Constitutive inheritance through oriented weighted C9 contacts

Sine is the unique current in the previously admitted analytic class that keeps the same profile, up to the complete chart below, throughout a nonempty open interval of the prescribed oriented contacts. This is a conditional selection theorem. Requiring that unchanged profile is an additional composition premise; the NFR ontology does not derive it. The previous nonsine family still has an exact collective law, with retained boundary-dependent sine and cosine coefficients.

The preceding classification owns the analytic class, its nonnegative potentials, target conditions and two symmetric controls. Here an explicitly supplied weighted fine law adds a distinct orientation-sensitive control. Neither source acquisition, contact identity nor event occurrence transfers to this extension.

Fixed support, complete weighted law and registered identity

Keep two marked C9 components, node order \(9r+\ell\), \(r=0,1\), \(\ell=0,\ldots,8\), with internal unit cycle edges and \(\alpha=2\pi/9\). For a fixed real \(s\in[0,1]\), add the undirected contacts

\[ (0,\ell)--(1,\ell),\quad w=1-s,\qquad (0,\ell)--(1,\ell+1),\quad w=s . \tag{1} \]

Indices are modulo nine. Zero-weight endpoint contacts are absent. Each node has total cross conductance one and final weighted degree three. For \(0<s<1\), there are eighteen cross edges; at either endpoint, nine. The same weight acts at both ends of each undirected edge. The orientation in (1) records the relative marked attachment, not a directed force.

The state is signed real form \(x\), primitive circular phase with declared continuous lifts \(\theta\), held unit capacities and the fixed support including \(s\). The structural clock is \(\tau=(1023/1024)t\). Supply a positive constant reciprocal coefficient \(\gamma\). With the actual weighted Laplacian \(L_s\), set

\[ \begin{split} S_{j,s,i}(\theta)&=\sum_{m\sim i}w_{im}j(\theta_m-\theta_i),\\ x'&=-\tfrac13L_sx+\tfrac{\gamma}{3}S_{j,s}(\theta),\\ \theta'&=\tfrac{\gamma}{3}L_sx,\\ H_s&=\tfrac12x^TL_sx+ \sum_{\{i,m\}}w_{im}V(\theta_m-\theta_i). \end{split}\tag{2} \]

Primes denote \(\tau\)-derivatives. Both rows have the same factor \(1023/1024\) in original time. The form row specifies the driver of the unit-capacity nodal equation; no pressure is inferred from an evaluated response. Structural units obey \([p]=[x]/([\nu_f][t])\); no laboratory unit or inverse-mass interpretation is inferred.

Equation (2) is a supplied weighted extension of the detached reciprocal comparison. It is neither the native pressure rule, whose phase channel uses unweighted unique neighbors, nor the general operator runtime. Its fixed weights are used consistently in form diffusion, phase current, the reciprocal phase row and storage. Oddness of \(j\), evenness of \(V\) and \(V'=j\) give

\[ H_s'=-\tfrac13\|L_sx\|_2^2,\qquad (\mathbf1^Tx)'=(\mathbf1^T\theta)'=0 . \tag{3} \]

The candidate registered identity is the invariant embedded family

\[ \mathcal M:\quad x_{r,\ell}=X_r,\qquad \theta_{r,\ell}=\Phi_r+\alpha(\ell-4). \tag{4} \]

Constituent membership, cyclic orientation, marked ports, relative phase registration and the weight \(s\) remain part of the description. The uniform phase-origin symmetry adds the same angle to every \(\Phi_r\); it does not erase their difference. A simultaneous cyclic relabeling preserves the contact rule. Reversing a constituent order or exchanging the two components also transforms the marked contact offsets. Holding the offsets unchanged while performing such a transformation need not be a symmetry.

The first phase resultant of a winding-one C9 vanishes, and its ninth moment loses registration modulo \(\alpha\). Neither is a substitute for the lifted \(\Phi_r\) with its known marked embedding. This note makes no claim that (4), or any particular attached equilibrium, is acquired or transversely stable for every admitted law.

