Repeated interaction of joined acquired classes¶
The retained class comparison encloses one nonlinear interaction under actual preparation and reading uncertainty. This theorem gives a finite return condition for repeating the same supplied word on the same joined system. Every branch keeps all 54 coordinates, its accumulated means and its own residual history. There is no repeated acquisition, state reset or new response evaluation.
The proof reuses the strict modified-energy method of isolated-class maintenance, with newly derived joined spectral constants and the degree-metric isometry. The new forward return family is not identified with the original formation image.
Complete state, branch means and repeated observation¶
Keep the three C9 rings, contacts \((4,13),(13,22)\), unit capacities and actual degree matrix \(D\). Write \(d=D\mathbf1\), so \(d^T\mathbf1=58\), and retain
The class targets \(\Theta_k\) have component windings \((1,k,1)\), \(k=1,2\), and local lifts \(2\pi k_c(j-4)/9\). Their bridges have zero phase gaps; their internal sine forces cancel. Common form and common phase translations therefore give equilibria on every conserved-mean leaf. All reference closing-edge integers remain fixed.
Let \(h=(A_h,B_h)\) range over \(00,10,01,11\). During each word, apply form increments \(aA_h e_4\) at time zero and \(bB_h e_4\) at time one, with
Read the receiver form at time \(T\), then leave that same full state unforced for a common dwell \(W\) derived below. The period of the supplied word is \(P=T+W\). The contact event occurs only once. Events at every later word retain their complete preevent states.
For a full state define
These are proof coordinates, not operations performed on the state. For Euclidean capture geometry use the different projection \(P_0=I-\mathbf1\mathbf1^T/27\). Its squared full-state quotient radius is \(\|P_0x\|_2^2+\|P_0(\theta-\Theta_k)\|_2^2\). The degree-centered \(P_D\) is orthogonal only in the degree metric; no Euclidean contraction property is attributed to it. Both means are conserved between jumps. A donor jump \(q\) changes \(\mu_x\) by \(3q/58\); it does not change \(\mu_\theta\). Consequently, before word \(n\), counting the initial word as zero,
The initial means can differ between the two acquired classes. They are common to the four branches within each class only at acquisition; (4) keeps all subsequent differences explicitly.
Use \(\sigma_{00}=\sigma_{11}=1\) and \(\sigma_{10}=\sigma_{01}=-1\). The repeated statistic is
Both \(\sum_h\sigma_h\) and \(\sum_h\sigma_h m_h\) vanish. Thus the accumulated common forms cancel exactly in (5), without resetting them or treating the branch residuals as equal. Eight fresh scalar reading errors contribute at most \(8\delta\) in each word.
Recurrent source family, work and trapping¶
On each branch's own mean leaf define the new return set
Products of these sets permit different residuals in all branches. The means retain (4). Since every degree is at least two, (6) implies coordinate errors at most \(\epsilon/\sqrt2<\epsilon\).
The initial acquired family is handled separately. Its componentwise zero-sum errors have Euclidean norms at most \(\epsilon\), as in the original source proof. Its joined means are bounded by \(4\epsilon/58\), so quotient centering only gives a coordinate bound \(31\epsilon/29\), not membership in (6). The already proved first-word work, trapping and comparison apply to that original family. The first unforced dwell will place it in (6). No stronger initial preparation is introduced.
For every subsequent word, \(L\preceq2D\) and criticality of \(\Theta_k\) give the initial excess-storage bound
For the phase term use Taylor's integral remainder and the global upper Hessian bound \(\nabla^2U\preceq L\). Componentwise zero sums are not assumed for the recurrent family. Common translations do not change this storage.
Put \(g=1/3000>\gamma\). For a first impulse \(u\in\{0,a\}\) and delayed impulse \(q\in\{0,b\}\), the carried-pressure estimate uses only the coordinate errors and the joined graph, and yields
The exact delayed work remains \(q(Lx^-)_{4}+3q^2/2\). Subtracting a constant initial form mean in a proof leaves that Laplacian unchanged. It does not replace the delayed state by its initial value. The spectral bound \((L\exp(-A))_{44}<9/8\) at unit time includes the actual donor degree three.
