Finite-aperture observations and mean-clock inference¶
Question and conditional result¶
A sensor that averages a response over time has a different observation law from the point sensor in the clock-drift owner. Replacing such an average by an endpoint value can introduce a first-order bias, large compared with the second-order curvature signal. This owner specifies four positive boxcar kernels and derives necessary inference from their averages without shrinking their widths relative to the probe.
The first three apertures each span one third of the first window; the fourth spans the complete second window. Exact polynomial moments give an algebraic reconstruction whose first-window error is cubic in duration. The second-window endpoint needs only a quadratic error bound. A separate clock-transfer bound permits a positive varying observation clock while retaining the first-window mean-rate target. The resulting adapter reuses one existing curvature inverse and preserves the original sensor errors explicitly.
These are conditional mathematical results and fresh implementation controls. They introduce no reserved response, physical calibration or reinterpretation of frozen point-reading evidence. The ontology retains the distinction between collective observation and physical identification.
F1-F2: complete dynamics and a fixed averaging law¶
Retain the complete source, support and clock class. On the eighteen-node two-port support, with degree matrix \(M\), \(A=M^{-1}L\), \(q=e_4-e_5\), held unit capacities and \(\gamma=1/(1023\pi)\), the structural rows are
Both rows multiply by the same positive \(C^1\) rate \(\rho(s)\) in observation time. Its supplied premises remain \(\rho_-\le\rho(s)\le\rho_+\), \(|\rho'(s)|\le\Lambda\) on \([0,2H]\), with \(H>0\), \(\Lambda\ge0\), and \(h_*:=\rho_+H\le1/2\). The target is \(\bar\rho_1=H^{-1}\int_0^H\rho(s)\,ds\). The full source retains arbitrary common means and centered residuals bounded by \(X,Y\), original nominal geometry priors and their acute admission. The actual angle remains \(B_{\rm initial}=b-(v_1-v_0)/8\).
There are phase-only jumps \(a_1q\) at zero and \((a_2-a_1)q\) at \(H\), for \(0<a_1\le a_2\le1\). Every form coordinate is continuous at these events. No additional event, prehistory, state reset, capacity change or support change is supplied.
Define the observed-time apertures and normalized densities
The four measured quantities are
with one held positive gain \(G\in[G_-,G_+]\) and offset \(O\). The primitive intervals \([\ell_i,u_i]\) enclose these four quantities. Their numerical widths differ from the supplied sensor-error allowance. There is one error per averaged channel; no pointwise noise process or independent error at each reconstructed point is assumed.
The kernels are nonnegative, normalized, fixed in observation time and independently supplied. The adapter does not calibrate them or accept arbitrary windows under the same name. All support lies inside the declared horizon. No aperture straddles the phase event; its measure-zero endpoint does not affect an average of continuous form.
Why a direct endpoint substitution is insufficient¶
A nonnegative normalized measure supported in \([0,2H]\) reproduces every affine function at zero only if its first moment is zero. Positivity then forces all its mass to be at zero. The corresponding statement at \(2H\) follows by reflection. Thus a genuine endpoint aperture generally has first-order timing bias; normalization alone does not remove it.
Nor does symmetry automatically remove event bias. Near a phase event, a continuous readout can have \(y(H+t)=y(H)+v_-t+(v_+-v_-)t_++O(t^2)\). A symmetric kernel cancels the affine term but retains \((v_+-v_-)\int t_+k(t)\,dt\). In (1), a nonzero slope jump can arise from the changed phase pressure while form itself stays continuous. The old smooth remainder must not be extended across that jump.
The finite-sample Peano and timing method is a useful mathematical precedent, with different source and event premises. Its coefficients and verdicts do not transfer to this sensor.
Exact moment reconstruction at fixed relative aperture¶
First consider a scalar function \(y(s)\) and its four noiseless averages under (2). In normalized time \(z=s/H\), the first three averages of \(1,z,z^2\) form the matrix
This invertible moment map gives the reconstruction
The first three rows reproduce quadratic values at \(0,H/2,H\). The last row is \(2\mathcal A_3-(P\mathcal A)_2\): it estimates the second endpoint by a trapezoid relation on its own continuous window. Every row sums to one, so a held offset remains one common offset. The negative reconstruction coefficients are algebraic operations on positive sensor averages, not negative physical kernel weights.
