is U3-compatible at the endpoint. No next selector request, repeated-call closure, persistence of the subset or autonomous RA occurrence follows.

The partial-subset controls freeze both association preparations, both backends and every forecast before four native responses. Independent rational formulas use represented source constants; finite binary64 observations retain a separate absolute 1e-14 allowance. Observers retain actual choices, live subsets, the ordered prefix, held pressure, all post-event coordinates, rates, histories, coordinator branch and adaptation/REMESH abstention. The proof establishes conditional geometry and ideal algebra; these four controls establish finite implementation agreement.

Local robustness requires subset and response margins

For an admitted primitive snapshot, let d_j be the circular distance from a target to neighbor j, and let the fixed U3 limit be L. Suppose every neighbor has a positive membership margin mu_j=abs(d_j-L). Under circular phase errors bounded by eta_0 at the target and eta_j at that neighbor, the triangle inequality gives abs(d_j_new-d_j)<=eta_0+eta_j. Hence eta_0+eta_j<mu_j for every neighbor preserves the complete selected subset K, including exclusions. This is a local ideal-real statement with fixed support and gate; numerical certification additionally needs a rounding budget. It does not certify the independent default selector or live grammar context.

For the same fixed nonempty K, form errors e_0,e_j>=0 give

abs(m_K_new-m_K) <= mean_(j in K) e_j = e_K,
abs(x_RA_new-x_RA) <= (1-r)*e_0+r*e_K,       0<=r<=1.

This bound assumes the same factor and scalar chart and applies before clipping, or after the same hard interval projection. A soft clipping branch need not be unit-Lipschitz and requires its own bound. Kind retention, sign compatibility and finite arithmetic remain independent checks. With the represented runtime threshold T=Q(RA_RUNTIME_AMPLIFICATION_TRIGGER) (materialized as binary64 1e-9), amplification is certain if [m_K-e_K,m_K+e_K] lies entirely in (T,infinity) or (-infinity,-T). It is certainly absent if that interval is contained in [-T,T]. Any overlap across a trigger boundary is unresolved; it must not be silently classified as absent. These rules follow from the implemented strict condition abs(m_K)>T; T is a runtime policy, not a fundamental constant.

Phase response needs an additional resultant margin. For phase error radii 0<=eta_j<=pi, the normalized selected resultant changes by at most E=mean_(j in K) 2*sin(eta_j/2). If its nominal magnitude is m>0 and E<m, the disk of radius E around that resultant excludes zero and bounds the mean-angle error by asin(E/m). For 0<=c<=1 and an unchanged shortest-arc branch, the RA phase error, circular or on the retained common lift, is at most (1-c)*eta_0+c*asin(E/m). Membership alone does not replace these phase regularity premises. The partial-subset source has coincident selected phases, so its normalized selected resultant is one and its membership margin is epsilon.

These bounds apply to one live action. Later adjacent-target actions must re-admit the modified state; the preceding successor geometry already shows that RA itself can change compatibility. Native sequential execution cannot freeze all event subsets from the initial snapshot merely because all requests were prepared together. A generic collective successor law, compressed predictor and repeated-call guarantee remain unproved.

Compatible form information also controls the capacity branch

The preceding comparison keeps amplification active in both preparations. A separate frozen family shows that phase-associated form information also decides whether Resonance (RA) increases capacity. This changes the subsequent nodal rate under the same held target pressure; it is not only a different instantaneous form blend.

Keep the same support, capacities, node order, supplied context, clock, seed, one-substep budget and defaults. Let A=15/16, target block form B=-21/32, and both middle blocks have form -A. Target phases are two; phases in blocks one and four are zero on even nodes and two on odd nodes; middle phases are one. In each adjacent block, the low/high-phase forms are (A,0) in the inactive preparation and (0,A) in the active preparation. Both retain the full-neighbor mean A/2 at the targets, the same unordered form inventories, all block means, capacities, phases and common kind seed.

Write e=0,1 for the inactive/active preparation and H=(B-A)/2=-51/64. The independent full-support fresh-pressure algebra gives

z = 9*k/8 + 3*b/4 - w/P,
p_target = z,
p_high = k*(H-e*A)-3*b/8-w/(2*P),
p_low = k*(H-(1-e)*A)-3*b/8+3*w/(2*P),
p_middle = 3*k*A/4,               peak_e=-p_high.

The target pressure is identical across preparations, with 8/1000<z<9/1000. The pressure peaks differ, so fresh target Si also differs:

Si_target(e)=alpha/4+beta*(1-1/P)+gamma*(1-z/peak_e).

Both target Si values lie strictly between 37/100 and 38/100, and both normalized target pressures lie below 3/100. All other Si values exceed alpha+beta*(1-3/(2*P))>85/100. The actual native selector thus requests RA on the first pair and IL elsewhere in both cases. This comparison does not claim identical Si or identical full-network pressure.

At either target, the two high-phase neighbors have zero phase gap and the two low-phase neighbors have gap two. Since 0<L<2, the live U3 subset is again exactly nodes (3,9). Its form mean is e*A, not the all-neighbor mean A/2. Both RA phase proposals equal the existing target phase two; the two nonadjacent events have unchanged propagation inputs. The event law gives

X_e=(1-r)*B+r*e*A < 0,             u_e=(1+e*a)/4,
rate_e=u_e*z,                     x_target_next=X_e+delta*u_e*z.

The selected zero mean leaves capacity unchanged; the selected positive mean exceeds the shared represented trigger and activates its default boost. Both events preserve sign and established kind. All endpoint forms are strictly inside the scalar rails. Other nodes retain their initial forms through IL and then use their own held pressure h*p_n with capacity one. Every held pressure magnitude exceeds the stability threshold, so no adaptation write can explain the different target capacities or rates. In particular,

rate_active-rate_inactive = a*z/4,
x_active_next-x_inactive_next = r*A+delta*a*z/4.

These ideal differences are approximately 0.00008275353679 and 0.1492129312447 respectively. Using the full-neighbor mean would incorrectly activate the boost in the zero-selected-mean preparation. The same initial target pressure can therefore produce distinct post-event nodal rates through RA's actual capacity law, without replacing pressure or rescaling the clock.

The unchanged post-event phases contain four nodes at two, four at one and two at zero. Their global resultant magnitude is

R = sqrt((4+6*cos(1))^2+(2*sin(1))^2)/10,
R_lo < R < R_hi.

There are no disruptors. The phase controller consequently takes its transition branch, not the stable branch of the earlier small-cone source. Alternating-series rational bounds for sin(1) and cos(1) establish these strict thresholds before execution. Both preparations share all phase inputs, so their coordinator outputs agree. The fresh neighbor angles are 1 at targets and middle nodes, and 3/2 at adjacent nodes. With global angle psi=1+atan2(2*sin(1),4+6*cos(1)), the ordinary common-lift coordinator row (1-kG-kL)*theta_n+kG*psi+kL*local_mean_n gives a separately budgeted finite phase forecast. No stable-branch coefficient is transferred from the earlier family.

The capacity-association controls freeze both preparations, configurations and all predictions before four native calls across scalar/NumPy paths. They check fresh pressure/Si, actual requests/subsets, all form/capacity writes, held-pressure rates, phase branch, history and adaptation/REMESH abstention, with the same declared 1e-14 absolute finite allowance. Together the two association studies resolve the single-event information question for their admitted sources. They do not establish a complete collective state, an interacting-target batch closure, formed identity, autonomous occurrence or physical identification.

An accepted Resonance prefix changes a later capacity branch

The native selector prepares all requests before the ordered event loop, but each Resonance (RA) reads its live compatible neighbors. These two boundaries cannot be identified. An earlier RA can change a later requested RA's capacity branch even when every phase stays zero and all neighbors remain compatible. This is more specific than the previously established dependence of sequential form mixing on target order.

Admitted source family. Retain doubled C5, node order 0..9, consecutive twins, common kind seed, supplied AL → IL → EN context, zero AL/EN lags and adaptation counters, strict grammar, seed 17, t=1, delta=1/16, live zero Gamma and the default laws. Initial rates remain the declared block values (i-2)/8. Let all phases be zero and set

q in [7/64,1/8],
M=(q,q,-q,-1/2,-q),               V=(1/4,1/4,1,1,1).

