Constitutive robustness of the source-specific winding selector

Every member of the declared matched-potential family acquires the same winding-one identity from the original uncertain Cayley corridor. This is a uniform statement for all preparation parameters \(s\in[3/4,4/5]\), all admitted latent errors and all fixed constitutive parameters \(\lambda\in[0,1]\). It is proved from each law's own source transport, current ordering, curvature and full storage. Matching the targets or their tangents alone would not imply this result.

The sine selector supplies the two-stage argument. The constitutive family supplies the alternative complete laws. This owner joins those mechanisms only after re-admitting their hypotheses. No trajectory, source sweep, coefficient producer or frozen response is evaluated.

Complete laws and the retained identity

Use one isolated unit C9, signed form \(x\), primitive phase lifts \(\theta\), degree matrix \(D=2I\), unnormalized Laplacian \(L\) and \(A=L/2\). Hold capacities at one and retain the same structural clocks and coefficients:

\[ e=1023/1024,\qquad \tau=et,\qquad \gamma=1/(1023\pi),\qquad \eta=\gamma^2. \tag{1} \]

For each fixed \(\lambda\in[0,1]\), the potential and current are

\[ \begin{aligned} V_\lambda(\delta)&=1-\cos\delta+\lambda W(\delta),& W(\delta)&=\frac{(1-\cos9\delta)^2}{6\,9^4},\\ j_\lambda(\delta)&=\sin\delta+ \frac{\lambda}{2187} \left(\sin9\delta-\tfrac12\sin18\delta\right),& f_{\lambda,i}(\theta)&=\tfrac12\sum_{j\sim i} j_\lambda(\theta_j-\theta_i). \end{aligned}\tag{2} \]

The actual complete rows and storage balance are

\[ \begin{aligned} x'&=-Ax+\gamma f_\lambda(\theta),& \theta'&=\gamma Ax,\\ H_\lambda&=\tfrac12x^TLx+ \sum_{\{i,j\}}V_\lambda(\theta_j-\theta_i),& H_\lambda'&=-\tfrac12\|Lx\|_2^2. \end{aligned}\tag{3} \]

Primes here denote \(\tau\). Both original-time rows are multiplied by \(e\), as is the loss. The first row independently supplies the pressure for the unit-capacity nodal equation. Both ordinary means remain zero. Support, capacity, loss, clock and source are supplied premises; there are no events, controllers, coefficient changes or state resets.

The organization is the same winding identity as in the sine proof: full constituent form and phase remain available, and a phase cell with an admitted full-storage margin is forward trapped. The conclusion concerns acquisition on the supplied substrate. It does not derive constituent birth, a support selector, an autonomous preparation, a collective event or realization of the thirteen operator contracts. The execution path is a detached continuous-law proof, not an operator word or the native relational runtime.

Sufficient constitutive conditions for the mechanism

For an even periodic potential \(V\) and odd current \(j=V'\), the same reciprocal rows admit the mixed coordinate

\[ z=\gamma x,\qquad y=\theta+z,\qquad \sigma=\eta(\tau-200),\qquad y_\sigma=f(y-z)=f(y)+r. \tag{4} \]

This is an identity on the actual full solution; \(y\) is not a replacement primitive phase. The following are sufficient conditions, not claimed necessary properties of every selector or every TNFR law:

1. Complete source transport. A globally admitted field with \(\|f\|_D\le F\), \(D\)-Lipschitz constant \(L_f\), the declared spectral gap and the original correlated source gives a uniform future form bound \(Z\), hence \(\|r\|_D\le\varepsilon=L_fZ\). Its own mixed initialization must satisfy the source cone below.
2. Passage ordering. The current increases on the ordinary gap interval. With \(k(q)=j(\pi-q)\), its actual closing current, require \(j(m)-k(q)>3/10+\varepsilon\) whenever the proof has \(-7/4\le q\le63/100\), \(m\ge0.899+q/2>0\), and the admitted ordinary gaps. This closes the cone and bounds the passage time.
3. Finite contraction. On the subsequent chart \(|\delta|\le7/5\), require \(V''\ge\mu>0\), an admitted contraction rate \(\alpha\le\mu/5\), and enough finite time to lower the full state storage. The bounded residual must keep both \(y\) and the actual \(\theta=y-z\) in this chart during that stage.
4. Permanent storage protection. The actual reciprocal loss must hold, with an upper bound on complete reached storage below every face of the enclosing winding-one cell. Uniqueness of its critical relative phase geometry must be proved for this current. Positive curvature over the entire open acute cell is one sufficient condition for that uniqueness.

