Collective Coupling and the post-event support law

This bounded admission uses the existing word source, the pair-support classification and the actual registered UM snapshot stage. A valid collective event, an admitted continuous law and subsequent geometric identity are separate claims.

1. Frozen source, action and comparison

Before evaluating a selected word, retain the following protocol:

• Use the ordered ten-node doubled C5, pairs (2*k,2*k+1), and all four unit cross edges between neighboring pairs, with no internal edges. Keep the complete conservative normalized-sine law e=0,w=beta=1 and original structural clock. The target winding has angle alpha=2*pi/5.
• Retain the word source's 9/32<m<11/32, centered full-state radius Z_f<z=1/512 and equal-member capacities 1/2<v_k<2. Capacities are held between events. Preserve the full constituent state and its associated AL-only history/lifecycle stratum. Supply Si=1/2 at every node as an auxiliary operator input; it is not reconstructed from the continuous law. Stored pressure is finite with abs(p_i)<=1/2, separately from evaluated sine pressure. No source or occurrence law is inferred from these inputs.
• Select pair 0 in fixed order (0,1) and the canonical classes with tokens emission,coupling,silence. Validate the whole word with the core compatibility profile and pass its contexts to the shared snapshot-stage dispatcher. Retain strict rejection and live precondition checks, the original EPI rails, capacity thresholds and minimum coupling degree. Explicitly set UM_MIN_EPI=UM_MIN_VF=1/8.
• Set AL_boost=1/1024, SHA_vf_factor=1/2, UM_theta_push=1/1024, UM_vf_sync=1/8 and UM_dnfr_reduction=1/8. All four UM flags (BIDIRECTIONAL, SYNC_VF, STABILIZE_DNFR, FUNCTIONAL_LINKS) are true. Use UM_COMPAT_THRESHOLD=3/4, UM_CANDIDATE_COUNT=0 (all candidates), UM_CANDIDATE_MODE="proximity", and no restricted node sample. Set DELTA_PHI_MAX=UM_MAX_PHASE_DIFF=pi/2; candidate and final checks both apply.
• Retain fixed execution seed 17, structural time 7, history window 8 and supplied per-stage timestamps. There are no optional diagnostics, monitors, callbacks or user extensions. Neighbor phase writes and actual edge attributes are part of the result; no weight or length is substituted. Authoritative theta agrees with its retained phase alias, if present. Graph phase/edge version counters are absent or valid nonnegative integers, and retained caches obey their shared-setter mutation scopes. These extra premises admit the phase and support writes absent from the AL/SHA source.
• First classify the admitted missing edges, the complete reset and its associated context under pair interchange. Then check the actual weight against the unit-conductance hypothesis of the current sine reader. A rejection must retain the completed event and its precise model boundary. A unit-edge subdomain needs its changed degrees, storage work and fresh continuation proof; the old no-internal-edge certificate cannot be copied.
• Use exact-real algebra and independent static represented controls. The two nominal probes have m=5/16, capacities one, common phase origin zero, and exact target midpoint phases. One has selected internal coordinates u_0=1/8192, delta_0=1/32768; the other has u_0=delta_0=0. Other internal coordinates vanish. Controls retain the represented phase materialization alpha_hat=674651885/536870912, also used by the prior word controls, and arithmetic defects separately from exact alpha. Unequal AL-only member histories are allowed with their physical association, and fixed timestamps use 2030-01-02T03:04:05+00:00 in the static controls.
• Stop with an admitted continuation subdomain or a precise obstruction. No trajectory, parameter search, changed source geometry, forced unit edge, new pressure law or event selector belongs to this protocol. This is a prospective analytical admission, not a numerical formation campaign.

The threshold and all-candidate policy are supplied operator parameters. The finite probes do not prove the open-domain theorem. Exact-real descent does not by itself establish arbitrary bitwise relabeling invariance of transcendental calls or order-dependent diagnostics.

