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Native relational exchange admission

This owner derives a conditional native joint law from explicit storage, locality and capacity premises on supplied support. Signed form, circular phase, held capacities and the declared structural clock remain part of the model. The opt-in implementation preserves its execution contract; these premises do not select a unique physical law.

Chapters and scope

Chapter Responsibility
Relational coefficient and capacity identification Spectral and finite-response discrimination, temporal acquisition and capacity-intervention evidence.
Sine law and constitutive information boundaries Global sine comparison, balance/locality premises, storage alternatives and phase-information restrictions.
Native local recovery and regional interaction Full-law recovery, equilibrium stiffness, regional transmission and work accounting.
Native phase domains and capture Regular-domain continuation, winding passage, reflected basins and full-state sector barriers.
Native validated transit and preparation controls Continuous transit, bounded law changes and preparation controls with their original evidence scope.

Shared form/phase variables do not make native and smooth-sine results interchangeable. A closure theorem, storage identity or contact response consumes its complete law and retained state; it neither selects pressure nor removes independent hidden initialization. Formation-to-maintenance handoffs must preserve those premises and the full environmental state. Continuous theorems, admitted finite steps and validated trajectory enclosures provide different evidence.

Section numbers remain stable. The execution plan alone assigns research work.

1. State, inherited geometry and the independent premise

Fix a finite connected simple reciprocal graph with at least two nodes, unit conductances, no loops, and held capacities N=diag(nu_i)>=0. Write D=diag(d_i), B=D-A and L=D^-1 B. All degrees are positive. Use signed real form x, primitive circular phase theta, and a regular phase chart: every neighbor resultant is nonzero and its displacement from the node phase lies in (-pi,pi). There are no inputs, events, clipping or capacity changes.

The existing phase metric and Dirichlet balance supply, without a new primitive,

\[ g_i=\frac{\operatorname{wrap}(\arg\sum_{j\sim i}e^{i\theta_j}-\theta_i)}\pi, \quad H_i=\pi\left|\sum_{j\sim i}e^{i\theta_j}\right| \operatorname{sinc}(\pi g_i)>0, \quad Hg=-\nabla V_\phi, \]
\[ V_\phi=\sum_{\{i,j\}\in E}(1-\cos(\theta_j-\theta_i)),\qquad E_D=\tfrac12x^TBx,\qquad q=\nabla E_D=Bx. \]

Within the explicitly edge-additive storage class, native-pressure alignment selects the phase cost above up to a positive factor. Continuity selects the metric's values at zero phase pressure. We use unit cost normalization, so the free positive factor is absorbed into beta. This restricted result does not itself derive Dirichlet form storage, joint separability or physical energy. Section 11.3 separately derives the quadratic/cosine split from an admitted joint edge cost, prescribed exact loss and uniform primitive-local completion with e,w>0. It does not replace the prescribed loss by mere passivity or identify physical energy.

Retain the native two-channel form row, with fixed e>=0, w>0:

\[ \dot x=N(-eLx+wg). \]

For an engine comparison, e,w must be the effective coefficients after its weight normalization; with only these channels active their sum is one.

The new independent premise is that E_D+beta V_phi, beta>0, measures joint storage and that phase/form transfer cancels at each nodal work port:

\[ \boxed{\quad w\nu_i q_i g_i-\beta H_i g_i\dot\theta_i=0 \quad\text{for every }i.\quad} \]

These are the coordinate contributions to the total storage chain rule; they do not assert an independent locally conserved energy at each node. The premise is stronger than zero total transfer. It chooses which nodal contributions cancel, a storage ratio and a phase reference for the exchange row. Global phase rotation makes total phase work unchanged, but adding an arbitrary common phase velocity generally breaks nodewise cancellation. Neither the nodal product nor the phase metric forces this extra premise. Its merit here is a precise, falsifiable dependency record. Section 11 gives an alternative admission: independent local capacity dependence and zero-capacity freezing derive the nodewise cancellation from global balance. Those capacity conditions are additional premises too. Section 11.1 gives another route using a uniform primitive-neighborhood information contract across supports, while retaining the zero-capacity phase condition and chosen storage. Section 11.2 removes individual phase freezing and classifies the remaining freedom as a common clock only. Adding gamma*1 to the phase velocity changes local allocations but not their sum, since 1^T Hg=0. It also leaves J_g*dot(theta) unchanged. The displayed representative fixes this common-rotation convention; it does not derive an absolute phase clock.

2. Forced phase row and its domain

At g_i!=0, cancellation fixes the phase velocity. Requiring that velocity to be continuous on the full regular phase chart gives

\[ \boxed{\quad \dot\theta=\frac w\beta H^{-1}NBx.\quad} \]

This extension is forced, not a division by zero at g_i=0. Since the support has no loops, holding neighboring phases fixed and varying theta_i gives partial_i g_i=-1/pi in a sufficiently small regular chart. The points g_i!=0 are therefore dense locally. Equality there fixes any continuous extension at its zeros. For nu_i=0 the chosen joint law freezes both local rows; the nodal identity alone would freeze only form.

Both rows use the node and its neighbors. The pressure is unchanged: there is no cotangent Kx correction, hidden mean projection or reconstructed pressure. The phase row is a new conditional law, not the default runtime's phase coordination or the separate independent-pressure extension.

3. Complete source, storage and mean balances

With J_g=Dg=(R-I)/pi from the shared phase response owner, the source and complete pressure satisfy

\[ \dot g=\frac w\beta J_gH^{-1}NBx,\qquad \dot p=-eL\dot x+w\dot g. \]

This derives source evolution from the proposed phase row, before observing an EPI response. It does not define pressure from an observed derivative. The storage identity is exact on this regular chart:

\[ \begin{aligned} \dot E_D&=-e q^TND^{-1}q+wq^TNg,\\ \beta\dot V_\phi&=-w g^TNq,\\ \boxed{\dot{\mathcal E}=-e q^TND^{-1}q\le0}, \qquad\mathcal E=E_D+\beta V_\phi. \end{aligned} \]

This admits heterogeneous and zero held capacities without inverse-capacity storage. It uses the actual nonnegative diagonal ND^-1; no Euclidean dissipation assumption about NL is needed. The storage is semidefinite in form, retaining the common-offset freedom, rather than controlling absolute EPI. The complete arithmetic mean remains

\[ \frac d{dt}\overline x= -\frac e n\mathbf1^TND^{-1}Bx+ \frac w n\mathbf1^TNg. \]

If all capacities are positive, the fixed diffusion weights rho_i=d_i/nu_i instead give

\[ \frac d{dt}\frac{\rho^Tx}{\rho^T\mathbf1} =\frac{w\,d^Tg}{\rho^T\mathbf1}. \]

The native Arg source need not make this last numerator zero. Relational storage and origin covariance therefore do not imply conserved form mean. At zero capacities use the complete raw mean formula, not rho. The existing passive-transfer budget explains why a nonincreasing finite storage cannot establish indefinite nonzero dissipative form contrast. This candidate supplies exchange from prepared phase structure, not an inexhaustible maintenance mechanism.

A conditional forward domain is available without simulation. If E(0)<beta, let a=acos(1-E(0)/beta)<pi/2. Nonincrease keeps every edge gap at most a, all resultants at least d_i*cos(a), and H_i>=pi*d_i*cos(a)*sinc(a)>0. Dirichlet storage bounds centered form on this connected graph. The source and complete mean velocity are bounded, so absolute form remains bounded on every finite interval. These estimates keep the smooth quotient field away from its singularities and give global ODE continuation under the stated held law. They do not prove persistence. The regular-domain theorem below enlarges this sufficient domain to E(0)<beta*max(2,d_min); that larger domain need not keep all edges acute.

4. Origin, units and exact replication

For a constant form shift x_new=x+c*1, both rows and storage are unchanged, because B*1=L*1=0. This is an actual symmetry of the stated joint model; it is not obtained by suppressing the mean equation.

For chart and clock conversion x_new=a*x, t_new=b*t, a,b>0, retain the same phase and dimensionless graph, and transform

\[ N_{new}=N/b,\qquad e_{new}=e,\qquad w_{new}=a w,\qquad \beta_{new}=a^2\beta. \]

Then both velocities transform correctly and E_new=a^2 E. In this convention capacity has inverse-clock units, w has form units and beta has squared-form units. The usual numerical coefficients can be dimensionless in a chosen normalized form chart. Changing form units while retaining every raw configured coefficient would change the model. This covariance does not determine the invariant ratio w^2/beta. Equivalently one can hold numerical capacity fixed and absorb the clock conversion into e_new=e/b, w_new=a*w/b; this is a representation choice, not a change to only one term of the nodal product.