The retained source correlations, both means and the full form energy are part of these conditions. The following sections verify them uniformly for (2). They do not infer them from equilibrium agreement.

Own-law transport of the original uncertain source

Keep the exact Cayley preparation, including \(M=512\), \(m=(2046/9)(355/113)^2\), \(h=e_3-e_4\), \(r_j=j-4\), \(a=(9M+M^{-1})h+r\), and

\[ x_s(0)=Q_s(ma+\xi),\qquad \theta_s(0)=\zeta, \qquad s\in[3/4,4/5]. \tag{5} \]

The two latent vectors have separate exact zero sums and coordinate bounds \(\rho_x=\rho_\theta=10^{-10}\). The form residual is \(Q_s\xi\), not an independently chosen box after transport. The full sixteen-dimensional relative source neighborhood and its zero-error member are included at every fixed \(s\).

The exact \(L\)-isometry gives equal initial form storage along this path for each latent realization. Since the same \(\zeta\) occurs for every \(s\), complete initial storage is equal along the path within each law. Storage across different potentials need not be equal. The old source upper bound acquires its own additional term:

\[ B_{\lambda,\mathrm{source}}= \frac12\sum_{\{i,j\}} (m|a_j-a_i|+2\rho_x)^2 +18\rho_\theta^2+6\lambda\rho_\theta^4 <320680599854722<10^{15}. \tag{6} \]

Indeed \(0\le W(v)\le v^4/24\) and \(|\zeta_j-\zeta_i|\le2\rho_\theta\) on nine edges. This positive quartic term is retained even below the interval grid scale. Every source has initial winding zero, independently of the chosen law.

For a uniform bound over \(0\le\lambda\le\Lambda\le1\), put

\[ a_\Lambda=\Lambda/1458,\qquad u_\Lambda=1+2\Lambda/243,\qquad F_\Lambda=\sqrt{18}(1+a_\Lambda). \tag{7} \]

Direct real derivative bounds for (2) give

\[ |j_\lambda-\sin|\le a_\Lambda,\qquad |j_\lambda'|\le u_\Lambda,\qquad \|f_\lambda\|_D\le F_\Lambda,\qquad L_f\le2u_\Lambda,\qquad \nabla^2U_\lambda\preceq u_\Lambda L. \tag{8} \]

They give global existence and uniqueness in the retained real lifts. The graph semigroup estimate \(e^{-\tau/5}\) is law independent. The raw error norms are still \(E_x=\sqrt{18}\gamma(1025/1023)\rho_x\) and \(E_\theta=\sqrt{18}\rho_\theta\); the enlarged forcing constant does not multiply those independent source errors.

Let \(N\) be the exact-source nominal norm bound and \(\rho\ge e^{-40}\), rebuilt as in the path owner. At \(\tau_0=200\), and uniformly for every later time, variation of constants gives

\[ \begin{aligned} R_{z,\Lambda}(200)&=\rho(N+E_x)+5\eta F_\Lambda,\\ Z_\Lambda&=\max\{R_{z,\Lambda}(200),5\eta F_\Lambda\},& \varepsilon_\Lambda&=2u_\Lambda Z_\Lambda. \end{aligned}\tag{9} \]

Integrating the exact mixed row instead gives

\[ y(200)-\gamma mQ_sa =\zeta+\gamma Q_s\xi+ \eta\int_0^{200}f_\lambda(\theta(v))\,dv, \qquad E_{y,\Lambda}=E_x+E_\theta+200\eta F_\Lambda. \tag{10} \]

There is no extra \(R_z\) in (10): the final fast form cancels exactly in this mixed identity. It remains fully present in (9) and later storage.

At \(\Lambda=1\), outward rational arithmetic gives

\[ Z_\Lambda<2.056\,10^{-6},\qquad \varepsilon_\Lambda<4.145\,10^{-6}<5\,10^{-6},\qquad E_{y,\Lambda}<8.221\,10^{-5}. \tag{11} \]

These are own-law estimates, not sine transit estimates reused as premises.

