Collective Reception on the weighted contact

Reception has an admitted collective action on the existing marked contact: it mixes form over the actual bare incoming neighbors while retaining the physical twin stratum. Its event work depends on the separate conductance weights and need not be nonpositive. The selected conservative sine law leaves fresh pressure and the instantaneous form row unchanged at this event, but can change the phase-rate row. These are conditional analytical results, not an evaluation of a new retained-source word or trajectory. The research program alone selects any subsequent investigation.

1. Complete model and event boundary

Use the twenty-node, 45-edge contact and contextual eleven-cell chart of the identity-transition owner. Its two individually marked ports are 0 and 1; the remaining cells are the nine exact pairs (2,3),...,(18,19). Bare support is fixed and reciprocal, with forty original unit-conductance edges, the internal edge {0,1} and four edges between {0,1} and {10,11}. Retain every actual conductance, capacity, geometric length and member association. A connected eleven-node simple graph cannot substitute for this chart.

The selected complete law has signed scalar form x, primitive circular phase with declared continuous lifts, positive held capacities and the original structural clock:

\[ q=L_Wx,\qquad S_i=\sum_{j\sim i}\sin(\theta_j-\theta_i),\qquad M=\operatorname{diag}\frac{\nu_i}{\pi d_i},\qquad p_i=\frac{S_i}{\pi d_i},\qquad \dot x=MS,\quad\dot\theta=Mq. \tag{1} \]

Here d is the number of distinct bare neighbors, not a weighted degree. Conductances enter q and structural storage

\[ H=\tfrac12x^TL_Wx+\sum_{\{i,j\}\in U} (1-\cos(\theta_j-\theta_i)). \tag{2} \]

The model supplies these laws; neither the nodal identity nor the Reception name selects them. There is no forcing or continuous loss. Storage is not identified with a measured physical energy. The event has no inferred duration, and all rate comparisons below use the same structural clock.

The EN contract, shared read snapshot, neighbor kernel and atomic stage own the executed map. Admission uses shared signed-scalar form, including uniform-real BEPI. Nonuniform or genuinely complex BEPI and magnitude projection are outside this scalar model. Every consumed form must be explicit; no absent-input fallback is used in this analytical domain.

2. Actual incoming mixture and collective descent

For a nonempty incoming neighbor set N_i, write

\[ u_i=\frac1{d_i}\sum_{j\in N_i}x_j,\qquad b_i=u_i-x_i,\qquad h_i=m_i b_i. \tag{3} \]

In the ideal unclipped map, each selected target changes to x_i+h_i and all other forms remain fixed. The selected configuration uses one common mix m; allowing target-specific m_i in the algebra does not add such a runtime interface. All proposals in an atomic stage read the same immutable source. Serially updating one port before reading the other gives a different map and must not replace this simultaneous event.

Equation (3) uses no conductance, phase compatibility or emission-source ranking. Directed execution reads predecessors; the present contact is undirected. EN acts on existing edges and does not require or create a new edge. Its optional source discovery is retained telemetry, not the weights or a filter of the numerical mixture. Kind selection uses the shared unclipped proposal and the associated ordered neighbor kinds; it can change even when a coarse numerical summary is unchanged.

Number the eleven cells in their displayed order. Their multiplicities are (1,1,2,...,2), and the two port means are exactly

\[ u_0=\frac{X_1+2X_2+2X_5+2X_6}{7},\qquad u_1=\frac{X_0+2X_2+2X_5+2X_6}{7}. \tag{4} \]

In particular, averaging four adjacent cell values equally is wrong. The exact port input consumes each member of a size-two cell. The contextual chart retains ordered fine neighbors, representations, kinds, source metadata, histories, persistent debt, supplied configuration and every other consumed or exposed record. Its declared lift reconstructs those inputs before applying the existing stage. Equal complete chart records therefore give the same admissions, proposals, context updates and outputs. No second operator implementation or lossy macro-node law is needed.