If the two-channel engine normalization e+w=1 must be retained, define k=e+a*w. The equivalent representation has e_new=e/k, w_new=a*w/k, N_new=k*N/b and beta_new=a^2*beta. The capacity rescaling is essential: simply renormalizing the transformed pressure weights changes both rates. This reparametrization belongs to the declared two-channel held-capacity family; it does not preserve arbitrary capacity source terms, adaptation policies or other engine channels.

The existing counted-support construction also supplies an exact comparison. Replace each base node by m replicas, with no within-block edges and all m^2 unit edges for each original edge. Prepare equal form, phase and capacity within each block. Permutation equivariance preserves that prepared submanifold. On it,

\[ g_{fine}=g,\quad L_{fine}x=LX,\quad (B_{fine}x)_{i\alpha}=m(BX)_i,\quad H_{fine,i\alpha}=mH_i. \]

Consequently both joint rows descend with unchanged beta, capacity and clock. Total E_D and V_phi each scale by m^2. This differs from the cotangent comparison's inherited exchange scale eta_eff=m; the difference comes from the storage/exchange premise, not from new fine-node capacity. The partition and synchronized preparation are supplied. No block formation or transverse stability is inferred.

6. Why changing cotangent storage alone cannot retain native pressure

There is an independent obstruction on unit P2 with unit capacity. Let delta=theta_1-theta_0 lie in (-pi,pi), h(delta)=pi*sinc(delta) and m=(x_0+x_1)/2. Native pressure gives m_dot=0. Suppose instead that pi_i=h*x_i is canonical momentum and the Hamiltonian is invariant under common phase rotation. Noether momentum is P=2h*m; the native diffusion contributes zero to its rate. Therefore

\[ 0=\dot P=2m h'(\delta)\dot\delta. \]

For delta!=0, h'(delta)!=0: the odd function sin(delta)-delta*cos(delta) has positive derivative delta*sin(delta) on (0,pi). On an open full-state domain, m!=0, delta!=0 is dense, so continuity forces delta_dot=0 everywhere, including zero-mean and consensus points. This does not assume a quadratic form storage.

More precisely, set q_theta=(theta_0+theta_1)/2 and J=h*(x_1-x_0)/2. The canonical one-form is P*dq_theta+J*ddelta. Rotation invariance and delta_dot=0 imply locally that the Hamiltonian is independent of q_theta and J. The native phase source then forces, on connected fibers,

\[ \mathcal H=w(1-\cos\delta)+F(P). \]

Thus this premise cannot retain native pressure and supply nontrivial relative-phase exchange by choosing a different relative-form storage. The isolated zero-mean sector is not an open full-state remedy. The cotangent P2 results remain valid because their corrected form row differs from native pressure. Other momentum maps, non-Hamiltonian interconnections and constrained models are not excluded.

7. Admission verdict and remaining information

The dependency is explicit: relational storage plus nodewise exchange forces the phase row; the existing nodal pressure then fixes source evolution, mean drift, loss and the spectral discriminator. The new premise avoids the cotangent pressure correction by making a different constitutive choice. Its skew power-canceling representation need not satisfy Jacobi; it is not promoted to a Hamiltonian law.

Remaining choices are the storage itself, beta, the capacity-composition or nodewise-cancellation premise, phase-reference convention and admitted support/capacity scope. Sections 10-12 distinguish what global balance forces, what capacity separability additionally selects and which alternatives remain. The nodal equation does not select those premises. Their independent physical justification remains open. The capacity-separable family is admitted for conditional implementation and testing; no autonomous NFR, physical energy or universal law is claimed.

Controls: relational exchange admission uses exact algebra for the premise, P2 obstruction, units, spectral ratios and replica reduction. A separate finite local response check reuses native pressure, the shared transport snapshot and derive_joint_nodal_response, with represented-coefficient errors distinguished from symbolic identities. It evaluates no formation trajectory and installs no phase-law default.

8. Comparing the premises before selecting a model

Neither completion follows from the nodal product alone. Their exact consequences should be compared against explicit requirements, not ranked by whether a pressure formula is already implemented:

Requirement Relational exchange Cotangent exchange
Retain the native two-channel pressure Yes Requires the derived Kx correction
Treat a common EPI offset as redundant Exact symmetry of both rows Absolute form enters momentum and storage; a coordinate shift must transform them too
Depend on primitive state within one hop Both rows do The correction generally reaches two hops
Admit held heterogeneous and zero capacities with nonincreasing storage Section 3 supplies the balance The original pullback/storage has the section 13.24 obstructions
Supply genuine Poisson geometry Power cancellation does not establish Jacobi The coordinate pullback establishes Jacobi and a conditional Noether charge
Reduce equal coherent complete replicas Same rows with unchanged beta Inherits eta_eff=m*eta

For preservation of the present relational pressure sector, the relational candidate requires a smaller change to that sector and admits more of its held-capacity domain. This is a reason to prioritize its conditional study, not a proof that its premises are physically preferable. If a canonical momentum and genuine symplectic geometry are independently required, the cotangent construction supplies an advantage the relational balance alone does not. Offset redundancy, strict one-hop locality and these particular storage choices are not universal TNFR axioms; the pressure-premise owner already distinguishes offset symmetry from changing a physically relevant form origin. Neither candidate has independent physical validation here.

A possible justification for relational storage, with its assumptions visible. Suppose form storage is additive over undirected support edges, uses the same scalar function U of the signed endpoint difference on every edge, has no other edge attributes, and does not depend on edge orientation. Require U(0)=0, nonnegative nontrivial storage and exact quadratic amplitude homogeneity U(a*r)=a^2*U(r) for every a>0. Orientation independence gives U(-r)=U(r); taking a=abs(r) gives

\[ U(r)=U(1)r^2=\frac{\kappa}{2}r^2,\qquad S(x)=\sum_{\{i,j\}\in E}U(x_j-x_i)=\kappa E_D(x),\quad\kappa>0. \]

Dividing the joint storage by kappa leaves the relative phase-storage scale beta. This selects the displayed form storage within that class, but the nodal equation does not impose edge additivity, identical edge contributions or quadratic homogeneity. Nor does the argument derive nodewise work cancellation or fix beta.

A nonlinear experiment that cannot distinguish the candidates. On P2 with unit capacity, let x=(a,-a) and delta=theta_1-theta_0, with h(delta)=pi*sinc(delta)>0. The invariant zero-mean cotangent sector has Kx=0. With its general exchange scale eta>0, both models obey

\[ \dot a=-2ea+w\delta/\pi, \qquad \dot\delta_{\rm cot}=-\frac{2a}{\eta h(\delta)}, \qquad \dot\delta_{\rm rel}=-\frac{4wa}{\beta h(\delta)}. \]

Their common phase velocities are zero. Thus beta=2*w*eta makes the complete nonlinear sector identical, including its diffusion, rather than merely matching a linear frequency. Relational storage is then twice the cotangent storage on this sector. A symmetric two-node exchange or one fitted response cannot select these premises. The two distinct graph modes in section 5 instead supply an invariant ratio after the common scale is eliminated; section 9 supplies the finite prepared-state domain and error bounds.

Counting edges is distinct from counting nodes. Disjoint copies add both relational storages linearly. The complete replica construction of section 4 multiplies every base edge into m^2 edges, so its m^2 storage scaling is consistent with edge additivity. Node-count extensivity under that different topology is an additional requirement, not a contradiction of the stated law. The inherited cotangent scale likewise cannot be reset merely by naming a coherent block one node.

10. What global exchange already forces

The consensus discriminator has weaker premises than the full nodewise exchange law. Retain section 1's finite connected simple reciprocal unit graph, held N>=0, native form row and storage E_D+beta*V_phi. Let F(x,theta) be a phase velocity continuous on the full regular phase chart. Instead of nodewise cancellation, require only the global exchange identity

\[ w(Bx)^TNg(\theta)+\beta\nabla V_\phi(\theta)^TF(x,\theta)=0 \]

for all admitted form and phase states. This is precisely the exchange condition remaining after the known diffusion loss is subtracted from the joint storage derivative. It does not impose a portwise decomposition.