Exact collective rows and retained boundary coefficients

The admissible fine current from the preceding classification is

\[ \begin{gathered} j(\delta)=\sum_{n\in\mathcal N}b_n\sin(n\delta),\qquad V(\delta)=\sum_{n\in\mathcal N}\frac{b_n}{n}(1-\cos n\delta)\ge0,\\ \mathcal N=\{n\ge1:n\equiv1\text{ or }-1\pmod9\},\qquad \eta_n=\begin{cases}1&n\equiv1\pmod9,\\-1&n\equiv-1\pmod9,\end{cases}\\ \sum_n b_n/n=1,\qquad \sum_n nb_n=1,\qquad \sum_n\eta_n n^2b_n=1 . \end{gathered}\tag{5} \]

The real coefficients decay exponentially. This is the previous real analyticity assumption and justifies the termwise operations below. It matches potential values, complete tangent and quadratic field at the earlier targets; it does not require equality of every target edge current. Nonnegative potential is a separate global premise.

Write \(u=X_1-X_0\), \(\Delta=\Phi_1-\Phi_0\), and define

\[ \begin{split} J_s(\Delta)&=(1-s)j(\Delta)+s j(\Delta+\alpha),\\ U_s(\Delta)&=(1-s)V(\Delta)+s V(\Delta+\alpha),\\ a_s&=1-s+s\cos\alpha,\qquad d_s=s\sin\alpha . \end{split}\tag{6} \]

Internal cycle currents cancel at each node. A component-zero node sees gaps \(\Delta,\Delta+\alpha\). At component one the same undirected contacts give the negative of that weighted current. Thus the exact rows on (4) are

\[ \begin{array}{ll} X_0'=\tfrac13[u+\gamma J_s(\Delta)],& X_1'=-\tfrac13[u+\gamma J_s(\Delta)],\\ \Phi_0'=-\tfrac{\gamma}{3}u,& \Phi_1'=\tfrac{\gamma}{3}u . \end{array}\tag{7} \]

Periodic analytic currents have bounded derivatives on the real circle, so the lifted fine vector field is globally Lipschitz. The tangent calculation and uniqueness therefore prove invariance of (4).

For every law (5), not only sine, the current and potential close exactly:

\[ \begin{split} J_s(\Delta)&=\sum_n b_n [a_s\sin(n\Delta)+\eta_n d_s\cos(n\Delta)],\\ U_s(\Delta)&=1-\sum_n\frac{b_n}{n} [a_s\cos(n\Delta)-\eta_n d_s\sin(n\Delta)]\\ &=1-a_s+a_sV(\Delta)+d_sK(\Delta),\\ K(\Delta)&=\sum_n\frac{\eta_nb_n}{n}\sin(n\Delta). \end{split}\tag{8} \]

The retained law coefficients are therefore the pairs \((a_sb_n,\eta_nd_sb_n)\), with the potential constant fixed as in (8). They depend on supplied boundary geometry, not on an inferred force. For a finite Fourier family there are finitely many such coefficients; for the analytic class the convergent sequence specifies the law. On the invariant family no extra evolving hidden coordinate is needed for fixed \(s\). Arbitrary off-family states retain their constituent information and need a separate closure or approximation argument.

Although the fine current is odd, \(J_s\) in the original relative coordinate need not be odd. The reverse-oriented contact has potential \(U_s(-\Delta)\) and the corresponding sign-reversed derivative. This is consistent with the same undirected reciprocal exchange; the marked interface carries the orientation.

Let \(B=18(1-\cos\alpha)\), the internal potential storage fixed by (5). The full storage and loss restrict to

\[ H_s|_{\mathcal M}=B+9[\tfrac12u^2+U_s(\Delta)],\qquad H_s'|_{\mathcal M}=-6u^2 . \tag{9} \]

The inherited potential \(U_s\) is nonnegative because it is a convex combination of nonnegative fine potentials. Its value at the original alignment is \(U_s(0)=s(1-\cos\alpha)\). Subtracting that alignment value does not generally leave a nonnegative function: alignment is not in general an equilibrium of (7), because \(J_s(0)=s\,j(\alpha)\). For the explicit family (20), \(j(\alpha)=\sin\alpha\), so original alignment is not an equilibrium at any strict interior contact.