Only singleton ports change form, so the nine physical twin equalities are preserved. Phase, capacity and support are held during EN. Thus the marked acute phase sector and its two constituent windings are immediately unchanged. This is event descent on the selected stratum, not a proof that generic perturbed twins reduce exactly or that the sector is maintained under subsequent flow.

The ideal numeric mixture commutes with a common form translation. This does not permit dropping the form origin from full execution: configured admission ceilings, clipping, magnitude-based kind selection and other operators can consume it. Preserve the existing contextual chart rather than inferring a stronger runtime quotient from (3).

3. Exact event work and its sign

For every admitted actual endpoint x^+=x+h with phase and support fixed, including a represented or clipped endpoint, direct expansion of (2) gives

\[ \Delta H=q^Th+\tfrac12h^TL_Wh =\sum_{\{i,j\}\in U}W_{ij} \left[(x_i-x_j)(h_i-h_j)+\tfrac12(h_i-h_j)^2\right]. \tag{5} \]

Thus the necessary and sufficient nonpositive-work criterion is q^T h <= -h^T L_W h/2. It is a condition on the full event, not an operator-name guarantee. This storage has no additional capacity term.

For one unclipped target, let s_i=sum_j W_ij>0 and let v_i=sum_j W_ij*x_j/s_i be its conductance-weighted neighbor mean. Then q_i=s_i*(x_i-v_i) and

\[ \Delta H=m_i b_i q_i+\tfrac12m_i^2s_i b_i^2 =\frac{s_i}{2}\left[(x_i+m_i b_i-v_i)^2-(x_i-v_i)^2\right]. \tag{6} \]

For m_i>0, this is nonpositive exactly when

\[ b_iq_i\leq-\tfrac12m_i s_i b_i^2. \tag{7} \]

The statement includes the zero-update case. When the update is nonzero, its direction must oppose q_i and its length must not overshoot the equal-storage endpoint about v_i. Reception moves toward u_i, whereas weighted form storage is minimized at v_i; these centers can differ.

With a fixed mix 0<m_i<=1, fixed positive incident weights and arbitrary target/neighbor forms in any nonempty open interval, single-target EN has nonpositive work for all such forms if and only if those weights are equal. For equal weights, u_i=v_i and (6) reduces to

\[ \Delta H=-m_i(1-m_i/2)\,q_i^2/s_i\leq0. \tag{8} \]

Conversely, unequal weights make the weighted and bare averaging linear functionals different. Choose neighboring forms within the interval for which u_i differs from v_i, and choose the target form x_i=v_i. Then q_i=0, the convex mixture stays in the interval and Delta H=m_i^2*s_i*(u_i-v_i)^2/2>0. The zero mix is the identity and is the explicit exception. This proves a universal-scope obstruction; a particular unequally weighted source can still satisfy (7).

This equivalence allows independent neighbor forms. If forms are constrained to be equal within fixed cells, equality of the two mean functionals instead requires, for every incident cell a with per-member bare count n_ia,

\[ \frac{\sum_{j\in a\cap N_i}W_{ij}}{n_{ia}}=\frac{s_i}{d_i}. \]

Only these per-cell average weights are then observable by the numeric mixture. The contact's admitted involution also makes the two weights from a port into each twin cell equal. Its old unit-weight cells and newly added subunit-weight cells can still violate this restricted condition, as the explicit witness below does.

For simultaneous targets T, let W_ij=0 for absent edges. Equation (5) can also be written

\[ \Delta H=\sum_{i\in T} \left(q_i h_i+\tfrac12s_i h_i^2\right) -\sum_{\{i,j\}\subset T}W_{ij}h_i h_j. \tag{9} \]

The final sum is the cross term omitted by adding isolated-target work. It is present at the actual internal port edge. Its sign depends on the two increments; single-target criteria do not decide a simultaneous stage. Whole-word work must additionally include IL's phase reset and any other actual intervening event, rather than attributing their sum to EN.