2. U3 admits the internal edge and excludes new interpair contacts

Write k=0 for the marked pair, b=h=1/1024, and use the source's signed representatives x_(j,+/-)=X_j+/-u_j and theta_(j,+/-)=Theta_j+/-delta_j. The AL stage changes only the two selected forms and their associated execution records. All four existing neighbors of each selected member have phase gaps bounded by

\[ \alpha+\sqrt2z<\pi/2. \]

Every missing edge from a selected member is either its partner or a member of a distance-two base pair. The partner gap is at most sqrt(2)*z, whereas the wrapped gap to every distance-two pair is at least

\[ 2\alpha-\sqrt2z>\pi/2. \]

The UM candidate reader applies U3 to the snapshot phases before its phase proposals. Thus each target has exactly one possible new neighbor, its partner. Moving other phases later in the same stage cannot introduce a candidate that failed this first gate. The two proposals concern the same undirected internal edge; the merge creates one edge, not two.

For each selected member s in {+,-}, the bidirectional consensus is

\[ C_s=\arg\left(e^{i(\Theta_k+s\delta_k)} +2R_{k-1}e^{i\Theta_{k-1}}+2R_{k+1}e^{i\Theta_{k+1}}\right), \qquad R_j=\cos\delta_j. \]

Choose its lift near theta_(*,k)+gamma. At the reference twist, the resultant is (1+4*cos(alpha))*exp(i*(theta_(*,k)+gamma)), of magnitude sqrt(5). The five constituent phase deviations give a chord error at most sqrt(5)*z. Consequently the resultant is nonzero and

\[ |C_s-\theta_{*,k}-\gamma| \le\arctan\frac{z}{1-z}<2z. \]

Target-to-consensus displacements are below 3*z. For each of the four neighbors, the displacement to either consensus on the nearest lift is below alpha+3*z<4/3. These statements fix the shortest-arc branches consumed by the actual stage, including the edge crossing the displayed 0,2*pi cut. A simultaneous phase-origin rotation changes none of the admission bounds.

After AL, all selected forms remain in (1/4,1/2). Capacities remain in (1/2,2) until their convex UM blend, and SHA then puts selected capacities in (1/4,1). The declared 1/8 form/capacity thresholds therefore have strict margins. All original fine degrees are four. This supplies the live preconditions independently of the whole-word grammar validation.

3. Complete primary map and associated execution state

Let C_bar=(C_++C_-)/2 on the preceding local lifts. The two target phase proposals read the same immutable post-AL snapshot. The common neighbors receive both proposals; the actual merge averages their shortest-arc displacements. Hence the exact-real map on the selected pair is

\[ \begin{aligned} X_k'&=X_k+b,&u_k'&=u_k,\\ \Theta_k'&=(1-h)\Theta_k+h\bar C,& \delta_k'&=(1-h)\delta_k+\frac h2(C_+-C_-),\\ v_k'&=\frac12\left[\frac78v_k+ \frac18\frac{v_{k-1}+v_{k+1}}2\right]. \end{aligned} \]

For each adjacent pair j=k-1,k+1, retain the lift of C_bar nearest its midpoint. Its map is

\[ X_j'=X_j,\quad u_j'=u_j,\quad \Theta_j'=(1-h)\Theta_j+h\bar C,\quad \delta_j'=(1-h)\delta_j,\quad v_j'=v_j. \]

Every other primary form, phase and capacity is unchanged. In all cases the unordered coordinates are obtained from the actual resulting members: R_j'=cos(delta_j'), U_j'=(u_j')^2, and Q_j'=u_j'*sin(delta_j'). In particular, bidirectional UM changes the neighboring pairs as well as the selected pair. It does not change their forms or capacities.

The pressure-relief branch writes, on each selected member only,

\[ p_{k,s}'=p_{k,s} \left[1-\frac18 A_s\right],\qquad A_s=\frac14\sum_{n\in N_{k,s}} \left(1-\frac{|\operatorname{wrap}(\theta_n'-\theta_{k,s}')|}{\pi}\right). \]

Here p denotes the supplied stored pressure consumed by UM. The branch uses the four original compatible neighbors and the merged phases, not the newly added partner. Neither these stored values nor their attenuation replace pressure evaluated from the post-event continuous law. Neighbor phase writes leave their stored pressures unchanged. AL and SHA preserve the stored pressure channel on this path.