Same-profile inheritance on an interface interval selects sine

Fix any nonempty open interval \(I\subset(0,1)\) before evaluating a law. The additional profile premise requires that for each \(s\in I\) there exist a positive common form chart \(X_r=A_sZ_r\), a positive constant common clock rescaling, a constant registration shift

\[ \Psi_r=\Phi_r+(2r-1)\beta_s/2 , \tag{10} \]

and one positive reciprocal coefficient making both complete rows the same normalized two-node reciprocal law with current \(j\). The shift and all charts may depend on the fixed interface, but not on state, trajectory or observation. No regularity in \(s\) is assumed for the chosen shift; its relation to the original marked ports is retained.

For \(z=Z_1-Z_0\), comparing diffusion terms fixes \(\sigma=\tau/3\). Comparing both reciprocal rows then requires

\[ J_s(\Delta)=\rho_sj(\Delta+\beta_s),\qquad A_s=\sqrt{\rho_s},\qquad \gamma_{\rm eff}=\gamma\sqrt{\rho_s},\qquad \rho_s>0 . \tag{11} \]

This is the allowed map class, not a claim about every conceivable nonlinear state conjugacy. A fixed reciprocal coefficient would be a different and stronger condition.

Define

\[ \mu(s)=1-s+s e^{i\alpha}=\rho(s)e^{i\phi(s)},\qquad \rho(s)=\sqrt{1-2s(1-s)(1-\cos\alpha)} . \tag{12} \]

Its real part is positive, \(\rho>0\), and the continuous argument obeys

\[ 0<\phi(s)<\alpha,\qquad \phi'(s)=\frac{\sin\alpha}{\rho(s)^2}>0\quad(0<s<1). \tag{13} \]

For an active Fourier mode \(b_n\ne0\), its multiplier is

\[ 1-s+s e^{in\alpha}=\rho(s)e^{i\eta_n\phi(s)}. \tag{14} \]

Uniqueness of complex Fourier coefficients in (11) consequently gives

\[ \rho_s=\rho(s),\qquad n\beta_s-\eta_n\phi(s)\in2\pi\mathbb Z \quad\text{for every active }n . \tag{15} \]

There is at least one active coefficient by the target moments. Suppose two distinct positive modes \(m,n\) are active. Eliminate \(\beta_s\) from (15):

\[ (m\eta_n-n\eta_m)\phi(s)\in2\pi\mathbb Z \quad\text{for every }s\in I . \tag{16} \]

The integer factor is nonzero. With equal residue signs it is a signed difference of distinct positive modes; with opposite signs it is a signed sum. Equation (13) maps every nonempty open interval to an interval of positive length, while (16) would map it to a discrete set. This is impossible. The argument permits arbitrary discontinuous choices of \(\beta_s\) and never assumes that the first harmonic is present.

Exactly one mode remains, say \(j=b_n\sin n\delta\). The first two moments of (5) require \(b_n/n=1\) and \(nb_n=1\). Hence \(n=1\), \(b_1=1\); the third moment also holds. Conversely sine satisfies every premise with \(\beta_s=\phi(s)\) modulo \(2\pi\), \(\rho_s=\rho(s)\). Thus

\[ \boxed{\text{same-profile inheritance on }I \quad\Longleftrightarrow\quad j(\delta)=\sin\delta} \tag{17} \]

within (5) and the map class (10)--(11). The proof uses full profile equality, not a finite sample of responses or the separately stronger first-resultant information premise.

The interval quantifier matters. At \(s=0\), every current has \(J_0=j\); at \(s=1\), every current has \(J_1(\Delta)=j(\Delta+\alpha)\). They use \(\rho=1\) and shifts zero or \(\alpha\). Even a strict interior singleton does not in general select sine: at \(s=1/2\), put

\[ \widetilde V_\epsilon(\delta)=1-\cos\delta+ \epsilon\cos\delta(1-\cos18\delta)^2,\qquad 0<|\epsilon|\le1/1296 . \tag{18} \]

Its current has only frequencies \(1,17,19,35,37\), all congruent to \(\pm1\) modulo eighteen. It satisfies (5), since the perturbation vanishes to order four at every \(k\alpha\). The bound

\[ |\cos\delta(1-\cos18\delta)^2| \le648(1-\cos\delta) \tag{19} \]

proves global nonnegativity in (18). At the midpoint \(\rho=\cos(\alpha/2)\), \(\phi=\alpha/2\), every one of its modes satisfies (15) with \(\beta=\alpha/2\). It therefore preserves its nonsine profile at that single interface, though not on an open interval.