Passage ordering with the correct closing current

Use the same fixed lift offsets as the sine proof:

\[ \widetilde y=y-2\pi Mh,\quad d_i=\widetilde y_{i+1}-\widetilde y_i\ (0\le i<8),\quad \beta=\widetilde y_0-\widetilde y_8+4\pi,\quad q=\pi-\beta. \tag{12} \]

Thus \(\sum_i d_i-q=3\pi\). The original source proxy and (10) prove, uniformly over the complete family,

\[ 0.3489<q(0)<0.6285<0.63,\qquad \min_i d_i(0)>1.2215,\qquad \max_i d_i(0)<1.2570, \qquad \min_i d_i(0)-q(0)/2>0.9073>0.9. \tag{13} \]

The primitive phase edges differ by at most \(Z_\Lambda\). They therefore have winding two at \(\tau=200\), with the same strict cut margins. This finite-time winding alone is not a capture certificate.

The actual continued closing current is

\[ k_\lambda(q)=j_\lambda(\pi-q) =\sin q+\frac{\lambda}{2187} \left(\sin9q+\tfrac12\sin18q\right). \tag{14} \]

In general it is not \(j_\lambda(q)\). The eighteenth harmonic changes sign under this reflection. The mixed edge equations are

\[ \begin{aligned} d_i'&=\tfrac12[j_\lambda(d_{i-1})-2j_\lambda(d_i) +j_\lambda(d_{i+1})]+r_i,\\ q'&=k_\lambda(q)-\tfrac12[j_\lambda(d_0)+j_\lambda(d_7)]+r_q, \qquad |r_i|,|r_q|\le\varepsilon_\Lambda . \end{aligned}\tag{15} \]

Primes now denote \(\sigma\), and missing endpoint neighbors have current \(k_\lambda(q)\). An edge functional has dual \(D\)-norm one.

Write \(m_d=\min_i d_i\), \(M_d=\max_i d_i\), \(C=m_d-q/2\), and \(\Delta_\lambda=j_\lambda(m_d)-k_\lambda(q)\). Where the ordinary current increases and \(\Delta_\lambda\ge0\), the same endpoint cancellation gives the one-sided inequalities

\[ C'\ge-\tfrac32\varepsilon_\Lambda,\qquad M_d'\le\varepsilon_\Lambda,\qquad q'\le-\Delta_\lambda+\varepsilon_\Lambda. \tag{16} \]

For eight slow units these retain \(C>0.899\), \(m_d>0.024\) and \(M_d<1.301\), until the target \(q_*=-7/4\). The sine comparison in this same cone gives

\[ \sin m_d-\sin q\ge \min\{\sin(1.214)-\sin(0.63),\sin(0.649),\sin(0.5)\}>1/3. \tag{17} \]

Both \(j_\lambda\) and \(k_\lambda\) differ from their sine terms by at most \(a_\Lambda\). Consequently

\[ \Delta_\lambda>1/3-2a_\Lambda,\qquad 1/3-1/729-5\,10^{-6}>3/10. \tag{18} \]

The positive curvature proved below also supplies the ordinary current ordering. The strict cone closes by first exit, and each actual source reaches \(q_*\) at its own time

\[ 0<\sigma_*<\frac{0.63+1.75}{0.3}=119/15<8. \tag{19} \]

Its closing principal mixed gap is then \(7/4-\pi\); the eight other gaps are positive and less than \(1.30004\). Subtracting the retained \(z\)-edges places the actual primitive phase in the winding-one acute cell. This passage is smooth full-state evolution, not a named event. No coincidence of mixed and primitive branch-cut times is asserted.

Own-law curvature, storage and permanent capture

Differentiate the supplied potential:

\[ V_\lambda''(\delta)=\cos\delta+ \frac{\lambda}{243}(\cos9\delta-\cos18\delta). \tag{20} \]

This is strictly positive throughout \(|\delta|<\pi/2\), as follows without extrapolating a target tangent. On \(|\delta|\le3/2\), \(V_\lambda''\ge\cos(3/2)-2/243>0\). On the remaining acute interval, \(17\pi/36<3/2\) ensures \(\cos9\delta\ge0\) and \(\cos18\delta\le0\); the added curvature is nonnegative and \(\cos\delta>0\). Evenness covers negative gaps.

For a further 128 slow units after each actual \(\sigma_*\), the edge maximum principle and (11) keep the mixed gaps within \([7/4-\pi-128\varepsilon_\Lambda, 1.3+136\varepsilon_\Lambda]\). Including the actual phase edge error \(Z_\Lambda\), their absolute magnitudes are less than \(7/5\). At \(\Lambda=1\) the rebuilt bound is less than \(1.392126\). This is a finite first-exit argument for both coordinates, including their connecting segments.