For the final existing-neighbor U3 check, each target differs from its neighbor by (1-h) times their initial gap plus at most h*|C_s-C_bar|<2*h*z. Therefore its magnitude is below alpha+sqrt(2)*z+2*h*z<pi/2. The partner also remains admitted. To see the latter without assuming exact initial synchrony, fix the shared external resultant T and differentiate the target map in its input angle:

\[ \frac{d}{dt}\big[(1-h)t+h\arg(e^{it}+T)\big] =1-h+h\frac{1+\Re(Te^{-it})}{|1+Te^{-it}|^2}. \]

Every external phase stays within alpha+sqrt(2)*z of every intermediate target angle, so Re(T*exp(-i*t))>0. The displayed derivative lies strictly between zero and one. The final selected phase separation is therefore smaller than its initial value unless both were zero, and delta_k'=0 holds exactly when delta_k=0.

The full collective event retains the pair-swap orbit of physical state and its associated constituent records, together with the action mark, support, all operator parameters and timestamps. Swapping a selected pair exchanges C_+,C_-, changes the signs of u_k,delta_k,delta_k', and leaves the common neighbor update and internal edge unchanged. Swapping a neighbor pair leaves its two-member resultant and mean capacity unchanged. Thus all displayed primary rows descend to that retained orbit. The homogeneous Si input and all-candidate policy introduce no hidden member selection.

Only the marked targets execute AL, UM and SHA lifecycle/history updates; a neighbor's phase write does not execute its own UM word. Existing AL origins and lineage stay attached to their constituent. UM appends its glyph with the fixed history window, advances the selected operator-step indices by one and writes their UM source glyph. Their U2 debt remains zero and their prior-IL flag false. The AL/UM and UM/SHA prefixes have no IL-pattern recognizer match, so this stratum appends no recognized pattern and emits no related warning. UM adds no emission or latency fields. SHA then writes the existing latency contract with the boosted form as preserved form and final selected step indices n_i+3.

Shared phase writes update both theta and phase aliases on the selected pair and its four neighbors and invalidate their declared trigonometric caches. Neighbor histories and source glyphs stay unchanged. Selected capacity and stored-pressure writes use the shared cache owners. The added edge has exactly {"weight": W} with no supplied length; support versions and affected caches are updated. The fixed execution seed is read, but the all-candidate configuration draws no random values. Its node-offset map is prepared on the detached proposal snapshot.

Graph stage status, actual edge attributes and these associated writes belong to the result. Every stage validates its complete proposal before committing graph-owned state. That stage contract does not assert rollback of the entire three-stage word or external resources. The primary equations alone do not establish lifecycle equivalence. An order-sensitive diagnostic or separately exposed cache witness requires its own transported mark, as in the source owner. Static controls exercise the represented implementation; the pair-interchange proof concerns exact real arithmetic.

4. Actual edge weight and the unit-support boundary

Let epsilon_E=1e-9 denote the implementation's positive form-similarity regularizer. With the homogeneous Si=1/2, the shared compatibility formula reduces in exact real arithmetic to

\[ \boxed{W=1- \frac{|\operatorname{wrap}(\theta_{k,+}'-\theta_{k,-}')|}{2\pi} -\frac{|x_{k,+}+b-x_{k,-}-b|} {4(|x_{k,+}+b|+|x_{k,-}+b|+\epsilon_E)}.} \]

The cancelling boosts in the numerator emphasize that UM leaves the post-AL form difference unchanged. Both directed candidate proposals have this same symmetric score. Phase contraction, x+b>1/4 and the source norm imply 1-W<sqrt(2)*z<1/4. Thus the internal edge is indeed added, with W>3/4; the finite candidate score does not merely leave its existence undecided. No interpair edge is added.

