Constitutive freedom in regional identity barriers and work

Matching the organized target, its storage, full tangent, quadratic field and the previously declared registered-composition conditions does not fix a C9 winding boundary's storage minimum or regional boundary power. The admitted analytic family below gives exact counterexamples to both claims. Its true first-boundary minima for winding zero and one vary affinely with the constitutive parameter on one explicit interval. Regional power also varies at one fixed complete state whose regional, contact and complete storage values are unchanged.

These are static and conditional complete-law results. A minimum over a phase-cell boundary is not the work consumed by an actual trajectory, an activation law or a proof that a source reaches that boundary. The frozen sine-source loss is preserved in its own model. The research program remains the sole queue.

Actual complete law and the matched properties

Use the fixed equal-contact support of the receiver model: 27 labelled nodes, three unit C9 regions, and contacts from oriented region pairs \((0,1),(1,2),(2,0)\) with offsets zero and one, each of weight one half. Every node has weighted degree four. All fine form and phase coordinates, region membership and boundary registrations are retained. Capacities are held at one and conductances remain fixed. There are no inputs, resets, support events or operator invocations.

Write \(\alpha=2\pi/9\), \(\gamma=1/(1023\pi)\), and use the structural clock \(\tau=(1023/1024)t\). The parameter \(\lambda\) below is the constitutive parameter of the composition-compatible counterfamily, not the slow-clock factor \(\gamma^2\). Restrict it to

\[ |\lambda|\le E:=1/10000, \qquad P(d)=\cos d(1-\cos9d)^2, \qquad V_\lambda(d)=1-\cos d+\lambda P(d), \qquad j_\lambda(d)=V_\lambda'(d). \tag{1} \]

With the actual weighted Laplacian \(L\), declare both rows:

\[ S_{\lambda,i}(\theta) =\sum_jw_{ij}j_\lambda(\theta_j-\theta_i),\qquad x_\tau=\tfrac14[-Lx+\gamma S_\lambda(\theta)],\qquad \theta_\tau=\tfrac\gamma4Lx. \tag{2} \]

The form row supplies pressure under unit capacity. It is not inferred from an observed form response. In the original clock both rows acquire the factor \(1023/1024\); structural units have no laboratory-unit bridge.

The current is odd, analytic and periodic. Its potential is even and periodic. The bounds proved for the original family give

\[ |P(d)|\le162(1-\cos d),\qquad |P(d)|\le4, \qquad V_\lambda(d)\ge(1-162E)(1-\cos d)\ge0. \tag{3} \]

The five-term Fourier expansion also gives

\[ |P''(d)|\le328, \qquad |j_\lambda'(d)|\le1+328E. \tag{4} \]

Thus the lifted vector field is globally Lipschitz. The fine law retains common form and phase origins, graph relabellings preserving all marked data, and simultaneous form/phase sign reversal. Both global means are conserved; individual regional means need not be.

For the actual support, the complete storage and its loss are

\[ H_\lambda=\tfrac12x^TLx+ \sum_{\{i,j\}}w_{ij}V_\lambda(\theta_j-\theta_i), \qquad (H_\lambda)_\tau=-\tfrac14\|Lx\|^2. \tag{5} \]

This follows directly from \(\nabla_\theta H_\lambda=-S_\lambda\) and both rows of (2). It re-admits the balance for the changed law rather than transferring an evaluated sine trajectory.

At every gap \(k\alpha\), the factor \((1-\cos9d)^2\) has a fourth-order zero. Hence

\[ P=P'=P''=P'''=0\quad\text{at }d=k\alpha. \tag{6} \]

The internal winding-one twist retains its storage, current, tangent and quadratic field. The same is true of the equal-contact joined target \(x_*=0\), \(\theta_{*,r,j}=\alpha(j-4)\): its actual fine gaps are zero or \(\pm\alpha\) modulo a turn. Substituting (6) preserves the fine-node current cancellation from the target proof. This concerns the full 54-dimensional target field. It does not identify the aggregate contact phase lag with a fine-edge gap or transfer the unequal-contact target.

The family also retains its previously proved profile-preservation properties under the matching and three-offset interfaces defined by its classification owner. That result does not assert profile preservation under every interface, including the two-offset contacts used here. Nor do the matching target data transfer source acquisition, nonlinear transport, capture radii or event contracts. This remains a detached complete-law comparison under the NFR admission, without a physical identity or a newly selected fundamental law.

The true C9 first-boundary minima

Retain the principal-branch integers of a marked region at release and continue its registered gaps until their first boundary contact. For initial winding \(w\in\{0,1\}\), define the compact set

\[ \mathcal B_w=\left\{\delta\in[-\pi,\pi]^9: \sum_j\delta_j=2\pi w,\quad \max_j|\delta_j|=\pi\right\}, \qquad B_w(\lambda)=\min_{\delta\in\mathcal B_w} \sum_jV_\lambda(\delta_j). \tag{7} \]

Every point of this set represents a consistent circular C9 phase configuration with the retained winding sum. At least one gap is on the boundary; winding at that boundary is not assigned. Minimization in (7) retains all other gaps and permits simultaneous boundary contacts.

