Native local recovery and regional interaction

Full-law local recovery, regular-equilibrium stiffness, regional transmission and work accounting; interacting environments remain retained.

Part of Native relational exchange admission. Section numbers are stable across the collection; hypotheses and model changes remain local to each result.

15. Local recovery under the complete relational law

Equilibria reuse the existing circulation classification

Fix the same connected simple unit graph, now with every held capacity strictly positive and e,w,beta>0. Let theta_* have strictly acute edge gaps and grad(V_phi)(theta_*)=0. The equilibrium family is

\[ \mathcal M_*= \{(x,\theta)=(c\mathbf1,\theta_*+\alpha\mathbf1):c\in\mathbb R, \ \alpha\in S^1\}. \]

These are all acute equilibria: the phase row and invertibility of N,H force Bx=0, then the form row forces g=0. Conversely those two conditions make both rows vanish. In the acute chart, zero g is equivalent to zero neighbor sine sums. Therefore the integral-cycle circulation classification in the existing owner applies to the geometry, without importing that owner's supplied sine evolution law or its common-capacity premise. Each admitted winding sector contains at most one such phase shape up to common rotation. Trees and graphs whose cycle space is spanned by triangles/quadrilaterals admit only consensus; a unit cycle admits twists 2*pi*ell/n with abs(ell)<n/4.

The resulting form is uniform, but the phase geometry need not be. In particular, winding-one C5 is an admissible nonconsensus geometry. Its existence is inherited from the supplied support and initial sector; it is not spontaneous formation or a differentiated stationary scalar form.

This also separates geometric identity from scalar summaries: at the exact winding-one C5 lock, the mean phasor is zero and every local mean absolute edge gap is 2*pi/5, while g=0 and both nodal rows vanish. A zero global phase-order parameter or a configured gradient warning cannot rule out this locally recoverable geometry. These observations do not select its dynamics.

Quotient linearization and the restoring mechanism

Choose an orthonormal matrix R spanning 1^perp and local lifted deviations x=c*1+R*u, theta=theta_*+alpha*1+R*v. These are local differences from the prepared circle-valued phase, not a common semicircle assumption on theta_*. Let K_* be the Hessian of V_phi there: its edge weights are cos(delta_*)>0. Write

\[ \begin{gathered} B_q=R^TBR>0,\quad K_q=R^TK_*R>0,\\ A_q=R^TND^{-1}R>0,\qquad M_q=R^TNH_*^{-1}R>0. \end{gathered} \]

At equilibrium, differentiating Hg=-grad(V_phi) gives Dg=-H_*^-1 K_*. The derivative of H^-1 contributes nothing to the phase row there because Bx=0. The exact quotient Jacobian is therefore

\[ J_q=\begin{pmatrix} -eA_qB_q&-wM_qK_q\\ (w/\beta)M_qB_q&0 \end{pmatrix}. \]

This includes arbitrary positive heterogeneous capacities and irregular degrees. Passing to storage coordinates (B_q^(1/2)*u, sqrt(beta)*K_q^(1/2)*v) gives the similar matrix

\[ \widetilde J_q=\begin{pmatrix}-\mathsf D&-\mathsf C\\ \mathsf C^T&0\end{pmatrix},\quad \mathsf D=eB_q^{1/2}A_qB_q^{1/2}>0,\quad \mathsf C=\frac w{\sqrt\beta}B_q^{1/2}M_qK_q^{1/2}. \]

The coupling matrix C is invertible. For a complex eigenvector (a,b), the real part of its work identity is Re(lambda)*(||a||^2+||b||^2)=-a^* D a. Thus no eigenvalue has positive real part. Equality would force a=0; the first block would then give C*b=0, hence b=0, contradicting an eigenvector. All quotient eigenvalues have strictly negative real part.

Conditional local theorem. Smoothness of the admitted field and this Hurwitz Jacobian imply local exponential recovery of M_*: sufficiently small joint form/phase perturbations remain acute and their deviations modulo the two common offsets decay exponentially. A pure phase deformation can initially have zero storage loss, but it induces form contrast through native pressure; exchange makes that deformation accessible to form dissipation. No added phase damping, telemetry selector or restorative operator is needed under this specified law.

