One collective Resonance word on the retained pair object

This result extends the admitted Emission--Silence word by one actual neighbor-dependent Resonance stage. It uses the unordered pair state, the RA contract, and the partial-transition criterion. The exact-real calculation is a finite supplied intervention followed by the same conservative normalized-sine law. It does not derive an event selector or identify a physical interaction.

1. Fixed protocol and complete execution domain

Keep the doubled C5, unit undirected support, original structural clock, and two continuous rows of the source-domain owner. In particular, capacity is held between events and pressure is independently specified by that law. Write alpha=2*pi/5, and keep its declared lifts

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

The source family and factors are fixed before evaluating a response:

\[ \frac9{32}<m<\frac{11}{32},\qquad Z_f<z=\frac1{512},\qquad \frac12<\nu_{j,+}=\nu_{j,-}=v_j<2, \]

\[ b_{\rm AL}=\frac1{1024},\qquad d_{\rm RA}=\frac18,\qquad c_{\rm RA}=\frac1{16},\qquad a_{\rm RA}=\frac1{32},\qquad a_{\rm SHA}=\frac12. \]

The supplied word is emission,resonance,silence on one marked whole pair k. Use the explicit core grammar profile and validated per-prefix contexts, the canonical public classes and the shared selected-target snapshot stage. Fix the hard U3 setting DELTA_PHI_MAX at its canonical pi/2. The RA policies are RA_MIN_SOURCE_EPI=RA_MIN_VF=1/8, RA_MAX_DISSONANCE=1/2, and RA_MAX_PHASE_DIFF=pi/2. Other rails, AL/SHA thresholds, execution seed, timestamps, window, plain graph ownership and excluded optional hooks are those of the source-domain owner. Set the homogeneous established epi_kind="pair-pattern" on every fine node. This makes semantic kind admission independent of a neighbor-magnitude tie.

Retain finite stored pressure with abs(p_stored)<=1/2 on each selected member. This is an additional runtime admission premise: the previous AL/SHA word did not consume this pressure threshold. One allowed initialization evaluates the declared sine pressure, for which abs(p_law)<=1/pi<1/2. Neither the threshold nor the stored number identifies the live law after a phase-changing event. Do not reconstruct pressure from the observed form response.

The source lifecycle/history stratum remains the associated one in the earlier word: each selected member has its separately owned AL-only history, zero U2 debt, no lifetime IL witness and no latency fields. Its count and origin can differ from its partner. Disable the optional RA metrics and network coherence tracking, as well as the optional generic monitors, metrics, pressure callback and jump certificate already excluded there. The graph retains the actual stage status, source glyph and last-operator label. This is a supported restricted execution environment, not a claim about arbitrary callbacks or all possible histories. The graph's _trig_version is absent or a valid nonnegative integer, and any retained trigonometric cache obeys the shared-setter mutation scope. Every authoritative source theta agrees with its phase alias if that alias is present. These are explicit phase-write premises: the preceding AL/SHA word did not consume the trigonometric version.

2. Uniform admission and a shared nonzero neighbor resultant

AL changes no phase. For every fine edge, its wrapped source phase difference is bounded by

\[ \alpha+|\eta_i-\eta_j| \le\alpha+\sqrt2\,Z_f <\alpha+\sqrt2\,z<\frac\pi2. \]

For example pi/2-alpha=pi/10>3/10, whereas sqrt(2)*z<3/1024. Thus every one of the four neighbors is individually U3-compatible for both selected members, with a strict margin. Their neighbor sets are the same, and neither target is its partner's neighbor. AL therefore changes none of the neighbor data consumed by this RA stage. This statement depends on the supplied complete doubled support and canonical limit; arbitrary pruning or a tighter limit needs a fresh calculation.