The curvature row and absolute row sums are
Consequently constant and linear terms cancel exactly in the reconstructed curvature while a quadratic retains its \(H^2\) signal. This moment identity does not claim exact reconstruction of an arbitrary trajectory.
Finite transfer of the averages to one mean-clock reference¶
Use the same source, phase events and held sensor with constant rate \(\bar\rho_1\) as a reference. Its full state agrees exactly with the actual history at zero and \(H\), including the post-event state. The drift owner's global contraction bound supplies, with \(g=1/3069>\gamma\),
The bounds cover both complete histories and the comparison arcs. Phase evolves throughout; its forcing is bounded globally.
For \(0\le s\le H\), write the exposure discrepancy as
For the second window, use \(\tau(H+t)-\tau(H)-\bar\rho_1t =H^{-1}\int_0^t\int_0^H[\rho(H+v)-\rho(u)]\,du\,dv\). The derivative and global range \(\Delta\rho=\rho_+-\rho_-\) therefore give
Integrate each bound against its nonnegative kernel. With \((c_0,c_1,c_2)=(7,13,7)/108\), sufficient averaged exposure bounds are
Taking the minimum after integration remains a valid upper bound. Since \(\|q\|_{M^{-1}}=1\), the actual/reference noiseless sensor averages differ by at most
This step transfers averages before reconstructing reference points. It does not substitute the mean rate into each actual window, differentiate \(\rho'\), or assume that a \(C^1\) clock has a bounded second derivative.
Complete-law reconstruction errors without crossing an event¶
The contraction bound, \(\|A\|_M\le2\), \(\|Df\|_M\le2\), and \(\|D^2f[h,k]\|_M\le2\|h\|_M\|k\|_M\) give global structural derivative bounds on each continuous window:
For example, differentiating (1) yields \(x_{\tau\tau}=-Ax_\tau+\gamma Df\,\theta_\tau\) and \(x_{\tau\tau\tau}=-Ax_{\tau\tau} +\gamma D^2f[\theta_\tau,\theta_\tau] +\gamma Df\,\theta_{\tau\tau}\). These bounds include every form and phase coordinate and use no frozen phase approximation. Their application below is to the constant-clock reference, whose observed derivatives introduce powers of \(\bar\rho_1\).
For the first three reconstruction rows, define the order-three Peano kernel on \([0,1]\) by
Exact polynomial integration gives
One explicit verification uses the nonnegative endpoint kernel
and \(K_2(t)=-K_0(1-t)\). For \(t\ge1/2\), \(K_1(t)=-(1-t)^3/48+(9/16)(2/3-t)_+^3\le0\); its reflection has opposite sign. Integration establishes (11), rather than bounding unrelated endpoint Taylor expansions independently.
On the second window the exact trapezoid error satisfies
Form continuity identifies the two values at \(H\); derivatives are taken on the appropriate side. Combining (10)-(12), sufficient recorded reconstruction allowances are
Only the first three entries enter curvature. Their combined error is \((67/3456)G_+M_3h_*^3\), retaining cubic order at the fixed aperture fractions in (2). The quadratic last-entry error affects the coarse two-input inference, not the first-window curvature directly.
Necessary inference and the original four errors¶
Re-admit all eleven primitives. Apply the exact rational matrix \(P\) to the four primitive average intervals before outward interval materialization. Let this linear interval image be \(J_i\). The constant-mean reference's noiseless point readings at \(0,H/2,H,2H\) belong to
The raw numerical half-width contributes through \(J_i\); the three terms in \(a_i\) respectively account for reconstruction, clock transfer and the original sensor errors. Offset cancellation occurs exactly in normalized rational arithmetic, even for a large common offset.
Invoke the existing curvature inverse
once with \(\widehat I_i\), the unchanged source/input/gain/rate priors,
observed duration \(H\), and child readout_error_bound=0.
The child's zero means no additional error is attached to these auxiliary
noiseless-point bands; the measured averages retain their original
\(\delta\). Its held clock denotes \(\bar\rho_1\), and
its effective gain denotes \(J_1=G\bar\rho_1\).
Every compatible averaged history induces such a reference history, so the resulting marginals are necessary bounds on the original geometry, held gain and first-window mean. A strict exclusion rules out the admitted averaged model. Nonempty rectangular constraints need not have a jointly realizable source, clock or error vector. The implementation retains the shared four-error map \(P\), the curvature row (5), and the coarse inverse rows composed with \(P\); independent interval projection can discard their remaining correlations.
