Constitutive profiles preserved by registered C9 composition

The two prescribed C9 interfaces restrict an analytic edge law, but do not select sine. Under the complete-law comparison below, the admissible current has Fourier harmonics congruent to either one or minus one modulo nine. Matching the organized target storage, tangent and quadratic field imposes three further linear conditions. Matching individual edge currents imposes a fourth, stronger condition. An explicit nonnegative potential satisfies all of them while changing the nonlinear law. These are conditional classification and counterexample results, not acquisition or physical identification results.

The registered inheritance owner defines the two interfaces and their source-specific certificates. This note changes the constitutive class and therefore re-establishes the consumed algebra and regularity. The original acquired sources, recovery dwell, finite contact bounds and evaluated evidence are not reused as certificates for these new laws.

Complete law, identity and the comparison class

Retain two labelled C9 components, node order \(9r+j\), with \(r=0,1\), \(j=0,\ldots,8\), nine unit cycle edges per component and \(\alpha=2\pi/9\). Choose exactly one supplied unit cross interface:

\[ \mathcal S_1=\{0\},\qquad \mathcal S_3=\{-1,0,1\},\qquad (0,j)--(1,j+s),\quad s\in\mathcal S_q. \tag{1} \]

Indices are modulo nine. There are \(9q\) cross edges and every final degree is \(d=2+q\). Constituent membership, local labels, orientation and attachment registration are retained boundary data. In particular, the first phase resultant of an ideal winding-one C9 vanishes, and its ninth moment loses the registration modulo \(\alpha\). Neither replaces the retained collective phase used here.

The complete fine state is signed real form and primitive circular phase with declared continuous lifts. Capacities are held at one, the support is fixed between any separately prescribed events, and the unforced clock is \(\tau=(1023/1024)t\). Supply an odd real-analytic \(2\pi\)-periodic current \(j\), with even periodic potential \(V(\delta)=\int_0^\delta j(s)\,ds\), normalized by \(V(0)=0\). Require \(V\ge0\) independently. The reciprocal coefficient is \(\gamma>0\), with \(1/(1023\pi)\) the earlier fine-model value. The comparison class permits the mapped positive coefficient derived below; holding it fixed would be a different equivalence question.

With the actual support Laplacian \(L\), let

\[ \begin{split} S_{j,i}(\theta)&=\sum_{\ell\sim i}j(\theta_\ell-\theta_i),\\ x'&=-d^{-1}Lx+\gamma d^{-1}S_j(\theta),\\ \theta'&=\gamma d^{-1}Lx,\\ H_V&=\tfrac12x^TLx+ \sum_{\{i,\ell\}}V(\theta_\ell-\theta_i). \end{split}\tag{2} \]

In original time both evolution rows acquire the factor \(1023/1024\). The first row supplies the pressure in the unit-capacity nodal identity; it is not reconstructed from a response. The structural chart fixes its units, with \([p]=[x]/([\nu_f][t])\); no laboratory time, sensor or energy unit follows. Pair antisymmetry and reciprocal cancellation give

\[ H_V'=-d^{-1}\|Lx\|_2^2,\qquad (\mathbf1^Tx)'=(\mathbf1^T\theta)'=0. \tag{3} \]

The candidate registered organization is the embedded family

\[ x_{r,j}=X_r,\qquad \theta_{r,j}=\Phi_r+\alpha(j-4). \tag{4} \]

It retains two forms, two registered phases, the internal winding-one identity and the specified boundary support. Common form-origin and phase-origin shifts, graph relabellings preserving the full supplied data, and simultaneous form/phase sign reversal preserve the corresponding law. The phase-origin symmetry does not discard relative registration. Away from (4), constituent means need not close; no arbitrary fine state is identified with an exact collective state.

This is the detached continuous/proof path under the NFR property admission. It neither runs nor changes the native relational engine or the general operator runtime. A supplied contact is not an automatically admitted Coupling event. Named operators still need their own live admission, collective descent and occurrence arguments. Continuous closure of (4) establishes none of those event claims.

