A computable forecast after finite observation¶

Frozen prospective comparison¶

The preparation theorem initializes one carried ten-coordinate reference without measuring hidden phases or resetting its post-formation state. This comparison asks whether that reference produces a certified finite forecast of the state retained after the unchanged observation word. The endpoint distinguishes retained observation backaction from omission of the intervention; it does not identify the same winding a second time.

Item	Fixed choice and scope
Fine preparation	Original independent class-one C9 children, zero-sum primitive errors at most `1e-6`, quarter formation 200, admitted full-state radius `1/12` and energy trap, uninterrupted recovery dwell `8e10`, decay bound `2^-80`; retain every original acquisition, support and work premise
Branch and attachment	Only `outer_one`: five recovered children, supplied common forms `4092*(c-2)`, zero common phase origins, unchanged-state same-slot attachment at `tau=0`; no hidden coordinate is set to zero
Fine law	Unit C9-by-C5, nodes `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 complete history	`tau=(1023/1024)*t`; unforced outer formation on `[0,200]`, original fifteen ramps and fifteen preceding plateaus on `[200,215.00015]`, then unforced continuation to `tau=216`; no donor, changed support or further event
Input word	Mark children `0,1,2,3,4`; each has plateaus of length `1e-5` at command levels `0,+epsilon,-epsilon`, each followed by a unit-duration cubic ramp with increment `epsilon,-2*epsilon,epsilon`, `epsilon=1/10`; use the existing waveform constants and ordering
Reduced initialization	At attachment, `U_c=4092*(c-2)`, `Phi_c=0`, with exact zero numerical radius; retain the analytically bounded fine-source discrepancy separately
Comparator	Identical preparation, law, support, capacities and clock, with the observation word omitted: `f=0` throughout `[0,216]`; it shares the actual formation history, not a target reset
Reserved observable	The five signed mean forms in child order `0,1,2,3,4` at `tau=216`; no in-word reading is fitted or used for initialization
Numerical method	Shared prescribed-time identity-metric validated Taylor steps, order 16, dyadic128 outward interval arithmetic, maximum step `1/8`, proved global same-time growth bound `g=1/3000`; separate steps at every input switch
Computation budget	At most 1,800 accepted steps per branch including its common prefix; planned grid: 1,600 formation steps, then 15 plateau steps, 120 ramp steps and 8 unforced-tail steps per branch. Compute the common prefix once and fork its unchanged center and retained radius: 1,743 steps per branch, 1,886 distinct steps in total
Numerical acceptance	Retained ten-coordinate Euclidean radius at most `1e-15` at every accepted step, successful whole-step domain and Picard checks, completed prescribed horizon; no adaptive step refinement, precision/order escalation, parameter scan or retry
Source and reading allowances	Each branch's intrinsic mean-form error is bounded by `E_source=49/31088000000000000000000000<1.6e-24`; each endpoint scalar has arbitrary correlated additive observation error at most `eta=1e-14`
Conditional exact quantities	Input, clock, support and capacities are stipulated exact; their additional error budgets are zero conditional premises, not instrument certifications
Decision	Compute the maximum coordinate separation of the two numerical endpoint interval vectors. Admit the distinction only if it is strictly greater than `2*(eta+E_source)` and both numerical certificates and all identity margins pass; retain all five intervals and the margin
Failure	Preserve an unavailable result if any declared numerical/domain/budget obligation or the separation fails; do not reinterpret it as a failed source formation or a physical falsification
Evidence	Freeze this owner, consuming implementation and driver before one new endpoint assessment. Preserve source/configuration, ordered stages, arithmetic/backend, all compact step witnesses and any unavailable reason. A losslessly compressed archive may retain the exact report bytes. No original source assessor or frozen response producer is rerun

The comparator is a declared counterfactual input history, not a competing constitutive law. Separation would establish a conditional structural forecast of finite backaction even though the final commanded offsets are zero and both histories retain winding one. It would establish neither a measured response nor resolved hidden constituent feedback or physical identification. Independent laboratory preparation and observation remain open.

1. Exact removal of the commanded offsets¶

Let L be the five-cycle Laplacian and let k_c=e_c-1/5. On a unit-duration ramp with signed increment a, local time r in [0,1] and preceding command level b, the existing waveform is

\[ s(r)=3r^2-2r^3,\qquad H=(b+a s(r))k_c, \qquad z_{cmd}=\frac{4a}{\gamma}L^+k_c s'(r), \qquad f=z_{cmd}'+Lz_{cmd}/4. \tag{1} \]

On a plateau H is constant and z_cmd=f=0. Both command offsets vanish before the word and after its release. Earlier child blocks finish at zero, so only the current marked child's direction is present in H. The command satisfies H'=gamma*L*z_cmd/4. Its phase and form offsets are continuous at the switches; bounded input switches introduce no state jumps.