Fix any admitted x, not necessarily small, and any common phase c. For an arbitrary real vector v, the regular perturbation theta=c*1+epsilon*v gives

\[ g(\theta)=-\frac{\epsilon}{\pi}D^{-1}Bv+o(\epsilon),\qquad \nabla V_\phi(\theta)=\epsilon Bv+o(\epsilon). \]

Divide the global identity by epsilon and use continuity of F. Since v is arbitrary and B is symmetric, the result is

\[ B\left[\beta F(x,c\mathbf1)-\frac w\pi D^{-1}NBx\right]=0. \]

Connectedness makes ker(B)=span{1}. Thus

\[ \boxed{F(x,c\mathbf1)=\frac{w}{\beta\pi}D^{-1}NBx +\gamma(x,c)\mathbf1.} \]

This conclusion includes heterogeneous and zero held capacities. The remaining common rotation need not pause at a zero-capacity node; pausing every phase row there was part of the stronger nodewise convention. At consensus J_g=-L/pi and J_g*1=0, so the source derivative is fixed:

\[ \dot g=-\frac{w}{\beta\pi^2}L D^{-1}NBx. \]

With unit capacity, the pure-mode phase projection and form acceleration of section 9 are therefore forced by this weaker identity too. Common rotation changes neither projection nor pressure response. The ratios 3 and 9 do not depend on selecting the nodewise completion away from consensus. These are consequences of the chosen storage and global balance, which remain constitutive postulates, not of the nodal product alone.

Away from consensus, let F_rel=(w/beta)*H^-1*NBx. Global balance allows F=F_rel+Z whenever grad(V_phi)^T Z=0. At a noncritical phase state this is one linear constraint on an n-component velocity. Common rotation already lies in that kernel; quotienting it leaves n-2 instantaneous relative directions before imposing locality, symmetry or compatibility across states. Counting these directions does not construct a continuous local law that realizes them. On P2 the regular noncritical domain has no remaining relative direction, and continuity fixes its consensus limit as well. Larger networks retain a separate admission question: whether the additional structural requirements remove or retain relative motion along phase-storage level sets. No longer trajectory can replace that premise test merely by confirming a selected completion's response.

11. Capacity separability can select the complete exchange row

Retain section 10's graph, native two-channel pressure, joint storage and global exchange identity. Regard held capacity as an independently variable vector nu in the full nonnegative orthant. Require the same constitutive phase law F(x,theta;nu) to obey that identity for every such vector, with the following additional admission conditions:

  1. At fixed form, phase and support, row F_i depends on capacity only through its own nu_i.
  2. F_i=0 whenever nu_i=0.
  3. The phase law is continuous across the full regular phase chart.

No linear dependence on form or capacity is assumed. These conditions force

\[ \boxed{F_i(x,\theta;\nu)=\frac{w}{\beta}\, \frac{\nu_i(Bx)_i}{H_i(\theta)}} \]

throughout that chart, not only at consensus. In particular, nodewise work cancellation is then a consequence of the admitted capacity structure and global balance, rather than a separate cancellation premise.

Proof. Write q=Bx. Hold x,theta fixed and set every capacity except nu_i to zero. Their phase rows vanish by condition 2. Row i has the same value as before the change, by condition 1. Global balance becomes

\[ g_i\left(w\nu_i q_i-\beta H_iF_i\right)=0. \]

For g_i!=0 this determines the displayed row. At g_i=0, hold neighboring phases fixed and vary theta_i inside a sufficiently small regular chart. The loopless support gives partial_i g_i=-1/pi, so nonzero g_i states are locally dense. Both the proposed row and F_i are continuous, while H_i>0; equality extends to the zero set. This also covers zero capacity.

A compositional formulation. Instead of assuming condition 1, require capacity additivity at fixed remaining state:

\[ F(x,\theta;\nu+\mu)=F(x,\theta;\nu)+F(x,\theta;\mu), \qquad \nu,\mu\ge0. \]

Together with the zero-row condition, decomposition into coordinate-axis capacity vectors gives F_i(nu)=sum_j F_i(nu_j*e_j)=F_i(nu_i*e_i). Thus additivity supplies the needed separability, and the same proof fixes the row. Additivity expresses an independent composition rule for capacity contributions; it is not implied by the nodal product. Common clock homogeneity F(a*nu)=a*F(nu) alone does not give this rule or exclude nonlinear dependence on capacity ratios.

Likewise an affine capacity law F_i=b_i+sum_j T_ij*nu_j plus the zero-row condition forces b_i=0 and T_ij=0 for j!=i, providing another sufficient admission. Affinity is not a weaker assumption than allowing arbitrary own-capacity dependence; the theorem derives that dependence's linearity.

Scope. The zero-row condition also fixes a common-rotation convention: an extra capacity-independent phase clock is not admitted. Neither phase freezing at zero capacity nor independence from neighboring capacities is already required by xdot=nu*p. Holding one capacity vector, or varying only a common capacity multiplier, is insufficient for the isolation proof. The existing form pressure's capacity-contrast channel is inactive here; restoring it changes the global work identity and falls outside this result. Capacity evolution likewise introduces separate closure and storage duties. The selected row uses only neighboring form and phase. Spatial dependence on one fixed support alone does not supply the capacity-composition rule; the graph-family locality premise in section 11.1 is a different sufficient restriction when combined with the same balance and zero-row conditions.

This is a conditional model-admission theorem, not a declaration that the new law is canonical or physically selected. The remaining foundational question is whether capacity really composes through independent local activity, or whether neighboring capacities mediate additional phase exchange. The distinction must be justified from a capacity mechanism or independent evidence, not by giving a preferred completion a new label.

11.1 Uniform primitive locality supplies a different selection premise

Section 11 assumes independence from neighboring capacities. The following alternative derives that independence from a uniform local information contract across supports, the same global storage balance, and zero-capacity phase freezing. It does not derive locality or freezing from the nodal product.

Information and domain. Fix the same positive w,beta and nonnegative e on every finite connected simple unit graph with at least two nodes. Admit arbitrary real forms, held nonnegative capacities and the full regular phase domain of section 1. A row can consume the induced rooted neighborhood

\[ \mathcal B_i=G[\{i\}\cup N(i)] \]

and the primitive (x,theta,nu) values on its vertices. This includes the root's degree and edges between its neighbors. It excludes total neighbor degrees, their external incidences, graph size, remote states and derived neighbor pressure/gradients that depend on those excluded data. Model coefficients and the structural clock are fixed and shared, not extracted from global diagnostics. Require the same row on isomorphic rooted data, regardless of how the graph continues outside that neighborhood. This is stronger than sparse dependence on state on each separately supplied graph: it also excludes hidden support-specific coefficients.

Suppose the phase law F is continuous in the admitted primitive phases, vanishes at row i whenever nu_i=0, and satisfies

\[ \beta\sum_i a_i F_i+w\sum_i\nu_i q_i g_i=0, \qquad a=\nabla V_\phi=-Hg, \]

at every admitted state on every such graph. Neighboring capacities may enter F_i arbitrarily at this stage; no capacity additivity, linearity, form covariance or capacity differentiability is assumed. Then necessarily

\[ \boxed{F_i=\frac w\beta\frac{\nu_i q_i}{H_i}} \]

on the full regular domain. This also holds if the law is posed only on strictly acute edges with all other premises unchanged.

Proof by a local completion. Retain any root neighborhood B_i, with all its forms, capacities, phases and internal edges, initially assuming that no internal edge has antipodal phases. For each neighbor j of the root and each k adjacent to j within B_i, add three private leaves attached only to j: one with phase 2*theta_j-theta_k and two with phase theta_j, all interpreted on the circle. Give every new leaf zero capacity and any finite form. No new vertex touches the root, so its entire admitted information is unchanged. At j, each old incidence and its three new leaves contribute the relative phasor sum

\[ e^{\mathrm i(\theta_k-\theta_j)} +e^{-\mathrm i(\theta_k-\theta_j)}+2 =2[1+\cos(\theta_k-\theta_j)]>0. \]

Thus every original neighbor has a_j=g_j=0 and positive metric. Each new leaf has a single nonantipodal neighbor, hence a nonzero resultant and a regular displacement; its phase and form velocities vanish by zero capacity. The root retains its original regular resultant. The completed graph is finite, simple, connected and wholly admitted, including for nonacute internal edges. The two aligned leaves prevent a negative resultant at a neighbor when a gap is obtuse. In the acute restriction all newly added edges remain acute as well.

Global exchange on this completed graph therefore reduces to

\[ g_i(w\nu_i q_i-\beta H_i F_i)=0. \]

For g_i!=0, this forces the displayed row. Uniform locality transfers it back to the original graph, since the root's data did not change. To cover the exceptional states, perturb the finitely many neighborhood phases arbitrarily slightly while remaining in the original open regular domain. Avoiding antipodal internal edges is dense. At a regular root with g_i=0, the neighbor resultant points along its own phase, so partial a_i/partial theta_i=|sum_j exp(i*theta_j)|>0; hence nonzero g_i is locally dense too. Continuity of the original root row and H_i>0 extend the equality to all exceptional states. No limiting regularity of the auxiliary leaves' metrics is required: the limit is taken only in the original root's admitted row. This proves the claim.

