Predictive sufficiency of finite collective form records¶

Fixed observation-to-future comparison¶

The retained information comparison removes the exact collective initialization consumed by the preparation-based forecast. It asks whether two complete histories compatible with one existing noisy record must have nearby future mean forms. The target is an output diameter, not an inverse for every phase or hidden constituent coordinate.

Item	Fixed scope
Complete fine law	Unit C9-by-C5 support, node order `9c+j`, 90 unit edges, held unit capacities, full-circle pairwise sine; `x'=(-L_fx+gammaS(theta))/4+Rf`, `theta'=gammaL_fx/4`, `gamma=1/(1023*pi)`
Clock and horizon	Structural `tau=(1023/1024)*t`; local time `v=tau-200` runs from 0 to `T=16`. Continue the actual reached state after the word; no reset, extra event or changed support
Observation-start class	Original `abs(x_i)<=X=1/100`, `theta=psi+RPhi+z`, `psi_j=2pi*(j-4)/9`, child-zero hidden means, `abs(z_i)<=1e-6`, principal collective gaps at most `7/5`, winding in `{-1,0,1}`; additionally retain `\|\|u\|\|2<=a0=31/10^9`, `\|\|z\|\|2<=b0=601/10^10`
Unknown collective state	All five initial mean forms remain unknown, including their conserved common mean in `[-X,X]`. Fix only the unobservable common phase gauge by choosing initial `Phi_0=0` after a common rotation; no relative phase, winding or attachment mean is supplied to an inverse
Input and observations	Unchanged fifteen cubic ramps and fifteen preceding plateaus, marked children `0,1,2,3,4`, increments `1/10,-2/10,1/10`, ramp duration 1 and plateau duration `1e-5`; all five mean forms at both endpoints of every plateau, in original order, 150 raw scalars
Raw errors	Each scalar has arbitrary correlated additive error at most `eta=1e-14`. Input waveform, support, capacities, gain and sample clock are exact conditional premises; no apparatus error is silently inferred to vanish
Future target	Five mean forms at `tau=216`, or local time 16. The last reading is at `214.00015`; prediction includes the final ramp and the unforced tail, a span of `1.99985`
Decision	Joint overlap of the complete 150-reading record sets must imply a coordinatewise future diameter at most `2e-13`, uniformly over the declared class. This admits an abstract coordinate-midrange radius at most `1e-13`; a future reading adds its own `eta`
Static certificate	Fifteen formal columns: five initial forms, five scaled initial edge sines and five scaled initial edge cosines. Enclose the resulting 150-by-15 sampled heat-response matrix and 5-by-15 future matrix; evaluate no nonlinear target trajectory or raw response
Coefficient computation	Order-16 Taylor coefficients, shared dyadic128 interval jets, maximum cell `1/8`, separate input stages. Partition each stage from its left endpoint into full cells and a final shorter cell: 15 plateau cells, 120 ramp cells, eight tail cells. Each coefficient's propagated radius must remain at most `1e-12`
Weight proposal	Convert matrix midpoints to IEEE binary64, form `Mhat.T @ Mhat`, and use one NumPy solve for `K_raw=solve(Mhat.T @ Mhat, Mhat.T)`. Independently round its last ten rows and `Nhat @ K_raw` to their nearest multiples of `2^-40`, ties to even, obtaining `K_b` and W. Record the proposal backend; validate the retained rational weights against the interval matrices
Numerical policy	One fixed matrix assessment, no row selection, parameter scan, refinement, precision escalation, alternate inverse search or retry. Floating-point least squares proposes weights only; exact/outward residual inequalities decide admission
Failure and evidence	A singular/nonfinite proposal, coefficient-budget failure, nonpositive bootstrap denominator or unmet diameter is a scoped unavailable certificate, not an impossibility theorem. Freeze this owner, consuming source and driver before matrix evaluation; retain source/configuration, all matrix enclosures, rational weights, residuals and verdict. No frozen producer, source assessor or reserved response is replayed

The class contains the previously acquired sources, but acquisition of every member is not asserted. This is a structural information comparison with an unchanged complete law. A numerical result cannot supply physical preparation, calibration, a laboratory clock, a unique fundamental law or a physical identity.

1. Fine-source uncertainty is retained on both sides¶

For each valid fine history introduce the carrier with its own actual initial means. Those means are unknown to the observer; this is a proof comparison, not an initialization procedure. Rebuild the shared hidden transport from the declared primitive bounds, with g=1/3000:

\[ s=2gT,\qquad Z=\frac{b_0+s a_0}{1-s^2},\qquad \rho_0=\frac{gZ^2}{9},\qquad E_0=\frac{T\rho_0}{1-(gT)^2}<2.3\cdot10^{-18}. \tag{1} \]

Both denominators are positive. The projected nonlinear sine remainder and heat contraction give this uniform Euclidean mean-form error, hence the same coordinate error, on the full 16-unit history. It includes the hidden initialization without identifying its variances. It is not the much smaller preparation-specific error requiring known attachment means.