This is a correlated source family: in particular M_2=M_4=-M_0=-M_1. It is not an unconstrained open set in the full native state. No source history, order or occurrence law is derived by specifying these coordinates.

With normalized form/capacity pressure weights k,b, define

z=3*b/8-k*q,                      P=k*(1/2-q),
p=(z,z,-P/2-z,P,-P/2-z).

Here P denotes this family's pressure peak, not the phase normalizer used above. All phase-pressure terms vanish. Exact rational bounds on the represented defaults give P>0 and abs(z)/P<27/1000. Consequently max_i abs(p_i)=P, because abs(-P/2-z)<(1/2+27/1000)*P<P. For separately resolved Si weights alpha,beta,gamma, the first two blocks have

Si_RA=alpha/4+beta+gamma*(1-abs(z)/P),
42/100 < Si_RA < 44/100.

The remaining blocks have Si at least alpha+beta>94/100. Thus the actual default selector requests RA at nodes zero through three and Coherence (IL) at nodes four through nine throughout this interval. All selector margins are strict. Si is retained and pressure is not recomputed during the event loop; RA retains its target pressure and IL applies its declared factor.

Live block recurrence. For pair-constant primitives and consecutive twin targets, twins see the same neighboring blocks. An action at the first twin does not change the second twin's inputs, since the twins are nonadjacent. An admitted RA therefore preserves that block's pair equality. In ideal-real arithmetic, each block action can be written using the current neighbor block forms and phases: re-admit the compatible blocks, take their equally weighted means/resultant, and apply the actual form, phase, identity and capacity laws. Later blocks consume these updated values. This conditional recurrence retains the admitted context and order; it is not a simultaneous initial-snapshot map or a complete collective state for arbitrary events.

For this family all four neighbors are compatible at every RA event. Write r,a for the default form-mixing and capacity-amplification factors and T=Q(RA_RUNTIME_AMPLIFICATION_TRIGGER) for the represented trigger. Initially the neighbor form means of both RA blocks are zero. The first RA block gives

m_0=(q-q)/2=0,
Y_0=(1-r)*q,                      u_0=1/4.

Its phase remains zero and its capacity does not increase. The later block instead reads

m_1=(Y_0-q)/2=-r*q/2,
Y_1=q*(1-r-r^2/2),                u_1=(1+a)/4.

Uniformly, abs(m_1)>8/1000>T. The negative selected mean activates the implemented absolute-value trigger; positive mean is not required. Both proposals remain strictly positive, retain kind seed and lie strictly inside the scalar rails. Therefore the full RA prefix is admitted. The later amplification is caused by the accepted earlier form jump, not by a new selector decision, capacity adaptation or an inserted operator.

Complete call and discriminator. IL retains the other pressures by its factor h and leaves their primitive form/capacity unchanged. The complete Euler form prediction is

x_0_next=(1-r)*q+delta*z/4,
x_1_next=q*(1-r-r^2/2)+delta*(1+a)*z/4,
x_2_next=x_4_next=-q+delta*h*(-P/2-z),
x_3_next=-1/2+delta*h*P.

The rates are the displayed time-increment coefficients and the capacities are (1/4,(1+a)/4,1,1,1). All endpoint forms satisfy abs(x)<3/5. The coordinator has resultant one and zero disruptor load, takes its stable branch and leaves every phase zero. The RA blocks fail capacity-adaptation qualification because their retained Si is below si_hi=1/2, even where their pressure lies inside the separate stability band. Other held pressure magnitudes exceed 1/50. All adaptation counts remain zero, with no adaptation write; short temporal histories also retain REMESH abstention.

A hypothetical pre-batch form snapshot would keep the later selected mean zero and incorrectly predict both Y_1_stale=(1-r)*q and u_1_stale=1/4. The actual-minus-stale differences are

event form:     -r^2*q/2,
nodal rate:      a*z/4,
terminal form:  -r^2*q/2+delta*a*z/4.

The terminal difference is strictly below -1/1000 on the whole source interval. At q=15/128, z is positive and nonzero, so the capacity change also gives a nonzero rate difference. At a source where z=0, the same capacity-branch distinction persists but its immediate form-rate difference vanishes; capacity does not override the nodal product.

The supplied block-one U2 debt remains one after RA; Resonance is neutral in that ledger. IL blocks discharge their supplied debt. Successful completion of this native call is not completion of every grammatical obligation or permission to discard its context on a subsequent call.

Form uncertainty without exact cancellation. Keep phase, capacity, support, kind, context, order and every law fixed, but perturb each initial nodal form independently by at most epsilon=T/2. For ideal-real arithmetic, every fresh pressure changes by at most eta=2*k*epsilon, and so does its graph-wide absolute maximum. With zeta=max(abs(z(7/64)),abs(z(1/8))) and P_min=3*k/8, the perturbed RA pressure ratio is bounded by (zeta+eta)/(P_min-eta)<27/1000; the same strict selector branches remain. The maximum-capacity IL blocks retain their independent Si lower bound.

Every sequential unclipped RA form update is a convex row, so its sup-norm error remains at most epsilon throughout the accepted prefix. The first block's selected mean is therefore in [-epsilon,epsilon], strictly inside the inactive trigger band. The later selected mean stays within epsilon of -r*q/2, whose magnitude remains above T. Positive target identity, scalar rails and adaptation rejection retain their separate margins. Thus capacities, phases and branch choices agree with the nominal source, and

max_n abs(x_n_next_perturbed-x_n_next_nominal)
    <= (1+2*delta*k)*epsilon < 41*epsilon/40.

This supplies a form-only enclosure with independent constituent errors. It does not preserve exact twin equality or produce a quotient of every perturbed source. Its narrow width is set by the implemented amplification threshold; it is not a physical preparation tolerance. The ideal enclosure and the finite nominal binary64 check remain separate claims.

The live-prefix controls freeze nominal q=15/128, both scalar/NumPy configurations and all forecasts before two native calls. Independent rational algebra certifies the family and nominal form/rate/capacity response; finite binary64 comparisons retain a separate absolute 1e-14 allowance. Observers record every prepared request, live subset, selected mean, ordered jump, held pressure, phase/capacity policy and history. This closes a bounded adjacent-RA capacity-branch question under supplied order. It does not prove autonomous ordering, general changing-U3 closure, repeated-call maintenance, formed identity or physical identification.

A restricted Resonance prefix cannot remove existing phase compatibility

There is also a geometric obstruction to one possible later rejection. Consider the ideal RA/IL event prefix, before the global/local phase coordinator, on doubled C5 with pair-constant phases, consecutive complete twin actions, fixed U3 limit 0<=L<=pi/2 and RA phase factor 0<=c<=1. Every action is assumed admitted in its other consumed state and grammar conditions. Each target has at most two distinct neighboring block phases, with equal multiplicities. No assumption about their form means replaces those conditions.

Put the target at phase zero on its local circular lift. If both neighbor blocks are compatible, write their offsets a,b in [-L,L]. Their normalized resultant is

cos((a-b)/2) * exp(i*(a+b)/2).

Unless it vanishes at opposite boundary offsets, its lifted angle is (a+b)/2. After RA the offsets relative to the moved target become

a_next=(1-c/2)*a-(c/2)*b,
b_next=(1-c/2)*b-(c/2)*a.

Each row's absolute coefficients sum to one, so both offsets remain in [-L,L]. If their previous clearances are mu_a=L-abs(a) and mu_b=L-abs(b), the first clearance is at least (1-c/2)*mu_a+(c/2)*mu_b, and symmetrically for the second. If only one block is compatible, its offset becomes (1-c)*a, retaining at least its old clearance. An undefined selected mean leaves the target phase fixed.

Each first twin leaves the second's neighbor inputs unchanged; completing the pair preserves pair-constant phases. Direct IL leaves phases unchanged. Induction therefore preserves every already compatible edge through this restricted event prefix. A later RA with an initially nonempty compatible set cannot lose its U3 admission solely through those earlier phase jumps. This does not prevent an independent identity or grammar rejection.

