Constitutive inheritance through registered organized patterns

Two prescribed interfaces between winding-one C9 patterns admit an exact four-coordinate collective restriction of the matched-potential family. Matching local nodes preserves every constitutive parameter. Adding the two neighboring registrations changes the first phase harmonic while preserving the ninth and eighteenth ones. The resulting law belongs to an enlarged algebraic family with a mapped reciprocal coefficient, under an explicitly derived form chart, clock and parameter map. This is conditional inheritance through one specified grouping, not selection of a fundamental law or formation of a universal hierarchy.

The exact restriction and the later acquired-state bounds have different domains. The restriction permits arbitrary collective coordinates on its registered invariant family. Actual recovered sources have small internal residuals and are handled through the full fine-state law, including the new support work. Neither support event is selected by the continuous equations.

Complete fine state and registered organizations

Retain node order \(9r+j\), with \(r=0,1\) and \(j=0,\ldots,8\). Each component has its nine unit cycle edges. Let \(\alpha=2\pi/9\) and choose exactly one of the two supplied sets of cross-edge offsets:

\[ \mathcal S_1=\{0\},\qquad \mathcal S_3=\{0,-1,1\}. \tag{1} \]

For each \(j\) and \(s\in\mathcal S_q\), add the undirected unit edge \((0,j)--(1,j+s)\), with indices modulo nine. The number of cross edges is \(9q\), every final degree is \(d=2+q\), and the full support is connected. No cross edge is repeated. Orientation, constituent partition, local indices and attachment registration are retained data.

Use signed real form, primitive circular phase and declared continuous phase lifts. Fine capacities stay one, and there are no continuous inputs unless explicitly supplied below. With the actual support Laplacian \(L\), keep

\[ \begin{split} W(\delta)&=\frac{(1-\cos9\delta)^2}{6\,9^4},\\ V_\lambda(\delta)&=1-\cos\delta+\lambda W(\delta),\\ j_\lambda(\delta)&=\sin\delta+ \frac{\lambda}{2187} \left(\sin9\delta-\frac12\sin18\delta\right),\\ S_{\lambda,i}(\theta)&=\sum_{\ell\sim i} j_\lambda(\theta_\ell-\theta_i). \end{split}\tag{2} \]

The source-admitted fine constitutive parameter is \(0\le\lambda\le1\), as in the acquisition theorem. Expressions (2) and the exact restriction and chart identities also apply to every \(\lambda\ge0\); that extension carries no automatic acquisition, capture or source-error certificate. The normalized pair comparison below permits the reciprocal coefficient \(\gamma>0\) to vary as a second parameter; the fine source-admitted model retains its fixed value.

The original clock is \(t\), and \(\tau=e t\), \(e=1023/1024\), \(\gamma=1/(1023\pi)\). In structural time the complete unforced rows and storage are

\[ x'=-d^{-1}Lx+\gamma d^{-1}S_\lambda(\theta),\qquad \theta'=\gamma d^{-1}Lx,\qquad H_\lambda=\tfrac12x^TLx+\sum_{\{i,\ell\}}V_\lambda(\theta_\ell-\theta_i). \tag{3} \]

In \(t\), both right-hand sides acquire the factor \(e\). The pressure in the unit-capacity nodal row is the independently specified first row in that clock, with the structural units retained. No laboratory clock or energy unit is inferred.

The reciprocal cancellation uses the actual complete law:

\[ H_\lambda'=-d^{-1}\|Lx\|_2^2,\qquad (\mathbf1^Tx)'=(\mathbf1^T\theta)'=0. \tag{4} \]

The phase mean here uses the declared lifts. The inherited state is the four-dimensional family

\[ \mathcal M:\quad x_{r,j}=X_r,\qquad \theta_{r,j}=\Phi_r+\alpha(j-4). \tag{5} \]

Its two collective phases are circular, with local continuous lifts where rates and means are evaluated. This is an embedding of a particular family in the full state space, not a quotient of arbitrary fine states. Away from \(\mathcal M\), constituent means alone need not close; the mean-closure obstruction and retained hidden-state requirements still apply.