If two fine histories admit one common raw record, their true forms differ by at most 2*eta at every one of the 150 coordinates simultaneously. Their two carrier histories consequently satisfy

\[ \|u(t_j)\|_\infty\le\delta:=2(\eta+E_0) \quad\text{at all 30 vector sample times}. \tag{2} \]

No independence or cancellation of errors is assumed. Conversely, (2) alone does not authenticate a raw record or prove joint realizability. At the future endpoint, a carrier-difference bound must again add 2*E0 for the two fine histories.

The original identity proof also extends to this containing class through local time 16. Substitute T=16 in its full fine-state bound E_f=2gX+2g^2T and retain the internal, outer and collar margins. The known command never shifts an edge by more than 1/10 and vanishes after release; all those margins remain strictly positive. This is a whole-history fine-state statement, independent of whether the new information certificate passes. Work, storage and the additional unforced loss retain their separate account.

2. A static design with unknown circular phases¶

Let L=L5, A=L/4, and orient edge e from child e to child e+1 mod 5. Let B have tail -1 and head +1, so L=B B^T and S5(Phi)=-B sin(B^T Phi). Remove the exact prescribed offsets as in the carried forecast:

\[ Y=U-z_{cmd},\qquad \Psi=\Phi-H,\qquad Y'=-AY+\frac\gamma4 S_5(\Psi+H),\qquad \Psi'=\gamma AY. \tag{3} \]

At every reading and at the future endpoint, z_cmd=0, so Y=U. The two compared histories share H and every input stage. Their difference u=Y^(1)-Y^(2) equals their form difference also between readings.

For each history let a_e=(B^T Psi(0))_e, understood modulo 2*pi, and put

\[ p=\left(u(0),b_s,b_c\right),\qquad (b_s)_e=\frac\gamma4\left(\sin a_e^{(1)}-\sin a_e^{(2)}\right),\qquad (b_c)_e=\frac\gamma4\left(\cos a_e^{(1)}-\cos a_e^{(2)}\right). \tag{4} \]

These fifteen formal coefficients need not be independently realizable. Allowing arbitrary coefficients enlarges the comparison and cannot invalidate a bound for the actual subset. Initial-edge phasors are independent of the common phase gauge and branch representatives. No real phase difference between the two histories, common winding or angle inverse is required.

Write h(v)=B^T H(v). Define the static heat response V by

\[ V'=-AV-B\left[b_s\mathbin\odot\cos h +b_c\mathbin\odot\sin h\right], \qquad V(0)=u(0). \tag{5} \]

It is linear in p. Let M stack its five coordinates at all thirty sample times, and let N be its five-coordinate map at time 16. Thus V_samples=M p and V(16)=N p. M and N depend only on the declared support, clock and known command; computing them neither chooses an unknown state nor evaluates a nonlinear future response.

3. A difference remainder rather than an absolute model bias¶

Use B_* = max(||b_s||infinity,||b_c||infinity) and U_* = sup_[0,T] ||u||infinity. Since the carrier has two incident sine currents per node, ||S5||infinity<=2. Equation (3) and heat contraction imply

\[ \|Y(v)\|_\infty\le X+gv/2,\qquad \Psi(v)-\Psi(0)=\gamma\{Y(0)-Y(v)\} +\frac{\gamma^2}{4}\int_0^v S_5(\Psi+H)\,ds. \]

\[ \|\Psi(v)-\Psi(0)\|_\infty\le2gX+g^2T, \qquad D:=4gX+2g^2T. \tag{6} \]

Hence each endogenous edge-phase drift d has magnitude at most D. For q_k=exp(i*a_e^(k)), the difference left after freezing the two initial phasors is

\[ q_1(e^{id_1}-1)-q_2(e^{id_2}-1) =(q_1-q_2)(e^{id_1}-1)+q_2(e^{id_1}-e^{id_2}). \tag{7} \]

Its modulus is at most D*abs(q1-q2)+abs(d1-d2). The scaled phasor difference obeys (gamma/4)*abs(q1-q2)<=sqrt(2)*B_*. Moreover ||B^T A||infinity=2, so the complete phase row gives abs(d1-d2)<=2g integral_0^v ||u(s)||infinity ds. There are two incident edges per node and 2*sqrt(2)<3. Consequently the exact pair row can be written

\[ u'=-Au-B[b_s\odot\cos h+b_c\odot\sin h]+r, \quad \sum_c r_c=0, \quad \|r(v)\|_\infty \le3D B_*+g^2\int_0^v\|u(s)\|_\infty\,ds. \tag{8} \]

This remainder vanishes with differences. Substituting an absolute single-history phase error here would create an unrelated error floor and would not prove a finite-noise continuity statement.