Meaning and limits. The extra leaves are mathematical comparison states, not events added to a live network or a physical mechanism that creates support. The proof uses zero capacities exactly, so a model defined only at strictly positive capacities needs an independently justified continuous zero-capacity extension. A single K3, a fixed degree class, a fixed graph size or one common capacity value does not provide the graph/state family used in the proof. Allowing neighbor degree or external-incidence metadata falls outside this theorem; no impossibility result for that larger class is asserted. Dropping zero-capacity phase freezing at least admits an additional common phase velocity, invisible to total phase storage.

The reference law already satisfies the information contract: q_i and H_i use only incident primitive form/phase and F_i uses own capacity. The theorem derives nodewise cancellation and capacity separability within this class; it does not fix chi, pressure weights, storage choice, capacity evolution, support birth or a physical interpretation. The section 12 countermodel retains its valid balance but consumes excluded neighbor gradients on general supports. Its one-hop K3 example therefore does not contradict this uniform result.

No exact closure for an unprepared coarse partition has been assumed. Hidden state elimination can still produce effective nonlocality and memory under the composition and memory contracts. Those reduced descriptions are not primitive rows in this theorem's class. The earlier primitive phase-lift obstruction concerns a different prescribed acceleration and is not overridden.

The exact locality controls check the completion, exceptional-domain boundaries and the old countermodel; the engine controls check actual remote-extension behavior and whole-graph rejection. These finite controls supplement the proof rather than replace its graph quantifiers. Mathematical locality concerns the row: graph-wide validation and storage aggregation remain necessary in the public executor, and backend-dependent floating arithmetic is not a promise of bitwise equality across all graphs.

11.2 Without phase freezing, only a common clock remains

Remove only the zero-capacity phase-row condition from section 11.1. Keep the same autonomous model, uniform induced-neighborhood information, all finite connected simple unit supports, arbitrary independent form/capacity states, full regular phase domain, phase continuity and global exchange identity. Model constants remain fixed across graphs and states; no global state-dependent rate is an admitted local input. Then

\[ \boxed{F=F_{\rm rel}+\gamma\mathbf1, \qquad F_{\rm rel}=\frac w\beta H^{-1}NBx,} \]

where gamma is a single state- and support-independent angular rate for that fixed model. It may depend on its declared constitutive constants. It is not the independent form forcing Gamma. In particular, dropping individual phase freezing cannot create a different relative phase law within this class. Common-phase covariance is a consequence here, not a further premise inserted into the proof.

1. Reduce to a degree-one row. Let Z=F-F_rel. The reference already satisfies the global identity, so a^T Z=0. Write h(u,v) for the correction at a degree-one node with primitive state u and its neighbor's state v. Their induced neighborhood contains just the edge; the neighbor's external degree is unavailable. On P2, with a nonzero nonantipodal phase gap delta, the balance gives sin(delta)[h(u,v)-h(v,u)]=0. Thus h is symmetric; phase continuity includes the zero-gap case. Antipodal P2 is not admitted.

2. Use phase-balanced stars. Fix a center phase theta and any finite list of nonantipodal gaps delta_1,...,delta_m satisfying sum_r sin(delta_r)=0. Add one leaf at each gap and m+1 further leaves aligned with the center. These are distinct vertices, even if some phases coincide. All primitive forms and capacities are arbitrary. The center's relative resultant is real and obeys

\[ \sum_{r=1}^m\cos\delta_r+m+1\ge1. \]

Every leaf is regular, and the center has a=g=0. Its unknown correction therefore performs no phase work. Aligned leaves also have zero gradient, without requiring their velocities to vanish. The global balance and pair symmetry imply

\[ \sum_{r=1}^m\sin\delta_r\,h(u,v_r)=0. \tag{L} \]

Taking the two gaps delta,-delta shows that h(u,v_+)=h(u,v_-) for arbitrarily and independently chosen leaf forms and capacities. Holding one leaf fixed while changing the other removes dependence on the latter's form and capacity. Pair symmetry removes those attributes of the center as well. Continuity extends this statement to zero gap. Denote the remaining phase function by h_theta(delta)=h(theta,theta+delta); it is even in the gap. No differentiability or continuity in form/capacity was used.

3. The balanced-star equation fixes the phase function. Define

\[ f_\theta(s)=s\,h_\theta(\arcsin s),\qquad -1\le s\le1. \]

The balanced pair delta,-arcsin(sin(delta)) in (L) gives sin(delta)*h_theta(delta)=f_theta(sin(delta)), including obtuse gaps; aligned leaves keep these comparison stars regular. The balanced triple with gaps arcsin(s),arcsin(t),-arcsin(s+t) then gives

\[ f_\theta(s+t)=f_\theta(s)+f_\theta(t) \quad\text{whenever }s,t,s+t\in[-1,1]. \]

Phase continuity makes f_theta continuous. Repeated addition gives f_theta(p/q)=(p/q)*f_theta(1) for rational points in the interval; continuity extends this to every real point. Hence f_theta(s)=c_theta*s and h_theta(delta)=c_theta for nonzero sine, with zero gap covered by continuity. Pair symmetry gives c_theta=c_(theta+delta) for every nonantipodal pair. An intermediate phase connects antipodal pairs, so all c_theta equal one constant gamma. Every degree-one correction therefore has this value, at any capacity.

4. Transfer to every root. Put Y=Z-gamma*1. Since sum_i a_i=0, a^T Y=0, and every degree-one row of Y vanishes. Reuse section 11.1's induced-neighborhood completion: all original neighbors have zero phase gradient, and the new private leaves have zero Y regardless of their capacities. The leaves need not have zero capacity in this proof. Thus a_i*Y_i=0 isolates the root. The same locality and continuity argument covers zero root gradients and antipodal internal edges of an originally regular graph. Consequently Y=0, proving the displayed classification.

Reference choice and the role of capacity. The theorem leaves an absolute common clock, not a second relative interaction mechanism. Any one of the following additional choices fixes gamma=0:

  • The individual zero-capacity phase-row condition of section 11.1.
  • The weaker all-at-rest condition F(x,theta;0)=0 for all admitted x,theta.
  • Same-model capacity homogeneity F(x,theta;lambda*nu)=lambda*F(x,theta;nu) for all positive lambda, with coefficients held fixed: the reference scales by lambda, so the remaining equality requires gamma=lambda*gamma.

Each is an explicit condition. Changing clock units alone does not impose the third one: an independently supplied rate gamma could transform with those units. A model admitting explicit shared time dependence would instead leave a common function gamma(t) by the same pointwise argument. It is outside the autonomous class stated here; no primordial clock is derived.

Exact relative equivalence. In the unforced held-support model, for any reference solution and the same initial state at t_0,

\[ x_{\rm alt}(t)=x_{\rm rel}(t),\qquad \theta_{\rm alt}(t)=\theta_{\rm rel}(t)+\gamma(t-t_0)\mathbf1 \]

is a solution of the alternative law throughout the same regular interval. All edge gaps, native phase pressure, metric, storage and form trajectories coincide. In particular J_g*1=0 makes the phase-source derivative identical. This is an exact transformation, not a small-step approximation or a claim that arbitrary absolute-phase observations are redundant. An external phase reference, forcing or event law must be transformed and admitted separately.

The exact controls exercise balanced stars, the common-clock null direction, root isolation and the homogeneity distinction. The existing executor keeps gamma=0 and its current phase chambers; this theorem introduces no runtime mode or expanded admission. Native pressure, joint storage and uniform primitive locality remain constitutive premises. The coefficients, their physical justification and primitive support origin are not selected by this result.

11.3 Joint edge storage is selected by the prescribed loss and locality

The preceding phase-cost classification assumed separate form and phase storage. The following result tests that split rather than imposing it. Retain the native form row, with fixed e>0,w>0, held independently variable nonnegative capacities, the same structural clock, and every finite connected simple unit graph on its full regular phase domain. There are no inputs or events. Let one twice continuously differentiable cost satisfy

\[ \mathcal S(x,\theta)=\sum_{\{i,j\}\in E} W(x_i-x_j,\theta_i-\theta_j),\qquad W(r,\delta+2\pi)=W(r,\delta), \]
\[ W\ge0,\qquad W(0,0)=0,\qquad W(-r,-\delta)=W(r,\delta). \]

The last condition expresses undirected edge reversal. Neither separate evenness nor either axis cost is assumed. Suppose there exists a continuous phase row F with section 11.1's uniform induced-neighborhood information contract, same-model capacity homogeneity

\[ F(x,\theta;\lambda\nu)=\lambda F(x,\theta;\nu),\qquad\lambda>0, \]

and the specified exact balance

\[ \frac{d\mathcal S}{dt}=-e\sum_i\frac{\nu_i q_i^2}{d_i}, \qquad q=Bx,\qquad \dot x_i=\nu_i\left(-e\frac{q_i}{d_i}+wg_i\right). \]

Homogeneity holds with the cost and all constitutive coefficients fixed. No individual zero-capacity phase condition, gradient-flow representation, native-aligned storage metric or previously selected phase row is imposed. Then one constant beta>0 necessarily gives

\[ \boxed{W(r,\delta)=\frac12r^2+\beta(1-\cos\delta),\qquad F_i=\frac w\beta\frac{\nu_i q_i}{H_i}.} \]

Thus the stated class selects the separated storage and existing phase completion together. The prescribed loss, native pressure, locality, capacity homogeneity and edge-additive cost remain premises of this conditional result.

