Causal composition of acquired sine-class interfaces¶
The collective interface already eliminates the 48 noncentral coordinates of the three-C9 system. Its block-diagonal hidden kernels implicitly contain a component decomposition. The earlier reduced-port composition also derives degree-aware assembly and storage balances for its different retained surrogate. Repeating either factorization would not establish a new composition result.
This owner gives a substitution theorem: component causal maps may be connected through the actual form and phase boundary inputs, with an explicit error bound after feedback. It also constructs a particular cubic component hierarchy with a certified complete-state residual, without evaluating another coefficient or response. The distinction is between a prescribed disturbance and an input generated by a neighboring pattern under the same law. The law, support and bridge occurrence remain supplied premises.
Fixed support and exact component boundary rows¶
Keep nodes \(0,\ldots,26\), three C9 rings, bridges \((4,13),(13,22)\), unit capacities and class tuples \((1,k,1)\), \(k=1,2\). The central nodes \(c_r\) have final degrees
all other degrees are two. Let \(D\) be this final degree matrix, \(L\) the whole graph Laplacian and \(A=D^{-1}L\). With \(\gamma=1/(1023\pi)<g=1/3000\) and structural clock \(\tau=(1023/1024)t\), the complete rows are
Continuous phase lifts are retained. Targets have aligned central origins and the original ring windings. Actual source deviations retain the acquired-family association; a Cartesian coordinate cover is not a claim of independently preparable corners. No source is reacquired or reset.
For ring \(r\), let \(L_r,S_r,U_r\) contain only its nine internal edges, where \(U_r=\sum_{\{i,j\}\in r}[1-\cos(\theta_j-\theta_i)]\). Its matrix \(D_r\) still uses the final degrees, including \(d_r\) at the center. Write \(X_r=x_{c_r}\), \(\Phi_r=\theta_{c_r}\), and define unnormalized boundary currents
The normalized two-channel input and the open component equations are
The local selector in (4) means the center in the ring's node order. Substitution of (2)-(3) reconstructs every row of (1) exactly. In particular, the two inputs in (3) are correlated consequences of the same neighboring forms and phases. Omitting the phase input changes the complete law even when the form row appears correct.
An isolated degree-two C9 has a different normalized central row. Its coefficients cannot replace (4). Here no edge is added or deleted: the composition is a description of the already admitted support.
Component charges, storage supply and events¶
Define the component weighted charges \(m_r^x=\mathbf1^TD_rx_r\) and \(m_r^\theta=\mathbf1^TD_r\theta_r\). Between events,
Both edge currents reverse sign at the other endpoint. Summing (5) therefore preserves the two full weighted charges. Component charges generally change: the neighboring organization supplies a real boundary input, not an independently chosen forcing value.
For internal storage put
Direct differentiation of (4) gives
The supply uses internal pressure and phase gradient, which depend on hidden state. It is not \(X_ru_r^x+\Phi_ru_r^\theta\).
Each existing bridge contributes its storage once:
Combining its derivative with (6), or substituting the recomposed rows in the whole storage, yields
The sum of internal losses alone is not the right side of (7). Bridge storage and the boundary supplies are required.
For simultaneous central form jumps \(x^+=x^-+B_pq\), phase is unchanged, and
The weighted form-charge jump is \(3q_1+4q_2+3q_3\); the phase charge does not jump. Cross terms in (8) cannot be discarded. Contact work from the original joining event retains its original certificate; repartitioning the equations is not another support event.
These are identities of the exact composed law. Approximate component substitutes do not acquire exact storage or charge conservation merely because their outputs are close. Their residuals enter the balances as additional terms.
Component maps and the closed causal problem¶
An exact component map \(\mathcal P_r\) takes its complete pre-input 18-coordinate state, its supplied central form jumps, and two locally integrable input histories \((u_r^x,u_r^\theta)\). It returns its central form and phase history from (4). It retains all sixteen hidden coordinates through their causal influence. There is no instantaneous input-to-output feedthrough; inputs enter time derivatives. Form jumps are applied separately and carried through subsequent flow.