For the exact carrier reference define Y=U-z_cmd, Psi=Phi-H. Substitution in both complete rows gives

\[ Y'=-LY/4+\gamma S_5(\Psi+H)/4,\qquad \Psi'=\gamma LY/4. \tag{2} \]

Thus large input terms cancel analytically rather than through subtraction of rounded numerical values. This is an exact time-dependent chart change, not a new constitutive law or an approximation to the input. At every plateau endpoint the form offset is zero; at tau=216 both offsets are zero, so Y=U and Psi=Phi. For the omitted-word comparator H is identically zero. Each branch continues its entire reached state across every boundary.

The consumer uses the shared full-sine field. Its five-cycle implementation has degree two; halving both returned structural-time rows reproduces the degree-four carrier inherited from the full product graph. This common factor changes neither gamma nor the declared tau clock. Direct comparison with (2) and the unchanged waveform checks this adaptation; resemblance of the two supports alone would not justify transferring their coefficients.

2. A global same-time numerical growth bound¶

For an oriented ring incidence B put K=B*diag(cos(B^T*(Psi+H)))*B^T. The state Jacobian of (2), with the same prescribed time in both compared solutions, is

\[ J=\frac14 \begin{pmatrix}-L&-\gamma K\\\gamma L&0\end{pmatrix}. \tag{3} \]

No acute-angle assumption is required. Since 0<=L-K<=2L in the quadratic-form order and ||L||2<=4, ||L-K||2<=8. For a real perturbation (a,b),

\[ (a,b)^T J(a,b) =-a^TLa/4+\gamma a^T(L-K)b/4 \le2\gamma\|a\|_2\|b\|_2 \le\gamma(\|a\|_2^2+\|b\|_2^2). \tag{4} \]

The identity-metric logarithmic norm is therefore at most gamma<g=1/3000 globally. H is a prescribed time function, not an extra uncertain state coordinate; its time derivative does not enter this same-time state Jacobian. Timing or input uncertainty would require a new error argument and is not absorbed in (4).

The shared metric Taylor kernel retains a full Euclidean ball. On each fixed smooth stage it establishes a whole-step Picard tube, bounds the Taylor polynomial and remainder with outward arithmetic, and propagates the radius using the proved growth bound. Prescribed-time jets include time derivatives of H. Switches are separate step endpoints; no Taylor interval straddles a different waveform branch. The small growth bound is not a replacement for the local remainder or whole-step existence checks.

The common formation prefix is one exact history shared by both comparisons. Reusing its computed center and complete radius preserves its uncertainty; neither tail starts from a newly exact endpoint. Per-branch interval bounds can discard correlations between the two numerical tails conservatively. The declared maximum-coordinate separation uses those full bounds, not a cancellation of unproved common numerical error.

3. Source uncertainty through the enlarged endpoint¶

The initialization proof retains separate conservative recovered child radii q_x=1e-9, q_theta=2e-12. The original zero-sum hypotheses give exact attachment means, while the five actual hidden states remain present. Their combined initial radii obey r0<3e-9, z0<6e-12.

The full-state heat comparison is valid for the entire horizon Horizon=216: identical uniform-child inputs cancel in the fine/carrier difference rows, including on the unforced suffix. Rebuilding from these primitives gives

\[ Z=2(6\cdot10^{-12}+g\,3\cdot10^{-9})=1.4\cdot10^{-11}, \quad \rho=gZ^2/9,\quad k=216g=9/125, \]

\[ E_U\le\frac{216\rho}{1-k^2} =\frac{49}{31088000000000000000000000} <1.6\cdot10^{-24},\qquad E_\Phi\le kE_U. \tag{5} \]

The quadratic collective sine remainder supplies rho. Its initial mean error is zero; no post-formation phase is observed or inserted. The source theorem's previous displayed interval ended with the word, but its actual 216-unit estimates and input-cancellation proof give (5) through the new endpoint. This extends the stated comparison interval without changing the preparation, dwell, recovery allowance, law or numerical precision. The same argument covers the omitted-word history. It asserts a uniform conditional source cover, not knowledge of an individual fine preparation.

If a numerical branch ends with center m_c and retained radius r, its actual mean form lies within m_c +/- (r+E_U). An admitted future scalar reading lies within m_c +/- (r+E_U+eta). Correlated reading errors are allowed; they are not reduced by averaging or by taking five coordinates. For the two numerical interval vectors I and J define

\[ d_\infty=\max_c\max(0,\inf I_c-\sup J_c, \inf J_c-\sup I_c). \tag{6} \]

The declaration's strict d_infinity>2*(eta+E_U) condition makes their admitted record boxes disjoint in at least one coordinate. Numerical radii are already included in I and J and must not be added again. The choice of the maximum over all five coordinates is fixed before evaluation; no coordinate, time, input or tolerance is chosen from the endpoint response.