Full-network stiffness at a regular equilibrium

The same quotient Jacobian applies at any exact regular equilibrium, with e,w,beta>0 and positive held capacities, even when some edge cosines are negative. Here K_q is symmetric but is not presumed positive. Criticality g=0 is essential to the identity Dg=-H^-1*K; uniform form alone does not establish it. Eliminating the form component of an eigenvector gives the symmetric quadratic pencil

\[ \begin{aligned} P(\lambda)&=\lambda^2\mathsf M+e\lambda\mathsf C+ (w^2/\beta)K_q,\\ \mathsf M&=M_q^{-1}B_q^{-1}M_q^{-1}>0,\qquad \mathsf C=M_q^{-1}A_qM_q^{-1}>0. \end{aligned} \]

For a nonreal eigenvalue and its phase vector v, the imaginary part of v^*P(lambda)v=0 gives 2*Re(lambda)*v^*M*v + e*v^*C*v=0; hence its real part is strictly negative. For real lambda>=0, P increases strictly in the positive-definite order and becomes positive definite for large lambda. Every negative direction of K_q therefore supplies a positive real eigenvalue, counted with multiplicity. If K_q>0, all quotient modes have negative real part, extending the local recovery theorem beyond the sufficient all-acute edge condition. If K_q has a kernel, those additional zero modes require a separate analysis.

These statements concern the full nodal quotient, including perturbations that break an imposed reflection. A negative cosine cut sum is a useful sufficient instability witness: the cut's indicator vector has that Hessian quadratic value and can be made orthogonal to the common phase by subtracting its mean. For two joined rings, opposite uniform rotations have stiffness 8*cos(a-A) when the matching bridge gaps agree in cosine. A negative value proves instability at an exact equilibrium; at a transient state it is only instantaneous phase curvature.

The reflected equilibrium classification uses this full-network result, with independent exact Hessian controls and the existing materialized uniform-tangent adapter. It does not infer ideal criticality or an irrational equilibrium from a small floating residual.

An explicit local domain and the offset limits

An energy barrier supplies a sufficient continuous admission neighborhood. Let m=min_edges(pi/2-abs(delta_*)), choose 0<r<m/sqrt(2), and define

\[ c_r=\min_{\{i,j\}\in E}\cos(|\delta_{*,ij}|+\sqrt2r)>0, \quad k_r=\frac{\lambda_2(B)}2\min(1,\beta c_r). \]

Within ||z||<=r, z=(u,v), every interpolated edge gap stays acute because abs((Rv)_j-(Rv)_i)<=sqrt(2)*||v||. Taylor's formula about the critical phase and the graph spectral gap give

\[ \mathcal E_{rel}=E_D(x)+\beta[V_\phi(\theta)-V_\phi(\theta_*)] \ \ge k_r\|z\|^2,\qquad \dot{\mathcal E}_{rel}=-e(Bx)^TND^{-1}Bx\le0. \]

If ||z(0)||<r and E_rel(0)<k_r*r^2, the trajectory cannot first exit this ball. The quotient stays in a compact interior sublevel, its metric remains nonsingular, and the smooth law continues for all future time. The largest invariant set with zero loss has Bx=0. To retain that equality, N*g must be a common vector; 1^T H*g=0 and positive H,N force it to be zero. Sector uniqueness then gives precisely the reference phase shape. LaSalle's argument consequently gives convergence in this sufficient basin; the local Hurwitz result supplies exponential convergence near its limit. These statements do not give a graph-independent decay rate or an error bound for a numerical integrator.

The common form and phase velocities are smooth functions of the quotient state and vanish at its equilibrium. Exponential quotient convergence makes both velocities integrable, so the actual offsets tend to finite limits c_infinity, alpha_infinity. They need not equal their initial values: the native Arg pressure does not generally conserve a form mean, and the phase row does not generally conserve a phase mean. The theorem recovers the prepared geometry modulo those symmetries, not a prescribed absolute offset. The limiting raw storage is beta*V_phi(theta_*), which is positive for a nonzero twist; only the local excess storage tends to zero. Unlike section 3's low-total-storage condition, this local excess-storage barrier can surround a winding equilibrium whose total storage exceeds beta.

Cycle rates, competing laws and excluded boundaries

For a unit C_n with common capacity nu>0 and uniform twist kappa=2*pi*ell/n, K_*=cos(kappa)*B and H_*=pi*D*cos(kappa). Each nonzero random-walk Laplacian mode lambda satisfies

\[ \boxed{\quad s^2+e\nu\lambda s+ \frac{\nu^2w^2\lambda^2}{\beta\pi^2\cos\kappa}=0.\quad} \]

At phase consensus this reproduces section 5. With e=w=1/2, beta=nu=1, winding-one C5 has damped oscillatory linear modes, whereas winding-one C6 has two negative real roots per mode. Those are predictions of this joint law, not the relaxation rates of the earlier supplied sine law and not a claim about particles or a physical clock.