The first phase resultant of either ideal C9 is zero. The ninth moment is \(e^{9\mathrm i\Phi_r}\), which loses \(\Phi_r\) modulo \(2\pi/9\). For example, changing only \(\Phi_1\) from zero to \(\alpha\) preserves both ninth moments but changes the matching-interface current from zero to \(\sin\alpha\). Thus a ninth moment without registration is insufficient even on (5). Here the retained labels and offsets determine \(\Phi_r\); no undefined circular-mean angle is used.

Invariant restriction and the interface harmonic map

Define

\[ \Delta=\Phi_1-\Phi_0,\quad u=X_1-X_0,\quad k_q=\frac q{2+q},\quad r_q=\frac1q\sum_{s\in\mathcal S_q}\cos(\alpha s). \tag{6} \]

Thus

\[ r_1=1,\qquad r_3=\frac{1+2\cos\alpha}{3}\in(0,1). \tag{7} \]

On (5), the two internal form differences vanish. The two internal phase currents cancel because \(j_\lambda\) is odd and periodic. Every node in component zero has the same cross form sum \(-q u\) and the same cross current sum \(\sum_s j_\lambda(\Delta+\alpha s)\). Wraps of local indices change the lifted gap by an integer multiple of \(2\pi\), leaving these currents and potentials unchanged. The reverse component has the opposite sums, since the offset set is symmetric.

The complete vector field is therefore tangent to (5) at every point. Its global Lipschitz property gives unique continuation in this invariant family. This applies to both rows and all collective coordinates, not just to a target equilibrium. Let

\[ \begin{split} J_{\lambda,q}(\Delta) &=\frac1q\sum_{s\in\mathcal S_q}j_\lambda(\Delta+\alpha s)\\ &=r_q\sin\Delta+ \frac{\lambda}{2187} \left(\sin9\Delta-\tfrac12\sin18\Delta\right) =r_q j_{\lambda/r_q}(\Delta). \end{split}\tag{8} \]

The first equality is the fine-state restriction. The second follows by pairing the first harmonics and using \(9\alpha s=2\pi s\) for each integer offset. In \(\tau\), the exact collective rows are

\[ \begin{array}{ll} X_0'=k_q[u+\gamma J_{\lambda,q}(\Delta)],& X_1'=-k_q[u+\gamma J_{\lambda,q}(\Delta)],\\ \Phi_0'=-\gamma k_q u,& \Phi_1'=\gamma k_q u. \end{array}\tag{9} \]

Both collective sums are constant. Equivalently, \(u'=-2k_q[u+\gamma J_{\lambda,q}(\Delta)]\) and \(\Delta'=2\gamma k_q u\). The factors \(k_1=1/3\) and \(k_3=3/5\) use the final degrees; retaining the old isolated degree two would give a different model.

This calculation extends the existing phase-offset partition criterion to this particular positive-loss, multiple-harmonic law. A first-harmonic criterion alone could not establish the second equality of (8).

Storage, dissipation and support-event work

Because \(W(\alpha)=0\), the 18 internal edges contribute \(18(1-\cos\alpha)\) on the whole invariant family. The averaged cross potential is

\[ \overline V_{\lambda,q}(\Delta) =1-r_q\cos\Delta+\lambda W(\Delta) =(1-r_q)+r_q V_{\lambda/r_q}(\Delta). \tag{10} \]

Set \(B_q=18(1-\cos\alpha)+9q(1-r_q)\). The exact pullback is

\[ H_\lambda|_{\mathcal M} =B_q+9q\left[\tfrac12u^2+r_qV_{\lambda/r_q}(\Delta)\right], \qquad \frac{dH_\lambda}{d\tau}=-18qk_q u^2. \tag{11} \]

Indeed, \(Lx\) equals \(-qu\) at every node of the first component and \(qu\) at every node of the second; substitution into (4) gives the loss. Differentiating the displayed collective storage using both rows of (9) gives the same result. The inherited storage is not an independently postulated pair energy.