1. A necessary condition at zero phase-storage gradient. Write f=partial_r W and h=partial_delta W. Simultaneous reversal makes both functions odd under (r,delta)->(-r,-delta); in particular f(0,0)=h(0,0)=0. At a root with incident differences (r_j,delta_j), define

\[ Q=\sum_jr_j,\quad P=\sum_j f(r_j,\delta_j),\quad J=\sum_j h(r_j,\delta_j). \]

Initially take each gap in (-pi,pi) and make the root regular by adding aligned zero-form leaves if needed. Give only the root positive capacity. For each neighbor j, add three private leaves: one with x_leaf=2*x_j-x_root and theta_leaf=2*theta_j-theta_root, and two with the neighbor's own form and phase. All these leaves have zero capacity. The old incidence and its reflected counterpart cancel h at j; the aligned incidences contribute zero. Every original neighbor therefore has zero phase-storage gradient. Its relative phase resultant is 2*(1+cos(delta_j))>0, and all private leaves have nonantipodal degree-one phase data. The completed finite tree is regular everywhere.

Each private leaf sees only zero capacities in its admitted neighborhood. Scaling the sole active root capacity leaves those local data unchanged, while capacity homogeneity would scale its phase velocity. Consequently its phase row is zero. This uses locality and homogeneity, not the unassumed rule that any node with zero own capacity must freeze. All nonroot form velocities also vanish. Global balance therefore reduces, whenever J=0, to

\[ \boxed{P\left(e\frac Qd-wg\right)=e\frac{Q^2}{d}.} \tag{JS1} \]

The phase row at the root cancels from this identity. Thus it is necessary for every candidate completion, regardless of its dependence on neighboring capacities. The auxiliary vertices are comparison states, not live support events or a proposed mechanism of formation.

2. Aligned padding resolves the native resultant. For a finite collection with J=0, put X=sum_j cos(delta_j), Y=sum_j sin(delta_j) and let d be its size. Append m additional incidences with (r,delta)=(0,0). They change neither P,Q,J,Y nor any cost derivative, but the degree becomes d+m and the real resultant becomes X+m. For all sufficiently large integers m, the root is regular with

\[ g=-\frac1\pi\arctan\frac{Y}{X+m}. \]

Constructing the preceding completion for each padded star gives, with k=w/pi>0,

\[ P\left[eQ+k(d+m)\arctan\frac{Y}{X+m}\right]=eQ^2. \tag{JS2} \]

If Q!=0, positivity of e forces P!=0, so (d+m)*arctan(Y/(X+m)) must be independent of m. Yet

\[ (d+m)\arctan\frac{Y}{X+m} =Y+\frac{(d-X)Y}{m}+O(m^{-2}). \]

Here d-X=sum_j(1-cos(delta_j))>=0, and it is strictly positive if Y!=0. A constant sequence has its limiting value Y, so its displayed first correction must vanish. It follows that Y=0, and (JS2) then gives P=Q. Only integer padding counts are used; no weighted-graph limit or uniform bound on the auxiliary phase velocities is assumed.

3. Derive an aligned form anchor. Fix any r!=0. The periodic smooth function delta->W(r,delta) has a stationary point away from the single excluded antipodal phase. Indeed, a nonconstant periodic function attains distinct minimum and maximum values, so at least one extremum is nonantipodal; a constant function has every point stationary. Apply step 2 to that singleton incidence. Since Q=r!=0, its sine must vanish, so its nonantipodal phase is zero. Moreover P=Q gives

\[ h(r,0)=0,\qquad f(r,0)=r. \]

Continuity includes r=0; integrating with W(0,0)=0 proves W(r,0)=r^2/2. These axis properties have been derived, not assumed.

Now consider any finite collection with J=0, including one with Q=0. Append a single aligned incidence (a,0), choosing a so that Q+a!=0. Its contributions are h=0, f=a and zero sine. Step 2 applies and gives Y=0 and P+a=Q+a. Consequently every such collection satisfies

\[ \sum_jh(r_j,\delta_j)=0 \quad\Longrightarrow\quad \sum_j\sin\delta_j=0,\qquad \sum_jf(r_j,\delta_j)=\sum_jr_j. \tag{JS3} \]

4. Classify the phase derivative without assuming separability. The image I of the continuous function h on the connected domain R times (-pi,pi) is an interval, symmetric about zero by simultaneous reversal. It is not the singleton {0}: that would contradict (JS3) for a single gap with nonzero sine. If two states have the same value of h, apply (JS3) to the first and the reversal of the second. Their sines agree. There is therefore a well-defined bounded function

\[ T:I\longrightarrow[-1,1],\qquad T(h(r,\delta))=\sin\delta. \]

Balanced triples in (JS3) give T(a+b)=T(a)+T(b) whenever a,b,a+b belong to I. This bounded additive function is linear. To see the needed regularity directly, choose [-eta,eta] inside I. For any positive integer n and sufficiently small t, repeated addition gives n*T(t)=T(n*t), hence |T(t)|<=1/n. Thus T is continuous at zero; rational addition and subdivision extend linearity across I. Its slope cannot be zero, since nonzero sine values occur. Accordingly, for one nonzero constant beta,

\[ h(r,\delta)=\beta\sin\delta. \]

Integrating in phase and using the derived form axis gives W(r,delta)=r^2/2+beta*(1-cos(delta)). Continuity and periodicity include antipodal endpoints. Nonnegativity at r=0 forces beta>=0; the nonzero slope excludes beta=0, so beta>0.

5. Select the phase completion. For this now-derived storage, section 11.2 applies to the candidate phase law: global balance and the uniform information contract leave only F=F_rel+gamma*1. The reference is homogeneous in capacity. Keeping all coefficients fixed, homogeneity of the candidate then requires gamma=lambda*gamma for every positive lambda, so gamma=0. Conversely, the existing reference storage and phase row satisfy all the stated requirements. No new default law or runtime parameter is introduced.

Scope and remaining premises. The prescribed exact loss is essential: its positive quadratic form term supplies the nonzero-form restriction in (JS2), and positive phase coupling supplies the padding constraint. Neither e=0 nor w=0 is covered. The argument uses arbitrary support sizes and degrees, independent capacities, and the full regular phase domain; a fixed P2, a fixed degree family, or acute-only observations do not supply these quantifiers. In particular the periodic extremum step cannot silently be restricted to acute gaps. Nonadditive storage, different losses, additional state variables, inputs and events require separate admission.

The arbitrary comparison leaves do not derive support, capacities or a physical clock. The positive coefficient beta remains free and is the existing storage-scale parameter. This theorem does not establish physical energy or the uniquely correct pressure of nature. Its normalization also retains a boundary: a constant per edge is excluded by W(0,0)=0; without that convention it is invisible to fixed-support evolution but affects support-event storage jumps. The older auxiliary-potential and cotangent models retain their different laws and balance premises.

Why smaller checks do not select the cost. At consensus let the edge Hessian have entries (a,b;b,c), with c>0, and put k=w/pi. For t<e/k, set sigma=e/(e-k*t), b=c*t and a=sigma+c*t^2. The positive quadratic tangent storage then has the mixed form sigma*E_D(x)+c*E_D(theta+t*x). With p=B*theta, the local tangent rows

\[ \dot x=-ND^{-1}(eq+kp),\qquad \dot\theta=ND^{-1}\big[(k\sigma/c+et)q+ktp\big] \]

satisfy the prescribed quadratic loss exactly. This uses a local real phase lift, not a globally periodic storage. The periodic candidate W=sigma*r^2/2+c*(1-cos(delta+t*r)) has that Hessian but need not satisfy the nonlinear balance: at e=w=1/2, c=1, t=-pi, r=-2/3, delta=pi/3 and total two-node capacity 3, its phase derivative vanishes while Sdot=-1/6 and required -L=-2/3. No finite phase completion can supply the missing work at that state.