The existing nonsine family remains closed but changes profile

The previously admitted potential

\[ V_\epsilon(\delta)=1-\cos\delta+ \epsilon\cos\delta(1-\cos9\delta)^2,\qquad |\epsilon|\le1/324 \tag{20} \]

has current coefficients

\[ b_1=1-\tfrac32\epsilon,\quad b_8=8\epsilon,\quad b_{10}=10\epsilon,\quad b_{17}=-\tfrac{17}{4}\epsilon,\quad b_{19}=-\tfrac{19}{4}\epsilon. \tag{21} \]

The exact boundary coefficients in (8) give its inherited complete law for every \(s\in[0,1]\), with no fitted parameters.

When \(\epsilon\ne0\), both modes one and eight are active. Their conditions (15) force \(9\phi(s)\in2\pi\mathbb Z\), whereas \(0<9\phi(s)<2\pi\) throughout the strict interior. This particular family fails same-profile inheritance at every \(0<s<1\), not merely somewhere in an interval. That failure changes neither the exact closure (7) nor the nonnegative storage (9).

Applying the sine phase shift \(\phi(s)\) to a nonsine law still gives a valid invertible chart, but generally not the same current profile or an equilibrium at zero shifted phase. Nor is \(1-\rho(s)\) necessarily the minimum of its actual \(U_s\). Equations (8)--(9) retain the actual potential and all its constants without that inference.

More explicitly, set \(\psi=\Delta+\phi(s)\) and retain the same positive form/clock chart \(X_r=\sqrt\rho Z_r\), \(\sigma=\tau/3\), \(\gamma_{\rm eff}=\gamma\sqrt\rho\). Every current (5) then has the exact normalized boundary law

\[ \begin{split} R_s(\psi)&=\frac{J_s(\psi-\phi)}{\rho} =\sum_n b_n\sin\{n\psi+(\eta_n-n)\phi\},\\ Q_s(\psi)&=\frac{U_s(\psi-\phi)}{\rho}\ge0,\qquad Q_s'=R_s . \end{split}\tag{21a} \]

Thus its transformed sine/cosine coefficient pair is \((b_n\cos((\eta_n-n)\phi),b_n\sin((\eta_n-n)\phi))\). Both transformed reciprocal rows have the form (23), with sine replaced by \(R_s\), while \(H_s=B+9\rho[\tfrac12z^2+Q_s(\psi)]\). The full potential constant is retained: \(Q_s\) need not vanish at zero. This exact chart changes the profile, rather than asserting that the nonsine law has become sine.

Complete sine map, work, sources and measured coordinates

For the uniquely selected profile under the additional premise,

\[ U_s(\Delta)=1-\rho(s)\cos(\Delta+\phi(s)) =1-\rho(s)+\rho(s)V(\Delta+\phi(s)). \tag{22} \]

Using (10) with \(\beta_s=\phi(s)\), \(X_r=\sqrt\rho Z_r\), \(\sigma=\tau/3\), \(\widehat\Delta=\Psi_1-\Psi_0\) and \(\gamma_{\rm eff}=\gamma\sqrt\rho\), both rows are

\[ \begin{array}{ll} dZ_0/d\sigma=z+\gamma_{\rm eff}\sin\widehat\Delta,& dZ_1/d\sigma=-z-\gamma_{\rm eff}\sin\widehat\Delta,\\ d\Psi_0/d\sigma=-\gamma_{\rm eff}z,& d\Psi_1/d\sigma=\gamma_{\rm eff}z . \end{array}\tag{23} \]

The full fine storage and loss become

\[ \begin{split} H_s|_{\mathcal M}&=B+9(1-\rho) +9\rho[\tfrac12z^2+1-\cos\widehat\Delta],\\ dH_s/d\sigma&=-18\rho z^2 . \end{split}\tag{24} \]

At strict interior contacts \(\rho<1\); thus the additive boundary baseline is positive and the reciprocal coefficient changes. The common factor \(1/3\) is a derived rate in a declared clock, not a newly derived capacity or physical mass.