The prescribed support event keeps all form and phase values unchanged. For a general fine state its work is exactly

\[ \mathcal W_q(x,\theta)= \sum_{j,s\in\mathcal S_q} \left[\tfrac12(x_{1,j+s}-x_{0,j})^2+ V_\lambda(\theta_{1,j+s}-\theta_{0,j})\right]\ge0. \tag{12} \]

On \(\mathcal M\), this becomes \(9q[\tfrac12u^2+(1-r_q)+r_qV_{\lambda/r_q}(\Delta)]\). At the aligned zero-form target its value is zero for matching edges and \(18(1-\cos\alpha)>0\) for the three-offset interface. The constant in (10) disappears from continuous derivatives but cannot be discarded when comparing the before and after support storage. Any positive event work requires a declared external supply; earlier continuous dissipation is not a reservoir. These formulas neither supply that reservoir nor cause the event.

Complete-law equivalence, coordinates and capacity scope

Specify the allowed comparison before removing coefficients. A positive common clock factor acts on both evolution rows. In the unchanged form chart, comparison with the unit-capacity normalized reciprocal pair law allows a common rate multiplier and a new reciprocal coefficient, but retains its normalized diffusion and first-harmonic coefficients.

No choice within that class puts the three-offset law directly into the original normalized family. Matching the diffusion coefficient fixes the common multiplier to \(k_q\). Matching the phase row then fixes the reciprocal coefficient to \(\gamma\). The first-harmonic form coefficient would consequently be \(k_q\gamma\), whereas (9) has \(k_q\gamma r_3\). Distinct Fourier harmonics cannot repair that discrepancy by changing \(\lambda\). Matching edges have \(r_1=1\), so they preserve the entire family under their derived common clock factor.

The regular form chart permitted by the form-chart contract does give an exact parameter map:

\[ \sigma=k_q\tau=k_q e t,\qquad X_r=\sqrt{r_q}\,Z_r,\qquad \lambda_{\rm eff}=\frac{\lambda}{r_q},\qquad \gamma_{\rm eff}=\gamma\sqrt{r_q}. \tag{13} \]

Keeping \(\Phi_r\) unchanged, both rows become

\[ \begin{array}{ll} \displaystyle\frac{dZ_0}{d\sigma} =Z_1-Z_0+\gamma_{\rm eff}j_{\lambda_{\rm eff}}(\Delta),& \displaystyle\frac{dZ_1}{d\sigma} =Z_0-Z_1-\gamma_{\rm eff}j_{\lambda_{\rm eff}}(\Delta),\\ \displaystyle\frac{d\Phi_0}{d\sigma} =\gamma_{\rm eff}(Z_0-Z_1),& \displaystyle\frac{d\Phi_1}{d\sigma} =\gamma_{\rm eff}(Z_1-Z_0). \end{array}\tag{14} \]

Writing \(z=Z_1-Z_0\), their inherited storage and loss are

\[ H_\lambda|_{\mathcal M} =B_q+9qr_q\left[\tfrac12z^2+V_{\lambda_{\rm eff}}(\Delta)\right], \qquad \frac{dH_\lambda}{d\sigma}=-18qr_q z^2. \tag{15} \]

The form chart transforms the pressure, the phase coupling and the storage scale together. Reusing the original pressure formula with the new form coordinate would violate (14).

More generally, a positive form-unit chart \(X_r=aZ_r\), after the diffusion coefficient has fixed \(\sigma=k_q\tau\), gives phase coefficient \(\gamma a\) and first-harmonic form coefficient \(\gamma r_q/a\). Their reciprocal equality forces \(a^2=r_q\) and hence the value of \(\gamma_{\rm eff}\) in (13). Consequently the three-offset restriction belongs to the enlarged two-parameter family \((\lambda_{\rm eff},\gamma_{\rm eff})\); it does not belong to the original fixed-\(\gamma\) line merely by extending the allowed \(\lambda\) interval. Keeping that original coefficient excludes exact same-law inheritance for this interface even under the stated positive form-unit charts and common clock maps.