For curvature specifically, there is no additional sensor support loss from the rectangular projection: its propagated allowance is \((10/3+2\cdot7/6+10/3)\delta=9\delta\), exactly the support of the composed row in (5). This equality does not extend automatically to the other rows or certify joint feasibility.
The report distinguishes original numerical widths, projected sensor
allowances, clock-transfer candidates and reconstruction candidates.
Certified upper bounds require the freshly rebuilt child's original-source
admission. bounded_candidate, incompatible and unavailable retain
their necessary-constraint meanings; no unavailable quantity becomes zero.
The inferred clock field explicitly names the first-window mean.
Informative finite-width regime without selecting a response¶
Use the previous positive-drift budget
Here \(t_i\) are numerical half-widths of the four original averages, separate from their four bounded errors. No source realization or response is selected. The apertures remain \(H/3,H/3,H/3,H\).
Put \(h_*=2H\), \(c=2U\Lambda H^2\), \(e=2M_3h_*^3/108\), \(e_m=2M_3h_*^3/2304\), and \(l=2M_2h_*^2/6\). Sufficient virtual-point half-widths are
These include both \(t_i\) and \(\delta\). For the separate coarse-inverse remainder, retain its own specialized quantities
Do not replace \(Q_s\) by the global \(Q_0\) merely because both are form bounds. For the two shared inverse rows, let \(S_j\) be the sum of their absolute coefficients. Their combined bound is \(\sqrt{S_1^2+S_2^2}<36000/h_*\). For a row \(w=(w_1,w_2)\), the three reading-radius contribution is \(|w_1|r_0+|w_1-w_2|r_2+|w_2|r_3\). Since \(r_0=r_2\le r_3\), it is at most \((|w_1|+|w_2|)(r_2+r_3)\). Hence
The child curvature remainder also keeps its own first-window bound: \(S_s=5+2Y+8gh_*Q_s<7\), \(M_{3,s}=8(1+2g^2)Q_s+4g(1+g^2)S_s+8g^3Q_s^2\), \(\varepsilon_2=4(1+g^2)X+4gY\). Exact rational substitution gives \(D=2\varepsilon_2+2HM_{3,s}<10^{-9}<1/800000000\). The normalized curvature numerator diameter obeys
The positive natural coefficient bound remains \(K_->k_0=1/18000\), with \(w_K\le11gw_b/8\). Compatibility and these error bounds preserve positive quotient numerators. Set \(\beta=w_K/k_0\) and \(t=2w_{J_1}+\beta+2w_{J_1}\beta\). The same endpoint-width argument yields
The conservative rational caps in (18)-(19) and the stated cap on \(D\) already suffice. Also \(w_b+Y/4<1/1024\) and \(w_{J_1}<1/2048\). These are conditional exact-arithmetic resolution budgets. Executable outward arithmetic and inverse availability remain separate checks; the stipulated precision is not a laboratory performance claim. Fixed relative aperture, rather than an arbitrarily tiny additional window, is what the moment calculation admits.
Implementation and remaining boundaries¶
The aperture adapter uses exact primitive admission, fixed moment coefficients and one fresh curvature calculation. The contract owns its API; the controls separate polynomial moment identities, complete-law transfer and averaged synthetic histories from any reserved evidence.
Point-sample exposure equivalence does not automatically give equal observed-time averages: the latter consume the response inside the intervals as well as their endpoints. Thus the earlier cosine-companion certificate cannot be transferred to this observation law. No converse claim of clock-profile identification follows either.
General kernels, calibration uncertainty in their timing/weights, event-straddling apertures, variable sensor gain/offset and physical clock units require their own premises and bounds. Source formation, support selection and future maintenance remain separate. The execution plan owns subsequent admission. No frozen response is averaged, replayed or reclassified by this theoretical extension.
Validated forward averages with an explicit affine clock¶
The inverse above accepts primitive average bounds; it does not generate them. A separate forward construction supplies such bounds from complete initial form and phase boxes. It reuses the source-box Picard/Taylor kernel and the complete structural field, without consuming an inverse report, target angle or inferred parameter. Endpoint exposure equivalence alone is insufficient for this task: the observation-time average also depends on the interior response.
Sufficient state, clock and instrument law¶
Keep the eighteen-node support, unit capacity, structural rows (1), dipole and four windows (2). Supply eighteen primitive intervals for each of \(x(0^-)\) and the continuous phase lift \(\theta(0^-)\). This forward domain does not require an acute chart or the inverse's nominal source family. Those are separate admission obligations when associating a produced response with an inference problem.