Starting from (7), the weaker already admitted ceiling \(22\epsilon^2+W_1+W_2\) is conservative. Likewise, for \(A_*=u+q\), a representative of the whole word obeys
Arithmetic-mean projection can only decrease the Euclidean norm in the last inequality. Exact rational substitution, inherited from the whole scale interval, gives
Each actual probe meets its \(2\,10^{-6}\) work allowance. The initial contact already met its separate \(10^{-12}\) allowance; it is not installed or charged again.
The joined acute barrier exceeds \(r^2/2700=1/388800\). Together with (9), (10) traps every postword state under uninterrupted relaxation in the same radius \(r\) quotient chart. All relevant edge cosines exceed \(c=1/20\). Indeed, each reference gap has magnitude at most \(4\pi/9\), and its perturbation is at most \(\sqrt2r\). Using the certified \(\pi>31/10\) and \(\sqrt2<17/12\), the remaining angle to \(\pi/2\) exceeds \(13/240\). Hence the edge cosine is greater than \(13/240-(13/240)^3/6>1/20\). The prescribed closing-edge integers specify the same acute circular gaps throughout. The events and flows throughout each word remain inside the admitted identity domain. No lower estimate of continuous dissipation is used as a work reserve.
Joined spectral constants and strict nonlinear decay¶
The fine graph has diameter at most ten. For \(d^Tz=0\), weighted bounded-range variance and a shortest path give
The normalized Laplacian is self-adjoint in the degree metric. On its mean-free subspace its spectrum therefore lies in \([\lambda,\Lambda]=[1/145,2]\). The upper bound follows from \(\sum_{ij}(z_i-z_j)^2\le2\sum_i d_i z_i^2\).
Set \(\widehat A=D^{-1/2}LD^{-1/2}\), \(\widehat x=D^{1/2}\bar x\), \(\widehat y=D^{1/2}y\), and remove the common null vector. The auxiliary coordinates and potential
obey exactly \(\xi'=v\), \(v'=-\widehat A v-\nabla W_k(\xi)\). The full phase and form rows are both used. This is a degree-metric isometry, not a change in the supplied law or clock.
On the trapped chart, \(cL\preceq\nabla^2U\preceq L\). The line from the target to any admitted state stays in that chart. Congruence by \(D^{-1/2}\widehat A^{1/2}\), without any commutation premise, bounds the transformed Hessian by
These gamma bounds follow from the same rational pi bounds as the selected complete law; no fitted rate is substituted.
With \(\beta=\lambda/4\), define
The shared modified-energy calculation gives
The cross terms cancel by the complete second row in (12). Exact rational constants are
The same constants hold for both classes and every branch.
One common finite dwell without an interval-grid floor¶
At the end of any admitted word the Euclidean quotient radius is less than \(r\). Weighted orthogonal projection minimizes the degree norm over common translations. Since the maximum degree is four, for either residual vector \(z\),
Thus both \(\|\bar x\|_D\) and \(\|y\|_D\) are at most \(2r=1/6\), and a common postword energy upper bound is
Fix the structural dwell and exact decay bound
Since \(\kappa W>256\) and \(\exp(1)>2\), \(\exp(-\kappa W)<\rho\). Keep \(\rho\) as an exact rational number. Materializing it first on a dyadic128 grid would destroy the required squared-norm resolution; no such operation is used.
Returning to the original degree norms gives
The last two strict comparisons are integer inequalities, not floating exponentials. Their respective upper bounds are below \(5.51\,10^{-70}\) and \(5.82\,10^{-74}\). The dwell in the original clock is \(1024W/1023\); this very large conservative time is not a laboratory feasibility claim.
The original first word is trapped by its own already proved certificate, so (18) places its actual endpoints in \(\tfrac12\mathcal K_k\). Thereafter (7)-(10) and (18) yield
on each successively shifted mean leaf. Here \(\mathcal W_{k,h}\) is the complete two-event word of duration two. Induction retains every actual branch for arbitrarily many supplied repetitions. It does not require equality of branch residuals, an exact equilibrium return or membership in the old formation image after the first word.
Recurrent response and the comparator's own memory allowance¶
For any word beginning in (6), compare each branch separately with its exact nominal reference, translated by that branch's initial common form and phase. Heat contraction and the real sine Lipschitz bound give original-coordinate errors at most \(\epsilon/(1-2gT)\) through the word. Identical jumps do not change these errors. Formula (4) cancels the reference translations in (5), even though the eight residuals are different. Hence
The exact retained nominal reference interval is
It is inherited from the independently integrated full nonlinear reference, not from a new coefficient or response evaluation. The retained comparison and its numerical premises remain explicit evidence dependencies. Its own target-rounding and numerical errors are already contained in (21).