Necessary and sufficient harmonic and target conditions

Define the interface action and geometric factors

\[ (A_q j)(\delta)=\frac1q\sum_{s\in\mathcal S_q} j(\delta+\alpha s),\qquad k_q=\frac q{2+q},\qquad r_1=1,\quad r_3=\frac{1+2\cos\alpha}{3}. \tag{5} \]

The internal currents on (4) cancel by oddness and periodicity. Every cross sum depends only on \(\Delta=\Phi_1-\Phi_0\), so both rows are tangent to (4). Writing \(u=X_1-X_0\), the exact restriction is

\[ \begin{array}{ll} X_0'=k_q[u+\gamma A_qj(\Delta)],& X_1'=-k_q[u+\gamma A_qj(\Delta)],\\ \Phi_0'=-\gamma k_qu,&\Phi_1'=\gamma k_qu. \end{array}\tag{6} \]

Analytic periodic currents and their first derivatives are bounded on the real circle. Together with the linear form rows this gives a globally Lipschitz lifted field and unique global continuation. Thus the tangent calculation proves invariance of (4), not merely instantaneous matching.

Here profile preservation means that a positive constant common clock rescaling, a positive common form-unit chart \(X_r=aZ_r\), unchanged collective phase, and one positive reciprocal coefficient put both rows into the normalized reciprocal pair law with the same function \(j\). Diffusion fixes the clock to \(\sigma=k_q\tau\); equality of the two reciprocal coefficients then forces

\[ A_qj=\rho_qj,\qquad a^2=\rho_q>0, \qquad\gamma_{\rm eff}=\gamma\sqrt{\rho_q}. \tag{7} \]

This defines the equivalence class; it does not exclude all nonlinear conjugacies with different inputs or observations. Matching contacts have \(A_1=I\), so their profile condition is automatic.

Require the organized storage and local derivative data to agree with sine at the consumed gaps \(\delta=0,\alpha,2\alpha\):

\[ V(\delta)=1-\cos\delta,\quad j'(\delta)=\cos\delta,\quad j''(\delta)=-\sin\delta. \tag{8} \]

Negative gaps then follow by parity. On the declared target supports, these conditions match target storage, the complete tangent and the full quadratic field. The target vector field itself vanishes for every odd current: opposite currents cancel at cycle nodes and symmetric contacts, and aligned bridge currents are zero. Matching those sums therefore does not require matching each nonzero edge current individually. Both nonzero winding gaps are included even though the composition in (4) uses winding one. They admit the same local target comparison on the earlier supports consuming winding two and aligned bridges; they do not transfer that model's acquisition proof.

Odd real-analytic periodicity supplies a sine series \(j(\delta)=\sum_{n\ge1}b_n\sin(n\delta)\) with real exponentially decaying coefficients. Equivalently, use a holomorphic extension to some open strip about the real circle and any smaller closed strip to bound the coefficients. This justifies the termwise shifts, integrals and all finite-order derivatives used below. Conversely, exponential decay supplies the required analytic current.

For each sine harmonic,

\[ A_3\sin(n\delta)=\frac{1+2\cos(n\alpha)}3\sin(n\delta). \tag{9} \]

Differentiating (7) at zero and applying (8) gives

\[ \rho_3j'(0)=\tfrac13[j'(0)+2j'(\alpha)] =\frac{1+2\cos\alpha}{3}=r_3. \tag{10} \]

Since \(j'(0)=1\), necessarily \(\rho_3=r_3\). Uniqueness of Fourier coefficients in (9) now gives

\[ b_n=0\quad\text{unless }n\equiv1\text{ or }-1\pmod9. \tag{11} \]

Let \(\mathcal N\) be those positive integers and put \(s_n=1\) for \(n\equiv1\pmod9\), \(s_n=-1\) for \(n\equiv-1\pmod9\). For \(k=0,1,2\), the retained frequencies obey \(\cos(nk\alpha)=\cos(k\alpha)\) and \(\sin(nk\alpha)=s_n\sin(k\alpha)\). The normalized potential is

\[ V(\delta)=\sum_{n\in\mathcal N}\frac{b_n}{n} (1-\cos n\delta). \tag{12} \]

Consequently (8) is equivalent to three moment conditions:

\[ \boxed{\quad \sum_{n\in\mathcal N}\frac{b_n}{n}=1,\qquad \sum_{n\in\mathcal N}nb_n=1,\qquad \sum_{n\in\mathcal N}s_nn^2b_n=1. \quad}\tag{13} \]

For necessity, use the potential value at \(\alpha\), the derivative at zero and the second derivative at \(\alpha\); \(1-\cos\alpha\) and \(\sin\alpha\) are nonzero. For sufficiency, substitution gives every value in (8), including the entire set at \(2\alpha\). The second winding therefore adds no independent moment after the interface restriction (11), although it remains part of the matched-target claim.