Specify the forward clock completely:
Require \(H>0\), \(r_{\min}=\min(r_0,r_0+2\kappa H)>0\), and the work-policy bound \(r_{\max}H\le1/2\), where \(r_{\max}=\max(r_0,r_0+2\kappa H)\). These exact endpoint inequalities prove positivity on the whole horizon because the clock is affine. They also give total structural exposure \(\tau(2H)\le1\). There is no separate bound on \(H\) alone. The constant-clock case is \(\kappa=0\); negative slopes are admitted when the final rate remains positive. Arbitrary positive \(C^1\) clocks from the inverse's larger class do not thereby supply the higher derivatives needed by a Taylor producer.
Let \(F(x,\theta)\) denote all thirty-six structural rows (1). Adjoin the selected rate \(r\) and a passive cumulative integral \(z\):
Both nodal rows multiply by \(r\). The integral row does not: \(z(s)=\int_0^s q^Tx(u)\,du\) integrates in observation time. Multiplying it by \(r\) would instead integrate against structural exposure and change the sensor law. Its units are \([z]=[x][s]\). The two auxiliary coordinates specify a supplied clock and an instrument; they are not additional nodal constituents or a derived controller. The accumulator does not feed back into any evolution row. The supplied rate converts both structural rows to observation time.
The phase events remain \(a_1q\) at zero and \((a_2-a_1)q\) at \(H\). The forward API admits \(0\le a_1\le a_2\le1\), so zero and equal inputs provide passive controls. The inverse retains its own stricter positive-input and rank obligations. Each event changes only phase; form, clock rate and cumulative integral remain continuous. There is no reset of \(z\) between apertures.
The extended field is smooth on all of \(\mathbb R^{38}\). The selected exact affine clock is positive by (21), even if an enclosing numerical clock interval includes an extraneous nonpositive value. The Picard domain certifies smooth evaluation of the extended field; it does not assert that every point of the artificial tube is a positive clock, an acute phase state or a retained winding identity. These distinct claims must not be conflated.
Four certified increments and their normalization¶
Use exactly four steps with observation-time starts \((0,H/3,2H/3,H)\) and durations \((H/3,H/3,H/3,H)\). The chosen ordinary integer order \(n\in[1,16]\) is held for all steps. Dimension thirty-eight is within the shared source-box policy of sixty-four; the comparison solver's separate dimension limit is unchanged. Each step must establish strict Picard inclusion and enclose all required Taylor derivatives on its whole-time tube. There is no adaptive subdivision, order change or retry of a failed step.
For step \(i\), let \(B_i\) be its complete post-event initial box, \(T_i\) its admitted tube and \(\Delta_i=|W_i|\) its exact positive duration. If \(c_{j,k}(B_i)\) encloses the normalized \(k\)-th solution derivative of coordinate \(j\), the shared kernel constructs an increment enclosure
The order-zero coordinate is omitted algebraically. For every trajectory starting in \(B_i\), Taylor's theorem and the tube certificate give \(w_j(s_i+\Delta_i)-w_j(s_i)\in D_{i,j}\). The endpoint may be tightened to \((B_i+D_i)\cap T_i\), but subtracting independent endpoint and initial intervals is not a substitute for the correlated increment (23).
The raw normalized average therefore belongs to
Divide both interval endpoints by the exact positive rational \(\Delta_i\) before outward materialization. An interval representation of a very small positive duration may contain zero; that numerical enclosure must not erase the independently admitted divisor. This normalization preserves an enclosure, but neither guarantees a small width nor removes the finite precision floor.
Carry the entire thirty-eight-coordinate endpoint box into the next step, then apply its declared phase jump. In particular, carry the evolved phase, the rate and the cumulative integral instead of reinstalling their initial values. Induction over the four steps proves that (24) encloses the four averages of the same complete hybrid trajectory for every admitted initial point. Interval widening can discard correlations but cannot justify choosing a different source or clock history for each window. The retained clock primitives and window boundaries determine exact values \(\rho(s_i)\), \(\rho(s_i+\Delta_i)\) and \(\tau(s_i+\Delta_i)-\tau(s_i)\); the report also retains cumulative structural exposures. This clock provenance does not change the observation-time denominator in (24).