3.1 A positive-work witness on the actual contact class

Take the original forty weights as one and all five added weights as 7/8. Let a=19/64, b=21/64, set the four old port neighbors 2,3,8,9 to a, the cross neighbors 10,11 to b, and both ports to

\[ v=\frac{16a+7b}{23}=\frac{451}{1472}. \tag{10} \]

All other forms may equal a. This respects every physical twin equality and lies strictly inside (1/4,3/8). Use the reference phases, any admitted positive symmetric capacities and the retained discrete context. Both port gradients vanish:

\[ q_0=q_1=4(v-a)+\tfrac74(v-b)=0. \]

Nevertheless their bare mixtures give, for any common m>0,

\[ h_0=h_1=\frac{4m(b-a)}{161}=\frac m{1288},\qquad \Delta H=\frac{23}{4}\left(\frac m{1288}\right)^2>0. \tag{11} \]

The internal edge has zero form change across it. Equivalently, the two isolated terms in (9) sum to (53/8)*h_0^2, and the missing cross term is -(7/8)*h_0^2, leaving (23/4)*h_0^2. This is an algebraic source, not a re-evaluation or retuning of the historical retained preparation. It proves that the native unit-support passivity result cannot be transferred to this weighted-contact EN stage, even with convex mixing and a preserved organizational class.

4. Fresh rows, means and pressure provenance

At the EN event, support, phase and capacity are fixed. Hence the complete selected law gives

\[ p^+=p,\qquad \dot x^+=\dot x,\qquad q^+=q+L_Wh,\qquad \dot\theta^+-\dot\theta=M L_Wh. \tag{12} \]

An untargeted neighbor can acquire a different phase rate through the corresponding off-diagonal entry of L_W h. No phase jump is being inferred from this rate change. On this connected positive support, a nonzero increment supported on the proper port subset is not a constant vector, so L_W h is nonzero; positive capacities make the phase-rate change nonzero somewhere. Equation (11) therefore gives a nontrivial subsequent phase-rate response even though both port gradients were initially zero.

There is also a local consequence for form motion. Let C(theta) be the bare Laplacian with edge coefficients cos(theta_j-theta_i). On either smooth continuation with held support and capacity, differentiation of (1) yields

\[ \ddot x=-M C(\theta)M q,\qquad \ddot x^+-\ddot x=-M C(\theta)M L_W h. \tag{12a} \]

Within the acute connected sector, C has only the constant null direction. If the acceleration difference vanished, M L_W h would be constant. Since 1^T L_W h=0 and M is positive, that constant must be zero. Connected positive-conductance support then forces h to be constant. Thus a nonzero port-only h changes both the phase rate and the form acceleration, although it leaves the instantaneous form rate unchanged. These are local derivative identities, not a finite-horizon response or maintenance prediction.

The EN primary map leaves stored pressure and the stored dEPI channel unchanged; optional diagnostics are separate observations, not a new evolution row. If the source pressure was fresh, it remains fresh for (1). If the source pressure was stale, EN preserves that same discrepancy; an explicit refresh is still needed before claiming the constitutive row. This differs from the pure-EPI realization bridge, whose pressure depends on form and generally changes after EN. The bridge's numerical kernels can be reused; its pressure-manifold defect and diffusion stability certificate are not certificates for (1).

Let rho_i=d_i/nu_i and R=sum_i rho_i. Capacity and support do not change, so the two means obey

\[ \mu_x^+-\mu_x=\frac{\rho^T h}{R},\qquad \mu_\theta^+=\mu_\theta. \tag{13} \]

The continuing conservative law preserves its new form-mean leaf. Local Reception is not generally a mean-conserving transfer; no compensating source decrement, reservoir or autonomous event selector is installed by the blend. A supply interpretation must account separately for (5).