Equations (11)--(13), real exponential coefficient decay and \(V(\delta)\ge0\) for every real \(\delta\) are jointly necessary and sufficient for this comparison. Nonnegativity is a separate global condition; these equalities do not imply it. This classification never assumes first-neighbor-resultant sufficiency. That different information premise would remove the higher harmonics by a separate argument.

The stronger edge-current requirement \(j(k\alpha)=\sin(k\alpha)\) for \(k=1,2\) is equivalent to the additional condition \(\sum_ns_nb_n=1\). It is not redundant. For example, the perturbation current

\[ h(\delta)=57\sin\delta-708\sin8\delta +185\sin10\delta+221\sin17\delta \]

has zero perturbations of all three moments in (13), but its signed current moment is 729. Its normalized potential \(Q=\int_0^\delta h(s)\,ds\) obeys \(|Q(\delta)|\le11328(1-\cos\delta)\): apply \(1-\cos(n\delta)\le n^2(1-\cos\delta)\) to each coefficient. Thus \(V=1-\cos\delta+eQ\), \(|e|\le1/22656\), is a nonnegative counterexample to deducing that fourth condition from the actual matched target fields. The next family satisfies even the stronger condition.

An explicit nonnegative nonsine family

For the exact real parameter \(|\epsilon|\le1/324\), define

\[ P(\delta)=\cos\delta(1-\cos9\delta)^2,\qquad V_\epsilon(\delta)=1-\cos\delta+\epsilon P(\delta),\qquad j_\epsilon=V_\epsilon'. \tag{14} \]

These functions are analytic, periodic and have the required parity. The elementary bound \(|\sin(9z)|\le9|\sin z|\) for real \(z\) gives \(1-\cos9\delta\le81(1-\cos\delta)\). Combining it with \(1-\cos9\delta\le2\) proves

\[ |P(\delta)|\le162(1-\cos\delta),\qquad \tfrac12(1-\cos\delta)\le V_\epsilon(\delta) \le\tfrac32(1-\cos\delta). \tag{15} \]

Thus the potential is nonnegative globally and vanishes only at full turns. Its positivity is proved without sampling phase or selecting a trajectory. Expanding its five harmonics gives

\[ P(\delta)=\tfrac32\cos\delta-\cos8\delta-\cos10\delta +\tfrac14\cos17\delta+\tfrac14\cos19\delta. \tag{16} \]

Frequency \(n\)	Current coefficient \(b_n\)	Residue sign \(s_n\)
1	\(1-3\epsilon/2\)	+1
8	\(8\epsilon\)	-1
10	\(10\epsilon\)	+1
17	\(-17\epsilon/4\)	-1
19	\(-19\epsilon/4\)	+1

All other coefficients vanish. Each is in (11), and direct substitution satisfies (13) and the additional edge-current moment. There is also a local verification: at every \(\delta=k\alpha\), the factor \((1-\cos9\delta)^2\) has a zero of order four. Hence

\[ P=P'=P''=P'''=0,\qquad P''''(k\alpha)=6\,9^4\cos(k\alpha). \tag{17} \]

The organized target, storage, full tangent and quadratic field therefore agree with sine. The cubic field generally changes:

\[ j_\epsilon'''(k\alpha) =[-1+39366\epsilon]\cos(k\alpha). \tag{18} \]

Every nonzero admitted \(\epsilon\) is genuinely nonsine in the declared chart. In particular, \(P'(\pi/2)=-1\) gives the exact static witness

\[ j_\epsilon(\pi/2)=1-\epsilon\ne1. \tag{19} \]

The unchanged derivative \(j_\epsilon'(0)=1\) excludes a mere global current multiplier. This is a different complete nonlinear vector field, not a finite measured response or a claim excluding every possible nonlinear state conjugacy.

The coefficient identities hold algebraically outside the admitted interval, but positivity need not survive. For example, taking \(\epsilon=2\) in (14) gives \(V_2(\pi)=2-8=-6\), while preserving all three target moments and the additional edge-current moment. This explicitly separates the global storage condition from the linear matching conditions; that value is rejected by the static family's admission.

This freedom has arbitrarily many independent directions. For each integer \(m\ge2\), the even trigonometric polynomial \(P_m(\delta)=\cos\delta(1-\cos9\delta)^m\) has only the same residue frequencies, and has a zero of order \(2m\) at every target gap. Any finite sum

\[ V=1-\cos\delta+\sum_{m=2}^M a_mP_m(\delta),\qquad \sum_{m=2}^M81\,2^{m-1}|a_m|\le\tfrac12 \tag{20} \]

obeys (11)--(15)'s nonnegative lower bound and all target conditions. The bound follows from \((1-\cos9\delta)^m\le81\,2^{m-1}(1-\cos\delta)\). The distinct polynomial powers are linearly independent. There is thus no finite count of alternatives selected by these constraints. The uniform derivative constants below apply only to (14), not automatically to every finite sum in (20).