Availability, evidence and sensor separation¶
The forward producer retains the primitive preparation, fixed support, phase jumps, clock law, all step certificates and the completed observation horizon. A failed step stops the sequence. Its last tube and reason remain available, but it yields no certified average, and later windows are not executed. A completed prefix is not a four-average response. The API contract owns the report's availability fields and the independent controls test the accumulator, clock transformation, event history and failure cases.
The output encloses the raw average \(\mu_i\) for every admitted source point. Its interval width includes supplied source uncertainty and numerical enclosure error; it is not automatically a numerical-error estimate or sensor tolerance. A separately declared held sensor would give \(\mathcal A_i=G\mu_i+O+\eta_i\), with its own calibration, error allowance and primitive interval projection. The producer selects none of these quantities and calls no inverse. Forward enclosures and fresh synthetic controls establish conditional software behavior. A reserved evaluation still requires a separately frozen preparation, explicit clock, sensor realization, observation horizon, numerical budget and acceptance criteria before its response is evaluated. No source formation, kernel calibration or physical identification follows from this forward admission.
Prospective software evaluation of the four averages¶
This protocol tests the admitted forward and inverse observation laws on new complete-flow averages. Preparation, clock profiles, sensor, numerical budget and decisions below are fixed before any reserved response. Earlier point-reading records are not observations for this sensor and are not inputs to the calculation.
F1-F2: fixed preparation and three clock histories¶
Retain the public source and observation budget (15), the full nominal geometry priors, the two phase inputs and the four boxcars (2). Select the same explicit preparation recipe as the earlier clock protocol: \(b=23/16\), \(c=4/5\), recipe index four, common form \(6/11\) and common phase \(-6/13\). In particular, independently specify
Construct the complete nominal phases from \(A_0=4\pi-8b\), \(d=(2\pi-c)/8\), \(\zeta=(A_0-c)/2\): the donor phases are \(0,A_0,A_0+b,\ldots,A_0+7b\), and the receiver phases are \(\zeta,\zeta+c,\ldots,\zeta+c+7d\). Subtract their full degree-weighted mean, then add the common phase and \(v\); form is the common form plus \(u\). Retain all thirty-six coordinates and an outward enclosure of mathematical \(\pi\). Both centered residual vectors have nonzero entries and norms below \(2^{-48}\); the coarse bound \(40\cdot114^2/2^{124}<2^{-96}\) suffices before materialization. Re-admit the actual constructed boxes separately. The original long-arc mean is \(B_{\rm initial}=23/16-5/2^{65}\); retain its complete initial box as well as that exact source value.
Declare a held sensor \(G=7/5\), \(O=5/13\), and four errors \(\eta=2^{-91}(-1,1,-1,1)\). All histories share this same sensor realization and original source. Their clock laws are
The global rate prior and derivative allowance hold exactly. Both actual first-window means are \(9/8\), with effective gain \(J_1=63/40\). Each actual/reference history agrees in nodal state at \(H\) by exact first-window exposure and the common source/event law. Their clock-rate and cumulative-integral coordinates need not agree. No oscillating companion or equality of interval averages is inferred from the older point-sample exposure certificate.
F3: independent integral response and public-only inference¶
Evaluate the new forward producer exactly once for each of (26), starting with the constant reference, then the positive and negative slopes. Each call uses four fixed thirty-eight-coordinate steps of order four, with whole-tube fifth-order remainders, sixteen Picard iterations per step and shared outward dyadic-128 arithmetic. The total cap is twelve steps. Carry all coordinates, including the clock and cumulative integral; the two supplied phase events alone change phase. Retain every successful certificate, attempted source, failed tube and achieved horizon. No midpoint source, integrator substitution, subdivision, precision increase, state reset or budget retry is admitted.
Form recorded average bands from the true raw averages using the held sensor and its fixed four errors. Require each recorded band's half-width to be at most \(r=\delta/16=2^{-94}\). This requirement concerns the combined source-materialization and numerical enclosure for this fixed preparation; the sensor-error allowance remains \(\delta=2^{-90}\). No producer-derived remainder may replace the inverse's independent bounds.
Send each actual history to a fresh inverse worker through only the eleven
documented primitives, with averaged_reading_bounds as its four observations.
The public clock inputs are the prior and derivative allowance. The exact
source, sensor, realized errors, forward clock profile, exposures, reference
history and certificates remain outside that packet. Retain each request
and hash. The reference has no inverse evaluation. This is software input
separation, not a secrecy or provenance-authentication guarantee.