Let the pre-AL pair variables be X,Theta,R,U,Q, with associated signed representatives u,delta, so R=cos(delta)>0, U=u^2 and Q=u*sin(delta). The shared neighbor form mean and complex resultant are

\[ A_k=\frac{X_{k-1}+X_{k+1}}2,\qquad S_k=\frac{R_{k-1}e^{i\Theta_{k-1}} +R_{k+1}e^{i\Theta_{k+1}}}{2}. \]

These are exact identities from summing the four actual neighbors, not replacement weights inserted into the continuous law. At the winding target, S_k=cos(alpha)*exp(i*(theta_*k+gamma)). For general source deviations the elementary chord bound gives

\[ \left|S_k-e^{i(\theta_{*,k}+\gamma)}\cos\alpha\right| \le\frac14\sum_{j\sim(k,+)}|\eta_j| \le\frac12\|\eta_{N_k}\|<\frac z2. \]

Since cos(alpha)=(sqrt(5)-1)/4>1/4, the resultant is nonzero. Define mu_k=arg(S_k) on the unique lift near theta_*k+gamma. Its rotated real part is positive and

\[ |\mu_k-\theta_{*,k}-\gamma| \le\arctan\frac{z/2}{\cos\alpha-z/2} <\frac{2z}{1-2z}<3z. \]

Both target-to-mean differences consequently have magnitude below 4z, so the runtime's shortest-arc interpolation uses precisely this same lift. The mean-alignment advisory also has a strict passing margin. The exact antipodal and zero-resultant branches are absent on this domain.

Every source form is in (143/512,177/512). AL remains unclipped and places the selected forms above 1/4. RA is a convex blend of positive values in (1/4,1/2), so it preserves positive sign and stays inside the hard rails. Its homogeneous kind remains pair-pattern. The shared A_k>1/4 exceeds the operational 1e-9 capacity trigger, strictly and independently of internal phases. Both targets therefore receive the same multiplier 33/32. Every strict source/capacity/pressure policy passes under the declared premises; the final SHA capacities are 33*v_k/64 in (33/128,33/32). Other pairs retain v_j.

3. Exact collective reset and retained internal information

Put s=7/8, t=15/16. The complete physical endpoint of AL--RA--SHA is

\[ \boxed{\begin{aligned} X_k'&=s(X_k+b_{\rm AL})+(1-s)A_k,\\ \Theta_k'&=t\Theta_k+(1-t)\mu_k,\\ u_k'&=s u_k,\qquad \delta_k'=t\delta_k,\\ v_k'&=\frac{33}{64}v_k. \end{aligned}} \]

All other physical pairs are unchanged by the word. The phase formula is on the declared nearby lift and is projected circularly at the boundary; it is not a global affine formula on arbitrary phase representatives. In the unordered coordinates this becomes

\[ \boxed{\begin{aligned} R_k'&=\cos(t\arccos R_k),\\ U_k'&=s^2U_k,\\ Q_k'&=s F_t(R_k)Q_k,\\ F_t(R)&=\frac{\sin(t\arccos R)}{\sqrt{1-R^2}},\qquad F_t(1)=t. \end{aligned}} \]

The ratio has the displayed continuous extension. These formulas follow because delta is in (-pi/2,pi/2): cosine is even, and the quotient of the two odd sines depends only on abs(delta)=arccos(R). They preserve the realizability identity Q'^2=U'(1-R'^2), including the synchronized tip. Pair-capacity equality survives the shared trigger and multiplier, so this intervention requires no unequal-capacity extension. It does not remove internal degrees of freedom or turn the object into a single bare nodal state.

A discriminating internal-geometry comparison

Start with uniform form m, zero u, equal-pair positive capacities, and the exact doubled twist. Compare it to the source with precisely delta_1=epsilon, where 0<abs(epsilon)<z/sqrt(2), and all other internal differences zero. Every X_j,Theta_j is identical between the two sources, and both are strictly within the fixed domain. Mark pair k=0. Relative to the common offset gamma, its two neighbors have means alpha and -alpha on the target-near lift. The perturbed neighbor resultant has argument offset

\[ \mu_0-\gamma =\arctan\left( \frac{(\cos\epsilon-1)\sin\alpha} {(\cos\epsilon+1)\cos\alpha}\right) =\arctan\left[-\tan^2(\epsilon/2)\tan\alpha\right]<0. \]

The unperturbed argument offset is zero. Thus the target collective phase after the word differs by (mu_0-gamma)/16, strictly, while its form mean and capacity outputs coincide. This exact sign argument proves that block means alone cannot determine the actual relational response. The difference can be arbitrarily small; a fixed floating-point tolerance cannot certify it for every nonzero epsilon.

The source U,Q do not enter the instantaneous target mean form or phase on this all-neighbor stratum. They are nevertheless retained: U determines the postword internal form amplitude, and Q determines its signed phase/form correlation. For example, changing u_k to -u_k at fixed nonzero delta_k, while retaining X,Theta,R,U, reverses Q_k and Q_k'. It changes the subsequent inherited R_dot=-nu_k*Q_k/pi under the same continuous law. This is a correlation of the associated internal coordinates, not an additional force or an event-selection rule.