Incompatible edges may enter the set. For example, the ideal local offsets 1 and 8/5 have only the first block admitted at the default gate. The RA shift is c; the second gap becomes 8/5-c<pi/2 for the default factor. This illustrates geometric possibility, not fresh native selection or an admitted complete call with that preparation.

The retained pair geometry, equal multiplicities and event boundary are essential hypotheses. This proof does not apply to internally dispersed neighbor phases, interleaved twins, arbitrary neighbor multiplicities, another phase-changing operator or the subsequent coordinator. It is an ideal-real statement; binary64 wrapping/resultants need an error allowance smaller than the propagated strict clearance. Equality at U3 does not acquire a universal floating-point guarantee from the affine inequality. The live-prefix controls also check the affine extremal and margin identities independently of native responses; no phase-loss trajectory is evaluated or inferred.

Resonance can enlarge a later target's live compatible neighborhood

An admitted Resonance (RA) phase jump can make an existing, previously excluded neighbor compatible with a later target. Its form then enters the later propagation mean. This changes the response even when both requests were prepared together and both capacity branches remain active. Support edges are retained; the changing object is the U3 propagation subset.

Source and selection. Retain the preceding doubled-C5 support, consecutive twins in node order 0..9, kind seed, supplied AL → IL → EN context, zero selector lags and adaptation counters, strict grammar, seed 17, t=1, delta=1/16, live zero Gamma and unchanged default laws. The prior block rates are (i-2)/8, with zero acceleration and empty temporal histories. Write P=Q(math.pi) for the represented pressure normalizer and L=P/2 for the U3 limit. The correlated pair-homogeneous preparation family is

A=15/16,                       epsilon in [1/2048,1/512],
M=(A,-A,A,0,-A),               V=(1/4,1/4,1,1,1),
theta=(0,L+epsilon,2*epsilon,3,L-epsilon).

These are supplied coordinates and grammar memories. Their formation or selection is not derived. The nominal epsilon=1/1024 has exactly materializable primitive coordinates. The family below uses ideal-real arithmetic on represented law coefficients; finite execution is checked separately with the frozen absolute 1e-14 allowance.

All initial phases lie in [0,3], narrower than a semicircle. Neighbor resultants are defined and their common-lift angles are (L,epsilon,(L+epsilon+3)/2,(L+epsilon)/2,3/2). With normalized pressure weights k,b,w for form, capacity and phase, respectively, fresh pressure is

p_0=-2*k*A+3*b/8+w/2,
p_1= 2*k*A+3*b/8-w/2,
p_2=-3*k*A/2-3*b/8+w*(L+3-3*epsilon)/(2*P),
p_3=w*((L+epsilon)/2-3)/P,
p_4=3*k*A/2-3*b/8+w*(3/2-L+epsilon)/P.

Exact rational endpoint bounds, extended by affine dependence on epsilon, place these in (.057,.059), (-.015,-.013), (.27,.28), (-.54,-.53) and (.21,.22), respectively. Thus the global absolute peak is -p_3. Both first-block pressure ratios are below .12, strictly below the default selector threshold. Independently resolved Si weights give both target indices in (.33,.34); the capacity-one blocks have Si at least alpha>.75. Therefore the actual prepared requests are RA at nodes zero through three and Coherence (IL) at nodes four through nine. Si is retained and pressure is not recomputed during the event loop; RA retains its target pressure and IL applies its factor h. Global selection does not supply U3 admission.

Two ordered live actions. Initially block zero admits only block four, while block one admits only block two. Their included gaps are L-epsilon and excluded gaps L+epsilon, with strict margins on both sides. For the default RA form, capacity and phase factors r,a,c, the first pair gives

selected_mean_0=-A,
Y_0=(1-2*r)*A > 0,              u=c*(L-epsilon),
capacity_0_star=(1+a)/4.

The negative selected mean activates the absolute-value capacity trigger. The twins are nonadjacent, so their consecutive actions read identical inputs. Block zero's new phase makes it compatible with block one: the clearance is u-epsilon>.12 throughout the source interval. When the later pair executes, its selected set has grown from two nodes to all four:

selected_mean_1=(Y_0+A)/2=(1-r)*A,
selected_phase_1=(u+2*epsilon)/2,
Y_1=-(1-r)^2*A < 0,
v=(1-c)*(L+epsilon)+c*(u+2*epsilon)/2
 =(1-c)*L+epsilon+(c^2/2)*(L-epsilon),
capacity_1_star=(1+a)/4.

Both selected form means are uniformly far above the amplification threshold in magnitude. The positive first and negative second target forms retain their signs, all neighbors retain kind seed, and the proposals lie inside the scalar rails. Each live RA therefore passes its separate identity and grammar conditions. The newly admitted forms and phases, not a changed request or refreshed pressure, account for the second response.

A hypothetical calculation that freezes only the initial compatible sets would retain block two alone for the later target. It predicts Y_1_stale=-(1-2*r)*A and v_stale=(1-c)*L+(1+c)*epsilon. Consequently

Y_1-Y_1_stale=-r^2*A,
v-v_stale=c*(u-2*epsilon)/2 > 0.

Both calculations amplify the same capacity. With the same held pressure, the terminal form difference remains exactly -r^2*A, approximately -.02374715 for the represented defaults. This discriminator isolates the changed live subset; it does not require a capacity-branch change.

Complete-call successor. If h is the direct IL pressure-retention factor, the post-event and integration rows are

Y=((1-2*r)*A,-(1-r)^2*A,A,0,-A),
V_star=((1+a)/4,(1+a)/4,1,1,1),
p_held=(p_0,p_1,h*p_2,h*p_3,h*p_4),
rate_i=V_star_i*p_held_i,        x_i_next=Y_i+delta*rate_i.

All terminal forms satisfy abs(x)<.96. Capacities remain admitted, all held-pressure magnitudes exceed the separate adaptation stability threshold, and the RA targets also have Si below si_hi=1/2. Counts stay zero without an adaptation write. Short histories retain REMESH abstention. RA is U2 neutral, so the supplied block-one debt remains one; IL blocks discharge their supplied debts. Successful execution does not close every grammatical obligation for a future invocation.

The post-event phases are Phi=(u,v,2*epsilon,3,L-epsilon), in the common arc [2*epsilon,3]. To certify the coordinator's own branch, pair the phasors at 2*epsilon and 3, and bound the other three by one:

R <= (3+2*cos(3/2-epsilon))/5 < 63/100 < R_lo.

The strict middle inequality follows from the degree-eight alternating Taylor upper bound for cosine at 3/2-1/512, which is below 3/40. There are no disruptors. The actual coordinator branch is dissonant; that policy label does not mean that Dissonance (OZ) executes. The global gain moves toward kG_max with up, while the local gain moves toward (kL_min+kL_max)/2 with down. Their resulting values K_G,K_L are positive with sum below one. For

Psi=arg(sum_j exp(i*Phi_j)),     ell_i=(Phi_(i-1)+Phi_(i+1))/2,
theta_i_next=(1-K_G-K_L)*Phi_i+K_G*Psi+K_L*ell_i,

every angle stays in the same common arc. The absolute-angle forecast uses an independent elementary-trigonometric evaluation with the declared finite allowance; the family branch inequality is an analytical certificate.

The gained block-zero/block-one compatibility also survives this coordinator. Put D=v-u, with 0<D<L. Their local-target difference has magnitude (D+L-3*epsilon)/2<L. The common global target cancels in their difference, giving the exact rational forecast

theta_1_next-theta_0_next
  =(1-K_G-K_L)*D+K_L*(ell_1-ell_0),
abs(theta_1_next-theta_0_next) < (1-K_G)*L < L.

This retains the gained edge only. It neither makes all edges compatible nor invokes another RA or selector decision.

The neighbor-gain controls freeze both nominal configurations and complete forecasts before two native calls, one scalar and one NumPy. They observe prepared requests, actual live subsets, ordered writes, held-pressure integration, coordinator and retained context against the independent predictions. The changing-subset mechanism belongs to the existing runtime; no selector, factor or evolution law is replaced. The result is a conditional one-call response, not general native closure, formation, autonomous ordering or physical identification.