For positive heterogeneous capacities on the same cycle, replace nu*lambda in this polynomial by an eigenvalue mu of T=(R^T N R)*B_q/2. This matrix is similar to a symmetric positive definite matrix, and mu_min>=nu_min*lambda_2(B)/2. All modes still have the same damped/critical/oscillatory class. For winding-one C5 with e=w=1/2, beta=1 and nu_min=3/4, their real parts satisfy Re(s)<=-3*lambda_2(B)/32. This linear spectral bound does not remove the nonlinear neighborhood restriction or supply an Euler error estimate.

The section 12 countermodel has the same equilibria and linearization here. Indeed its skew correction is Z=S*grad(V_phi), with S=0 at uniform form and grad(V_phi)=0 at the phase equilibrium; locally Z=O(||u||*||v||). Its ideal storage balance is unchanged. Thus it too has the stated local recovery: stability alone cannot select the independent capacity premise. The finite capacity-intervention discriminator remains necessary to separate their nonlinear responses.

The explicit acute-basin theorem has concrete limits. At e=0 its local positive excess storage is conserved, so a nonzero quotient perturbation cannot converge to the reference equilibrium family; with positive quotient stiffness the linearized nonzero modes are purely imaginary. If w=0, phase is frozen and arbitrary phase damage cannot recover (that boundary is outside the admitted production model). Zero capacities require new observability conditions: for example, two inactive nodes with distinct frozen forms prevent convergence to common form, however small their initial difference. That explicit basin does not cover nonacute equilibria; the separate regular-stiffness theorem above supplies a local criterion for them. Neither result covers changing capacity/support, forcing, operator events, global attraction across winding sectors or indefinite stability of finite Euler execution.

Bounded C5 recovery control and first retained evaluation

The shared-engine instrument uses one supplied winding-one C5 with e=w=1/2, beta=1 and held capacities (1,5/4,3/4,3/2,1). At epsilon=2^-8, prepare

\[ x(0)=\epsilon(1,-1,0,0,0),\qquad \theta_i(0)=2\pi i/5+\epsilon(0,1,-1,0,0)_i. \]

The ideal preparation lies in the sufficient continuous basin before any trajectory is computed. Take r=pi/(20*sqrt(2)) in the preceding barrier. Then c_r=sin(pi/20)>1/10, lambda_2(B)=(5-sqrt(5))/2>1, and k_r*r^2>9/16000. The initial quotient norm is 2*epsilon=1/128<r. Using the unweighted graph Hessian as an upper bound on the cosine Hessian along the initial perturbation gives E_rel(0)<=6*epsilon^2=3/32768<9/16000. This is an ideal continuous admission argument, not a floating-point trajectory enclosure.

The prediction was frozen before the two positive-capacity executions. Its finite hypotheses are: at structural time T=32, joint quotient distance is at most half its initial value on both grids; endpoint grid separation is at most 1/64 of the initial distance; the acute margin stays at least pi/20; actual work residual stays at most 1e-12; and support, capacities, pressure path and winding one are retained. Grids use dt=1/64 and 1/128. These tolerances are numerical acceptance policies, not new dynamics or rigorous ODE error bounds. The common form and phase offsets are retained separately.

The response, first evaluated on 2026-09-27, passed every frozen criterion without retuning:

Euler steps	Initial joint distance	Final joint distance	Final / initial
2048	0.0078125	3.8753314601e-5	0.0049604243
4096	0.0078125	3.9089447971e-5	0.0050034493

The two endpoints differ by 3.9732190974e-7 in the quotient, approximately 5.086e-5 of the initial perturbation. Both traces retained winding one and minimum acute margin 0.3102530154 radians. Maximum observed actual work residual was about 1.008e-18. No positive storage increment was observed; nonzero Euler storage defects remain in the record and are distinct from instantaneous work residuals.

On the fine grid the final form mean is 0.00016167095, and the common phase-offset change is approximately -0.00042482317. The relative geometry recovers while these common offsets move, as the theorem permits. Neither the final distance nor these finite offsets are asserted to be exact limits. The same perturbed state with all capacities zero stays exactly frozen in one shared engine step despite nonzero pressure; the declared clock advances. That control distinguishes inactive retention from restorative dynamics.

