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Forced support balance and model boundaries

Fixed-support form balance is the common starting point. Regional response, phase clocks, locking and native writer results retain separate complete laws. Source-bound executed comparisons are archived; mathematical restrictions remain maintained.

Chapters and scope

Chapter Responsibility
Retained forced-support regional observations Frozen finite regional comparisons and their original preparation and numerical limitations.
Forced regional response and environmental readouts Regional balances, represented phase reduction and environmental-input geometry.
Forced-source closure and phase clocks Phase/capacity tangency, source constraints and the limits of synchronization as a clock.
Conditional phase locking and circulation geometry P2 and network locking, capacity adaptation, circulation and cycle periods.
Winding generation and native writer boundaries Acute winding retention, chart escape and native Mutation/capacity writer obstructions.
Geometric identity and capacity-feedback bounds Local restoring response, geometric identity and prospective capacity tubes.

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

1. The reference comes from the existing nodal channels

The connection studied in THOL birth and transport changes both the weighted EPI walk and the unweighted capacity/phase neighborhoods. Its next evolution can be analyzed by retaining that connected support, positive capacities and phases, then refreshing the canonical pressure before each declared physical step.

Let W be fixed symmetric nonnegative conductance, connected through its positive off-diagonal entries. Let d_i=sum_j W_ij>0, D=diag(d_i), B=D-W, and nu_i>0. Loops follow the shared conductance convention and zero-weight support edges contribute no EPI transport. The existing normalized EPI channel coefficient is e>0. Write the held remaining canonical channels as

\[ F=w_{\nu}g_{\nu}+w_{\phi}g_{\phi} +w_{\mathrm{topo}}g_{\mathrm{topo}}. \]

Here g_nu and g_topo use the actual unique-neighbor support; g_phi uses the canonical neighbor-phasor argument. Holding these source fields and their channel coefficients makes F constant. It is a decomposition of the existing pressure, with no additional feedback coefficient. Its phase component need not be a linear phase Laplacian.

The nodal equation on this retained state is therefore

\[ \dot x=\operatorname{diag}(\nu)(-eD^{-1}Bx+F). \]

An exact rational companion can treat finite materialized coefficients as exact real inputs. A live pressure refresh generally differs from this reference by a recorded numerical residual. Neither a stored pressure value nor an operator label establishes that the other channels have been independently identified or held fixed.

2. Compatibility, mean drift and the unique relative profile

Define the positive metric vector h_i=d_i/nu_i, its diagonal matrix H=diag(h_i), total weight Z=sum_i h_i, and weighted mean

\[ m(x)=\frac{h^\top x}{Z}. \]

Symmetry gives 1^T B=0, hence

\[ \frac{d}{dt}(h^\top x)=\mathbf1^\top DF=:b, \qquad \dot m=\bar v:=b/Z. \]

The initial mean is unrestricted. Its exact reference trajectory is m(t)=m(x0)+bar_v*t; the mean is conserved only when b=0.

A zero-pressure stationary EPI exists precisely when

\[ b=\sum_i d_iF_i=0. \]

Indeed its equation is e*B*x=D*F, and the range of a connected symmetric Laplacian is the subspace orthogonal to 1. Positive capacity makes zero rate equivalent to zero pressure in this fixed model. Zero capacity, disconnected positive conductance and changing channels require different conclusions.

For every F, including b!=0, there is a unique centered relative profile z satisfying

\[ eBz=DF-\bar v h,\qquad h^\top z=0. \]

The right-hand side sums to zero by the definition of bar_v, so a solution exists and is unique after fixing its weighted mean. A useful exact computation is the nonsingular rational system

\[ (eB+hh^\top)z=DF-\bar v h. \]

Multiplication by 1^T enforces h^T z=0; the remaining equation is the original profile equation. The rank-one term enforces the algebraic choice of mean and is not part of the nodal pressure or evolution. The existing exact matrix inverse suffices for this bounded solve.

Let

\[ A=e\operatorname{diag}(\nu_i/d_i)B, \qquad u=x-m(x)\mathbf1-z. \]

Then h^T u=0 and direct substitution gives

\[ \dot u=-Au, \qquad x(t)=[m(x_0)+\bar v t]\mathbf1+z+e^{-At}u(0). \]

This separates a retained spatial form, uniform mean drift and relaxing deviations without adding an independent dynamical law. A nonzero z alone does not establish localization or a zero-pressure equilibrium. On the exact drifting profile the pressure is bar_v/nu_i, while the raw Dirichlet energy stays at z^T*B*z/2 because B*1=0. A constant spatial energy therefore does not establish stationarity.

3. Exact relaxation and the finite-chart boundary

The identity H*A=e*B makes A self-adjoint in the positive H metric. Its only zero mode on connected conductance is the uniform vector. The centered error variance and Dirichlet error satisfy

\[ V(u)=\tfrac12u^\top Hu, \qquad \dot V=-e\,u^\top Bu\leq0, \]
\[ E(u)=\tfrac12u^\top Bu, \qquad \dot E=-e(Bu)^\top\operatorname{diag}(\nu_i/d_i)(Bu)\leq0. \]

The positive nonuniform spectrum implies u(t)->0 in the exact held model. Thus the relative profile attracts every initial field after matching its initial weighted mean. When b!=0, the complete EPI field continues to drift and does not converge to a stationary state.

