A nonlinear collective interface for acquired sine classes¶
The repeated interaction theorem retains a class-dependent nonlinear response under complete carried histories. This owner derives an interface through the three central ports. Its retained observations have six coordinates; the remaining 48 coordinates enter causal kernels and initialization terms. The interface is a controlled nonlinear history functional, not a closed six-dimensional Markov law.
The earlier mediator memory retained 36 donor/receiver coordinates and an exact tangent memory term. Here the quadratic hidden correction is eliminated explicitly into a cubic port functional. It cannot be discarded merely because its direct port output vanishes. The proof reuses the complete amplitude hierarchy and its analytic remainder, without evaluating another coefficient, source or nonlinear trajectory.
Complete model, input family and retained observations¶
Keep the 27 nodes, three C9 components, contacts \((4,13),(13,22)\), unit capacities, actual degrees and class tuples \((1,k,1)\), \(k=1,2\). In the original continuous phase lifts,
The nominal target \(\Theta_k\) has aligned central origins and each component's stated winding. Let \(y=\theta-\Theta_k\) and \(z=(x,y)\), retaining all 54 original coordinates in the law.
Let \(B_p=(e_4,e_{13},e_{22})\). A finite signed central-form impulse measure consists of fixed times \(t_j\in[0,H]\) and vectors \(q_j\in\mathbb R^3\), with jumps
Simultaneous impulses are combined into one vector. Values at an observation/event endpoint are right-continuous unless a preevent value is explicitly requested. No future input is inferred from an evaluated response. This input family includes the four supplied donor words with impulses \(7/10000\) at zero and one.
The visible vector is
All other forms and phases are hidden in \(h\in\mathbb R^{48}\). The hidden coordinates include both reflection parities. They are not replaced by component means or stationary values. There is no claim that these observations are a minimal sufficient interface.
Approximation bounds below hold over each admitted finite window. The large return dwell and indefinitely many accumulated impulses do not lie in the domain \(H\le2\) of one cubic approximation. For repetition, the full-law return supplies the next actual source; the next description retains that source and its accumulated means. Changing a description's time origin is not resetting the state.
Six-port causal equations with hidden quadratic recoupling¶
Use the complete original-coordinate maps
where \(C_k\) is the degree-normalized cosine Laplacian and
The normalization includes the Taylor factorials. Both evolution rows use the same original clock. No phase residual is divided by \(\gamma\).
Partition the original generator according to (3):
Here \(C_k^{hv}\) is a generator block, not the phase Laplacian \(C_k\). The hidden propagator has an explicit finite modal form. In each ring order its eight hidden nodes as \((5,6,7,8,0,1,2,3)\). They form a grounded path with normalized Laplacian diagonal one and adjacent entries \(-1/2\); every hidden degree is two. For that ring's cosine \(c\), set
The normalized spatial sine vectors have entries \(\sqrt{2/9}\sin(jm\pi/9)\). Each modal form/phase pair therefore has generator \(-\lambda_mT_c\). In an interleaved form/phase block notation, the hidden propagator entries are
The full hidden propagator consists of the three such ring blocks. This diagonalizes only their spatial part; both complete dynamic coordinates remain in every \(2\times2\) exponential.
The port meets both grounded-path endpoints. With actual port degree \(d\in\{3,4\}\), the visible-to-hidden and hidden-to-visible factors are \(T_c/2\) and \(T_c/d\), respectively. The resulting self-memory block is explicitly
Thus \(\mathcal K_{c,d}(0)=T_c^2/d\). The three self-memory blocks are diagonal by component; instantaneous bridge coupling remains in \(E_k\). Reflection acts on mode \(m\) with sign \((-1)^{m+1}\). Four spatially odd modes disappear from the linear port kernel (5b), but all eight modes in (5a) remain available to the nonlinear forcing and initial-state source.
For a port history \(w\) and full forcing \(f\), define
These fixed kernels derive from the entire hidden block; they are not fitted from the response. Time-zero initialization terms are kept separately below.
For the zero-residual nominal reference, scale the whole measure in (2) by an auxiliary amplitude parameter. Its first level obeys the six-port Volterra equation
Set \(z_1=(v_1,h_1)\), in the partitioned order. The second forcing is \(f_2=\mathbb Q_k(z_1,z_1)\).