The internal diffusion is a Markov generator in maximum norm: \(-D_r^{-1}L_r\) has nonnegative off-diagonals and zero row sums. The normalized internal sine and phase-Laplacian Lipschitz bounds are at most two. Thus the open equations are globally Lipschitz in state for prescribed integrable inputs.
Let \(\mathcal B(v)\) denote the map (2)-(3) from the six central coordinates to all physical input channels. The exact composed interface solves
The complete pre-input source is a fixed argument, not determined by the six current ports. Substitution proves that (9) is equivalent to (1) with the same initial state and events. The globally Lipschitz full field therefore gives a unique solution on every finite window; equivalently one can solve the Volterra equation successively on short intervals. No contraction on the entire window is assumed.
For identical sources and jumps but different open inputs, heat contraction and the phase row give the component response estimate
It also bounds the complete local state difference. All input components in (10) are the normalized physical channels (3). Since a center has \(d_r-2\) bridge neighbors and \(\max_r(d_r-2)/d_r=1/2\),
The sine estimate is global on real phase lifts.
A portable output-map substitution theorem¶
Suppose replacement component maps \(\widetilde{\mathcal P}_r\) have independently certified discrepancy at most \(\eta\), uniformly over the entire declared two-channel input-history domain, with the same sources and jumps. Suppose both exact and replacement feedback solutions exist and their inputs stay in that domain. In particular, existence of a replacement feedback solution is not implied by its pointwise output tolerance alone.
Applying (10) to the exact maps and then (11), the closed output error \(e(t)\) satisfies
Its scalar resolvent gives, for \(0\le H\le2\),
For \(\eta=0\), the error is zero. The rational sufficient estimate \(e^{(1+3g)H}\le e^{2+1/500} <(11/4)^2(500/499)<8\) proves the last constant.
This is a modular substitution result, rather than a fit of one global output. Its hypothesis is stronger than an error for a single prescribed impulse. The existing central-impulse \(R_5\) bound does not certify arbitrary continuous form-and-phase inputs, and is not inserted as \(\eta\) in (12). A component model that drops hidden initialization must separately satisfy the uniform input/source hypothesis; linear invisibility of an initial mode does not prove nonlinear invisibility.
A second substitution contract using reconstructed state¶
There is a useful alternative when each replacement supplies a complete 18-coordinate causal lift. After recomposition, suppose the lifted state has the full-law defects
The maximum initial discrepancies are \(\eta_x,\eta_\theta\). Include cumulative event mismatches in those budgets if the event maps do not match exactly. Let \(R_x\ge\int_0^H\|r_x(t)\|_\infty dt\) and \(R_\theta\ge\int_0^H\|r_\theta(t)\|_\infty dt\). For example, uniform residual bounds \(\rho_x,\rho_\theta\) give \(R_x=H\rho_x,R_\theta=H\rho_\theta\).
The exact complete-law solution and the lift then obey
Writing \(a_x=\eta_x+R_x\), \(a_\theta=\eta_\theta+R_\theta\) and \(\alpha_H=2gH<1\), one obtains
Diffusion contraction is retained, rather than replaced by a growing absolute row-sum estimate. Both phase and form are needed to close these inequalities.
Since a central Laplacian row has absolute row sum \(2d_r\), the same bound gives \(|(Lx)_{c_r}-(L\widetilde x)_{c_r}|\le2d_rX\). For an identical simultaneous jump \(q\), its quadratic work term in (8) cancels between the two histories, so
This is a work enclosure, not an energy identity for the substitute. Indeed its continuous storage derivative includes the additional terms \((L\widetilde x)^Tr_x+ \nabla U(\widetilde\theta)^Tr_\theta\). Exact dissipation and identity remain properties of the complete law under its own premises.