Name the alternative before applying a null threshold: a fixed full tangent law about each joined class target, with the same graph, gamma, clock and repeated donor events. Its phase Hessian is the fixed cosine-weighted Laplacian at that target. Admit independent branch residuals in (6), with the compatible mean progression (4), at the beginning of its repeated protocol. This family contains the exactly correlated tangent histories from one shared initial state in (6); it does not assert that every original acquired source lies in (6).
The tangent model retains all 54 coordinates. Its quadratic potential has the Hessian bounds (13) globally. Explicitly, in its target-phase coordinates it obeys \(x'=-Ax-\gamma C_k y,\ y'=\gamma Ax\), where \(C_k=D^{-1}\nabla^2U(\Theta_k)\); its weighted quadratic form satisfies \(cA\preceq_D C_k\preceq_D A\). Its normalized phase operator has maximum-norm bound two, so the same word envelopes (9) hold. In particular, the postword degree norms are bounded by \(\sqrt{58}X\) and \(\sqrt{58}Y\), both less than \(1/6\). Weighted centering can only reduce these degree norms. Equations (16)-(19) then prove its own common finite return. No nonlinear trapping theorem, reset to an actual nonlinear state or assumed observation zero is used to supply that comparator continuation.
For its nominal histories, linearity makes the mixed response zero. Independent returned residuals supply instead the necessary allowance
If the tangent alternative starts from one common source and carries the four complete repeated schedules, exact linearity preserves their additive correlation and its mixed response is exactly zero. That narrower fact does not justify replacing (22) by zero on the larger independent-residual comparator family used here. The correlated case also covers an original acquired common source outside (6), by linearity alone; it does not need a new source preparation.
The mean condition is necessary. For example, add the constant form one only to class one's \(00\) branch and leave every phase and all other branches unchanged. Every quotient residual, work increment and return bound is unchanged, but the class contrast acquires the constant one. Thus quotient balls alone cannot bound (5); the actual mean progression and its mixed cancellation must be supplied and retained.
The nonlinear recorded band adds its \(8\delta\) allowance to (20); the independent alternative's recorded band is \([-B_L-8\delta,B_L+8\delta]\). Exact rational comparison gives
Thus every repeated word separates the two eight-record sets through the fixed statistic. Both source-memory allowances appear explicitly. The original first word has the same \(B_N\) allowance from its original source proof, so the comparison also covers \(n=0\) without asserting initial membership of the acquired family in (6). This conclusion is uniform in the number of repetitions, not an assertion that the branch states or their individual readings recur.
Scope and continuing costs¶
The first word retains its original acquired-source result. Later words use the new proved return family and its independent branch errors. Absolute means are retained by (4); they can grow without bound while the relative storage and geometry stay trapped. No periodic full state or unique periodic quotient orbit is claimed.
Every positive impulse is an externally supplied event. Over finitely many words its cumulative work is the sum of its actual preevent work; the uniform per-event allowances do not supply a finite reservoir for indefinite repetition. Continuous dissipation is not used to pay a jump. The chosen support, event schedule, long dwell, tiny source and reading allowances and the complete constitutive law remain premises.
This is a conditional repeatability theorem for an interaction property of organized patterns. It is not a new acquisition theorem, an autonomous intervention selector, an independently calibrated physical measurement or identification of a fundamental-particle property. No frozen evidence is changed and no new scientific trajectory is evaluated.
Implementation and retained-evidence boundary¶
The private proof calculator accepts only a supplied nominal contrast interval. It rebuilds the complete graph, weighted spectral certificates, recurrent work and radius bounds, the rational Lyapunov return and both residual/noise allowances. It reuses shared scalar admission, coordinate envelopes and the exact modified-energy kernels. The supplied interval is a conditional premise; scalar admission does not establish its scientific provenance, entry into the return family or mixed-mean compatibility. There is no new SDK execution path or solver.
The independent controls check the actual graph and metric, carried event work, return arithmetic, mean counterexample and exact error margins. Their retained-response association uses the existing read-only comparison audit to reconstruct (21) from frozen complete histories. Stored derivative and Picard generation remain explicit numerical premises; this arithmetic audit neither regenerates them nor evaluates a new response.