The controls compare the production tangent with independently derived cycle blocks, check common offsets and the zero-capacity boundary, and reuse the saved checkpoints and decisions. They do not regenerate the full two-grid campaign. The artifact contains four checkpoints per trace and summaries over every executed step; saved snapshots alone do not independently authenticate omitted steps.

Disposition. The named F3 local-recovery obligation is complete for this conditional family: a proof covers positive held capacities on the stated acute equilibria, and the finite control exercises the existing integrated law on a nonuniform phase geometry. It adds no solver, stabilizing controller or default policy. Supplied support and winding, formation of that sector, interaction with other regions and physical identification remain distinct.

16. Interaction of two recoverable regions under the same law

Joined support changes the shared geometry, not the evolution rule

Prepare two oriented unit cycles A=(0,1,2,3,4) and R=(5,6,7,8,9) with the fixed bridge (0,5), all capacities one and the same e,w,beta as above. For k=0,...,4, let theta_*[k]=theta_*[k+5]=k*kappa, where kappa=2*pi/5. The bridge endpoints agree. Existing acute circulation classification requires zero phase gap on a bridge at equilibrium; the two ring circulations balance independently. Uniform form and these phases are therefore an equilibrium of the complete joined law.

Only the geometry and balance accounting of the historical contact study are reused. Its supplied sine-law trajectories and maximum principle do not apply to this joint law. There is no support event during the experiment, imposed interaction force, phase driver, selector or retuned pressure. Joining is a supplied preparation.

The joined graph has degrees three at ports 0 and 5, and two elsewhere. Writing c=cos(kappa) and h=1+2*c=(1+sqrt(5))/2, its equilibrium phase metric is H_ports=pi*h, H_other=2*pi*c. Both native diffusion normalization and the phase metric must be evaluated on the joined support. An isolated ring's row cannot be copied to a port after joining.

The disconnected reference admits two independent form offsets and two independent phase rotations. The connected graph admits only one of each: relative regional offsets are now part of the dynamical state. Section 15 supplies local recovery of this joined equilibrium modulo the remaining global symmetries. Prepared regional winding is retained in that local domain; neither the bridge nor the winding sectors are generated by the law.

A transmitted response that changes internal geometry

Perturb only an internal left node: x(0)=epsilon*e_1, with both phases unchanged. The receiver's complete form and phase velocities initially vanish. At the donor port,

\[ \dot x_0(0)=e\epsilon/3,\qquad \dot\theta_0(0)=-w\epsilon/(\beta\pi h). \]

At the receiving port q_5=(Bx)_5=0, so differentiating H_5^-1 supplies no initial term in its phase acceleration. The native pressure derivative uses partial_0 g_5=1/(pi*h) at the prepared lock. Consequently

\[ \boxed{\quad \ddot x_5(0)=\epsilon\left(\frac{e^2}{9} -\frac{w^2}{\beta\pi^2h^2}\right),\qquad \ddot\theta_5(0)=-\frac{ew\epsilon}{3\beta\pi h}. \quad} \]

All other receiver accelerations vanish initially. The form response contains competing diffusion and phase-mediated contributions, both from the existing law. Its sign is not universal over storage scales: for e=w=1/2, beta=1 the displayed form coefficient is positive and the phase coefficient negative.

On consistent local lifts define the receiver's port-versus-rest contrasts

\[ \chi_x=x_5-\tfrac14\sum_{i=6}^9x_i,\qquad \chi_\theta=(\theta_5-\theta_{*,5}) -\tfrac14\sum_{i=6}^9(\theta_i-\theta_{*,i}). \]

These remove a common regional shift and have the same initial accelerations. They therefore detect internal deformation, not merely a translated regional mean. In the ideal disconnected comparison the recipient remains at its equilibrium for all time. Let D_chi be joined recipient minus separately evolved disconnected recipient; its leading coefficients are D_chi(T)=T^2*chi''(0)/2+O(T^3). The order is a smooth local response order, not a finite signal delay or physical propagation speed. Floating-point drift of the disconnected preparation is measured separately rather than assigned the ideal zero value.