Every deficit in this expression is nonnegative. Therefore

\[ \boxed{W=1\quad\Longleftrightarrow\quad u_k=0,\ \delta_k=0.} \]

Without the frozen homogeneous Si premise, equality of the two supplied Si values would be a third necessary condition. The exact unit-edge family is a synchronized-member stratum. It retains positive relative width in the other pairs' internal states, all mean states and paired capacities, but it is not open in the full source family. Every source with nonzero selected form or phase split produces a strictly subunit edge under this exact-real protocol.

The current complete normalized-sine reader requires unit conductances. The generic weighted result is consequently an admitted UM event followed by an unsupported continuation request under that reader. Replacing its weight by one, dropping the edge or importing unmodified degree-four coefficients would change the evaluated model. Pair-swap symmetry of the new internal edge does not remedy this missing constitutive premise. This unit-model assessment introduces no weighted law; Sections 7-10 separately admit a completion retaining the actual conductance.

These exact-real equalities are not authentication rules for rounded data. The production score uses its prescribed normalized form arithmetic and binary64 phase affinity; a sufficiently small positive deficit can round to a stored weight of one. The represented continuation reader must admit the actual stored graph and coordinates. Conversely, the exact theorem requires its exact hypotheses; rounded weight equality alone does not prove exact member equality. Static controls below keep the nominal mathematics and represented evaluation distinct.

5. Unit-edge continuation: new rows and an identity barrier

Restrict now to u_k=delta_k=0. The two consensuses coincide, the new edge has unit weight, and the selected members retain identical forms, phases and capacities. Write E'=E union {{k,+},{k,-}}, let L' be its unit Laplacian and D_i its actual fine degrees. They are five on the selected pair and four elsewhere. The post-word capacities are the positive held values from Section 3. The admitted complete continuation is

\[ \dot x_i=\frac{\nu_i}{\pi D_i} \sum_{j\sim_{E'}i}\sin(\theta_j-\theta_i),\qquad \dot\theta_i=\frac{\nu_i}{\pi D_i}(L'x)_i. \]

Pressure is the independently specified sine sum in the first row. These are original-clock rows on the new support; neither stored UM pressure nor the old degree-four normalization is substituted.

The existing internal-edge derivation extends directly from common capacity to the held paired capacities, which multiply each pair's two fine rows equally. With e_j=1 for j=k, zero otherwise, d_j=2, D_j=4+e_j, and

\[ J_j=\sum_{\ell\sim j}R_\ell\sin(\Theta_\ell-\Theta_j),\quad C_j=\sum_{\ell\sim j}R_\ell\cos(\Theta_\ell-\Theta_j),\quad q_j=\sum_{\ell\sim j}(X_j-X_\ell), \]

the exact pair rows are

\[ \begin{aligned} \dot X_j&=\frac{2v_j}{\pi D_j}R_jJ_j,& \dot\Theta_j&=\frac{2v_j}{\pi D_j}q_j,\\ \dot u_j&=-\frac{2v_j}{\pi D_j}\sin\delta_j(C_j+e_jR_j),& \dot\delta_j&=\frac{2v_j}{\pi D_j}(2+e_j)u_j. \end{aligned} \]

Every independent pair interchange is still a support automorphism. The exact unordered description therefore persists. At the selected tip, u_k=delta_k=0 makes both internal rates zero, so that tip remains invariant. As a useful exact reuse, throughout this invariant stratum the internal edge contributes zero to both fine sums. The entire new-support flow then equals the old doubled-C5 flow with auxiliary effective target capacity (4/5)*v_k, and all other capacities unchanged. This is a restricted equivalence of complete rows, not a modification of the actual stored capacity or an admission of the new graph by the old strict replica reader. At the same fine state, the actual selected pressure itself is 4/5 of the old-support pressure; the auxiliary-capacity representation instead keeps that old pressure and puts the same factor into its capacity. The two representations agree on both continuous rates, not on every primitive label. Representing the auxiliary rational capacity in binary64 introduces its own rate defect and must not be asserted exactly equal by floating-point coincidence.