Sufficient state and a native five-node realization of one RA/IL call

The preceding Resonance (RA) and Coherence (IL) results compose into a conditional complete-call reduction. On the domain below, one existing native call on a five-node unit C5 realizes the ideal block response of doubled C5. Each reduced node represents two nonadjacent twins, and each reduced edge lifts to the four unit edges between its endpoint blocks. No new selector, integrator or operator is needed. This is a symmetry of a supplied execution family, not formation of a collective NFR or a claim that every organized object admits this reduction.

Execution domain. Fix the existing default pressure/Si laws, default selector, direct glyph path, hard clipping, strict grammar, live built-in zero Gamma, default Euler integration and one substep delta=1/16 at a declared structural time, with use_Si=True and apply_glyphs=True. Process reduced nodes in order 0..4, and fine nodes in the corresponding consecutive-pair order 0..9. Keep the seed and the actual node/neighbor orders for reproducible finite execution. There are no external nodes, custom callbacks, optional mathematics engine or optional RA coherence/metric tracking. Automatic REMESH remains enabled but its separate stable_frac history has fewer than the required default 25 samples; this call produces no additional such sample, so the gate abstains.

Use signed scalar forms inside their rails, positive capacities inside their rails and a common nonempty semantic kind. Every RA proposal must avoid a strict nonzero sign reversal, retain that kind and have finite post-amplification capacity inside the rails. All physical scalars, clocks, grammar records and consumed histories pass their own admission. Concrete floating execution also requires representable produced arithmetic; finite inputs alone do not guarantee a finite terminal acceleration. The phase representatives admit one common circular lift in an arc of width less than pi. Equal positive multiplicities then give defined full-neighbor, selected-neighbor and global resultants. U3 still requires a nonempty compatible subset at each live RA; the common arc alone does not supply that condition.

The prepared mark at every block must be RA or IL, including the actual lag policy. An AL or EN override, or a base request for OZ, ZHIR or NAV, exits this two-operator domain. Such an exit is not a grammar rejection of that mark. All subsequent RA/IL grammar and identity gates remain live. This defines a partial one-call map; its successor need not satisfy the next call's selector or lag conditions.

Retained state and output. The reduction keeps associated coordinates, not independent marginal inventories:

Retained input	Role in the call
Per-block form x_i, phase theta_i, capacity v_i, kind	Fresh pressure/Si, live U3 and RA propagation/identity, flow and phase/capacity policies
Per-block AL/EN lag counts	Increment-before-comparison forcing; a base RA or IL request alone is insufficient
Normalized grammar trace, U2 debt, prior-Coherence flag and operator ordinal	Live admission and successor context; RA retains debt and IL reduces it
Associated stability count s_i	Capacity qualification duration and the actual write decision
Prior stored form rate r_i and admitted temporal EPI history	Terminal acceleration and separate before-event/after-event/flow records
Current global phase gains, clock and invocation ordinal	The actual coordinator, time update and runtime bookkeeping
Retained output archives and insufficient REMESH stability history	Record continuation and the independently justified absence of delayed mixing

Support, multiplicity, order and complete configuration are fixed model inputs. Initial stored pressure and Si are refreshed before selection; they are not independent causal coordinates for this path. Initial acceleration is read and normalized during selector preparation, although the default base-choice formula does not use it. Its admission, including represented normalization, cannot be skipped. Within that admitted domain the successful endpoint is independent of its old value: integration replaces it using the retained prior rate. A rejected call can retain that old acceleration, so reconstructing its complete failed-state telemetry requires retaining it.

Previous phase-controller telemetry is output history, not feedback in the next branch decision. The evolving gains and post-event glyph load are feedback. Rebuildable caches, wall-clock timings and array workspaces are excluded observations, not extra physical coordinates. This projection is neither a checkpoint format nor a proof of a minimal state for other paths.

A bounded grammar memory. Default glyph retention is seven entries. Every accepted RA/IL action gives

H_i_next = suffix_6(H_i) concatenated with g_i,
debt_i_next = debt_i                  if g_i=RA,
              max(0,debt_i-1)         if g_i=IL,
prior_IL_i_next = prior_IL_i or (g_i=IL),
operator_step_i_next = operator_step_i+1.

Thus six retained old glyphs suffice for the successful successor trace and the coordinator's subsequent load observation, whose window is fifty but sees at most seven entries per node here. For RA/IL, U1a needs only whether the incoming trace is empty, U2 needs authoritative debt, and U4 is inapplicable. Empty history with exactly zero form rejects either mark; RA cannot preserve debt above the configured capacity, whereas IL may reduce an over-capacity debt. U3 reads current physical state separately.

This memory reduction applies after all original identifiers and ledger values have been admitted. Preserve the normalized lifetime operator step, which is at least the original trace length, and the prior-Coherence flag; truncating first would change reconstruction or hide invalid history. Older entries remain necessary if the requested output includes the original rejected trace. No average of constituent histories replaces their required within-block association.

One explicit block map. Let ell_i be the common-lift mean of the two neighboring block phases, d_i=wrap(ell_i-theta_i), V=max_i v_i and P=max_i abs(p_i). With independently resolved normalized pressure and Si weights, fresh rows are

p_i = w_phase*d_i/pi
      + w_epi*((x_(i-1)+x_(i+1))/2-x_i)
      + w_vf*((v_(i-1)+v_(i+1))/2-v_i),
Si_i = alpha*v_i/V + beta*(1-abs(d_i)/pi)
       + gamma*(1-abs(p_i)/(P or 1)).

Both supports are regular, so their degree-contrast pressure is zero despite their different degrees, two and four. Equal multiplicity preserves every displayed mean and maximum. With no lag override, the actual default selector chooses IL when Si_i>=si_hi, or RA when si_lo<Si_i<si_hi and abs(p_i)/(P or 1)<=dnfr_hi. Other branches exit the stated sector. These are configured policies, not consequences of the nodal differential identity.

Initialize live variables (y,phi,u,q)=(x,theta,v,p). Let h be IL_dnfr_factor and L the retained default U3 limit. At the next block i, IL writes q_i=h*q_i and leaves y_i,phi_i,u_i unchanged. For RA let

J_i = {j in {i-1,i+1}: abs(wrap(phi_j-phi_i))<=L},
m_i = mean_(j in J_i)(y_j),     a_i = Arg(sum_(j in J_i) exp(i*phi_j)).

Require J_i nonempty and the separate live grammar/identity gates. Using the actual default factors r,c,a and represented amplification trigger T, the accepted RA rows are

y_i <- clip((1-r)*y_i+r*m_i),
phi_i <- phi_i+c*wrap(a_i-phi_i)          modulo 2*pi,
u_i <- (1+a)*u_i if abs(m_i)>T, else u_i,
q_i <- q_i.

These means use the updated prefix, including newly compatible neighbors. Neither preparation nor selection freezes J_i. Apply the history rows above and increment both nonforced lag counts once. Twins have equal local context and no internal edge; acting on the first changes none of the second's inputs. Its identical accepted action therefore restores paired state. Induction over complete pairs proves this ordered recurrence. It does not assert pair constancy at intermediate singleton prefixes.

After the complete batch, the default one-substep form update is

f_i=u_i*q_i,                    x_i_next=clip(y_i+delta*f_i),
stored_dEPI_i=f_i,              stored_d2EPI_i=(f_i-r_i)/delta.

Pressure remains q_i and Si remains the pre-batch value. The canonical clamp centers phase representatives before coordination. The rail hypothesis on u matters: without it, integration consumes the amplified capacity before the later capacity clamp, and a second capacity stage must be modeled.

The coordinator computes R=abs(sum_i exp(i*phi_i))/5 and the normalized disruptor load from the appended block traces. Duplicating every phasor and every trace leaves both ratios unchanged. It therefore selects the same branch, smooths/clamps the same retained gains and applies the same wrapped global/local phase update. Its default gain rails imply positive gains with sum below one, so the common lifted phase arc is preserved. Stored raw phase representatives still follow the canonical-centering and wrapped-difference conventions; a circular equivalence is not permission to invent a new lift.