The zero-sum remainder benefits from the heat spectral gap. For zero-sum w, lambda2(L)=(5-sqrt(5))/2>4/3, sqrt(5)<9/4, and norm conversion yield

\[ \|e^{-vA}w\|_\infty \le\min\{1,(9/4)e^{-v/3}\}\,\|w\|_\infty. \]

Split the integral at 3 and use exp(1)>8/3. The integral of this majorant on the positive half-line is at most 3+(27/4)*exp(-1)<177/32<6. Applying it to (8) gives

\[ R_*:=\sup_{[0,T]}\|u-V\|_\infty \le a B_*+\beta U_*,\qquad a=18D,\quad\beta=6g^2T. \tag{9} \]

All sampled difference vectors satisfy (2), including local time zero. For any time up to T, its most recent sample is at most two time units earlier. This includes the last ramp and the release tail; there is no unobserved two-unit interval mistakenly shortened to the tail alone. The static force in (8) has norm at most 3*B_*. Heat contraction from that most recent sample, followed by (8), therefore gives

\[ U_*\le\delta+6(1+D)B_*+2g^2T U_*. \]

\[ c=1-2g^2T>0,\qquad A_*:=a+\frac{6\beta(1+D)}c,\qquad B_0:=\frac\beta c,\qquad R_*\le A_* B_*+B_0\delta. \tag{10} \]

The symbols B_*, B and B_0 respectively denote a scalar coefficient norm, the incidence matrix and a scalar remainder coefficient; none is a hidden-state observation.

4. Verifiable weights and the prospective diameter inequality¶

Retain rational proposed matrices K_b of shape 10-by-150 and W of shape 5-by-150. The former proposes recovery of the ten scaled phasor coefficients. The following nonnegative quantities are maximum absolute row sums, with interval matrix products and upper endpoints used where M and N are enclosed:

\[ k=\|K_b\|_\infty,\qquad w=\|W\|_\infty, \qquad J=[\,0_{10\times5}\ I_{10}\,]-K_bM. \]

\[ e_0=\|J_{[:,0:5]}\|_\infty,\quad e_b=\|J_{[:,5:15]}\|_\infty,\quad F=N-WM, \quad n_0=\|F_{[:,0:5]}\|_\infty,\quad n_b=\|F_{[:,5:15]}\|_\infty. \tag{11} \]

No inverse identity is taken on trust. Since the first sample is at zero, ||u(0)||infinity<=delta. Applying K_b to u_samples=M p+(u-V)_samples and retaining the residual J gives

\[ B_*\le k(\delta+R_*)+e_0\delta+e_b B_*. \]

If the declared assessment proves

\[ d:=1-e_b-k A_*>0,\qquad Q:=\frac{k(1+B_0)+e_0}{d}, \tag{12} \]

then B_*<=Q*delta and R_*<=(A_* Q+B_0)*delta. The future identity u(T)=W u_samples+F p+(u-V)(T)-W(u-V)_samples then yields

\[ G:=w+n_0+n_b Q+(1+w)(A_*Q+B_0), \qquad \boxed{\|U_{fine}^{(1)}(216)-U_{fine}^{(2)}(216)\|_\infty \le\delta G+2E_0.} \tag{13} \]

The fixed decision is delta*G+2*E0<=2e-13, in addition to every numerical, domain and evidence obligation. An unavailable bound is not a constructed pair whose future separation exceeds the target.

This is a theorem about the set of future states jointly compatible with one complete record. If nonempty, its coordinate midranges have error at most half the proved diameter. Computing that nonlinear feasible set is a separate operational problem. In particular, W applied to a single raw record is not automatically such a forecast: (7)-(13) bound pair differences, not the single-history nonlinear bias.

5. Fixed validated computation of the static matrices¶

Each of the fifteen coefficient columns solves q'=-A*q+f_H(v). A first-form column starts at one basis vector with zero forcing. A phasor column starts at zero and has forcing equal to an incidence column times the known sine or cosine command, so ||f_H||infinity<=1. All exact columns obey ||q(v)||infinity<=1+v<=17.

For each fixed cell, form the degree-16 jet of the exact local command polynomial using the shared interval Taylor owner. Its sine and cosine jets give the forcing coefficients F_j. The normalized solution coefficients obey the linear recurrence

\[ q_{j+1}=\frac{-Aq_j+F_j}{j+1}. \tag{14} \]

Use point-center coefficients through degree 16. For the degree-17 Taylor remainder, let F be the maximum absolute zeroth forcing coefficient over the entire cell. Heat contraction supplies the center-started local bound ||q||infinity<=||center||infinity+duration*F. Use the symmetric box from that bound as the zeroth state in the same recurrence, and let the local clock's zeroth coefficient equal the entire cell; its first clock coefficient is one and the others zero. The recurrence therefore encloses the required normalized seventeenth derivative everywhere on that local solution. This analytically derived local tube avoids an independent existence assumption. No scalar trigonometric error is substituted for a derivative or ODE remainder.