Put \(c=\cos(\pi/8)\). The exact result on the full interval (1) is

\[ \boxed{B_0(\lambda)=B_1(\lambda) =2+8(1-c)+\lambda\beta,\qquad \beta=8c(1+c)^2-4.} \tag{8} \]

In particular, the number \(9/4\) in the regional sine theorem remains a valid conservative bound in its stated sine domain; it was not claimed to be the sharp value now computed at \(\lambda=0\).

To prove (8), select one boundary gap. Its potential is always

\[ V_\lambda(\pi)=V_\lambda(-\pi)=2-4\lambda. \tag{9} \]

If the other eight have sum \(\pi\) or \(-\pi\), their equal-gap candidate has cost

\[ C_\lambda=8V_\lambda(\pi/8) =8(1-c)+8\lambda c(1+c)^2. \tag{10} \]

The following elementary rational bounds will isolate every minimizer:

\[ c>23/25,\qquad 1/4<\cos(5\pi/12)<1/3. \tag{11} \]

For the first, \(\sqrt2>7/5\) gives \(c^2=(2+\sqrt2)/4>17/20>(23/25)^2\). For the second, use \(\cos(5\pi/12)=(\sqrt6-\sqrt2)/4>1/4\), since \(\sqrt6>1+\sqrt2\), and \(\cos(5\pi/12)=\sin(\pi/12)<\pi/12<1/3\).

By (3) and (11),

\[ C_\lambda<16/25+32E. \tag{12} \]

If any of the remaining eight gaps has magnitude at least \(5\pi/12\), its potential alone is at least \(2/3-4E\). The other seven potentials are nonnegative. Thus their total exceeds the candidate by more than

\[ 2/3-4E-(16/25+32E) =2/75-36E=173/7500>0. \tag{13} \]

Every minimizing remaining gap consequently lies in \((-5\pi/12,5\pi/12)\). Throughout that interval,

\[ V_\lambda''(d)=\cos d+\lambda P''(d) >1/4-328E=543/2500>0. \tag{14} \]

Strict convexity and Jensen's inequality now prove that eight gaps with sum \(\pm\pi\) have their unique minimum at eight identical gaps \(\pm\pi/8\). This is a global minimum over the full constrained face, because (13) excluded every omitted large-gap point.

For winding zero, either selected boundary sign leaves the opposite sum \(\mp\pi\), so this proves (8). For winding one, a selected \(+\pi\) leaves sum \(\pi\). The remaining possibility is a selected \(-\pi\) with sum \(3\pi\). The global inequality \(1-\cos d\ge2d^2/\pi^2\), followed by Cauchy--Schwarz and (3), bounds that remaining cost below by

\[ \frac2{\pi^2}\frac{(3\pi)^2}{8}-32E=9/4-32E. \]

Its excess over the candidate in (12) is greater than

\[ 9/4-16/25-64E=4009/2500>0. \tag{15} \]

It cannot minimize. This also excludes configurations with further boundary gaps. The minimizing configurations are therefore one \(+\pi\) and eight \(+\pi/8\) for winding one; for winding zero they have one \(\pm\pi\) and eight \(\mp\pi/8\), with arbitrary choice of the marked boundary edge. Substituting (9)--(10) gives (8).

A fixed constitutive discriminator and the donor/receiver trade-off

The expression for \(\beta\) is increasing in \(c>0\). Equation (11) gives \(23<\beta<28\). In particular the two already declared endpoint laws satisfy

\[ B_w(1/10000)-B_w(-1/10000)=\beta/5000>23/5000. \tag{16} \]

This is an exact unequal-minimum result, not a sampled comparison or a comparison of arbitrary boundary-state energies. It refutes selection of the regional first-boundary minimum by the matched target and registered composition properties alone.

For a zero-form winding-one internal twist, the internal baseline remains \(9V_\lambda(\alpha)=9(1-\cos\alpha)\). For a flat zero-form receiver it remains zero. Their geometric storage differences to the first boundary are therefore

\[ R_D^*(\lambda)=B_1(\lambda)-9(1-\cos\alpha),\qquad R_R^*(\lambda)=B_0(\lambda). \tag{17} \]

Both have the same slope \(\beta\), and \(R_R^*-R_D^*=9(1-\cos\alpha)\) is unchanged. Raising this common barrier increases both donor headroom and the receiver's necessary entry storage. It does not selectively guarantee transmission with retention.

These baselines are ideal internal states, not source-release certificates. For independently established bounds \(\|x_r\|\le X\) and \(\|\theta_r-\vartheta\|\le b\), with \(\vartheta=\alpha(j-4)\) or zero, (4) and the stationary internal gradient do give the own-law static upper bound

\[ H_{\lambda,r}\le U_{\lambda,r}(\vartheta) +2X^2+2(1+328E)b^2. \tag{18} \]

The cycle Laplacian bound \(L_I\preceq4I\) proves both terms. Establishing that a changed-law trajectory reaches such a state is a separate obligation; the sine source bounds, velocity defect, release time and capture margins cannot supply it. A boundary minimum also omits positive form storage, accumulated loss, contact work and the direction of passage.