Finally, with the retained fresh Si and post-event pressure,

s_i_next = s_i+1 if Si_i>=si_hi and abs(q_i)<=EPS_DNFR_STABLE, else 0,
v_i_next = (1-mu)*u_i+mu*(u_(i-1)+u_(i+1))/2   if s_i_next>=tau,
           u_i                               otherwise.

This is the immutable-snapshot capacity proposal, with the implementation's fixed-point and hull guards. It neither refreshes pressure nor changes the already stored rate to v_i_next*q_i. Equal counts and neighbor snapshots give equal twin writes, including when adaptation is active. The temporal recorder admits the old samples, restarts on a same-time form jump and appends the flow endpoint; all these operations commute with duplication. Clock and invocation ordinal advance once, not once per represented constituent.

Native realization and limits. Let D duplicate the retained five-node state and lift its ring edges to the doubled support, and let F_5,F_10 denote the ideal complete native calls on this domain. The preceding rows prove, for the declared physical/local-memory observations,

F_10(D(s)) = D(F_5(s)).

Graph-wide phase records agree without duplication. Node-keyed temporal and REMESH EPI archives lift their keys, while extensive glyph counts and node counts scale by two. Fine labels, _node_sample, support and cache shapes are not literally the five-node values. Both graphs use the full node sample because their sizes are below fifty. Default graph validation retains its range/provenance conditions; uniform duplication also preserves the mean sigma vector. Its normalized norm gate is unchanged.

At a live grammar or identity rejection, both ideal paths reach the same rejected block after the same completed block prefix. Neither rolls that prefix back. The accepted-call equation does not reconstruct discarded old telemetry or claim transactional behavior on failure. Selection outside RA/IL, unsupported histories or another execution path requires separate admission, even if the native engine can execute it.

The equality is an ideal-real theorem, not a universal binary64 identity. In particular, the bare ring's pressure owner takes its certified two-neighbor phase path, whereas four-neighbor accumulation takes another floating path. Strict branch margins and an independently specified numerical allowance remain necessary; retaining a seed does not equate those arithmetic reductions.

The native block-descent controls reuse the live-prefix, neighbor-gain, active-capacity-adaptation and U3-rejection preparations. They fix all fine/reduced inputs, unchanged independent forecasts, observation projection and budgets before execution. This checks the existing motor as its own reduced realization against the earlier independent formulas, including changing U3 subsets and active later policies. No benchmark speedup, repeated-call invariant domain, autonomous source or physical interpretation follows merely from the smaller representation.

Four-operator composition and the first selected exit

The preceding reduction extends conditionally to Emission (AL), Reception (EN), Coherence (IL) and Resonance (RA). This extends the admitted event map; it does not prove that native selection keeps its successors in that domain. Retain the same complete law, support, ordered pairs, associated memory and numerical admission. Re-admit each call from its actual predecessor, including the evolving phase gains and fresh pressure/Si.

Additional event rows. With the resolved default boost b and Reception mix m, the direct native glyphs add the live-prefix rows

AL: y_i <- clip(y_i+b),
EN: y_i <- (1-m)*y_i+m*(y_(i-1)+y_(i+1))/2.

Reception reads the current full-support neighbors, not the U3 subset, and does not require a preceding Emission source. Here its snapshot uses track_sources=False. Direct Emission does not install the public class's lifecycle metadata. Both rows preserve phase, capacity, stored pressure and the common nonempty kind; append the actual glyph with its provenance and increment the operator ordinal. Both retain U2 debt. Their live grammar and state admission remain necessary. Unlike Resonance's identity gate, these rows do not impose a general prohibition on a nonzero form sign change.

Two consecutive nonadjacent twins read identical inputs, so the pair induction and subsequent flow/controller reduction still apply. Coherence retains its pressure factor; Resonance retains its live U3 and amplification branches. Temporal recording follows the actual form change: an unchanged clipped Emission or no-op Reception does not create a jump merely by its label. The common phase arc persists under these four events and the default coordinator, but this alone does not admit a later Resonance subset.

Actual lag policy. For incoming Emission/Reception lags a,e, first set A=a+1,E=e+1. The native effective request and successor lags are

A>5:           AL, (0,min(E,3)),
A<=5 and E>3:  EN, (A,0),
otherwise:     prepared base request, (A,E).

Emission has priority over Reception. A forced request replaces the base choice for this purpose; an unused outside-sector base choice is not an exit. From zero lags, completed sector calls give (1,1), (2,2), (3,3), forced Reception with (4,0), (5,1), then forced Emission with (0,2). This is conditional invocation bookkeeping, not a physical occurrence law. Six calls are an observation horizon, not a closed grammatical word.

Prospective second-call certificate. Keep the unchanged nominal neighbor-gain preparation, seed 17, initial time 1, step 1/16, zero lags and the default complete law. Its first-call request is RA RA IL IL IL. Let X,V,Phi be the independently derived final forms, final capacities and pre-coordinator event phases, and K_G,K_L its new gains. Those quantities are rational in the represented coefficients. Write ell_i=(Phi_(i-1)+Phi_(i+1))/2. The first coordinated phase is

theta_i_next = z_i+K_G*Psi,
z_i = (1-K_G-K_L)*Phi_i+K_L*ell_i.

The common global angle Psi cancels from every next neighbor offset:

d_i_next = (z_(i-1)+z_(i+1))/2-z_i.

All phases remain in one arc narrower than pi, so each circular neighbor mean is its declared lifted midpoint. Therefore the next fresh pressure and Si follow exactly from the previous block formulas with X,V,d_next, without evaluating an inverse trigonometric function or measuring a native response. With P=max_j abs(p_j) for this new pressure, exact rational inequalities give

0.380 < Si_0 < 0.381,    abs(p_0)/P < 0.005,
0.376 < Si_1 < 0.377,    0.235 < abs(p_1)/P < 0.237,
min(Si_2,Si_3,Si_4) > 0.84,
abs(p_1)/P-dnfr_hi > 0.1.

The first two Si values lie strictly between the configured cuts 1/4,1/2; the remaining three exceed the upper cut. Lags increment to two, below both forcing limits. Thus the exact ideal second request is RA NAV IL IL IL: Transition (NAV) exits the four-operator sector with a strict normalized-pressure margin. This calculation uses independent source and law data; it does not reconstruct pressure from observed motion.

Frozen finite protocol. The composition controls prepare fine and five-node graphs for scalar and NumPy paths before any native response. They retain a six-attempt horizon and a maximum of 24 invocations. The first call and second fresh diagnostics use the unchanged independent forecast with absolute allowance 1e-14. Both coarse trajectories and their declared observations are frozen before either fine trajectory; cross-support comparison allows 2e-14 per ordinate and requires exact decisions/discrete memory. No source sweep, forced word or tolerance adjustment is part of this protocol.

A test-local observer captures the actual batch preflight, forced requests and every nonforced selector result. Immediately before the final outer resolver returns, it checks the complete ordered effective batch. Any outside-sector mark stops the whole batch before its first event, including an earlier supported mark. This is earlier than the Mutation prelude and live grammar gates, but after fresh pressure/Si and lag increments. An all-forced batch contains only admitted-sector AL/EN requests. The stop is separate from grammar rejection, numerical-domain failure or a solver defect; it changes no production selector or operator.

Finite result and retained boundary. All four trajectories complete call one and stop on call two with the predicted RA NAV IL IL IL block requests: eight native invocations, below the frozen maximum. First-call outputs and second fresh diagnostics agree with the independent forecasts within 1e-14; fine/coarse comparisons pass the fixed 2e-14 allowance. Decisions and retained discrete memory agree exactly. No second-call Resonance prefix, Transition, forced Reception or forced Emission is executed by this protocol.

At the stop, fresh pressure/Si and both lag counts of two remain. The invocation ordinal is two, while physical time remains 17/16. Form, phase, capacity, rate, acceleration, stability counts, grammar, evolved phase gains and their archives retain the completed first-call values. The same-time left-boundary flow record does not alter temporal samples. Selector cleanup runs; no second integration, coordination, adaptation or REMESH append occurs. An empty stable_frac still supplies no automatic REMESH event, independently of the existing EPI archive.