Another candidate W=r^2/2+(beta+alpha*r^2)*(1-cos(delta)), with beta>0, alpha>0, passes the regular two-node phase-critical check. On a six-cycle with phase gaps pi/3 and alternating form +a,-a, however, both native phase pressure and the entire phase-storage gradient vanish. Its form-storage gradient is (1+alpha)*q, so Sdot=-(1+alpha)*L, independently of the phase row. This violates the specified equality even though storage decreases at that state. Exact loss and passivity are distinct conditions; this example alone does not give a complete passive alternative.

The joint-storage controls check the reflected completion, aligned-padding restriction and named mixed counterexamples with independent exact arithmetic. A compatible mixed quadratic tangent law shows why a consensus expansion alone cannot establish this theorem; its natural periodic continuation can fail even on an acute two-node state. A form-dependent phase-cost amplitude can pass a two-node critical-point check and fail on a phase-balanced cycle. These are controls of the proof's obligations, not a substitute for its all-state argument. The existing phase-law locality controls and engine execution tests retain the continuous-law versus represented-evaluation boundary.

11.4 Passive phase loss is a distinct admissible completion

The exact loss in section 11.3 is an independent premise. To examine it, retain the already specified comparison storage E=E_D+beta*V_phi, rather than asserting that its classification still holds after weakening that premise. On the same unforced held-support regular domain, with e,w,beta>0 and held N>=0, declare the complete family

\[ \dot x=N(-eD^{-1}Bx+wg),\qquad F_\rho=\dot\theta=\frac w\beta H^{-1}NBx+\rho Ng, \qquad \rho\ge0. \]

Here rho is a supplied constant coefficient, not a diagnostic, an inferred reservoir or a consequence of the nodal product. The case rho=0 is the existing reference; other values are comparison laws, not production modes. The added row uses only primitive neighbors and own capacity. It is linear in capacity and preserves common form shifts, common phase rotations, relabeling and simultaneous (x,theta)->(-x,-theta) reversal. It also vanishes at a node with zero capacity.

The older passive-transfer comparison already used an additional multiple of g. That result employs centered Euclidean form storage and unit capacity on a supplied prism. Its cancellation method is reused here; its storage, rates and numerical bounds are not transferred to this Dirichlet-storage family.

Exact work and unchanged form equation. Since grad(V_phi)=-H*g,

\[ \boxed{\dot E=-L-R_\rho,\qquad L=e\sum_i\frac{\nu_iq_i^2}{d_i},\qquad R_\rho=\beta\rho\sum_i\nu_iH_i g_i^2\ge0.} \]

Thus passivity, locality, capacity homogeneity and the listed symmetries do not select the exact reference loss. The extra phase motion dissipates storage directly whenever rho>0 and Ng!=0; the reference transfers phase/form work before dissipation through form contrast. This accounting does not identify either loss with a physical bath or make it available to fund an event.

The complete form-mean formula is still

\[ \dot{\bar x}=\frac1n\mathbf1^TN(-eD^{-1}q+wg). \]

It agrees at the same prepared state, but its later values generally differ because the phase trajectory changes. The mean angular velocity acquires rho*mean(Ng); it is not generally a common rotation. Likewise the pressure derivative remains pdot=-eD^-1B*xdot+w*J_g*F_rho, with the chosen phase row included. No additional term has been silently inserted into xdot.

Units and replicas. The new coefficient obeys [rho*nu_f]=1/[time] for dimensionless angular phase and native g. In section 4's convention, capacity has inverse-clock units and rho is dimensionless. Under x_new=a*x, t_new=b*t, N_new=N/b, it stays unchanged. If capacity is held numerically fixed instead, rho_new=rho/b. When the pressure weights are renormalized with k=e+a*w and N_new=k*N/b, the equivalent coefficient is rho_new=rho/k. Failing to transform it changes the comparison law. The ratio rho*sqrt(beta)/w is an additional invariant under these form/clock conventions; their covariance does not choose it. Equal complete replicas retain g and each prepared capacity, so the added row descends under the same supplied synchronized replication as the reference. This is a compatibility check, not a derivation of rho.

Equilibria and inactive nodes. For every held nonnegative capacity vector, both laws have precisely the equilibrium conditions

\[ Nq=0,\qquad Ng=0. \]

At an equilibrium the nonnegative losses must vanish. Since e>0, this gives Nq=0; the form equation then gives Ng=0. Conversely these two conditions annul both complete rows. With all capacities positive, connectedness gives uniform form and a regular critical point of phase storage, which can include winding patterns. Zero capacities retain inactive frozen coordinates and do not permit replacing these conditions by global consensus. Equal equilibria do not imply equal trajectories, attraction basins or rates.

One prepared analytic discriminator. On P2 set e=w=1/2, beta=1, capacities (1,2), uniform form, and the phase gap theta_0-theta_1=pi/3. Both nodes lie strictly inside the acute domain. Then

\[ g=(-1/3,1/3),\qquad H_0=H_1=3\sqrt3/2,\qquad \dot x=(-1/6,1/3). \]

The reference initially has F_rel=0 and Edot=0. Relative to it, the declared alternative predicts

\[ \Delta F=(-\rho/3,2\rho/3),\qquad \Delta(\dot\theta_0-\dot\theta_1)=-\rho,\qquad R_\rho=\rho\sqrt3/2, \]
\[ \Delta\ddot x=wN J_g\Delta F =\left(\frac{\rho}{2\pi},-\frac{\rho}{\pi}\right), \qquad J_g=\frac1\pi\begin{pmatrix}-1&1\\1&-1\end{pmatrix}. \]

The initial form response is identical; the relative phase velocity, storage loss and form acceleration distinguish the laws. These are exact conditional initial-rate predictions, not a sampled response, a fitted coefficient or a physical validation. No previously frozen acquisition is rerun or reinterpreted.

Local recovery survives with different damping. Take a strictly acute phase equilibrium, uniform form and positive held capacities. Let K=Hess(V_phi) and evaluate H at that equilibrium. For form/phase perturbations (u,v), the linearization is

\[ \begin{aligned} \dot u&=-eND^{-1}Bu-wNH^{-1}Kv,\\ \dot v&=(w/\beta)NH^{-1}Bu-\rho NH^{-1}Kv. \end{aligned} \]

On the quotient by the two common offsets, Q=(u^TBu+beta*v^TKv)/2 is positive definite and

\[ \dot Q=-e(Bu)^TND^{-1}(Bu) -\beta\rho(Kv)^TNH^{-1}(Kv). \]

For rho>0 this is strictly negative on every nonzero quotient state, so the linearization is Hurwitz and the smooth nonlinear law has local exponential recovery. The rho=0 case uses the reference's separately proved exchange observability argument. This establishes existence of a local recovery neighborhood; it does not preserve a previous numerical rate or capture tube.

At phase consensus with common capacity nu, a nonzero random-walk Laplacian mode lambda has characteristic polynomial

\[ s^2+\nu\lambda(e+\rho/\pi)s +\nu^2\lambda^2\left(\frac{e\rho}{\pi} +\frac{w^2}{\beta\pi^2}\right)=0. \]

The reference polynomial is recovered at zero rho; extra dissipation changes both coefficients. For the explicit comparison e=w=1/2, beta=1, its discriminant divided by (nu*lambda)^2 is 1/4-1/pi^2>0 at rho=0, but 1/4-1/pi<0 at rho=1. Thus the same equilibrium and local-recovery verdict coexist with real decaying modes in one law and damped oscillatory modes in the other. This is a tangent response distinction, not a perpetual nonlinear pulse or a change in the supplied graph. On an acute uniform cycle twist, replace the last w^2/(beta*pi^2) term by w^2/(beta*pi^2*cos(kappa)). No monotonic optimal decay claim, global attraction, zero-capacity recovery or unchanged validated formation/capture evidence follows from these tangent formulas.

A sufficient selection premise under total passivity. One can recover the reference without assuming its exact loss directly, but another condition is needed. Keep the fixed comparison storage and native form row, and suppose the phase row is exactly homogeneous of degree one in signed form:

\[ F(ax,\theta;\nu)=aF(x,\theta;\nu),\qquad a\in\mathbb R, \]

with coefficients, phases and capacities held fixed. This is a condition on different preparations of the same model, not a conversion of form units. For an arbitrary proposed phase law write the signed exchange residual

\[ \mathcal R(x,\theta;\nu)=wq^TNg-\beta g^THF, \qquad \dot E=-L+\mathcal R. \]

Under this form condition, R(a*x)=a*R(x) while L(a*x)=a^2*L(x). Requiring only total passivity for every signed amplitude therefore gives a*R-a^2*L<=0. Both signs and arbitrarily small amplitudes force R=0. The uniform-locality theorem then leaves only a common angular rate, and same-model capacity homogeneity removes it. The resulting law is F_rel.