For completeness, the identity barrier can be checked directly on the actual new graph. Its storage is

\[ H_{E'}=\frac12 x^TL'x+ \sum_{\{i,j\}\in E'}[1-\cos(\theta_j-\theta_i)]. \]

Differentiating cancels the two rows term by term, so dH_(E')/dt=0. The old winding target remains a critical point: the new internal edge has zero target form and phase differences. It adds zero target storage, so the same target value H_* applies. With l=e_(k,+)-e_(k,-),

\[ L'=L+ll^T,\qquad \lambda_2(L')\ge5-\sqrt5,\qquad \lambda_{\max}(L')\le2\max_iD_i=10. \]

The phase changes of Section 2 have Euclidean norm below 3*h: there are two selected displacements below 3*h*z and four neighboring displacements below 4*h/3. The AL centered form increment has norm sqrt(8/5)*b. Centering can only reduce these norms. Hence the post-word full centered norm satisfies

\[ Z_f' < z+\sqrt{8/5}\,b+3h <\frac{13}4z<\frac1{128}. \]

Take r=1/16. All old edge target gaps are alpha, and the new target gap is zero. On the radius-r chart the storage Hessian therefore gives

\[ H_{E'}-H_* \ge\kappa (Z_f)^2,\qquad \kappa=\frac{5-\sqrt5}{2}\cos(\alpha+\sqrt2r)>\frac18, \]

with alpha+sqrt(2)*r<pi/2. The global Hessian upper bound and criticality of the reference state also give

\[ H_{E'}(z')-H_*\le5(Z_f')^2 <\frac5{16384}<\frac1{2048}<\kappa r^2. \]

Conservation prevents a first radius exit, which would require storage excess at least kappa*r^2. Thus the exact continuation retains the winding neighborhood, acute fine edges and nonantipodal pair chart for all time. This is geometric identity retention, not a waveform-return, synchronization deadline or formation theorem.

The actual conserved means use rho_i=D_i/nu_i after both support and capacity resets. Indeed sum rho_i*xdot_i=sum rho_i*thetadot_i=0, with the declared continuous phase lifts. Thus the conserved form and lifted phase means are sum rho_i*x_i/sum rho_i and sum rho_i*theta_i/sum rho_i. Arithmetic means need not be conserved; the barrier uses instantaneous centered deviations and does not assume that they are. The finite positive capacities and these mean bounds also preclude a finite-time escape of the fine solution.

6. Reset work, static controls and remaining boundary

On the unit-edge stratum, account for each prefix on its actual support. Writing x for the source form, AL contributes the known exact amount

\[ \Delta H_{\rm AL} =4b\sum_{j\sim k}(X_k-X_j)+4b^2. \]

UM preserves the post-AL forms but changes phases on six nodes. Its old-edge storage contribution is exactly

\[ \Delta H_{\rm UM,old} =\sum_{\{i,j\}\in E} [\cos(\theta_j-\theta_i)-\cos(\theta_j'-\theta_i')]. \]

The new unit edge contributes (x_(k,+)'-x_(k,-)')^2/2+1-cos(theta_(k,+)'-theta_(k,-)')=0. Capacity synchronization, stored-pressure attenuation and SHA do not enter this storage directly. Therefore H_after-H_before=DeltaH_AL+DeltaH_UM,old. No sign of the UM phase work is assumed: unlike the target-only RA reset, its bidirectional writes also alter neighboring external edges. The continuation proof uses the full post-event storage bound rather than silently treating the word as passive. This accounting applies to the unit-support continuation. Section 10 separately accounts for generic subunit UM outputs under the declared weighted completion.

The static controls exercise the frozen complete word, all-candidate gate, actual edge and neighbor writes, member-associated context, unit/nonunit reader boundary and changed continuous rates. They evaluate no trajectory. Independent fine-state formulas check the complete-word map and storage, while the source-domain and all-time statements above are exact conditional proofs. Finite represented probes are neither a proof of arbitrary bitwise relabeling nor a substitute for the full theorem's premises.

The outcome separates three obligations. The supplied UM word has an exact collective description on this source. It creates one internal link, with generic subunit weight. Its synchronized-member subdomain has an admitted unit-support continuation and a retained geometric identity. The broader weighted continuation is outside that unit-only model; the independently admitted completion below addresses it without repairing the UM output. The original organization was already connected, so this operation does not join two separate objects, derive acquisition of a common identity, or select when an event occurs.