This closes the six-attempt sector question by an early selected exit. The conditional four-operator reduction remains valid; the unchanged source does not reach its conditional fourth/sixth-call lag events in that sector. It is neither a grammar failure nor evidence against every collective description. The existing Transition obstruction and mean/contrast prediction identify why target-associated innovations and internal contrasts may matter, but their different source coefficients and verdicts cannot be transferred here.

A genuine second-call continuation must start from the completed first-call predecessor with its retained histories, gains and random-stream progress. Calling the motor again on the protocol-stopped graph instead consumes a third ordinal and advances the lags again. Keep that stopped response distinct; it is not a restartable representation of the unevaluated second call. The observation records are not complete checkpoints. A later protocol must freeze the actual predecessor or explicitly account for its preparation replay.

A collective form prediction beyond pair equality

The selected Transition (NAV) exit need not prevent a collective observation from being predicted. Continue the unchanged neighbor-gain source from its completed first call, retaining the full native law and associated state. This question concerns its genuine second RA NAV IL IL IL block batch, not a third invocation on the preceding protocol-stopped graph. The declared observations are the five form means and signed within-pair contrasts at time 9/8, together with the physical and memory channels needed to audit their mechanism. No bare-C5 default execution is assumed to realize them.

Admission before response. Write X_i,V_i,Theta_i for the first-call block endpoint and p_i,Si_i for the next independently calculated fresh diagnostics. The second-selection certificate fixes the actual requests without using a new response. Retain strict grammar, pair-contiguous order, seed 17, node-associated random streams, duration delta=1/16, live zero Gamma and the existing inactive REMESH gate.

Resonance (RA) on block zero admits both adjacent blocks, one and four. Their U3 clearances exceed 0.42. Its live selected mean and proposed form are

m_0 = (X_1+X_4)/2,            -0.796 < m_0 < -0.795,
Y_0 = (1-r)*X_0+r*m_0,         0.411 < Y_0 < 0.412,
U_0 = (1+a)*V_0,               0.270 < U_0 < 0.271.

Thus Resonance preserves the positive target form sign and common kind, amplifies capacity within its rails, and retains pressure p_0. The twins are nonadjacent and see the same inputs, so this accepted prefix remains pair-equal. Other blocks keep Y_i=X_i,U_i=V_i. Transition then changes only the two block-one pressures. Its preceding Resonance neighbor action does not refresh their prepared pressure or alter their own capacity.

Let xi_- and xi_+ be the actual innovations at nodes two and three, respectively. With default eta=1/2, NAV_STRICT=False, NAV_RANDOM=True, positive fresh p_1, and jitter amplitude j,

B = (1-eta)*p_1+eta*V_1,
q_(2) = B+xi_-,                 q_(3) = B+xi_+,
q_(0)=q_(1)=p_0,
q_(2i)=q_(2i+1)=h*p_i          for i=2,3,4.

Here h is the retained Coherence (IL) pressure factor. Uniformly for |xi_-|,|xi_+|<=j, the block-one held pressures lie in (0.134,0.215) and differ from the incoming pressure by more than 0.045. This separately admits Transition's nonidentity postcondition. Every resulting Euler form stays inside its rails; no clipping changes the following equations.

Mean and contrast rows. Define

xi_bar=(xi_-+xi_+)/2,          xi_delta=(xi_+-xi_-)/2,
M_i=(x_(2i)+x_(2i+1))/2,      d_i=(x_(2i+1)-x_(2i))/2.

The complete second-call form observation is

M_0_next = Y_0+delta*U_0*p_0,             d_0_next=0,
M_1_next = X_1+delta*V_1*(B+xi_bar),      d_1_next=delta*V_1*xi_delta,
M_i_next = X_i+delta*V_i*h*p_i,           d_i_next=0   (i=2,3,4).

The five form means consume the innovation mean, while the internal form contrast consumes its signed difference. Hence the admitted mean prediction does not require equal constituent responses. At the event, Transition creates a pressure contrast xi_delta; subsequent nodal evolution creates rate contrast V_1*xi_delta and form contrast delta*V_1*xi_delta. Equal incoming rates give acceleration contrast V_1*xi_delta/delta. These are different observations at different stages, not a direct Transition form jump.

Two symbolic innovation pairs with the same mean and different differences produce the same five form means but different internal forms. This is a conditional algebraic distinction within the admitted amplitude box; it does not assert that one fixed seed realizes every pair or assign a physical noise distribution. Replacing both innovations by one common sample removes this contrast and can also change the mean. A bare node's single seeded draw is not the original two-target innovation mean.

The generated means and contrasts also retain a joint budget. If M_1_det=X_1+delta*V_1*B, then

abs(xi_bar)+abs(xi_delta)=max(abs(xi_-),abs(xi_+))<=j,
abs(M_1_next-M_1_det)+abs(d_1_next)<=delta*V_1*j.

The image of the innovation amplitude box is a diamond-shaped enclosure for these mean/contrast observations. Independent marginal bounds would admit combinations outside that joint image. This correlation is available for subsequent admission; the fixed seed selects one point and does not establish reachability of the entire enclosure or an invariant-domain theorem.

The subsequent policies remain part of the model. Let z be the rational relative phase from the preceding certificate. After Resonance the phase vector has a common origin plus

psi=(z_0+c*((z_1+z_4)/2-z_0), z_1,z_2,z_3,z_4).

Transition and direct Coherence do not change phase. Pairwise differences therefore give the origin-independent exact identity

R^2=(5+2*sum_(i<k) cos(psi_i-psi_k))/25.

Rational alternating-series cosine bounds certify 0.719<R<0.720, strictly between the configured phase-order cuts. The block traces are EN RA RA, EN RA NAV, and EN IL IL on each of the three remaining blocks. Transition counts toward the coordinator's disruptive-glyph load, giving 1/15; it remains neutral for U2 debt. That load is below the lower load cut. The coordinator consequently selects its transition branch and moves the retained gains toward their configured midpoints. Its resulting gains are approximately K_G=0.06141231186574749, K_L=0.14166901183551323. The final phase diameter is below 2.024, and the pair phases remain equal. Absolute phase readouts still require a declared global-angle evaluation; the rational relative-phase certificate does not replace it.

All held pressure magnitudes exceed the stability tolerance, so inherited zero stability counters remain zero and no capacity adaptation write occurs. In particular, the rate observation uses the post-Resonance capacity. Each node retains its common kind, actual grammar history and prior-Coherence flag; block one retains U2 debt one and the other blocks retain zero. Operator ordinals advance to five, AL/EN lags to two, and the runtime ordinal to two. Only block zero has a same-time form jump at 17/16 and resets its temporal form history. Transition retains its prior form samples and appends the flow endpoint; its label alone cannot supply a measured form jump.

Frozen numerical protocol. The Transition contrast controls declare two first-call preparation replays followed by two genuine second-call responses, one scalar and one NumPy. Both complete forecasts and configurations precede every native call. Each preparation is checked against its unchanged independent forecast before the actual completed graph continues; copied observation records are evidence, not replacement checkpoints. No stopped graph is reused, and no coarse trajectory or third call is evaluated.

The first draw at each of offsets two and three is reconstructed independently with the existing standard-library BLAKE2b/Random oracle. Expected values do not call the production jitter implementation. The equations above are ideal-real identities over represented inputs; the finite native comparison uses a fixed absolute allowance 1e-14 for each ordinate and mean/contrast observation, including acceleration. Actual draws, marks and discrete memory must agree exactly. Alternative innovation associations are analytical controls, not additional native responses or replacement laws.

Finite result. Both genuine second calls complete the predicted native batch. The independently reconstructed first draws are approximately xi_-=0.02188843323, xi_+=-0.01430477745; their actual binary64 values and target/stream associations agree exactly. The form difference is predicted as x_3-x_2=-0.0005880201996622686, with signed half-contrast d_1=-0.0002940100998311343. Both native paths match all declared ordinates and mean/contrast observations within the unchanged 1e-14 allowance. Phase/capacity remain paired; the observed coordinator branch, disruptive load, grammar, histories and absent adaptation writes match their forecasts.