Exact degree-one form response is an additional constitutive premise, not a consequence of the nodal product or of mere oddness under form reflection. The supplied rho*Ng term violates it away from zero phase pressure. The storage remained fixed throughout this comparison; passivity has not replaced the loss hypothesis in section 11.3's storage-classification theorem. Higher order form dependence and other passive completions require their own test, rather than being excluded by calling the reference canonical.

The shared admission controls reuse the reference phase geometry and verify the added work, inactive rows, unit/replica conversion, exact P2 jet, consensus spectrum and signed-amplitude argument. Detached production field/tangent readings retain their represented defects separately from these ideal identities. No comparison coefficient is added to the engine or SDK, and no frozen response is regenerated.

11.5 Nonlinear signed form response survives passivity and local recovery

The degree-one form response in section 11.4 is sufficient for selection, but the other retained properties do not force it. Keep the fixed comparison storage E=E_D+beta*V_phi, native form pressure, e,w,beta>0, held N>=0, and the full regular phase domain on supplied connected simple unit support. Declare one constant 0<eta<2 and the complete comparison law

\[ \dot x=N(-eD^{-1}q+wg),\qquad F_\eta=F_{\rm rel}+Z, \]
\[ \boxed{Z_i=\frac{\eta e}{\pi}\nu_i h(s_i)g_i^2,\qquad s_i=\frac{q_i}{d_i\sqrt\beta},\qquad h(s)=\frac{s^3}{1+s^2}.} \]

This is an explicit countermodel, not a derived coefficient or an engine default. The interval for eta is a sufficient declared bound; no optimal range or search over alternatives is claimed. The denominator is positive for every finite signed form, so the row is smooth wherever the native phase geometry is regular.

Information, parity and units. The correction uses own capacity and primitive incident form/phase only. It is capacity-linear, vanishes at zero capacity, and preserves relabeling, common form shifts and common phase rotations. Since h is odd and g^2 is even under phase reflection, both the complete phase row and its correction obey signed-form oddness and simultaneous (x,theta)->(-x,-theta) reversal. At uniform form the correction and reference phase row vanish. However h(a*s) is not generally a*h(s): oddness and degree-one form response remain distinct requirements.

The coordinate s_i is dimensionless. Under section 4's transformations x_new=a*x, beta_new=a^2*beta, N_new=N/b, it stays unchanged and Z_new=Z/b, with eta unchanged. If the pressure weights are renormalized, e_new=e/k and N_new=k*N/b compensate in the prefactor as well. Thus no form or clock conversion is mistaken for a change of the nonlinear response. On the supplied synchronized complete-replica construction, q_fine=m*q, d_fine=m*d and g_fine=g; consequently s and the phase row descend unchanged. This prepared compatibility is not closure of an arbitrary coarse observation or a mechanism of block formation.

A global pointwise work bound on the regular domain. Write the signed extra exchange work as

\[ \mathcal R=-\beta\sum_i g_iH_iZ_i =-\frac{\eta e\beta}{\pi}\sum_i\nu_iH_i h(s_i)g_i^3, \qquad \dot E=-L+\mathcal R. \]

The native geometry and elementary scalar inequality give

\[ 0<H_i\le\pi d_i,\qquad |g_i|<1,\qquad |h(s)|\le\frac{s^2}{2}. \]

The last inequality follows from 2*|s|<=1+s^2. Since L=e*beta*sum_i nu_i*d_i*s_i^2, these independent bounds imply

\[ \boxed{|\mathcal R|\le\frac\eta2L,\qquad \dot E\le-\left(1-\frac\eta2\right)L\le0.} \]

The residual can have either sign. It must not be called a nonnegative added loss: the alternative can dissipate less storage than the reference at one preparation and more at another, while retaining total passivity. The bound holds whenever the ideal state is regular; by itself it is not a claim of global domain invariance or a discrete-step storage guarantee. No loss is made available to fund a support event without its own reservoir law.

Equilibria and the retained local mechanism. For every held nonnegative capacity vector, the equilibrium conditions remain exactly

\[ Nq=0,\qquad Ng=0. \]

Indeed, zero velocity implies zero storage derivative, and the strict factor 1-eta/2>0 forces L=0, hence Nq=0. The form row then forces Ng=0. Conversely these conditions annul the reference and added rows. Inactive coordinates retain their frozen-value obstructions; this statement does not infer global consensus or positive-capacity recovery when some capacities are zero.

At uniform form and a regular phase equilibrium, q=0 and g=0. For small joint form/phase perturbations (u,v),

\[ Z=O(\|u\|^3\|v\|^2)=O(\|(u,v)\|^5). \]

Consequently the reference Jacobian is unchanged. This is a statement about the vector field's expansion in state amplitude; it does not assert that the first four temporal derivatives agree at a finite perturbed preparation. At a strictly acute phase equilibrium and positive held capacity, the existing Hurwitz result therefore gives local exponential recovery for the comparison as well. Neither the equilibria, signed symmetries nor that linear response select the complete nonlinear phase law.

An exact finite-amplitude discriminator. Fix P2 with eta=1, e=w=1/2, beta=1, capacities (1,2), forms (1/2,-1/2) and phases (pi/3,0). The state is strictly acute. At this same preparation,

\[ q=(1,-1),\quad g=(-1/3,1/3),\quad H_0=H_1=3\sqrt3/2, \quad \dot x=(-2/3,4/3), \]
\[ Z=\left(\frac1{36\pi},-\frac1{18\pi}\right),\qquad \mathcal R=\frac{\sqrt3}{24\pi}>0,\qquad L=\frac32. \]

Relative to the reference, the predictions are

\[ \Delta(\dot\theta_0-\dot\theta_1)=\frac1{12\pi},\qquad \Delta\ddot x=wN J_gZ =\left(-\frac1{24\pi^2},\frac1{12\pi^2}\right). \]

Reflecting the form while holding phase fixed reverses Z and R, while leaving the storage and L unchanged. Both preparations remain passive. These are analytic initial-rate distinctions, not sampled trajectories or physical evidence. The full form-mean equation remains mean(xdot)=mean(N*(-eD^-1q+w*g)): the two laws agree on its initial value, but their later phase responses change it. In this preparation the difference of mean form acceleration is 1/(48*pi^2). The mean of Z is -1/(72*pi) and its relative component is nonzero, so the correction is not a removable common angular clock.

The same local geometric basin, with a new proof. A stronger local statement can be recovered without transferring a stored trajectory. Use the local recovery owner's strictly acute critical phase theta_*, its quotient coordinates z=(u,v), and positive held capacities. Choose

\[ m=\min_{\{i,j\}\in E}(\pi/2-|\delta_{*,ij}|),\qquad 0<r<m/\sqrt2, \]
\[ c_r=\min_{\{i,j\}\in E}\cos(|\delta_{*,ij}|+\sqrt2r)>0, \qquad k_r=\frac{\lambda_2(B)}2\min(1,\beta c_r). \]

The unchanged storage and phase Hessian give, inside ||z||<=r,

\[ E_{\rm rel}=E_D+\beta[V_\phi(\theta)-V_\phi(\theta_*)] \ge k_r\|z\|^2. \]

For ||z(0)||<r and E_rel(0)<k_r*r^2, the newly proved inequality Edot_rel<=-(1-eta/2)*L<=0 prevents a first exit from that same ball. The quotient stays in a compact interior sublevel; the phase metric is bounded away from zero and the smooth law continues. Thus the same analytically specified continuous energy sublevel is a sufficient domain for this comparison, by its own argument.

Within it, the largest invariant subset of L=0 has q=0. There Z=0 and F_rel=0. Keeping q=0 requires Ng to be common across the connected graph; sum_i H_i*g_i=0 and positive H,N force that common value to vanish. Strict convexity in the acute quotient neighborhood then selects the reference phase shape. LaSalle's argument gives convergence, and the unchanged Hurwitz Jacobian gives exponential convergence near the limit. Common-offset velocities are smooth functions of the quotient state and vanish there, so their time integrals converge and the offsets approach finite limits.

This result does not retain a previous quantitative decay constant, validated trajectory tube, arrival time or proof that a remote formation preparation enters the basin. Those concern the changed vector field and require their own evidence. Zero capacities, nonacute targets, events and changing support are outside this local capture statement.