Complete partial action and its boundary

AL and SHA retain their established lifecycle updates. RA leaves the emission lineage and stored pressure untouched, updates the stated physical channels and kind, and appends RA to each associated history. The shared phase setter writes both theta and its legacy phase alias, including creation of the alias if absent, and advances the graph's trigonometric invalidation version once per selected target. This version changes by two even if a particular phase value happens to stay fixed. It invalidates a rebuildable cache rather than adding a physical coordinate. SHA then records the actual post-RA form in preserved_epi; its was_initial_on_silence is false. The final operator-step index is n_i+3, while AL alone increased the emission lineage count to n_i+1. Debt remains zero and the prior-IL flag remains false. The successive two-token suffixes AL/AL, AL/RA and RA/SHA match no registered IL recognition pattern, so the declared empty pattern sink receives no append. Optional RA observation streams were explicitly excluded.

For the exact-real transition, every pair swap preserves the neighbor sets, form mean and resultant. Swapping the selected pair swaps its two local proposals; swapping a neighbor pair only permutes terms in the shared means. Associated histories and origins move with their members. Admission and all declared successor data therefore factor through the joint physical/history orbit at each reached prefix. The shared snapshot stage makes both target proposals before either write, and its stage status is the same two-phase schedule with two targets. The partial-transition criterion yields one collective three-token word on this typed domain. It supplies neither arbitrary catalog closure nor whole-word rollback: a failed later stage leaves a previously committed prefix. Numerical reductions and represented outputs require the separate checks in section 5.

4. Actual work and all-time postword geometric identity

Let h=x'-x and q=theta'-theta on the declared lifts. The total structural storage jump is

\[ \Delta H=(L_fx)^Th+\frac12h^TL_fh +\sum_{\{i,j\}\in E_f} \left[\cos(\theta_j-\theta_i) -\cos(\theta_j-\theta_i+q_j-q_i)\right]. \]

The phase term is essential. The earlier phase-preserving AL/SHA formula cannot be transferred unchanged. Equivalently the exact unordered-pair expression, with primes from section 3, is

\[ \begin{aligned} \Delta H={}&2\sum_{j\sim k} \left[(X_k'-X_j)^2-(X_k-X_j)^2\right] +4(U_k'-U_k)\\ &-4\sum_{j\sim k}R_j \left[R_k'\cos(\Theta_j-\Theta_k') -R_k\cos(\Theta_j-\Theta_k)\right]. \end{aligned} \]

It follows from the existing fine/coarse storage decomposition and counts each of the two incident base edges once. SHA does not change this storage. A declared cost is not a reservoir, and a contraction of internal u,delta alone does not prove a global storage inequality.

Conditional passivity of this RA stage

There is a stronger, model-specific inequality on the fixed all-neighbor domain. Write y=x+b_AL*1_pair for the committed post-AL form and d=d_RA. The RA-only form increment on the selected pair is h_RA,i=-(d/4)*(L_f*y)_i. The two targets have no mutual edge, so the selected principal block of L_f is 4*I. Direct substitution into the quadratic work identity gives

\[ \Delta H_{\rm RA,form} =-\frac{d(2-d)}8 \sum_{i\in\{k,+;k,-\}}(L_fy)_i^2\le0. \]

For phase let d_s=theta_{k,s}-mu_k on the admitted lift. Each selected target has neighbor phasor sum 4*S_k, and no phase edge is incident to both targets. Its incident potential is therefore 4-4*abs(S_k)*cos(d_s). RA sends d_s to t*d_s, yielding

\[ \Delta H_{\rm RA,phase} =4|S_k|\sum_{s=\pm} [\cos d_s-\cos(t d_s)]\le0, \qquad |d_s|<4z<\pi. \]

Cosine is decreasing with absolute argument on this interval, and 0<t<1. The capacity multiplier changes no term of this structural storage. Hence H_after_RA<=H_after_AL exactly. This is conditional passivity of the selected full-neighbor, nonadjacent-target snapshot stage. It is not a theorem for arbitrary U3 subsets, simultaneous adjacent targets, clipping, or a diagnostic named coherence. The complete word still contains supplied AL work, so the inequality does not make the word autonomous or globally cost-free.