7. Independent protocol for conservative weighted continuation

This separate admission retains the original unit-law boundary above. Before evaluating the weighted reader, freeze the following extended protocol:

• Reuse exactly the source family, complete word, factors, event contexts, nominal probes and represented phase materialization of Section 1. Retain every actual UM edge and its generated conductance, including the generic nonunit result. Do not renormalize weights, remove edges or alter the source.
• After the supplied word, hold its actual simple undirected support U, symmetric nonnegative dimensionless conductances W, positive capacities and node ordering fixed. Read d_i as the number of bare neighbors, including supported zero-weight edges, rather than the conductance strength.
• Select the explicitly conservative completion e=0,w=beta=1 in the original structural clock. Its pressure is p_i=S_(U,i)/(pi*d_i), where S_(U,i)=sum_(j in N_U(i)) sin(theta_j-theta_i). Its phase row is theta_dot_i=nu_i*(L_W*x)_i/(beta*pi*d_i). This is a supplied complete law; the nodal identity alone does not select its reciprocal phase row.
• Predict before evaluation that both frozen post-word outputs are admitted under this model, that its unit-weight limit agrees with the earlier sine model, and that its full storage derivative cancels. Independently test weighted form gradients, bare phase sums, supported zero-weight edges, degree normalization, pair interchange and source/report readmission.
• Prove geometric retention for the entire exact frozen source by its uniform post-word norm and storage bounds. Use static interval/rate controls only: no trajectory, parameter search, positive-loss extension, varying-weight law, unrecorded event or physical identification belongs to this protocol. Numerical results retain represented input and enclosure provenance; they do not turn a pointwise balance observation into a future certificate.

The shared weighted owner is relational_sine_support.py, with independent static controls. Its separate admission does not weaken the earlier unit-only reader or the strict replica consumer. Sections 8–10 give the complete mathematical scope of this explicit continuation and its application to actual UM outputs.

8. Complete weighted law, balance and retained state

For this section let beta>0; the word application fixes beta=1. Let U be fixed simple connected undirected support without loops, with bare degrees d_i>0. Supply a finite symmetric nonnegative conductance W_ij on every edge, with its declared dimensionless scale, and fixed positive capacities nu_i. A supported zero-weight edge remains in U. Write

\[ q=L_Wx,\qquad S_i=\sum_{j\in N_U(i)}\sin(\theta_j-\theta_i),\qquad V_U=\sum_{\{i,j\}\in U}[1-\cos(\theta_j-\theta_i)],\qquad M=\operatorname{diag}\frac{\nu_i}{\pi d_i}. \]

The supplied complete rows and storage are

\[ \boxed{ p_i=\frac{S_i}{\pi d_i},\qquad \dot x=MS,\qquad \dot\theta=\beta^{-1}Mq,\qquad H_W=\frac12x^TL_Wx+\beta V_U.} \]

This retains signed real form, primitive circular phase, the original structural clock and zero forcing. Capacity, support and conductance have zero derivative on each declared continuous segment. The independently evaluated pressure is not the UM-stored pressure after relief. No pressure is reconstructed from a response, and no supported zero edge is discarded. Conductance enters the form gradient in the phase row; the conservative form row has bare phase pressure and no separate diffusive form-pressure term. Positive-loss transport would need its own normalization and complete admission and is outside this result.

Since grad_x H_W=q and grad_theta H_W=-beta*S, symmetry of the diagonal M gives the exact cancellation

\[ \dot H_W=q^TMS-S^TMq=0. \]

Equivalently, these rows have the constant skew structure

\[ \begin{pmatrix}\dot x\\\dot\theta\end{pmatrix} =\begin{pmatrix}0&-M/\beta\\M/\beta&0\end{pmatrix} \begin{pmatrix}\nabla_xH_W\\\nabla_\theta H_W\end{pmatrix}. \]

This identifies a conditional conservative completion; it does not identify H_W with physical energy. Its form and phase units inherit the source's declared structural chart and beta conversion. The method reuses the weighted balance completion: that owner's native resultant pressure and phase metric also cancel through H_U*g_U=-grad V_U. Here the sine model instead specifies g_U=S/(pi*d) and metric pi*d. The common cancellation method does not make these different complete laws identical.