This asymptotic statement assumes an unrestricted scalar chart with the same fixed coefficients for all times. If every EPI coordinate must stay within finite bounds [l,r], its positively weighted mean must also stay in that interval. The ideal affine mean therefore cannot remain inside the chart beyond

\[ t>\frac{r-m(x_0)}{\bar v}\quad(\bar v>0), \qquad t>\frac{m(x_0)-l}{|\bar v|}\quad(\bar v<0). \]

Individual nodes may reach a bound earlier; the mean does not control every coordinate. Even b=0 does not ensure that the limiting profile lies inside the configured chart. Clipping changes the endpoint map and contributes a measurable defect. A clipped stationary endpoint need not have zero canonical pressure.

4. Finite refreshed Euler observations and their exact defects

For one declared duration dt>=0, let p be the captured pressure before integration and x_plus the observed endpoint. Define

\[ p_*(x)=-eD^{-1}Bx+F, \qquad \varepsilon_p=p-p_*(x), \]
\[ \delta_x=x_+-x-dt\operatorname{diag}(\nu)p, \qquad \xi=dt\operatorname{diag}(\nu)\varepsilon_p+\delta_x. \]

The first defect measures pressure realization relative to the declared held model. The second measures the implemented endpoint relative to its actual held pressure, including arithmetic, clipping or a different solver. These definitions give the exact finite identity

\[ x_+=(I-dtA)x+dt\operatorname{diag}(\nu)F+\xi. \]

Consequently the mean budget is

\[ m(x_+)-m(x)-dt\bar v =\frac{dt\,d^\top\varepsilon_p+h^\top\delta_x}{Z}. \]

Neither defect is inferred from the other. Their signed mean contributions can cancel, so a small mean error alone is insufficient evidence of an accurate pressure realization or endpoint.

Let P_H=I-1*h^T/Z, Q=I-dt*A, and chi=P_H*xi. Centering the observed endpoint by its own weighted mean gives

\[ u_+=Qu+\chi, \qquad \chi=dtP_H\operatorname{diag}(\nu)\varepsilon_p+P_H\delta_x. \]

The exact Dirichlet error budget is

\[ \begin{aligned} E(u_+)-E(u) ={}&-dt\,e(Bu)^\top\operatorname{diag}(\nu_i/d_i)(Bu)\\ &+\tfrac12dt^2(Au)^\top B(Au)\\ &+(BQu)^\top\chi+\tfrac12\chi^\top B\chi. \end{aligned} \]

The first term is the dissipative nodal contribution. The next is the finite Euler correction. The remaining terms retain the signed interaction with the observed defect and its quadratic contribution. They must not be discarded merely because the underlying continuous reference is dissipative.

Likewise the weighted variance has the exact identity

\[ V(u_+)-V(u) =-dt\,e\,u^\top Bu+\tfrac12dt^2(Au)^\top H(Au) +(Qu)^\top H\chi+\tfrac12\chi^\top H\chi. \]

Over a finite sequence with the same held model, each mean identity and each energy identity telescopes. Centered errors also propagate through the complete matrices Q_k, so defects need not stay in one spatial eigenmode. These are finite observations; a future error bound or a runtime asymptotic theorem needs additional uniform hypotheses.

For the ideal Euler map, the sufficient step condition dt*e*max(nu)<=1 places the spectrum of dt*A in [0,2]. Hence neither V nor E increases when chi=0. Equality can preserve an alternating mode on an even cycle; nonincrease is not automatically strict decay. The observed defect terms remain necessary under this step condition.

5. Reuse, realization and the evidence boundary

forced_support.py supplies derive_forced_support_balance, observe_forced_support_state and observe_forced_support_step. They rebuild public cached fields before using them and check the held node order, conductance, unique support and capacities. Their detached records contain no execution seal. The profile computation reuses _exact_linear_algebra.py. It introduces no alternative EPI integrator. The finite error budget reuses support_transport.py on detached error coordinates with the same conductance and capacity. Its reference error pressure is -e*D^-1*B*u, and its endpoint defect is exactly chi. This is an algebraic observation of transformed data; the error coordinate is not written into live nodes or interpreted as a newly created physical node.

The held non-EPI source must be obtained from the actual canonical channels independently of a fitted pressure residual. In particular, the nonlinear phase contribution must use the shared phasor kernel, retain its represented coefficients and distinguish its realization from the subsequent full multichannel pressure arithmetic. A direct operator pressure write or a stale stored value belongs in the pressure defect and must not silently redefine F.

capture_non_epi_forcing materializes the unit phase channel with the existing fused NumPy kernel, then combines its represented values with the exact support capacity/topology gradients and the effective channel coefficients. It reads the engine's actual cached weight mix without normalizing it again. The fresh full-kernel pressure defect and the stored-minus-fresh pressure residual are reported separately. Their sum is epsilon_p in the finite budget above.

This bridge is restricted to the default NumPy, non-JIT pressure branch with at most 100 directed unique-support entries. It rejects a disabled NumPy branch or a custom pressure callback. Neighbor insertion order and zero-weight phase-support edges are preserved. The materialized phase values define exact reference coefficients; the observer does not prove exact transcendental evaluation, identify another backend or validate future graph states.

The bounded experiment begins with the causally born and canonically connected child from the preceding block. It retains the resulting support, capacities and phases during explicitly refreshed Euler segments. A separately prepared compatible control uses uniform capacity and aligned phase on the same support. A separate prepared boundary control starts near a scalar EPI bound. These comparisons distinguish nonzero forcing, homogeneous relaxation and clipping without changing the birth history of the causal case.

The tests must reject a claimed zero-pressure profile when b!=0, verify the profile equation and its centered gauge, and account for both pressure and endpoint defects at every captured step. They must also keep a clipped mean change distinct from dt*bar_v and preserve all fixed-support assumptions. Agreement over the measured horizon establishes neither spontaneous selection of the held fields nor restoration under later structural events. Those require an admitted policy that actually changes the support, capacities or phases.

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