4. Identity and work on the unforced continuation¶

Extend the finite-word comparison to observation-relative time T=16. Set y=x-R*z_cmd and e=theta-theta(200)-R*H. The exact carried rows of the finite-word owner still hold after release, with both command offsets zero. For X=1/100, g=1/3000, they imply

\[ \|y\|_\infty\le M=X+16g=23/1500,\qquad \|e\|_\infty\le E=2gX+32g^2=23/2250000. \tag{7} \]

The command changes every outer edge by at most epsilon=1/10 throughout the entire interval and every internal edge by zero. With zeta=1e-6, A=7/5, the sufficient margins remain

\[ 2\pi/9-2\zeta-2E,\quad \pi/2-2\pi/9-2\zeta-2E,\quad \pi/2-A-\epsilon-2\zeta-2E,\quad \pi/2-2\zeta-4E. \tag{8} \]

All are strictly positive; pi>3.14 alone puts the outer margin above 314899/4500000. Hence the five internal and nine outer winding-one identities and the circular observation collar persist through tau=216. The unforced comparator satisfies the same conservative margins with zero command. This continuation uses the already proved actual-source inclusion at tau=200; it does not claim that every admitted norm-box member forms.

The unchanged fine storage satisfies

\[ H_f'=-\|L_f x\|_2^2/4+9(LU)^Tf. \tag{9} \]

On the suffix f=0, so supplied work is exactly zero and storage cannot increase. The unit product graph has degree four and therefore ||L_f x||infinity<=8M, not 4M. For suffix duration delta=216-215.00015=19997/20000, its additional loss is bounded by

\[ Loss_{suffix}\le(45/4)(8M)^2\delta =720M^2\delta =\frac{10578413}{62500000}=0.169254608. \tag{10} \]

All earlier preparation, attachment and finite-word work obligations retain their existing owners. Zero suffix work does not erase their costs or make the preceding intervention autonomous. The continuation introduces no support-event storage jump or reservoir. These are structural units, not calibrated laboratory energy or time.

5. Instrument and evidence boundary¶

The fixed consumer reuses the shared full-sine field, interval arithmetic, prescribed-time Taylor and metric-ball owners. The exclusive driver records one assessment after the declaration revision is fixed. Compact step witnesses retain each time, duration, tube, Picard margin, polynomial, remainder, growth, local error, center and radius; initial state and the ordered preceding endpoints reconstruct each incoming ball. They permit inspection without discarding the numerical uncertainty history. Any compressed evidence must decompress to the preserved exact report bytes.

The independent controls check the waveform, shared-field adaptation, source/identity/resource arithmetic, budget decisions and error accounting without evaluating this reserved endpoint. They supplement, rather than replace, the conditional enclosure proof. A point trajectory, two-solver agreement, or a cached passing flag is insufficient evidence for this declaration.

The sole execution queue owns any subsequent investigation. This owner changes no original frozen protocol, response or source producer and activates no physical campaign.

Result: the carried endpoint separates the two input histories¶

The single assessment under declaration/source revision 8f5a28c5 completed all 1,886 distinct steps: 1,600 shared formation steps and 143 steps in each continuation. Both branches reach tau=216 without a state reset or a changed numerical budget. The retained Euclidean radii are 1.48883e-21 or less for the finite-word branch and 1.47905e-21 or less for the omitted-word branch, below the declared 1e-15 limit.

The certified maximum coordinate separation of their numerical intervals is greater than 6.32174272689e-7. The required separation, including both source and reading allowances, is less than 2.000000001e-14. Hence the two admitted endpoint-record boxes are disjoint by the predeclared rule. Each complete endpoint-record radius is below 1.000001e-14; reading uncertainty dominates the numerical and source errors.

Child	Finite-word mean-form center	Omitted-word mean-form center
0	`3.2564699029529785e-6`	`3.4353965238619072e-6`
1	`1.6457621619214067e-6`	`1.7177138516426278e-6`
2	`-1.2539430823153328e-7`	`0`
3	`-1.9736155054707554e-6`	`-1.7177138516426278e-6`
4	`-2.803222251172097e-6`	`-3.4353965238619072e-6`

These centers are display approximations. The lossless report archive retains exact rational centers, all step witnesses, numerical intervals and expanded future-record bounds. Its receipt associates the archive and original report bytes; the attempt records the declaration, source inventory, runtime and fixed arithmetic. The local uncompressed report remains in artifacts/organized_computable_prediction/.

All four continued identity margins are positive; the smallest exceeds 0.07077. Both histories retain winding one after formation. The result therefore distinguishes finite retained input backaction even though the command offsets have returned to zero and the final suffix is unforced. It resolves neither a nonzero hidden correction nor a competing constitutive law, and it evaluates no physical response. Exact input/clock and the conditional independent preparation remain necessary premises.

The read-only evidence controls rebuild every stored Picard inclusion, time/step chain and metric-error recurrence, then the complete source/reading separation. They do not regenerate Taylor coefficients or rerun a trajectory; those coefficients remain supported by the shared enclosure proof, frozen implementation and independent analytic kernel tests. Byte receipts are content associations, not authentication of physical acquisition or execution.