For rho_i=d_i/nu_i, edge antisymmetry gives

\[ \frac{d}{dt}\sum_i\rho_i x_i=\frac1\pi\sum_iS_i=0, \qquad \frac{d}{dt}\sum_i\rho_i\theta_i =\frac1{\beta\pi}\sum_iq_i=0. \]

The second statement uses declared continuous lifts. Thus both weighted means use bare degree divided by capacity. Conductance strength cannot replace this weight. Form-origin shifts, common circular phase rotations and joint relabeling of state, U, W and capacities preserve the law. At W_ij=1 it recovers both rows and storage of the earlier unit model. If a bare edge has zero weight, it still changes S, V_U and degree; deleting it is a different support event. No regular limit of conductance alone establishes the absence of that phase relation.

Conservation does not uniquely select this completion. With the displayed form row fixed, an additional supplied phase row z satisfying S^T*z=0 would leave this scalar balance unchanged. Extra restrictions would be needed to exclude every such row. Conversely, weighting the sine pressure and phase storage by W would define another conservative model but would change the bare phase-channel premise retained here. The selected reciprocal diagonal phase row is an explicit constitutive choice, not a law inferred from the nodal equation, unit-limit agreement or conservation alone.

9. Exact pair quotient with the actual internal conductance

Retain the complete or empty unit interpair blocks of the frozen graph. For each pair j, write epsilon_j for the presence of an internal bare edge and omega_j for its conductance. Absent edges have both zero; present zero-weight edges have epsilon_j=1, omega_j=0. Let d_j count adjacent base pairs, D_j=2*d_j+epsilon_j be the fine bare degree, and v_j be the held equal-member capacity. Use the neighboring sums J_j,C_j,q_j from Section 5. Direct averaging of both fine rows gives

\[ \boxed{\begin{aligned} \dot X_j&=\frac{2v_j}{\pi D_j}R_jJ_j,& \dot\Theta_j&=\frac{2v_j}{\beta\pi D_j}q_j,\\ \dot u_j&=-\frac{2v_j}{\pi D_j} \sin\delta_j(C_j+\epsilon_jR_j),& \dot\delta_j&=\frac{2v_j}{\beta\pi D_j} (d_j+\omega_j)u_j. \end{aligned}} \]

The internal edge contributes -sin(2*delta_j) to the plus-member bare sine sum whenever present, but 2*omega_j*u_j to its weighted form gradient. Thus its weight changes the internal phase row without multiplying its bare phase-pressure contribution. Its contributions to both pair means cancel; the new degree still normalizes those means. Putting omega_j=epsilon_j recovers the unit internal-edge rows of Section 5.

For

\[ A_j=\frac{2v_j}{\pi D_j}(C_j+\epsilon_jR_j),\qquad c_j=\frac{2v_j}{\beta\pi D_j}(d_j+\omega_j), \]

the unordered internal rows are smooth:

\[ \dot R_j=-c_jQ_j,\qquad \dot U_j=-2A_jQ_j,\qquad \dot Q_j=-A_j(1-R_j^2)+c_jU_jR_j. \]

They preserve the derivative of Q_j^2-U_j*(1-R_j^2) exactly. More strongly, evolving any admitted fine representative and projecting gives these same rows, so realizability is retained up to the pair-chart boundary, including synchronized tips. The derivative identity by itself would not have established those inequalities or chart retention.

The collective state includes the actual U,W and held capacities along with the unordered member state. Every independent pair interchange preserves both bare support and conductance in the present graph; consequently the full weighted fine field commutes with those interchanges. On more general weighted attachments, preserving bare adjacency alone is insufficient: the consumed conductances must also be preserved. A weighted asymmetric external block needs its own state and symmetry admission.