If exact bridge currents are evaluated on the replacement ports, the local lifted row defects assemble directly into (13). If bridge currents are themselves approximated or truncated, their defects must also be included. An open-loop port tolerance alone supplies neither of these full-row residual budgets. Equations (12) and (14) are different admission contracts.
Constructing the local cubic interfaces without a full-state solver¶
The following witness supplies an explicit bound for (14). It uses component kernels and the complete amplitude hierarchy, not a numerical trajectory or a newly evaluated coefficient.
For each ring with cosine \(c=\cos(\kappa_r\,2\pi/9)\), define
The sixteen hidden coordinates have the same grounded eight-node path generator, propagator \(G_r\), port-to-hidden factor and hidden-to-port factor as the explicit kernel owner. Only the two-coordinate local central generator in (15) is internal. Adding the linear bridge feedback reconstructs the old full six-port generator exactly. In particular, replacing (15) by a degree-two isolated center fails this recomposition.
Let \(A_b\) be the three-port bridge Laplacian divided rowwise by \((3,4,3)\). Around the aligned central target, the bridge feedback has linear part and cubic form part
The bridge quadratic term is zero because its target phase difference is zero. This does not eliminate the internal ring quadratic term.
Admit external central form impulses with total variation
Each component solves its two-coordinate first-level Volterra equation using \(E_r^{\rm int}\), its own self-memory kernel, and the first-level input \(\mathcal B_1(v_1)\), besides its external jumps. Its first hidden level is the explicit convolution of that port history. The second hidden level is the convolution of the internal quadratic force. Reflection gives \(v_2=0\), but \(h_2\) is retained. The third port equation receives \(\mathcal B_1(v_3)+\mathcal B_3(v_1)\) and its internal forcing
Hidden third levels are reconstructed by the same fixed convolutions. Only first-level forms jump at an external impulse. All levels carry between events. The linear initialization source is independently composed with \(\mathcal B_1\), retaining every ring's complete pre-input source.
These are three two-coordinate component equations at each retained visible level. The hidden levels are explicit causal integral functionals; there is no new simultaneous 54-coordinate integration. The construction is a triangular succession of linear Volterra equations: the second and third forcing terms are polynomials in already determined lower levels. Finite input variation and complete finite initialization therefore give a unique solution of each level on every finite horizon. The envelopes below then establish the stated approximation domain; existence is not inferred from closeness alone. Adding the component equations and (16) gives exactly the first, second and third equations of the complete amplitude hierarchy. Uniqueness at each successive linear Volterra level proves equality with that hierarchy. Thus independently assembling the components does not omit a path through a hidden odd mode or a bridge.
This construction composes the bridge order by order. Evaluating the untruncated sine of the approximate port phases would add higher orders and would be a different approximation. The old reflection remainder transfers to the constructed hierarchy by the proved equivalence; it does not automatically transfer to such a different feedback algorithm.
An explicit complete-state defect and finite bound¶
Let the maximum initial form and phase residuals about the declared common origins be at most \(\epsilon\ge0\). The uniform numerical statement below additionally assumes \(\epsilon\le10^{-32}\). The linear source retains those residuals exactly. Original acquired sources use their original common origins; any later centered description must retain its accumulated means separately. No parity is assumed for the actual initial residuals.
Use the complete lift \(\widetilde z=z_s+z_1+z_2+z_3\), including hidden \(z_2\). Let \(D_H=1-2g^2H^2\), \(\ell_H=1-2gH\), and set
These are envelopes, not measured coefficients. The first bound uses the heat-phase formula for the signed central input measure. The normalized quadratic and cubic maps have bounds \(\|Q(a,b)\|_\infty\le2\|a\|_\infty\|b\|_\infty\) and \(\|T(a,b,c)\|_\infty\le(4/3)\|a\|_\infty \|b\|_\infty\|c\|_\infty\). Integrating their form forcings and the linear phase feedback gives the common denominator \(D_H\) and the remaining bounds in (19).