The degree-16 polynomial plus this remainder times the cell duration to power 17 encloses the local endpoint. Choose the polynomial's exact rational midpoint as the next center. Add the maximum sum of polynomial radius and absolute remainder to the carried column radius: the homogeneous heat flow is contractive in the infinity norm, so this retains initialization and rounding errors without box wrapping or an uncertain auxiliary clock. Check the 1e-12 radius cap through the final radius of each column. The propagated radii are nondecreasing, so this checks every cell and sample endpoint as well.

Both plateau endpoints are stored. Stage boundaries preserve coefficient centers and radii, and the future matrix follows the final unforced tail. Record all enclosures, exact grid times and each column's polynomial, remainder and carried radius for read-only recurrence checks. Those checks retain the shared kernel's derivative-enclosure premises; cached residuals alone cannot establish arbitrary supplied matrix provenance. The first five initial-form columns carry the common form mode; no projection discards it.

The midpoint normal-equation solve only proposes fixed weights. Its rank, conditioning, rounding and backend cannot certify (12)-(13). Those statements follow only from the retained rational K_b and W, outward interval products against all coefficient enclosures, and the exact scalar comparisons. The interval matrices do not authenticate their source; the evidence receipt retains the frozen implementation and configuration separately.

6. An unavoidable common-form floor¶

The common form has not been fixed to zero. Translation by a constant multiple of the all-ones vector changes neither the Laplacian, pressure currents nor the phase row. Take the admissible exact carrier with zero initial mean forms and equal collective phases, and compare its common form translations by +eta and -eta. They satisfy the same complete law, hidden bounds and initial class. Their form difference is exactly 2*eta at every node and time under the same neutral word.

The midpoint of their true 150-reading vectors is one common admitted record, with opposite errors of magnitude eta at every scalar. Their future mean-form separation is also 2*eta. This proves a uniform diameter floor of 2e-14, or a forecast-radius floor of 1e-14, without evaluating a trajectory. It does not decide the looser declared 2e-13 target.

All conclusions concern this law, class, word, ideal clock and finite observation allowance. The independent physical preparation and measurement bridge remains open. The sole execution queue owns any subsequent work; this owner creates no parallel campaign.

Retained static certificate¶

The declaration above preserves its pre-evaluation wording, including the historical active-gate reference. This assessment is closed; current task selection belongs to the execution plan.

The single assessment declared at de5b500f70b1626d75f12484af0e6381fc9d93af certified the prospective diameter. Its attempt, exact response archive and content receipt retain all 150-by-15 observation coefficients, the future matrix, rational weights, 2,145 compact step witnesses and source/backend associations. There are 143 fixed cells per column. No nonlinear target history, source producer or physical response was evaluated.

The displayed decimals summarize retained outward/exact rational quantities:

Certificate quantity	Result
Maximum coefficient radius	`4.38345956e-20`, below `1e-12`
Phase-weight row norm k	`499.5434372`
Future-weight row norm w	`3.025122144`
Inverse initial / phasor defects	`1.55604e-11` / `3.48244e-10`
Future initial / phasor defects	`6.36647e-12` / `2.01118e-11`
Positive bootstrap denominator d	`0.8161673611`
Carrier future gain G	`3.9317888454`
Fine future diameter upper bound	`<7.866e-14`
Margin below the fixed `2e-13` target	`>1.2134e-13`

Thus the coordinate midrange of any nonempty jointly compatible future set has radius less than 3.933e-14. A future sensor reading adds eta, giving less than 4.933e-14 against that abstract forecast. This is a uniform conditional statement over the declared unknown-initialization class, not only the earlier known attachment state. It neither identifies all hidden coordinates nor selects one constitutive law.

The certified diameter is also below 1e-13. Consequently any certified member of the true feasible-future set would itself be accurate within that policy radius; finding its exact coordinate midrange is unnecessary. Computing and certifying such a member from a raw record is a separate obligation. A fitted phase vector or the dual expression W*y does not discharge it. The operational record-only result instead validates one source-tolerant carrier forecast with explicit model and numerical error; it does not claim exact fine-feasible membership.

Read-only evidence tests verify the source associations, fixed grid, sample/endpoint reconstruction from compact witnesses, error recurrences and independently rebuilt dual residuals. The Taylor derivative enclosure continues to rely on the proved shared kernel and declared forcing law; receipts are content associations, not execution authentication. Preserve this response and the declaration prefix without replaying the producer.