Let \(\mathcal R\) reflect each local index \(j\mapsto8-j\) in both channels. It fixes every central port and commutes with \(J_k\). Central form inputs are reflection even; the nominal amplitude hierarchy has \(\mathcal Rz_n=(-1)^{n+1}z_n\). In particular,
To verify (8), \(f_2\) is reflection odd, its visible part vanishes, and \(B_kG_k(t)(f_2)_H=0\). The homogeneous second port equation has the unique zero solution. The hidden response itself need not vanish and is a known causal quadratic functional of \(v_1\).
The complete cubic port equation is
Every coefficient carries through every event. Only the first form level jumps. Equations (7)-(9) require two six-coordinate port histories. The hidden first and second levels are explicit integral functionals, not another 48-coordinate unknown evolution. The reconstructed third hidden level is needed only when a hidden-state or boundary-pressure observation is requested.
This is a nonlinear input/output reduction with memory. Substituting (6) successively expresses the cubic drive as nested causal integrals of the input measure, retaining both the direct trilinear term and the quadratic-then-quadratic term. The latter cannot be erased by the identity \(v_2=0\). For example, at mediator center 13, take the reflection-even phase vector \(e_{13}\) and the odd vector \(e_{14}-e_{12}\). Then \(Q_k\) at that port is \(\sin(k\alpha)/4\ne0\), \(\alpha=2\pi/9\). Odd internal information can therefore feed an even cubic port output. A six-state instantaneous law omitting that functional has not been derived.
Complete initialization without a reset¶
Let \(z_0=(v_0,h_0)\) be the complete actual pre-input deviation \(z(0^-)\), relative to its declared reference origins. A time-zero impulse belongs to (7), not to this source again. The exact linear-source port response \(v_s\) obeys
It is precisely the port projection of \(\exp(J_kt)z_0\). Its argument is the complete hidden initial vector \(h_0\), rather than a prescribed equilibrium or component mean. Some components, including reflection-odd hidden data, cancel in this linear port projection. They can nevertheless affect the nonlinear response; their full admitted uncertainty remains in the initialization defect below. Class and present port values do not determine the source. A known source can be retained explicitly, while an unknown source must retain its admitted uncertainty rather than being assigned zero.
Choose reference origins together with the source bound, not by an unproved change of gauge. In the first acquired word, use the original zero common origins: its raw coordinate errors are bounded by \(\epsilon\), and (10) retains their small possibly nonzero joined means. Degree-centering that initial family would instead give the weaker coordinate bound \(31\epsilon/29\).
For later words, the proved return family bounds each small degree-centered representative, while its accumulated common form and phase means are retained as exact reference origins. Equation (10) consumes that representative, and the common output constants are added back exactly. Those absolute means are not assumed to be \(\epsilon\)-small. This is the same mean bookkeeping as in the repeated interaction.
The linear source need not be mean zero if other admitted origins were used. The collective approximation is
Its source dependence includes a rigorous nonlinear initialization defect, derived next. Equation (11) alone does not assert that a linear source captures all nonlinear dependence on initialization. No reflection parity is assumed for the actual source.
Uniform finite error for ports, phase and pressure¶
Let \(g=1/3000\), \(\ell_H=1-2gH\), \(D_H=1-2g^2H^2\), and suppose each initial form and phase coordinate differs from its chosen nominal reference by at most \(\epsilon\). The domain (2) has \(\ell_H>0\), \(16g^2H^2<1\) and \(2gA_*<1\).
For the nominal reference, the Markov heat phase satisfies
The \(\ell^1\) variation in (2) is important for simultaneous multichannel events. The full real nominal phase consequently satisfies \(\|y_{\rm nom}\|_\infty\le gA_*/D_H\). The actual-to-nominal full source difference satisfies \(\max(X_\Delta,Y_\Delta)\le\epsilon/\ell_H\), by heat contraction and both evolution rows under identical jumps.
Write the normalized nonlinear sine residual as \(N_k(y)=D^{-1}[S(\Theta_k+y)+L_{c,k}y]\). Its derivative has maximum-norm bound \(\|DN_k(y)\|_\infty\le4\|y\|_\infty\). Thus
Subtract the complete linear-source solution \(\exp(J_kt)z_0\) from that full source difference. Its remaining form error has zero initial value and no event jumps. The linear phase feedback gives denominator \(D_H\), so the entire original-coordinate form defect is bounded by
Its phase defect is at most \(2gH E_{\rm init}\). At \(A_*=0\) the small quadratic initialization term in (14) remains; it must not be silently replaced by zero.