This gives a source-specific collective form prediction after exact pair equality is lost. It retains the actual two-target innovation law and the joint enclosure. It does not supply a complete reduced default-engine state, prove closure under another selected batch, or establish acquired collective identity. In particular, future selection consumes normalized individual pressures, so a successful mean forecast does not authorize discarding the contrast before re-admitting the next call.

Uniform admission beyond the first internal contrast

The preceding innovation diamond admits one further complete ideal-real call uniformly. Transition (NAV) is selected on blocks zero and one, and Coherence (IL) on the other three. This is an analytical consequence of the same supplied law, selector and grammar context. No third native response is evaluated.

Joint source state. Let C be the second endpoint's deterministic form means, obtained by setting its innovation mean to zero. Retain its paired capacities V, phases, support, order, clock, gains, grammar and zero stability counters. At time 9/8, the admitted forms are

M_1=C_1+u,             M_i=C_i (i!=1),
d_1=d,                d_i=0   (i!=1),
x_(2i)=M_i-d_i,       x_(2i+1)=M_i+d_i,
abs(u)+abs(d)<=rho,    rho=delta*V_1*j,     delta=1/16.

Here j is the unchanged Transition amplitude and 0.000646<rho<0.000647. This is the exact affine image of the prior amplitude box. The ideal-real seed-17 forecast supplies one member, not reachability of every member with that seed. Its earlier finite native comparison remains separate. The input law and associated random-stream progress remain declared.

Associate prior rates and histories with each member. If B_previous is the previous deterministic block-one held pressure, its incoming fine rates are V_1*B_previous+(u-d)/delta and V_1*B_previous+(u+d)/delta. Their accelerations use the preceding first-call rates. Temporal endpoints are C_1+u-d and C_1+u+d; their earlier samples remain associated with the same preparation. These are forward-map identities, not pressure inferred from a response. Keeping the nominal rates at nonnominal diamond points would invalidate the next acceleration and temporal evidence.

Phase frame and selection. Write N_i(a)=(a_(i-1)+a_(i+1))/2 on C5. From the previous phase certificate, with relative event phases psi and its transition-branch gains K_G,K_L, the second endpoint has relative phases

z_i=(1-K_G-K_L)*psi_i+K_L*N_i(psi).

Both global-angle contributions are common origins and cancel. The exact rational frame has diameter below 1.605<pi; each two-neighbor resultant therefore uses the midpoint on that lift. Relative offsets and phase order need no approximate absolute-angle evaluation. With the represented pressure weights k=w_epi, b=w_vf, w=w_phase and normalization pi_rep,

B_i=w*(N_i(z)-z_i)/pi_rep+k*(N_i(M)-M_i)+b*(N_i(V)-V_i),
p_(2i)=B_i+k*d_i,       p_(2i+1)=B_i-k*d_i,
P=max_i(abs(B_i)+k*abs(d_i)).

Regular degree contrast vanishes. Each primitive pressure is affine in (u,d). At all four diamond vertices, its block signs are (-,+,-,-,+) and block three has the strictly largest magnitude. The same strict linear inequalities therefore hold on the convex hull. Thus P=-B_3 is constant, in (0.268,0.269), and individual normalized magnitudes are affine throughout. Capacity and relative-phase terms of Si are fixed, so Si is affine as well. Vertex bounds give

0.13843 < abs(p_j)/P < 0.13888  (j=0,1),
0.58473 < abs(p_j)/P < 0.58562  (j=2,3),
0.399 < Si_j < 0.400           (j=0,1),
0.377 < Si_j < 0.379           (j=2,3),
Si_j > 0.874                  (j=4,...,9).

The smallest Transition pressure-cut clearance exceeds 0.0054. These bounds give the uniform base batch NAV NAV IL IL IL. Incremented lags are three, so neither strict lag override applies. This default selector applies no parametric soft-repeat filter; live grammar permits repeated Transition. Transition is U2-neutral and consumes no new U3 coupling gate; Coherence retains the existing stabilization context. The effective requests are therefore admitted without substitution throughout the diamond.

Event and flow map. Each Transition pair receives its actual two target-associated innovations xi_(i,-),xi_(i,+). Set their mean and signed half-difference to xi_bar_i,xi_delta_i; each pair obeys abs(xi_bar_i)+abs(xi_delta_i)<=j. With default mix 1/2 and Coherence factor h, the held-pressure means and contrasts are

qbar_0=(B_0-V_0)/2+xi_bar_0,    qdelta_0=xi_delta_0,
qbar_1=(B_1+V_1)/2+xi_bar_1,    qdelta_1=-k*d/2+xi_delta_1,
qbar_i=h*B_i,                  qdelta_i=0       (i=2,3,4).

These events change pressure only. They consume the prepared rows; there is no post-event pressure refresh. The held-pressure Euler step gives

M_i_next=M_i+delta*V_i*qbar_i,
d_i_next=d_i+delta*V_i*qdelta_i,
d_0_next=delta*V_0*xi_delta_0,
d_1_next=(1-delta*V_1*k/2)*d+delta*V_1*xi_delta_1.

Rate means/contrasts are V_i*qbar_i,V_i*qdelta_i. Acceleration subtracts the associated incoming rates before division by delta. Once admitted, the mean rows consume the incoming means and innovation means. Exact internal differences determine fine state; their joint bounds remain part of this admission certificate.

Every row above is affine on the product of the old diamond and four future innovation intervals. Its extrema are therefore certified by their 64 input vertices. The Transition held pressure and its change from prepared pressure are both negative on block zero and positive on block one; Coherence held pressures have signs (-,-,+). These fixed signs make their absolute-value lower bounds affine too. Every Transition nonidentity gap and held-pressure magnitude exceeds 0.011, above the configured stability tolerance. All unprojected Euler forms remain inside their rails; block ranges fit respectively within (0.408,0.410), (-0.659,-0.655), (0.954,0.955), (-0.056,-0.055) and (-0.911,-0.910). Stability counters stay zero and capacity adaptation writes nothing. The exact affine image retains correlations; these convenient marginal ranges are not a replacement source for future admission.

Phase and memory. Exact alternating cosine bounds on the fixed z yield 0.830<R<0.831. Post-append block traces are EN RA RA NAV, EN RA NAV NAV and EN IL IL IL on each remaining block. Their disruptive load is 3/20, below the configured lower cut. The coordinator consequently selects its stable branch and moves both gains toward their configured minima, giving approximately K_G=0.05365395977, K_L=0.12545426230. The final phase diameter is below 1.519<pi; phases and capacities remain paired. Absolute phase is retained through the inherited common lift: if Theta_i=Omega+z_i and A(z) is its resultant angle, the declared absolute row is Theta_i_next=Omega+(1-K_G-K_L)*z_i+K_G*A(z)+K_L*N_i(z), using the new gains. The prior strict common arc admits this row. Relative certificates do not evaluate or discard its common origin, or replace phase by a form-derived angle.

At 19/16, runtime ordinal is three, operator ordinals six and both lags three. Prior-Coherence flags remain true; block-one debt is one and other debts zero. No event resets form history: append the endpoint with the existing four-entry retention cap. Block zero has three samples, the others four. The REMESH form record receives its third snapshot; absent stable_frac history still prevents automatic REMESH. Phase-policy records append their actual branch, order, load and gains. Targets zero/one consume draw index zero, targets two/three index one; their resulting draw counts are one and two, respectively. No other target consumes jitter.

The analytical controls use exact represented coefficients, independent affine rows and cosine bounds. They block production runtime, pressure, selector, operator and integrator evaluation. They certify the ideal-real domain and formula, not a universal binary64 enclosure or a newly observed trajectory. The resulting image is generated by 64 input vertices; these need not be distinct extreme output vertices. This finite admission result supplies neither an invariant family, an autonomous collective occurrence law, a bare-node stochastic realization nor physical identification.