Relative to the original complete-bipartite storage, the optional internal edges add exactly

\[ \sum_j\left[2\omega_jU_j+ 2\beta\epsilon_j(1-R_j^2)\right]. \]

The two coefficients retain the distinct conductance and bare-support roles. An equal-member tip remains invariant, and on that restricted stratum the old-support effective-capacity equivalence still holds because both added fine sums vanish. A generic internally split pair does not admit that substitution: its restoring sine term and weighted phase term have the different coefficients displayed above.

10. Entire-source continuation and complete event work

For every exact source in Section 1, the completed word produces one edge with actual weight omega in (3/4,1]. It leaves every old edge at unit weight. Let l=e_(k,+)-e_(k,-) and L_0 be the original doubled-C5 Laplacian. The weighted form and bare phase Laplacians are respectively

\[ L_W=L_0+\omega ll^T,\qquad L_U=L_0+ll^T. \]

Both dominate L_0 as quadratic forms, while both have largest eigenvalue at most ten because omega<=1 and the largest bare degree is five. The same uniform-form winding reference remains critical for H_W and has the same value H_*: its internal form and phase gaps are zero. The norm estimate in Section 5 used only the complete word's uniform phase and form bounds, not the selected tip condition. Hence it applies to all sources here:

\[ Z_f'<\frac1{128}. \]

At r=1/16, the two Hessian bounds therefore give

\[ H_W-H_*\ge\kappa Z_f^2\quad\hbox{on the radius-}r\hbox{ chart}, \qquad \kappa=\frac{5-\sqrt5}{2}\cos(\alpha+\sqrt2r)>\frac18, \]

\[ H_W(z')-H_*\le5(Z_f')^2 <\frac5{16384}<\frac1{2048}<\kappa r^2. \]

The lower form bound uses L_W>=L_0; the lower phase bound uses the positive old and internal edge cosines together with L_U>=L_0. The upper bound uses each actual Hessian rather than replacing omega by a silently repaired unit edge in the dynamics. Conservation then excludes the first radius exit exactly as before. Positive held capacities and the conserved D_i/nu_i means control the common coordinates; smooth fine flow exists globally. All frozen-source outputs thus retain the winding neighborhood, acute support and nonantipodal pair chart under this supplied weighted continuation. No selected internal split needs to vanish.

This is a conditional theorem about the newly specified complete law. The old unit-only reader's rejection of a nonunit UM output remains correct; it is neither a solver defect nor an obstruction to the separately admitted model. The current bound proves geometric retention, not a period, return deadline, spontaneous attachment or acquisition of a new organization.

For full hybrid accounting retain the actual output conductance rather than its unit limit. The original AL work and UM old-edge phase work from Section 6 still apply. The new edge adds its full endpoint cost,

\[ \boxed{\Delta H_{\rm attachment} =2\omega u_k^2+2\sin^2\delta_k',\qquad H_{\rm after}-H_{\rm before} =\Delta H_{\rm AL}+\Delta H_{\rm UM,old} +\Delta H_{\rm attachment}.} \]

This is the existing weighted reset functional evaluated at the actual post-UM state. SHA and capacity synchronization do not enter it directly, but their actual capacities determine the subsequent rates. The old-edge and attachment terms need not vanish or have cancelling signs. Continuous conservation does not make this supplied event passive. The attachment cost vanishes on the earlier synchronized-member subdomain and is positive for either nonzero selected split.

During continuation omega is the fixed value generated at the event. It is not recomputed from the evolving UM compatibility score. Recomputing it would add a weight law and the term omega_dot*(x_(k,+)-x_(k,-))^2/2 to the storage derivative, requiring a separate admission. No such law, past-loss reservoir or autonomous selector is supplied here.

The weighted reader's independent controls check the declared edge sums, rate normalization, storage balance and unit limit, including supported zero conductance and invalid-domain rejection. The complete-word controls retain actual UM weights and their original source/configuration. The exact all-source theorem does not depend on their finite verdicts. This completes the compatibility step for a supplied UM action followed by an explicit conservative weighted flow, while law selection, event occurrence, formation and physical identification remain distinct open obligations.