Own-law regularity and local target identity

For (14), summing absolute Fourier derivative coefficients gives

\[ \begin{split} |j_\epsilon'|&\le1+328|\epsilon|\le163/81,\\ |j_\epsilon''|&\le1+\tfrac{8913}{2}|\epsilon| \le3187/216<15. \end{split}\tag{21} \]

At any edge target \(0,\pm\alpha,\pm2\alpha\), its potential curvature is exactly the sine curvature, whose minimum is \(\cos(4\pi/9)>1/6\). One algebraic proof of the latter bound sets \(c=\cos(4\pi/9)\in(0,1/2)\); it is the root there of \(8c^3-6c+1=0\), a strictly decreasing polynomial with positive value at \(c=1/6\). The mean value estimate from (21) consequently gives

\[ V_\epsilon''(\delta)>\frac{11}{120}>0 \quad\text{when the edge is within }1/200 \text{ of one of those target gaps}. \tag{22} \]

This re-admits a local convex cell for the actual potential. On any of the connected target supports with these edge gaps, opposite cycle currents and zero aligned bridge currents still make the target an equilibrium. On the two-C9 three-offset support the extra opposite cross currents likewise cancel at alignment. Its full tangent is

\[ \mathcal J= \begin{pmatrix} -D^{-1}L&-\gamma D^{-1}L_c\\ \gamma D^{-1}L&0 \end{pmatrix}, \qquad Q=\tfrac12x^TLx+\tfrac12y^TL_cy, \tag{23} \]

where \(D\) is the actual positive degree matrix and \(L_c\) the target curvature-weighted Laplacian. Both \(L\) and \(L_c\) have only the common-origin kernel on a connected support, because every target curvature is positive. Restricting to fixed conserved \(D\)-weighted form and lifted phase means removes those two origins. For regular interfaces these are the ordinary means. Along the linear tangent, \(Q'=-x^TLD^{-1}Lx\). The only trajectory in its zero-loss set has \(Lx=0\), then \(L_cy=0\), so the relative tangent is asymptotically stable and, in finite dimension, Hurwitz. Smooth nonlinear continuation therefore gives local asymptotic stability for the actual law; (22) also provides its own strictly convex storage neighborhood.

This is an independently admitted local identity, conditional on starting sufficiently close with sufficiently small full excess storage. It gives neither the old radius \(1/12\), a source-to-cell route, a uniform recovery dwell, nor a contact certificate for a previously acquired fine state. Nonlinear remainders, source transport and event bounds must be rebuilt before asserting any of those properties. The entire invariant family (4) need not be transversely stable at every collective phase.

Both evolution rows, storage and observations inherit together

For every current admitted by the classification, not only (14), \(A_qj=r_qj\). Integrating this equality and evaluating at zero gives

\[ A_qV(\Delta)=(1-r_q)+r_qV(\Delta). \tag{24} \]

Indeed, \(A_1V(0)=0\), while \(A_3V(0)=2V(\alpha)/3=1-r_3\) by target matching. Let \(B_q=18(1-\cos\alpha)+9q(1-r_q)\). The full fine storage and loss on (4) are exactly

\[ H_V|_{\mathcal M}=B_q+9q[\tfrac12u^2+r_qV(\Delta)], \qquad\frac{dH_V}{d\tau}=-18qk_qu^2. \tag{25} \]

The inherited chart and clock are forced within the declared map class:

\[ \sigma=k_q\tau,\qquad X_r=\sqrt{r_q}Z_r, \qquad\gamma_{\rm eff}=\gamma\sqrt{r_q}. \tag{26} \]

For \(z=Z_1-Z_0\), both complete rows and the full storage become

\[ \begin{array}{ll} \displaystyle\frac{dZ_0}{d\sigma}=z+\gamma_{\rm eff}j(\Delta),& \displaystyle\frac{dZ_1}{d\sigma}=-z-\gamma_{\rm eff}j(\Delta),\\ \displaystyle\frac{d\Phi_0}{d\sigma}=-\gamma_{\rm eff}z,& \displaystyle\frac{d\Phi_1}{d\sigma}=\gamma_{\rm eff}z, \end{array} \tag{27} \]

\[ H_V|_{\mathcal M}=B_q+9qr_q[\tfrac12z^2+V(\Delta)], \qquad\frac{dH_V}{d\sigma}=-18qr_qz^2. \tag{28} \]

For (14), \(\epsilon\) is unchanged by either interface. The reciprocal coefficient is unchanged only for matching contacts. Three-offset contacts still require its change and the form chart; a fixed-\(\gamma\) family is not closed under this complete comparison. Unit fine capacities are retained, and \(k_q\) is a derived common rate, not an inferred capacity law or an inverse mass.