Reception on the generated correlated collective family

Reception (EN) admits the entire preceding generated image at its actual lag-forced boundary. Its five terminal form means factor through the incoming five means, with the retained phase/capacity state and event order. Fresh individual diagnostics and acceleration still consume additional information. This is a finite ideal-real composition theorem under the unchanged complete law, not a new native trajectory or a complete mean-only engine state.

Source and preparation. At 19/16, take the exact affine image from the contrast-domain theorem. Its six input coordinates are the preceding (u,d) diamond and four subsequent innovations, each in [-j,j]. Keep their joint domain and associated rates, histories, grammar, stream progress, support and node order. The 64 input vertices generate its affine form image; independent marginal intervals do not replace that image. The ideal-real seed-17 forecast is one member. The amplitude-domain theorem does not assert that every member occurs under that fixed seed.

Write the incoming form means/contrasts as M_i,d_i, and paired capacities as V_i. Only d_0,d_1 can be nonzero. If z is the preceding relative phase frame and K_G3,K_L3 its stable gains, the present relative frame is

z3_i=(1-K_G3-K_L3)*z_i+K_L3*N_i(z).

Its common origin is retained separately on the inherited lift. Its diameter is below 1.290<pi; each neighbor-resultant offset is N_i(z3)-z3_i. The prepared pressure is independently specified by the unchanged mixed law:

B_i(M)=w*(N_i(z3)-z3_i)/pi_rep
       +k*(N_i(M)-M_i)+b*(N_i(V)-V_i),
p_(2i)=B_i+k*d_i,              p_(2i+1)=B_i-k*d_i.

The affine pressure rows have fixed block signs (-,+,-,-,+) throughout the domain, and every magnitude exceeds 0.072. Both block-one pressures exceed every other block's pressure magnitude by more than 0.00018. Consequently the normalization peak is now

P=B_1+k*abs(d_1),               0.2014<P<0.2021.

Its maximizing twin can change within the family. Individual Si therefore contains a piecewise-rational normalization; it is not an affine image of the input coordinates. Positive peak and capacity bounds admit finite fresh diagnostics. Bounds on normalized ratios must retain the maximum or use justified enclosures, rather than assuming affine Si extrema at the 64 vertices.

Actual occurrence and admission. Selector preparation precedes the lag override. Both lags increment from three to four. Emission's higher-priority condition remains false (4<=AL_MAX_LAG=5), and Reception's strict condition holds (4>EN_MAX_LAG=3), so all ten effective requests are Reception. The retained grammar admits these requests and preserves prior Coherence and U2 debt. The direct-glyph route consumes finite signed target/neighbor forms; it adds neither the public class's separate ceilings nor an emission-source or U3 prerequisite. These configured lag rules justify the requests on this path; they are not a derived autonomous occurrence law.

Ordered event and held flow. Reuse the Reception identity, re-admitting its premises on this nonzero-phase source. With m=EN_mix, initialize Y=M, then perform in block order i=0,...,4

Y_i <- (1-m)*Y_i+m*(current Y_(i-1)+current Y_(i+1))/2.

Consecutive twins are nonadjacent and share the same live neighbor mean at their own events. Thus the event form means are Y and contrasts (1-m)*d. This equality holds at completed pair boundaries; the intermediate single-node prefix is retained in the fine proof. With all operands inside their rails and 0<m<1, each event blend remains interior. The event changes no pressure, phase, capacity or stored rate. In particular, the later flow retains the prepared pressure contrast -k*d, not the post-event form contrast.

The complete form/rate map is

M_i_next=Y_i+delta*V_i*B_i(M),
d_i_next=(1-m-delta*V_i*k)*d_i,
rate_bar_i_next=V_i*B_i(M),
rate_delta_i_next=-V_i*k*d_i.

Here delta=1/16. Subtract the associated incoming rate means/contrasts and divide by delta to obtain acceleration. Each contrast factor lies strictly between zero and one; this single contraction does not establish indefinite retention. No new innovation is consumed.

The ordered fine prefix and the five-block sweep give identical affine coefficient rows. Signed vertex inequalities then certify every event jump and unprojected flow endpoint over the joint domain. All jumps have block signs (-,+,-,-,+) and magnitude above 0.015. Terminal forms fit within (0.119,0.121), (-0.367,-0.364), (0.667,0.668), (-0.084,-0.083) and (-0.675,-0.674), respectively. These displayed ranges summarize admission; the calculation retains the exact correlated rows.

Two wrong timing substitutions have uniformly separated predictions on this same domain. At block one, the ordered terminal mean is more than 0.034 below the simultaneous-mixing prediction. At block zero, refreshing pressure after the Reception batch would raise the terminal mean by more than 0.00168. Neither substitute is the admitted native map. In particular, multiplying (1-m) by (1-delta*V_i*k) gives the wrong contrast coefficient.

Sufficient observation and retained memory. The terminal mean formula uses only M, fixed V,z3, the declared coefficients, order and clock once the family is admitted. Two members with the same means but different internal contrasts therefore have identical terminal means and different fine pressure, form and diagnostic records. The internal domain remains part of the admission certificate; mean prediction alone does not authorize arbitrary lifts.

Even all instantaneous forms do not determine the next stored acceleration. Let D=1-delta*V_1*k/2, and choose s=rho/2. Compare the zero input with the admissible alternative

u=0, d=s, xi_(0,-)=xi_(0,+)=0,
xi_(1,-)= D*s/(delta*V_1),
xi_(1,+)=-D*s/(delta*V_1).

The prior Transition map gives identical incoming forms because its block-one contrast is D*s+delta*V_1*xi_delta_1=0. Its incoming rate contrast differs by -s/delta. Reception and its flow consequently give the same forms and rates, but acceleration contrasts differ by s/delta^2. This is an exact amplitude-domain witness, not a claim that both inputs occur at seed 17. The consumed rate/history association cannot be replaced by the form snapshot.

Post-event policies and records. Reception leaves primitive phase fixed. Exact alternating cosine bounds give 0.888<R<0.889. Appending Reception to the retained traces yields six disruptive glyphs among fifty, hence load 3/25, below the lower cut. The coordinator again selects its stable branch, with new gains approximately K_G4=0.04713038776, K_L4=0.11182017030. For incoming phase Omega3+z3_i, the complete absolute row remains

Theta_i_next=Omega3+(1-K_G4-K_L4)*z3_i
             +K_G4*Arg(sum exp(i*z3))+K_L4*N_i(z3).

The inherited common lift admits the resultant angle. Relative certificates need not evaluate it; the common origin remains retained. Final phase diameter is below 1.229, and phases remain paired. Every held-pressure magnitude exceeds the stability tolerance, so counters remain zero and capacities stay unchanged without a write. No joint phase/capacity closure is inferred beyond this admitted boundary.

All form jumps are nonzero, so each temporal history restarts at its own right-hand Reception form at 19/16, then appends the flow endpoint at 5/4. The resulting secant excludes the jump and equals the held flow rate. Operator ordinals become seven, runtime ordinal four, and lags become since_AL=4,since_EN=0. Prior-Coherence flags, common kind and U2 debt remain; only block one has debt one. Target-associated random records are unchanged: targets zero/one retain one draw, targets two/three two, and others none. The fourth REMESH form snapshot is appended; absent stable_frac history still prevents automatic REMESH. Phase-policy records append their new values.

The analytical controls reuse the prior guarded source certificate and independently compare fine prefixes, collective rows and counterpredictions. Production execution remains disabled. The result distinguishes a sufficient finite mean observation from the larger state needed for diagnostics and history. It does not certify a universal binary64 enclosure, complete collective NFR identity or autonomous formation. The affine form/rate image must not be confused with the generally non-affine normalized diagnostics.

Conditional collective calculus for five native operators

The consolidated five-operator calculus owns the shared event/held-flow identities and their memory and admission premises. This published anchor is retained for existing references.

Finite internal-contrast retention with admission margins

The finite retention criterion owns the joint request, event, flow and memory margins, with the existing positive-width source and zero-peak counterexample.

Joint form and capacity calculus across the adaptation exit

The joint form/capacity calculus owns covariance, capacity-input association and post-flow adaptation. Exact coordinates remain distinct from a smaller closed collective state.