Regional work changes at identical complete state and storage

For each region retain the notation of the regional balance, now using the actual changed current:

\[ \ell_r=L_{I,r}x_r,\quad b_r=(L_Cx)_r,\quad s_{\lambda,r}=(S_{\lambda,I})_r,\quad c_{\lambda,r}=(S_{\lambda,C})_r. \]

Direct differentiation of the internal storage gives

\[ \begin{split} H_{\lambda,r}&=\tfrac12x_r^TL_{I,r}x_r +\sum_{\text{internal }e}V_\lambda(\delta_e),\\ D_r&=\tfrac14\|\ell_r\|^2,\\ P_{\lambda,r}&=-\tfrac14\ell_r\cdot b_r +\tfrac\gamma4(\ell_r\cdot c_{\lambda,r} -s_{\lambda,r}\cdot b_r),\\ (H_{\lambda,r})_\tau&=-D_r+P_{\lambda,r}. \end{split} \tag{19} \]

The internal reciprocal terms cancel by (2); the contact terms retain their state dependence. Including the contact storage recovers (5). The first-exit proof applies with the actual \(B_w(\lambda)\) and the actual integrated power in (19). Neither the sine power history nor a larger static barrier determines the changed-law history.

There is an exact static witness on the same weighted support. Number nodes \(9r+j\), and choose

\[ x=e_9,\qquad \theta=(\pi/2)e_0. \tag{20} \]

Subtracting the common form and phase means puts this witness on the global mean-zero leaf without changing any following value. Every consumed phase gap is zero or \(\pm\pi/2\). At these gaps the potentials agree with sine, while

\[ j_\lambda(\pi/2)=1-\lambda, \qquad S_\lambda=(1-\lambda)S_0. \tag{21} \]

The exact internal storage, internal loss, contact storage and total loss are independent of \(\lambda\):

\[ (H_{\lambda,0},H_{\lambda,1},H_{\lambda,2})=(2,1,0), \quad (D_0,D_1,D_2)=(0,3/2,0), \quad H_{\lambda,C}=3, \quad H_\lambda=6, \quad D_{\rm all}=19/4. \tag{22} \]

For example, the sole nonzero form node has degree four, so its complete form storage is two. Its Laplacian column has squared norm \(4^2+2\cdot1^2+4\cdot(1/2)^2=19\). The phase node has two internal and two units of contact conductance, giving internal phase storage two and contact phase storage two. The remaining contact storage is form.

Nevertheless, evaluation of (19) gives

\[ (P_{\lambda,0},P_{\lambda,1},P_{\lambda,2}) =\left(-\frac\gamma8(1-\lambda), -1+\frac\gamma8(1-\lambda),0\right), \qquad P_\lambda-P_0=(\gamma\lambda/8,-\gamma\lambda/8,0). \tag{23} \]

For region zero, \(\ell_0=0\); its contact form gradient is \(-1/2\) at nodes zero and eight, and its internal current is \((1-\lambda)(-2,1,0,\ldots,0,1)\). Their dot product gives the first entry. The other entries follow from the same retained contacts. The regional storage rates sum to \(-5/2\), and contact storage has rate \(-9/4\), yielding the unchanged total rate \(-19/4\).

Thus identical complete state, every regional and contact storage, and complete instantaneous loss do not determine regional power across this admitted constitutive class. The sign of the difference in (23) changes with \(\lambda\); this does not claim that the regional power itself changes sign. The witness is a supplied mathematical state, not a state shown reachable from the original two-ramp source.

The branch-normal velocity at any identical complete state remains \(\gamma b_e^TLx/4\), independently of \(\lambda\), because the phase row is unchanged. The changed current instead changes the form row and subsequent evolution. A different barrier or instantaneous work value therefore does not by itself establish a different crossing outcome.

These counterexamples identify nonlinear constitutive information left free by the matched target and composition requirements. They neither select a preferred member of the family nor transfer the evaluated sine source to it. Acquisition, donor maintenance, complete joined capture and physical identification retain their separate obligations.

Implementation and independent controls

The boundary assessment rebuilds the primitive constitutive law before evaluating (8), its strict proof margins and the ideal donor target margin. Laws outside (1) but inside the family's admitted interval remain valid laws; this narrower theorem returns unavailable minima for them. It imports no release or source report. The independent controls evaluate the factored potential, actual equal-network target gaps, boundary configurations and the inequalities consumed by the global proof.

The shared regional observer accepts the entire already admitted family, \(|\lambda|\le1/324\), using the single potential/current owner. It retains the admitted law and both complete rows. Its default zero member preserves the sine arithmetic; the separate source-retention assessment remains sine-only. Signed form follows shared scalar materialization, while primitive phase lifts retain their admitted represented values. A rounded quarter turn is not the exact state in (20). The regional controls therefore use a represented nonzero angle and independently evaluate its actual potential/current. The shared exact-phasor controls check (21). None of these controls evaluates a trajectory or regenerates frozen evidence.