Uniform fine form inputs \(f_r(\tau)\) and independent phase inputs \(g_r(\tau)\) preserve (4). For any current (5), their exact balance is

\[ H_s'=-6u^2+ 9[u(f_1-f_0)+J_s(\Delta)(g_1-g_0)]. \tag{25} \]

In the sine chart they become \(3f_r/\sqrt\rho\) and \(3g_r\), respectively, with original time arguments \(\tau=3\sigma\). Dividing (25) by \(1/3\) and applying the chart agrees with (24)'s storage derivative and source work. A uniform form jump \(h_r\) becomes \(h_r/\sqrt\rho\); a phase jump keeps its angle while the same fixed registration offset remains in the chart. Its work is the difference of the full storage, not just its dissipation term.

Measured original coordinates remain

\[ X_r=\sqrt\rho Z_r,\qquad \Phi_r=\Psi_r-(2r-1)\phi(s)/2,\qquad \theta_{r,\ell}=\Phi_r+\alpha(\ell-4). \tag{26} \]

Thus the contact-dependent phase shift is not an unobserved change of physical registration. Any other observation is composed with this same embedding and chart. Sensor calibration is an independent model.

Attaching (1) at fixed state changes storage by the exact nonnegative support work

\[ \mathcal W_s=\sum_{\ell=0}^{8}\left[ (1-s)\{\tfrac12(x_{1,\ell}-x_{0,\ell})^2+ V(\theta_{1,\ell}-\theta_{0,\ell})\} +s\{\tfrac12(x_{1,\ell+1}-x_{0,\ell})^2+ V(\theta_{1,\ell+1}-\theta_{0,\ell})\}\right]. \tag{27} \]

On (4), \(\mathcal W_s=9[\tfrac12u^2+U_s(\Delta)]\). For all laws (5), attachment at \(u=\Delta=0\) requires \(9s(1-\cos\alpha)\). For sine at its shifted minimum \(u=0,\Delta=-\phi(s)\), it requires \(9(1-\rho(s))\). Both are positive for strict interior \(s\). The additive constant in (24) therefore has an observable role in the support-event budget. A positive cost needs a declared supply; continuous loss does not provide a reservoir or select the event.

All these charts hold \(s\) fixed. A time-dependent contact weight changes the fine storage explicitly, introduces registration and form-chart derivatives, and requires its own weight evolution and work law. Neither (23) nor (25) supplies those missing terms.

The composition premise and the ontological boundary

The NFR ontology requires an inherited description of the actual organization and its relations, with sufficient internal/boundary state, derived memory or controlled error. Equation (8) meets that continuous requirement on the invariant family even when (11) fails. An object can retain its constitutive coefficients and marked attachment data while its effective interaction profile depends on its boundary.

Consequently the requirement that every weighted contact in a declared interval reduce to the same scalar profile under (10)--(11) is an independently supplied additional hypothesis. It is mathematically consistent and sharply restrictive, but it is not deduced from nesting, phase circularity, positive storage, the nodal identity or continuous closure. Sine selection under that hypothesis does not establish that nature or all TNFR organizations satisfy it.

The supplied contact is likewise not an automatically realized Coupling operator. Its live admission, marked targets, retained history, collective descent and occurrence need their own contracts. This result derives neither the objects' acquisition under the weighted law nor an autonomous attachment, universal hierarchy or physical identification. The execution plan alone owns the next research decision.

The three-object network comparison separately tests simultaneous use of these pairwise charts. Its cycle-phase and amplitude conditions can fail even though every pair has a valid sine description; the complete network then retains its boundary coefficients. Neither pairwise equivalence nor this retained network closure supplies an acquired-source identity certificate.

Implementation and independent controls

The detached static owner _oriented_interface_constitutive.py represents the supplied weighted contacts, actual boundary coefficients, complete collective rows and storage of the explicit family (20). It admits primitive state and parameters before constructing these reports. It fixes the fine reciprocal coefficient to \(1/(1023\pi)\); the proof allows any positive fine coefficient and its explicitly mapped value. It does not install a native law, execute a support event or transfer an acquired-source certificate.

The independent controls rebuild the eighteen-node weighted field and storage, the registration chart, source work and harmonic obstructions. They check static implemented behavior under the stated proof; no trajectory, source producer, physical dataset or frozen evidence is evaluated here.