The ideal preparation with epsilon=1/256 lies within the sufficient continuous basin before any simulation. The joined graph has ten nodes and diameter five. For zero-mean v, summing pathwise Cauchy inequalities gives 10*||v||^2=sum_(i<j)(v_i-v_j)^2<=45*5*v^T Bv, hence lambda_2(B)>=2/45. With r=pi/(20*sqrt(2)) and c_r>1/10, section 15's barrier exceeds pi^2/360000>1/40000. The preparation has E_rel(0)=epsilon^2=1/65536 and quotient norm epsilon*sqrt(9/10)<r. This proves continuous admission and eventual local recovery, not the finite Taylor band or an Euler error enclosure.

Regional accounting and the information that cannot be discarded

For the fixed receiver, the form total weighted by full-graph degrees has the exact balance

\[ M_R=3x_5+2\sum_{i=6}^9x_i,\qquad \dot M_R=e(x_0-x_5)+w\sum_{i\in R}d_i g_i. \]

The phase row likewise implies

\[ \sum_{i\in R}H_i(\theta)\dot\theta_i =\frac w\beta(x_5-x_0). \]

The latter is an instantaneous weighted rate, not the derivative of a weighted phase total: H changes with phase. The former is an EPI accounting identity, not conservation of physical mass. It does not discard the internal phase source or imply that the ordinary arithmetic mean obeys a pure cut-flux law. The existing regional support observer owns the represented form-total and variance decomposition. Its forcing argument is supplied as F=w*g directly from the phase state, independently of the form rate; native pressure split defects remain separate.

Shared signed-work and regional-rate integration

The engine now retains the exact represented graph gradient q=Bx in field.work, alongside the older floating gradient. Using represented capacity, phase source, phase gradient and materialized model rates, define

\[ D_i=e\nu_iq_i^2/d_i,\qquad J_i=w\nu_iq_i g_i,\qquad F_i=q_i\dot x_i,\qquad P_i=\beta(\nabla V)_i\dot\theta_i. \]

In exact real arithmetic the admitted law has F_i=-D_i+J_i and P_i=-J_i. In the represented engine these are independently retained defects:

\[ \delta F_i=F_i+D_i-J_i =q_i(\nu_i\,\epsilon_i+\eta_i),\qquad \delta P_i=P_i+J_i, \]

where epsilon=pressure_split_residual and eta=nodal_rate_rounding_defect. Thus sum_i(delta F_i+delta P_i)=field.balance_residual exactly. Positive J_i transfers phase storage toward form storage; D_i>=0 is dissipation. P_i uses the retained trigonometric gradient and represented rate, so its defect includes their failure to satisfy the ideal metric/source identity. Exact rational arithmetic here does not certify the trigonometric evaluation error. The existing global loss and work now sum these same nodal arrays; neither rates, pressure nor constitutive premises change.

These are nodal contributions to the derivative of total storage, not derivatives of independently assigned node energies. Regional sums retain that distinction. They sum globally for a partition; overlapping regions count shared nodes repeatedly. Uniform form has q=0, hence zero immediate work and exchange, even if g!=0 generates nonzero form rates. Zero work therefore does not imply equilibrium or absence of a subsequent response.

For arbitrary positive capacities on a supplied region, the two ideal weighted balances generalize the unit-capacity equations above:

\[ Q_R=\sum_{\substack{i\in R\\j\notin R}}A_{ij}(x_i-x_j) =\sum_{i\in R}q_i, \qquad \sum_{i\in R}\frac{d_i}{\nu_i}\dot x_i =-eQ_R+w\sum_{i\in R}d_i g_i, \]

\[ \sum_{i\in R}\frac{H_i}{\nu_i}\dot\theta_i=\frac w\beta Q_R. \]

The shared support owner computes the outward cut once for both balances. The actual represented form rate includes sum_R d_i*epsilon_i and sum_R (d_i/nu_i)*eta_i; subtracting them gives an exact zero accounting residual. The actual weighted phase rate retains its difference from (w/beta)*Q_R. Its weights use the captured H, so it remains an instantaneous rate, not the derivative of a weighted phase total. For the full support the cut is empty. Zero diffusion weight removes the form boundary term without removing phase exchange. A zero capacity inside the region makes divided rates unavailable; all nodal work and undivided cut/model terms remain defined. A zero capacity outside the selected region does not invalidate its divided rates. These local requirements are separate from the older regional variance observer's positive-full-capacity contract.

observe_relational_pattern captures one fresh field, reuses this cut owner and exposes regional work and boundary through Network.relational_pattern and exact JSON projection. No diagnostic chooses an operator, evolves a graph, or closes regional dynamics. The API contract owns field names and availability; static analytic, sign-reversal, exact gradient and retained interaction checks exercise the shared implementation without rerunning a trajectory. This integration adds explanatory accounting, not new evidence for physical identity or another constitutive law.