For the zero-residual nominal response let \(P_+=(I+\mathcal R)/2\). It is a maximum-norm contraction and commutes with \(A\). The entire projected form is odd in the auxiliary amplitude. The complex-amplitude proof therefore applies to \(P_+x\), not just one central scalar:
The proof uses the disk radius \(1/(2gA_*)\), the full nonlinear minus tangent form bound \(8gH/D_H\), and odd Cauchy tails. For \(A_*=0\), set \(R_5=0\). It requires no new coefficient calculation. Applying (15) to arbitrary actual residuals would be invalid; (13)-(14) are their separate treatment.
Central selectors and central Laplacian rows are reflection invariant. Combining (14)-(15), the interface errors over the whole window obey
Here the auxiliary full-coordinate \(x_{\rm int}\) is reconstructed from the form parts of \(z_s+z_1+z_3\), plus the common form. It is used only for these reflection-invariant observations; an unprojected full-state error of size \(R_5\) is not asserted. The hidden quadratic level remains in the cubic forcing even though its direct contribution to these observations vanishes. For phase, integrate \(\gamma A P_+\) on the full projected form remainder; a central-form bound alone would not justify this step. For pressure, use the exact Laplacian row norm \(2d_p\), equal to six at donor/receiver and eight at the mediator. The reconstructed full coefficient/source functionals supply that row observation.
Boundary work and limits of the general input domain¶
The work of one simultaneous central form jump is exactly
The cross terms cannot be replaced by a sum of independent scalar works. Equation (16) gives a preevent pressure/work enclosure from the same causal interface, with error at most \(\sum_p2d_p|q_p|(R_5+E_{\rm init})\) on its linear term. The exact phase row also gives \((Lx)_p=(d_p/\gamma)\theta'_p\), but differentiating a retained pointwise phase enclosure does not produce a justified derivative enclosure. No such shortcut is used.
The interface retains an interaction port, not a storage determined solely by six present values. Hidden energy and initialization remain relevant to balance and future output. The general variation domain (2) is an approximation domain, not a universal work or identity certificate. For example, a single donor impulse \(7/5000\) at the exact target costs \(2.94\,10^{-6}\), above the original per-event allowance. The fixed two-event word alone inherits its separate full-law work, trapping and return proof.
Preservation of the repeated class-dependent interaction¶
Use the retained complete cubic contrast enclosure
This is the nominal cubic output of (7)-(9) for the already declared word, by uniqueness of the eliminated hierarchy. Its generating execution remains the retained coefficient premise. Equation (18) is not a new observation and is not inferred from the full nonlinear forward report.
Let \(a=b=7/10000\), \(H=2\), \(\epsilon=10^{-32}\) and \(\delta=10^{-30}\). For each history use \(A_{00}=0,A_{10}=a,A_{01}=b,A_{11}=a+b\), and set
The factors two count the two classes; the two single-input histories remain distinct even when \(a=b\). The retained linear-source term of (10) has contrast magnitude at most \(B_s\), with no cancellation assumed for the distinct recurrent residuals. The actual nonlinear contrast therefore lies in
There is one nominal truncation allowance. Starting instead from a full-response interval and composing an interface approximation in both directions would require its own additional error accounting; that two-step construction is not used in (18)-(20).
The persistent tangent alternative has its own \(B_s\) residual allowance. Both mean progressions cancel in the mixed statistic and each model contributes its own eight reading errors. The sufficient recurrent separation margin is
Exact rational substitution gives
These inequalities use the retained coefficient premise. The tiny initialization defect is additional to the retained linear source, not a claim that arbitrary source uncertainty has disappeared. The separate full-law return establishes the next word's small centered source and retains its means; no cubic evolution is asserted over the long dwell.
The result is a nonlinear collective interface with quantified memory, initialization and observation errors. It supplies neither an autonomous input selector nor a physical calibration, an intrinsic particle label or a new fundamental interaction law. Finite-dimensional Markov closure, minimal memory and inheritance by different support or constitutive laws remain separate questions.
Implementation and evidence boundary¶
The private collective-interface owner provides the fixed structural blocks, edge factors, modal-kernel description and error certificates. It reuses the admitted amplitude tail and repeated-interaction bounds. The separate changed-input prediction executes the causal time-coefficient calculation with its own fixed input, source, numerical and observation budgets. The structural descriptor itself does not evaluate a kernel or forward response.
The independent structural and budget controls check the complete partition, hidden modes, nonlinear recoupling, initialization bounds and simultaneous-event work. The retained cubic evidence reader reconstructs the existing coefficient enclosure and its association with (18)-(22), without regenerating coefficients or trajectories. Supplying a cubic interval to the private certificate remains conditional on that interval's independently established evidence. No new response archive is needed for this theorem.