The factor \(k_q\) is a derived common response factor of this complete restricted law. One may represent it as a common effective capacity only after specifying that the capacity-dependent complete model multiplies both reciprocal rows by that factor. The nodal identity alone specifies only the form row and does not justify such a phase law. Allowing arbitrary independent form and phase mobilities is a larger complete family; the unchanged-chart obstruction above does not exclude that different comparison. Fine capacities here remain the declared value one.

For matching edges, all constitutive parameters in (13) are unchanged. For three offsets, every \(\lambda>0\) increases and \(\gamma\) decreases: \(\lambda_{\rm eff}\gamma_{\rm eff}^2=\lambda\gamma^2\). This last equality is an algebraic property of (13), not a physical conservation law. If \(\lambda>r_3\), the effective parameter is outside the source-admitted interval \([0,1]\), although it is still in the larger algebraic family. Even when \(\lambda\le r_3\), a pair of collective nodes does not inherit a C9 acquisition proof merely by sharing a parameter value. The fixed point \(\lambda=0\) of this particular parameter map does not select sine: the supplied matching interface preserves every \(\lambda\), and the larger family accommodates the three-offset map.

Even imposing an unchanged constitutive profile under both interfaces does not select sine. The analytic classification and counterfamily retain the same target storage and local derivatives while preserving a different nonlinear profile. Its new laws have their own admission scope; the acquired-state certificates below remain specific to (2).

Inputs, observations and event boundaries

Suppose an independently supplied continuous form input adds \(b_r(\tau)\) to every fine form row in component \(r\). It preserves (5) and adds \(b_r/k_q\) to \(dX_r/d\sigma\), or \(b_r/(k_q\sqrt{r_q})\) to \(dZ_r/d\sigma\). If a phase input \(c_r(\tau)\) is separately declared and constant within each component, it transforms to \(c_r/k_q\) in the phase row. An input specified in original-time units first transforms by the additional factor \(1/e\). No phase input is inferred from a form input.

The corresponding continuous boundary work follows from the same full storage:

\[ \frac{dH_\lambda}{d\tau} =-18qk_q u^2+ 9q\left[u(b_1-b_0)+J_{\lambda,q}(\Delta)(c_1-c_0)\right]. \]

The transformed input coefficients and storage (15) reproduce this balance after division by \(k_q\). A common input to both components shifts its retained mean without doing work on the difference-only storage.

Similarly, a prescribed uniform form jump \(X_r^+=X_r^-+a_r\) becomes \(Z_r^+=Z_r^-+a_r/\sqrt{r_q}\). A uniform phase jump keeps its angular increment. Its event work is the difference of the full pulled-back storage (11) or (15), with identical result in either chart. A single fine-node input generally leaves \(\mathcal M\); its subsequent full dynamics cannot be replaced by (14). The registered fine event and its collective admissibility remain separate from any grammar, history or occurrence rule of a named engine operator.

For an independently specified observation \(y=h(x,\theta)\), the exact collective observation is \(h\) composed with (5). In the new chart it is the same map with \(X_r=\sqrt{r_q}Z_r\). In particular, a raw constituent form reading \(X_r\) is represented by \(\sqrt{r_q}Z_r\), not by \(Z_r\). A calibrated scalar sensor \(g(X_r)\) becomes \(g(\sqrt{r_q}Z_r)\); replacing it by \(g(Z_r)\) would change the sensor. Neither this coordinate map nor the derived clock factor calibrates a laboratory observer.

Both prescribed support events map the exact family (5) to itself while changing its vector field and storage. Their specified event rule therefore has an exact restriction on this family. This is not a quotient descent claim after discarding registration or residual fine state. Admission and occurrence still require the collective event criterion.