A concrete obstruction prevents replacing each ring by only its mean and storage. Move the same impulse from node 1 to node 2. The donor mean, form norm, Dirichlet storage, phase geometry, capacities and port form are unchanged. Both receiver accelerations are now zero, because the donor port has zero initial velocity in both rows. Thus identical aggregate observations can have different transmitted responses. A reduced interaction description must retain sufficient internal/port information or justify an appropriate memory term; this example does not prove that every reduction is impossible. The two donor mean derivatives already differ (-e*epsilon/30 versus zero), so a description that retains those rates has additional information. The obstruction concerns the stated instantaneous means/storage, not every augmented observation or a sufficiently resolved boundary history.

The local composition theorem now fixes six regional mean/port observations on this same single-bridge equilibrium. They require ten coordinates for exact first-variation closure; an explicit mixed form/phase counterexample excludes nonlinear closure of that minimal linear observation. The theorem owns its symmetry, rank and coefficient scope; it does not replace the full nonlinear engine state. Its nonlinear regional budget identifies the missing mobility/form covariance: even the same zero outward cut can coexist with different mean phase response. The shared observer retains that contribution, its squared bound and actual rate-rounding defect.

Prospective finite control

The instrument keeps epsilon=1/256, e=w=1/2, beta=1, and structural horizon T=1/16. It compares the joined graph with both disconnected cycles executed separately through the same connected-graph engine owner, at 64, 128 and 256 steps. The prior criteria require each signed contrast divided by its quadratic prediction to lie in [3/4,5/4]; successive signal differences must contract by at least the declared factor 3/4 plus 1e-12, and each finest signal must exceed 32*(last_grid_difference+1e-12). Winding one, an acute margin of at least pi/20, held support/capacities and work residual at most 1e-12 are separate checks. Disconnected receiver drift is bounded by the separately declared 1e-12 numerical gate.

These are prospective numerical hypotheses, not a proved Taylor remainder, an autonomous regional closure or empirical physical evidence. The complete prediction must be frozen before finite-horizon campaign execution; a failed criterion remains a failed result. No prior recovery or capacity-response artifact is changed by this control.

First retained transmission evaluation: 2026-09-27

The frozen prediction preceded every finite-horizon campaign trace. Detached field/Jacobian checks and a one-step Euler implementation control preceded that freeze; none evaluated the reserved finite-horizon response. The retained response contains nine traces (three scenarios on three grids), with initial, midpoint and final checkpoints, runtime/source provenance and numerical decisions. All five gates passed on the first evaluation, without retuning:

Finest-grid contrast, joined recipient minus disconnected recipient	Observed response	Ratio to quadratic prediction	Last grid difference
Form	1.3432193095e-7	0.9725660824	5.0174490386e-10
Relative phase error	-1.2133275087e-7	0.9700781649	4.5079212794e-10

The successive differences approximately halved. Every trace retained its declared support, unit capacities and winding; minimum acute margin was 0.3139754258 radians and maximum actual work residual was 6.661e-19. The largest disconnected-recipient full-state drift was 2.769e-18; it was subtracted as observed rather than assigned zero. These finite checks are neither a full continuous error enclosure nor independent validation of every omitted step from the retained checkpoints alone.

At the fine-grid endpoint, the receiver's degree-weighted form-total rate decomposes into positive cut contribution 1.9770232650e-5, negative phase source contribution -6.8497774475e-6 and native pressure-split defect about 4.203e-17. The existing regional observer checks the exact bookkeeping on represented values. This is a concrete balance of the two transmission channels, not a physical-mass measurement or a new regional evolution law. The ordinary form mean (2.7112836143e-8) and full-graph transport-weighted mean (3.6881703848e-8) differ and retain their distinct meanings.

The controls independently check initial accelerations, the equal-summary obstruction, regional accounting, first-step Euler ordering and frozen-protocol admission. They bind saved preparations and capacities, replay 27 detached checkpoint fields and reconstruct the finite contrasts without repeating the trajectories.

Interaction and formation have different scope. Recovery and transmission use one unchanged conditional law. Acute continuous fixed-support execution preserves cycle winding, so this interaction experiment does not create its supplied winding sectors. The regular-domain analysis separates that execution restriction from genuine constitutive singularities; the validated transit supplies the separate prepared formation result. Neither result selects the initial support or law.