On that smaller chart,

\[ \mu_\Lambda:=\cos(7/5)-2\Lambda/243 >1/6-2/243=77/486>5/32. \tag{21} \]

Combined with the graph quotient gap \(1/5\), this admits the uniform contraction rate \(\alpha=1/32\). The sine rate \(1/30\) is not silently carried into the new law. The target in this same lift is \(\Theta_1=2\pi Mh+(2\pi/9)r\); its mean is zero and its currents cancel for every law. Let \(R=\|y-\Theta_1\|_D\). The same geometric Poincare bound gives \(R(0)<15\) at the passage endpoint, and the actual perturbed mixed row gives

\[ R(v)\le15e^{-v/32}+32\varepsilon_\Lambda, \qquad 0\le v\le128. \tag{22} \]

The positive Taylor sum through degree nine proves \(e^4>54\), so \(R(128)<15/54+32(5\,10^{-6})<0.278\).

The full storage upper bound uses the actual primitive phase and the global own-law Hessian bound (8):

\[ H_\lambda\le H_1^*+ u_\Lambda[R(128)+Z_\Lambda]^2+Z_\Lambda^2/\gamma^2, \qquad H_1^*=9[1-\cos(2\pi/9)]. \tag{23} \]

The target value is unchanged because \(W(2\pi/9)=0\). Since \(W\ge0\), the sine all-face acute boundary bound remains a valid lower bound on this law's own boundary storage:

\[ B_{\lambda,1}\ge B_1=9-8\cos(3\pi/16)>2.348. \tag{24} \]

No equality of boundary minima or later losses between laws is required. Using \(H_1^*<2.106\), \(u_\Lambda<1.009\), \(R+Z<0.279\), \(Z<2.5\,10^{-6}\) and \(\gamma>1/3300\), the storage excess in (23) is below \(0.079\). Thus \(H_\lambda<2.185<B_1\), a strict full-state margin greater than \(0.163\). Tighter outward bounds at \(\Lambda=1\) are

\[ R(128)<0.274868,\qquad H_\lambda<2.181819,\qquad B_1-H_\lambda>0.166424. \tag{25} \]

Full storage nonincrease now traps every member in the acute winding-one cell. Equation (20) proves strict convexity of its potential on the relative lift chart, so the regular winding-one phase is its unique relative critical geometry. On the fixed zero-mean leaf the trapped sublevel is compact. Zero loss forces \(Lx=0\); remaining in the zero-loss set then requires \(f_\lambda(\theta)=0\). LaSalle's argument therefore gives convergence of the actual full state to \((0,\Theta_1)\) and permanent winding \(+1\).

Every member is captured by its own time \(\sigma_*+128<136\). Only the subsequent storage trapping carries that result to a common deadline and all later times:

\[ \tau_{\mathrm{cap}}=200+136/\eta,\qquad t_{\mathrm{cap}}=(1024/1023)(200+136/\eta). \tag{26} \]

The numerical budgets coincide with those of the sine proof because these new bounds suffice, not because the old deadline transfers automatically. They remain conservative structural times without a laboratory unit bridge.

What this robustness establishes and leaves free

The captured identity agrees throughout this already declared law family on the entire original uncertain source corridor. This particular formation observation consequently does not discriminate \(\lambda\). The laws still have different currents, transient histories and higher nonlinear fields. Their nominal distinct-neighbor interaction onset can even have opposite signs. That separate interaction theorem retains its own joined support, nominal preparation, supplied inputs and local time scope; acquisition here does not install its contact or replace reached states with nominal targets.

The result is conditional robustness over a bounded class of complete laws. It establishes neither one attractor shared by all TNFR dynamics, absence of fixed natural laws, unique constitutive selection nor physical identity. The earlier 57-state sine critical census is neither transferred to \(\lambda>0\) nor needed here: convergence follows from this law's own acute Hessian, loss and means after full-storage trapping. Other source-path intervals remain unresolved. Capture at (26) means a protected acute identity cell; it does not assert the older \(1/12\) radius, a specified recovery precision, contact readiness or a closed collective response at that deadline.

The private own-law calculator re-admits primitive deformation/source intervals and error budgets, reuses the shared exact source geometry, and evaluates the bounds above. Its independent controls check current reflection, source errors, uniform margins, full storage, representation boundaries and explicit abstention when sufficient conditions fail. An incoming report, cached verdict or sine transit result cannot replace an actual-law premise. The execution plan alone owns subsequent research admission.