Actual acquired sources, new support work and full-state identity

Use two copies of the original class-one uncertain ramp preparation, with each original form and phase coordinate error at most \(10^{-10}\) and the separate exact zero-sum constraints. The own-law recovery theorem gives, for every fine \(\lambda\in[0,1]\), the common uninterrupted handoff time \(\tau_h=100+10^{13}\) and

\[ \|x_r\|_2\le\varepsilon,\qquad \|y_r\|_2\le\varepsilon,\qquad y_{r,j}=\theta_{r,j}-\alpha(j-4),\qquad \varepsilon=10^{-70}. \tag{16} \]

The component means of \(x_r,y_r\) are still exactly zero. Their complete reached coordinates and absolute origins are retained; (16) does not replace either source by its target. Only isolated recovery is reused. The previous three-C9 bridge event, its four input branches and their work or identity certificates are not premises of this composition.

The present event adds exactly the chosen cross support and changes no coordinate. Let \(\Theta_{r,j}=\alpha(j-4)\), \(u_\lambda=1+2\lambda/243\), and

\[ W_q^*=9q(1-r_q),\qquad H_q^*=18(1-\cos\alpha)+W_q^*. \tag{17} \]

All deformation potentials vanish on the target edges, whose circular differences are \(0,\alpha\) or \(-\alpha\). The target cross-edge gradient is zero: matching currents vanish separately, and the two offset currents cancel at every node. Thus the first variation of the complete added potential is zero, even though its individual offset terms need not be stationary.

The cross-support Laplacian is bounded above by \(2qI\); the full one is bounded above by \(2dI\). Also \(|V_\lambda''|\le u_\lambda\) globally. On the cell

\[ \mathcal B=\{(x,y):\mathbf1^Tx=\mathbf1^Ty=0,\ \|x\|_2^2+\|y\|_2^2<R_0^2\}, \qquad R_0=1/12, \tag{18} \]

every edge Hessian is strictly greater than \(1/2\). Indeed,

\[ V_\lambda''(\delta) \ge \cos(\alpha+\sqrt2R_0)-2/243 > 47/72-2/243>1/2. \tag{19} \]

Here \(\alpha+\sqrt2/12<5/6\), using \(\pi<22/7\) and \(\sqrt2<3/2\), and \(\cos z\ge1-z^2/2\). Periodic edge representatives handle the cycle seam without changing the fixed lifts or the winding.

Taylor's formula for the whole added potential, including cancellation of its first variation, and (16) now give the actual event work

\[ W_q^*\le \Delta H_{\rm support} \le W_q^*+2q(1+u_\lambda)\varepsilon^2. \tag{20} \]

The lower bound uses convexity along the segment from the target to the actual state, which lies within (18). The added form storage is nonnegative. For matching edges \(W_1^*=0\); for three offsets \(W_3^*=18(1-\cos\alpha)>0\). This event therefore requires an explicit external work supply covering its actual cost. The statement is conditional on that supplied event and work, not a passivity result or a deduction of an occurrence rule. No previous continuous loss is credited as an available reserve.

The matching support is the Cartesian product \(C_9\mathbin{\square}K_2\). Its smallest nonzero Laplacian eigenvalue is \(2-2\cos\alpha>2/5\). For example, \(\pi>3\) gives \(\cos\alpha<\cos(2/3)\le191/243<4/5\). The three-offset support adds positive semidefinite edge Laplacians, so it keeps the same lower bound. Together with (19), this proves the full transverse coercivity, throughout (18),

\[ H_\lambda(x,\Theta+y)-H_q^* \ge \tfrac1{10}\bigl(\|x\|_2^2+\|y\|_2^2\bigr). \tag{21} \]

This is positivity on the full zero-mean quotient, including all internal residual directions; it is not a test only along the collective family. At the actual unchanged state after contact,

\[ \|x\|_2^2+\|y\|_2^2\le4\varepsilon^2,\qquad E_q:=H_\lambda-H_q^*\le2d(1+u_\lambda)\varepsilon^2 <\frac{201}{10}\varepsilon^2<\frac1{1440}. \tag{22} \]