Uniform constituent form inputs \(b_r(\tau)\) and independently supplied phase inputs \(c_r(\tau)\) preserve (4). They transform respectively to \(b_r/(k_q\sqrt{r_q})\) and \(c_r/k_q\) in (27), with their original time argument transformed as well. The full continuous balance is

\[ \frac{dH_V}{d\tau}=-18qk_qu^2+ 9q[u(b_1-b_0)+r_qj(\Delta)(c_1-c_0)]. \tag{29} \]

Dividing by \(k_q\) and using (26) reproduces the same work in (28). A uniform form jump \(a_r\) becomes \(a_r/\sqrt{r_q}\); a uniform phase jump keeps its angle. Their work is the difference of the full storage. A raw form observation reads \(X_r=\sqrt{r_q}Z_r\), not \(Z_r\). Any more general observation is composed with the same fine embedding and chart. Independent sensor calibration and discarded constituent information remain separate obligations.

Finally, attaching the specified support without changing state requires the exact event work

\[ \mathcal W_q=\sum_{j,s\in\mathcal S_q} \left[\tfrac12(x_{1,j+s}-x_{0,j})^2+ V(\theta_{1,j+s}-\theta_{0,j})\right]\ge0. \tag{30} \]

On (4) it is \(9q[\tfrac12u^2+(1-r_q)+r_qV(\Delta)]\). At the aligned target it is zero for matching and \(18(1-\cos\alpha)>0\) for three offsets. The additive constant in (24) is therefore retained even though it disappears from continuous derivatives. Positive work needs a declared external supply, and neither passivity nor the continuous law selects event occurrence.

What the classification establishes

The two named interfaces constrain the harmonic content and the matched targets constrain three coefficient moments. Even imposing the fourth moment for individual edge-current equality, together with nonnegative storage, these requirements still admit the explicit nonsine laws above. Thus same-profile organized composition does not select a unique current in the stated analytic class. Repeating either proved algebraic interface map leaves those profiles unchanged; it does not construct the nested geometries, sources or events needed to repeat the physical organization.

Composition at different phase registrations, other cycle sizes, every possible interface, a first-resultant information restriction or a different operator-descent requirement would add premises. None follows from this two-interface calculation. Any further selection principle must be justified against retained NFR and boundary state rather than chosen to remove the counterexample. The execution plan owns that decision; this proof creates no parallel research queue.

The oriented weighted-interface comparison now isolates one such extra premise: unchanged profiles across its declared open interface interval select sine within this class. The other currents still inherit exact boundary-dependent laws, so that conditional selector does not follow from the NFR ontology or invalidate the present counterfamily.

The regional barrier and work comparison gives a further consequence of this same counterfamily on its admitted interval: even its true winding-boundary storage minima are not fixed by the matched target and composition properties. Regional powers can also differ at identical complete state and storage. This does not transfer a source outcome or select a preferred law.

Implementation and independent controls

The detached static owner _composition_compatible_constitutive.py admits the exact parameter domain of (14), derives its finite harmonic coefficients and observes edge jets and both complete collective rows. The static pair instrument fixes the fine reciprocal coefficient to \(1/(1023\pi)\), while the proof permits any positive fine coefficient with its correspondingly mapped comparison value. It reuses the shared registered-interface geometry and complete chart data without transferring a sine acquisition certificate. It represents the explicit stronger counterfamily, not a decision procedure for global nonnegativity of every analytic series in the classification. Its interval-angle adapter reuses the same finite harmonic evaluator for the full-state regional work observer. The sine member retains its original observation arithmetic; phase intervals enclose phasors rather than asserting that every corner of the sine/cosine rectangle lies on the unit circle.

Its independent controls rebuild the potential and derivatives, target moments, exact nonsine witness, full fine-state restriction, storage and transformed observations. These algebraic and static tests check implemented behavior; they do not evaluate a trajectory, form an organization under the new law, freeze a physical protocol or regenerate previous scientific evidence.