Taylor expansion of the real sine about each target edge has a normalized fourth-order remainder at most \((2/3)\|y\|_\infty^4\). Put \(w=y_2+y_3+y_s\). The lifted form equation includes \(Q(y_1,y_1)\), \(2Q(y_1,y_2)\) and \(T(y_1,y_1,y_1)\). Subtracting these terms from \(Q(y_1+w,y_1+w)+T(y_1+w,y_1+w,y_1+w)\) and bounding the fourth-order remainder gives
All terms are nonnegative rational expressions in the admitted budgets. This includes internal and bridge truncation, as well as nonlinear source interactions. In particular \(Y_s\) is not silently set to zero for a zero-input history.
The phase row of every level is linear and exact, so its defect vanishes. Sources and jumps of the lift and complete law agree. Consequently (14) gives
Unlike the sharp central reflection remainder, this is a bound on all reconstructed coordinates.
Every expression in (19)-(21) is nondecreasing in the nonnegative budgets on (17). At their upper endpoints and \(\epsilon=10^{-32}\), exact rational substitution gives
This checks a whole finite input/window domain, not a coefficient or trajectory sweep. An arbitrary new approximate local map still needs its own (12) or (13) admission; (22) certifies only the explicitly constructed hierarchy. The hierarchy here is a mathematical causal functional. Numerical convolution, coefficient or rounding errors of any executable approximation require their own additional allowance; they are not included silently in (21).
A named interaction distinction survives composition¶
Use the already declared single mediator jump \(q=7/10000\), \(H=1\), and read \(x_{13}(H)\) separately for each class. The response-free comparison gives nominal full-law minus grounded tangent port output strictly above \(1/40000\). The comparator retains its own visible-source uncertainty and has no hidden memory. Each model's actual-source allowance is at most \(B_s=\epsilon/(1-2g)\); each reading error is \(\delta=10^{-8}\).
The constructed component hierarchy has its actual linear source, and (21) bounds its difference from the actual full response. Thus its recorded output remains above the separately recorded grounded comparator by at least
The coarser uniform \(E_x\) in (22) already suffices. No fresh coefficient is needed to establish this strict inequality. The retained independent full-law result is compatible finite evidence for the same supplied law and word; it is not a new evaluation of component feedback.
The sharper reflection-based \(R_5+E_{\rm init}\) bound also remains available after the hierarchy equivalence proof. It can preserve the older delicate class-dependent mixed statistic under its own word, source and recording budgets. Bound (22) alone is not claimed to resolve that much smaller contrast.
The structural consequence is that the neighboring patterns can supply both continuous boundary channels while the component maps retain their internal dynamics, rather than treating those inputs as independently prescribed scalars. This does not derive a unique interaction law. It does not show that nonlinear correction is necessary for the particular memory-loss comparison in (23). Formation, identity and work certificates belong to the exact complete state; output fidelity alone does not transfer them to every substitute.
The general domain (17) remains an approximation domain. It does not assert that every allowed impulse schedule satisfies the old work ceiling or acute trapping condition. The two specific existing words retain their separately established event and identity premises. There is no additional support event, response campaign, source acquisition or physical measurement in this composition proof.
Implementation and evidence boundary¶
The private structural and bound owner reuses the admitted complete-law geometry and shared scalar admission. It exposes internal degree-aware partitions, both bridge rows and their linear/cubic factors, the conditional two-row residual bound, and the explicit cubic residual certificate. It evaluates no causal kernel, finite-time coefficient or trajectory. The generic output-map theorem (12) is an analytic contract; the implementation does not infer its uniform domain or tolerance from a supplied endpoint report.
The independent component controls compare recomposition with complete edge equations, check charges and storage supply, reject degree-two and form-only substitutions, retain hidden initialization, and exercise exact residual arithmetic and representation boundaries. The retained forward-evidence audit reconstructs the existing nominal endpoints and comparison allowances without regenerating the response. Applying the new bound to those retained intervals is a compatibility consequence, not another reserved evaluation or a proof of coefficient generation.