Minimal premise ledger and the next scope. The joint-cost theorem selects storage and phase response using its prescribed exact loss and edge-cost class. With storage fixed, signed degree-one form response and total passivity are another sufficient route to exact exchange and the reference row. The present nonlinear alternative meets the listed parity, locality, capacity, equilibrium and local-recovery requirements while violating that degree-one condition. Those properties therefore do not derive it. The fixed storage used for the comparison is not circularly promoted to a theorem about all passive models, and the supplied eta has not become a selected mechanism.

The shared exact exchange controls check the bound, scaling, equilibrium conditions and finite-amplitude witness. This closes the declared necessity check. The next formation/interaction question is whether the full-state two-C5/two-bridge protected-sector capture argument extends to this admitted comparison. That is a separate gate: the local-ball proof above neither establishes its larger sector barrier and capture obligations nor repeats the original formation computation.

12. A capacity-mediated alternative and a discriminating intervention

The missing capacity premise is consequential. Reuse the earlier power-null construction's method, but retain this owner's Dirichlet storage and exact native form row. Write a=grad(V_phi), ell=D^-1 Bx and define the symmetric pair mobility

\[ \mu_{ij}=\begin{cases} \nu_i\nu_j/(\nu_i+\nu_j),&\nu_i+\nu_j>0,\\ 0,&\nu_i=\nu_j=0. \end{cases} \]

For adjacent nodes set

\[ S_{ij}=\frac w\beta\frac{\mu_{ij}}{d_i d_j} (\ell_i+\ell_j)\sin(\theta_j-\theta_i),\qquad S_{ii}=0,\qquad Z=Sa. \]

Nonadjacent entries vanish. The factors other than the sine are symmetric, so S^T=-S and a^T Z=0 identically. Both F_rel and F_rel+Z therefore have the same complete storage loss and unmodified form pressure. In general they redistribute nonzero nodal phase work while its sum cancels. This is an explicit countermodel, not a selected correction or a new production phase rule.

This alternative uses only existing form, phase, capacity and support:

  • It is equivariant under node relabeling, common form shifts, common phase rotations and simultaneous (x,theta)->(-x,-theta) reversal.
  • nu_i=0 makes row S_i zero. The pair mobility is homogeneous of degree one in all capacities, but not additive in independent capacity vectors. It is continuous at joint zero and smooth on positive strata; no capacity derivative at the joint-zero corner is claimed. Held capacities give a smooth field in evolving form and phase throughout the regular chart.
  • The section 4 unit/clock conversions preserve this field, including the compensating capacity change when channel weights are renormalized.
  • Equal complete replicas preserve the correction: ell is unchanged, a_fine=m*a, each pair coefficient becomes S_ij/m^2, and each base neighbor supplies m identical contributions. Disjoint admitted components are also independent. Thus these composition checks do not imply capacity additivity at a fixed state and support.

Locality boundary. On K3 every primitive consumed is within one hop. On general graphs, reading a neighbor's ell and a reaches two primitive neighborhoods. The replica identity does not turn that full fine law into a globally one-hop implementation. This example disproves uniqueness from the stated fixed-K3 balance, symmetry and zero-capacity conditions, and from their general finite-locality counterpart. It does not disprove a stronger theorem demanding one uniform primitive-neighborhood law on all admitted graphs. Section 11.1 now proves that conditional result without assuming closure of unprepared coarse refinements.

Exact nonconsensus witness. On unit K3 take x=(1/3,0,-1/3), theta=(0,pi/6,-pi/6), nu=(1,1,1). All resultants and phase displacements lie in the admitted chart. With k=w/beta,

\[ Z=\frac{k}{64} \begin{pmatrix}1+\sqrt3\\-3-\sqrt3\\-3-\sqrt3\end{pmatrix}, \qquad J_gZ=\frac{k(2+\sqrt3)}{64\pi} \begin{pmatrix}-2\\1\\1\end{pmatrix}\ne0. \]

Both laws have identical x, pressure and instantaneous xdot. Their phase-source rates differ by J_gZ, and their form accelerations by w*N*J_gZ. This is relative evolution, not a removable common rotation. At phase consensus the correction vanishes, so the frozen section 9 discriminator cannot separate these two members of the relational-storage family. The new witness addresses the missing premise directly.

Hold that form and phase fixed, keep nu_0=nu_2=1, and change only nu_1 from 1 to 1/2. The selected capacity-separable law leaves F_0 unchanged. The alternative changes its extra row by

\[ \Delta F_0=\Delta Z_0=-\frac{w(1+\sqrt3)}{192\beta}\ne0. \]

The same intervention admits a relative-phase observation, independent of a common phase reference. Put R=F_0-F_2. Both unchanged-capacity baseline rows are invariant under this intervention, whereas the countermodel also has Delta Z_2=k*(3+sqrt(3))/192. Therefore

\[ \Delta R_{\rm separable}=0,\qquad \Delta R_{\rm mediated}=-\frac{k(2+\sqrt3)}{96},\qquad k=w/\beta>0. \]

These are ideal differences at the same declared nonconsensus preparation, not a finite-sample acquisition. Relative phase removes a common rotation; it does not remove rate units or observation uncertainty. A uniform positive separation from zero requires a positive lower bound on k in that clock, or an admitted normalized observation cancelling the unknown scale. Identifying chi alone does not supply such a bound. For instance, (e,w,beta)=(1/2,1/2,1) and (1/3,2/3,4) both have chi=1, but k=1/2 and 1/6, respectively. Other retained coefficient/clock information must justify any absolute prediction. Both candidates must keep the same admitted preparation and parameter information through the reserved comparison.

No pressure is reconstructed from a response and no scale is refitted. This exact constitutive intervention distinguishes local independent activity from one admissible pair-mediated activity. Independent physical preparation/observation of such an intervention remains unestablished.

Controls: the capacity-identification test solves for unknown nonlinear own-capacity coefficients using the complete storage chain rule on an irregular P3. The selection counterexample tests reuse the shared exact phase geometry and check the full work identity, zero capacities, nonadditivity, source-response difference, capacity intervention, transformations and complete replicas. These controls add no temporal campaign or default engine dynamics.

13. Shared engine execution of the conditional reference

The admitted capacity-separable family is implemented in dynamics/relational.py. Evaluation and one simultaneous Euler step share that owner; the SDK delegates to it through Network.relational_exchange and Network.step_relational. The required RelationalExchangeModel.storage_scale declares beta rather than deriving or fitting it. Native pressure resolves the normalized EPI/phase weights. The implementation admits fixed connected simple unit-conductance support and held nonnegative capacities. Its default domain requires strict acute represented edge phases; section 18 defines the opt-in positive-resultant chamber. Both reject unsupported forcing and states before any live commit.

The field computes H from the local phasor resultant and its displacement, using the continuous sinc limit at zero. It never obtains H by dividing a possibly zero pressure by a phase gradient. Native pressure and the separately materialized form/phase split retain their difference. The phase row and shared nodal product are evaluated from the same supplied state; no telemetry selects an operator, partitions the network or supplies a missing capacity law.

Exact-rational work on materialized values retains the measured residual r=storage_rate+continuous_loss. For one admitted step, with D its recorded Euler storage defect, the accounting identity is

\[ E_{next}-E_{old}=-h\,\mathrm{loss}_{old}+h\,r_{old}+D, \qquad D=E_{next}-E_{old}-h\,\mathrm{storage\_rate}_{old}. \]

The positive or negative value of D is retained; it is not an a priori error bound or a new physical term. Mathematical continuous dissipation, represented instantaneous work and numerical endpoint nonincrease remain separate. In particular a lossless Euler step can increase storage. Keeping both endpoints and the numerical proposal segment acute does not prove that an exact future ODE trajectory stays there for arbitrary duration.

The execution contract owns scalar/clock admission, atomicity, caches, aliases and immutable report semantics; the SDK guide owns invocation. Example 180 executes four small declared steps, including a zero-capacity node. The routine regression owner checks analytic rates, the neighbor-capacity intervention, shared pressure, domain failures, zero-capacity behavior, atomic commit and numerical defects. The symbolic conditional theorems remain in their research controls. This integration does not install the countermodel, replace default runtime policies or turn successful finite execution into physical validation.

The shared pattern observer and Network.relational_pattern now retain the regional information used by the recovery and transmission studies below: complete centered form and reference-lift phase coordinates, separate squared norms and common offsets. They reuse winding and full-support regional transport accounting with the independent phase source, preserving explicit unavailable domains and pressure defects. A supplied frame is not a demonstrated equilibrium or temporal identity. The observation/export contract owns admission and exact JSON projection. Routine controls compare selected retained checkpoints with this owner without altering or rerunning the frozen campaigns; their historical benchmark wrappers remain immutable evidence.

These aliases route existing citations to their substantive owner.