The constant degree makes the new conserved weighted means proportional to the total means, which remain zero. The boundary of (18) has excess at least \(R_0^2/10=1/1440\). The exact loss law (4), (21) and (22) therefore keep every actual source inside (18) for all future unforced time under its own constitutive law. Both constituent cycle windings remain one. In particular,

\[ \|x(\tau_h+T)\|_2^2+\|y(\tau_h+T)\|_2^2 <201\varepsilon^2,\qquad T\ge0. \tag{23} \]

Thus the distance to the aligned target on \(\mathcal M\) is below \(15\varepsilon\) for all future time. On this compact trapped mean leaf the only invariant set with zero loss is the target: \(Lx=0\) forces \(x=0\), invariance of the form row forces a critical potential, and the strictly convex potential has only \(y=0\) in the cell. The usual invariant-set argument consequently also gives convergence. This proves local full-state identity and attraction for these actual sources, not transverse attraction of every arbitrarily excited point of the entire family (5).

Finite carried-state comparison and collective row defect

Fix the prospective comparison interval \(0\le T\le1/10\) in \(\tau\), with original duration \(T/e\). No input is applied during it. The full vector field (3) has global raw-coordinate sup-norm Lipschitz constant at most

\[ \max\{2+2\gamma u_\lambda,2\gamma\}<3. \tag{24} \]

Comparing the actual source with the exact aligned equilibrium, the same actual law and support therefore give

\[ \|(x,y)(\tau_h+T)\|_\infty \le e^{3T}\varepsilon \le \frac{\varepsilon}{1-3T} \le R_T:=10\varepsilon/7. \tag{25} \]

This bound carries every fine coordinate, including the asymmetric internal errors that need not lie on \(\mathcal M\).

For actual collective observations define \(\bar X_r=\frac19\sum_jx_{r,j}\) and \(\bar\Phi_r=\frac19\sum_jy_{r,j}\), retaining the original lifts. Their centered phase fluctuations have magnitude at most \(2R_T\). Since both internal and cross degrees are regular, summing the fine linear rows closes the diffusion contribution and the entire phase row exactly. Only the nonlinear cross current has a remainder:

\[ \begin{split} \bar X_0'&=k_q\{\bar X_1-\bar X_0+ \gamma J_{\lambda,q}(\bar\Phi_1-\bar\Phi_0)\}+\rho,\\ \bar X_1'&=-k_q\{\bar X_1-\bar X_0+ \gamma J_{\lambda,q}(\bar\Phi_1-\bar\Phi_0)\}-\rho,\\ \bar\Phi_0'&=\gamma k_q(\bar X_0-\bar X_1),\qquad \bar\Phi_1'=\gamma k_q(\bar X_1-\bar X_0),\\ |\rho|&\le4\gamma k_q u_\lambda R_T. \end{split}\tag{26} \]

For each cross edge the actual phase difference differs from the registered mean-plus-offset difference by at most \(4R_T\). The global current derivative bound \(u_\lambda\) and the \(9q\) edges give the last inequality in (26); internal currents cancel exactly. The opposite remainders retain total form conservation. Formula (26) is a controlled defect of collective closure, not an additional autonomous law for hidden fluctuations or a memory-free quotient.

Here the actual component means at handoff are zero, so the exact collective reference is the aligned equilibrium. These estimates do not claim that this particular source launches a nonzero collective excitation. Arbitrary collective initial coordinates still obey the exact restriction proved above, but their physical preparation and full transverse stability require their own premises.

Implementation and independent controls

The inheritance owner rebuilds both prescribed supports, their actual degrees, harmonic and coordinate maps, full storage and static collective rows. Its acquired-state consumer reuses isolated recovery before admitting the new event, full-state barrier and finite closure defect. The prior three-component handoff uses that same recovery owner while retaining its own contact contract. Neither consumer accepts an incoming verdict as source evidence.

The acquired-state controls check independent asymmetric residual work, transverse coercivity, exact subgrid corrections, unsupported sources and invalid primitives. The static algebra controls compare the full eighteen-node edge sums with both collective rows and transformed storage. The shared recovery controls preserve original handoff behavior and separate isolated admission from later application events. These calculations provide conditional proofs and static certificates; no trajectory, reserved response or frozen producer is evaluated.