The following conservative estimate certifies the entire fixed source family without evaluating or tuning a trajectory. Let D be the two-row linear map from fine form deviations to A_k-x_{k,+}, A_k-x_{k,-}. Each row has four entries 1/4 at the neighbors and one target entry -1. Consequently

\[ DD^T=\begin{pmatrix}5/4&1/4\\1/4&5/4\end{pmatrix}, \qquad \|D\|=\sqrt{3/2}<5/4. \]

The fullword form reset satisfies

\[ \|h\| \le\sqrt2\,\frac78 b_{\rm AL}+\frac18\|D\|\|\xi\| <\frac54 b_{\rm AL}+\frac5{32}z. \]

The phase reset is supported on the same pair. The resultant estimate in section 2 gives

\[ \|q\|\le\frac1{16} \left(\sqrt2\,|\mu_k-\theta_{*,k}-\gamma| +\|\eta_{\{k,+;k,-\}}\|\right) <\frac38z. \]

Arithmetic centering is an orthogonal projection. With the source norm and the triangle inequality this proves

\[ Z_f'<z+\|h\|+\|q\| <\frac{49}{32}z+\frac54 b_{\rm AL} =\frac{69}{16384}<\frac1{128}<r=\frac1{16}. \]

The RA passivity calculation gives the sharper bound by the already derived post-AL work:

\[ H'-H_*\le H_{\rm after\ AL}-H_* <\frac{17}{524288}<\frac1{16384} <\frac1{2048}<\kappa_f r^2, \]

where the existing acute-tube barrier has kappa_f>1/8. All endpoint forms remain in (1/4,1/2), so the new positive-capacity weighted form mean is in that slab too. These are the premises of the existing capacity-independent first-exit proof, with excess ceiling 1/16384. Its reuse is justified by the new phase-inclusive RA work calculation, not by assuming the prior word's verdict transfers.

Continue both original normalized-sine rows with these endpoint coordinates, held positive postword capacities and the original clock. The fine state retains its acute winding-one geometric identity for all subsequent times and remains within every admitted nonantipodal pair chart. This establishes neither waveform recovery nor sustained activity, attraction, acquisition or autonomous repetition. RA changes the live sine pressure through phase, even though the event leaves stored pressure unchanged. The continuation must evaluate its own pressure and cannot read the stale event record as a complete law.

5. Representation, verification and remaining scope

The formulas and positive-width theorem above use exact real arithmetic. AL has the exact binade addition property proved by the earlier word, but that property does not make the RA blend or its circular mean exact. Even the rational capacity multiplier 33/32 can round on an arbitrary binary64 capacity. Equal input capacities still undergo the same calculation, whereas a floating-point primary result must be admitted on its actual stored value.

For a represented execution, independently record the actual total form increment and circular phase displacement, actual capacities, kind, histories, and pressure provenance. Rebuild the consumed collective state and structural work. If an error enclosure against the exact map is used, retain the separately bounded centered form and phase defects e_x,e_theta; a sufficient replacement norm bound is

\[ Z_{\rm represented}' \le\frac{69}{16384}+\|e_x\|+\|e_\theta\|. \]

Recheck the strict source, U3, positive-capacity, radius and energy premises using certified values. A static successful control is not a symbolic proof that all transcendental calls are exact, and a conservative bound failure does not demonstrate loss of identity. Neighbor reduction order and circular projection are part of the represented execution protocol; exact-real permutation invariance alone does not establish bytewise equality for all floating results. The associated history quotient also cannot be authenticated by equal rounded summaries.

The static selected-stage controls exercise the supplied complete word against independently reconstructed collective rows, associated histories, the internal-neighbor distinction, graph-edge work, rejected-prefix preservation and the existing same-law identity reader. They run no trajectory and retain finite precision and source admission separately from this exact-real proof.

This completes a scoped analytical relational-action result: internal neighbor phase geometry affects a collective TNFR operator response, and an explicit admitted intervention preserves the previously constructed object's geometric identity under its subsequent law. The source preparation, support, target, event times, factors and grammar policies remain supplied. Autonomous occurrence, resource realization, origin of the substrate and physical identification remain separate questions. The autonomous relative-phase corollary uses the same internal split under the unchanged continuous law and proves a finite response with small preparation/readout errors. It reuses the earlier persistent-response bounds; it neither executes RA nor equates its factors with a continuous elapsed time. The execution plan owns the next investigation.