5. One grammatical word on a positive-width domain

Reuse the analytical source domain of the previous contextual word: old weights one, new weights in (3/4,1], the exact twin symmetry, 1/4<x_i<3/8, port capacities in (3/8,5/8), other positive capacities at most one, and

\[ \theta_i=\tau+\theta_{*,i}+\eta_i, \qquad \max_i|\eta_i|<1/8. \tag{14} \]

Set stored pressure initially to the independently specified fresh value in (1). At the reference phases S_i=0 for every node: old-cycle sine terms cancel and added edges have zero phase difference. The sine Lipschitz bound therefore gives

\[ |p_i|<\frac1{4\pi}<\frac3{20}. \tag{15} \]

This meets the selected default Reception pressure ceiling 0.15, and its form ceiling 0.9 is also strict. The ceilings are admission policies, not derived universal constants. Use the configured default EN mix 1/(pi+1); the ideal argument also holds for any fixed 0<=m<=1. Hard form rails [-1,1] are inactive, since every mixture remains in (1/4,3/8). IL consequently retains its positive-form admission; capacities remain in (3/8,5/8) until SHA, whose selected factor 1/2 leaves them in (3/16,5/16) after its live minimum-capacity check.

The complete selected word is EN -> IL -> SHA. Retain the original associated histories AL,UM,SHA,IL,OZ at port 0 and AL,AL,AL,UM,SHA,IL,OZ at port 1, each with maximum length eight, persistent debt one and prior coherence. Use strict rejection, live preconditions, atomic target order (0,1) and ValidatedSequence context {"initial_epi_nonzero": true, "compatibility_profile": "core"}. Existing nonzero form supplies the U1 start premise; EN is not assigned a generator or closing role. Its U2 debt progression is

\[ 1\xrightarrow{\mathrm{EN}}1 \xrightarrow{\mathrm{IL}}0 \xrightarrow{\mathrm{SHA}}0. \tag{16} \]

EN followed by SHA alone would retain the outstanding debt. The selected word uses neither coupling/resonance phase admission, temporal mutation evidence, bifurcation nor recursive actions. Passing these applicable finite grammar checks does not assert unrestricted U1-U6 stability.

EN leaves phase fixed, so the preceding owner's IL phase-sector proof applies unchanged with the same phase coefficient 0<=lambda<=3/10. Retain its pressure-retention premise 0<=kappa<1 and the declared refresh after each completed stage; the default configured IL factors meet these conditions. SHA also leaves phase fixed. Thus the ideal word retains the original acute sector and both constituent windings. The successor domain is its actual ordered prefix images, not a promise that the source box admits indefinite repetitions.

For the canonical finite U6 observations, retain geometric lengths equal to conductances, the same baseline convention and threshold pi/2. EN's before/raw/refreshed pressure comparisons vanish, because phase and stored pressure are held and initial pressure is fresh. The later IL/SHA raw/refresh comparisons inherit the existing bound strictly below 3/2. This covers only those ideal event snapshots. It is not a future potential, small-radius or all-time maintenance certificate.

6. Representation and verification boundaries

The ideal equations use exact real arithmetic. Runtime neighbor means, blends, BEPI scalarization and clipping have their declared represented arithmetic. A binary64 coefficient matrix is not automatically identical to the two-stage mean/blend computation. Use the actual endpoint h in (5), (9), (12) and (13); retain the difference from the ideal h in (3) when comparing those models. A shared numerical kernel does not make floating-point sums permutation invariant or authenticate their source.

The contextual reconstruction keeps per-member kind, source and history associations even where physical coordinates coincide. Lossless physical compression therefore does not eliminate the records consumed by grammar or exposed by execution. Synthetic controls of the shared runtime are separate evidence from the positive-width analytical proof. Neither those controls nor the work identity executes the historical frozen producer, selects event occurrence or establishes a physical interpretation.

The collective Reception controls exercise the existing atomic stage, immutable simultaneous reads, bare multiplicities, kind/source association, live rejection, exact weighted work and complete-word context. They compare independent edge accounting with actual endpoints, and keep ideal rational witnesses separate from their represented calculations. Testing owns the maintained verification selection.