Prospective transfer, curvature and alternative bounds¶
The preceding complete-law bounds apply unchanged. Set \(T=4H\), \(Q_0=X+7gT\), \(V=Y+2gTQ_0\). The supplied-source slope bound uses the actual two pulse values and gives, on both continuous windows and their comparison arcs,
This is a source/law bound, not a derivative measured from the new response. For \(\kappa=\pm\Lambda/2\), the first-window exposure difference is \(-\kappa s(H-s)/2\), and the second is \(\kappa t(H+t)/2\), where \(s=H+t\). Therefore all three first-window positive-clock averages strictly exceed their reference averages; the last is strictly smaller. Negative-clock signs reverse. Using (8), the minimum true magnitudes in units of \(\delta\) exceed
Shared sensor errors cancel in these comparisons. A difference of two recorded bands can extend by at most \(4r=\delta/4\) beyond its true difference, so their entire intervals retain these strict signs. Also, each true difference has magnitude at most \((7/20)b_i\) from (9), while \(\min_i b_i/\delta>300\). The same numerical allowance leaves each whole difference interval strictly within its independently reconstructed \([-b_i,b_i]\).
Use the composed curvature \(C=(9/4,-9/2,9/4,0)\mathcal A\). Writing \(v_0=q^Tx_\tau(0^+)\), the reference slope varies by at most \(M_2T\); exact substitution gives \(M_2T<10^{-8}\). Linearizing only the actual/reference difference around that common slope, the integrated absolute coefficient sum is \(5/6\). Consequently
The two curvature interval enclosures can lose at most \(36r=9\delta/4\), leaving strict ordering \(C_+<C_-\). This reuses the original four averaged errors; no new derivative or noise channel is introduced.
The explicit phase-blind alternative is \(x_\tau=-Ax,\ \theta_\tau=\gamma Ax\), with the same source, clocks, phase events and sensor class. Contraction gives \(|q^Tx|\le X\); hence any two recorded average differences lie in \([-B_{\rm heat},B_{\rm heat}]\), where \(B_{\rm heat}=2G_+X+2\delta\). Under (27) and \(\rho\ge1/2\), the first adjacent-third average decrease is at least \(GH/120000\); the final adjacent-window decrease is at least twice that. The smaller exceeds \(2G_+X+4\delta+\delta/4\) by exact primitive arithmetic. Thus both selected recorded differences \(\mathcal A_1-\mathcal A_0\) and \(\mathcal A_3-\mathcal A_2\) must strictly exclude that complete alternative band. No endpoint samples substitute for these averages.
F4: fixed decisions and first-outcome retention¶
Pass only if all source, sensor, clock, complete-horizon, handoff and recorded-width checks pass, together with the following fixed decisions:
- Each actual primary inverse returns
bounded_candidatewith source, drift transfer, aperture reconstruction, finite curvature, rank and positive-division evidence admitted. Its necessary marginals cover the nominal angle, original actual angle and its initial source box, \(J_1=63/40\), \(G=7/5\) and first-window mean \(9/8\). - Strict widths are below
1/1024for original actual angle,1/2048for effective gain,1/16for gain and1/64for mean rate, matching the separately derived aperture budget. Marginal coverage is not joint parameter realizability. - All four actual/reference difference bands have the predicted signs and lie inside their transfer allowances. After the interval loss accompanying (29), the two composed curvature intervals remain strictly ordered: the positive-clock upper endpoint is below the negative-clock lower endpoint. The thirty-six nodal coordinate boxes at \(H\) intersect the reference boxes before and after the common event; this is numerical consistency with exact exposure equality, not its proof.
- False clock prior
[25/32,13/16]and false gain prior[31/16,2]leave their coarse children available but yield a strict curvature exclusion. False angle prior[11/8,353/256]is excluded; equal cumulative inputs(1/4,1/4)give rank-based unavailability. These controls change declarations while retaining the actual observations. - Both selected adjacent average differences exclude the complete phase-blind alternative with its admitted uncertainty and sensor band.
Freeze this prospective text, machine-readable protocol, runtime source, evaluator and isolated worker before the response. An exclusive attempt ledger precedes the first flow. Retain the first outcome, including a partial response, failed criterion, unavailable calculation or exception. After an incomplete forward history, do not acquire remaining histories. No second attempt or alteration of the evaluated budget is admitted; corrections require separate evidence. Success would establish conditional software robustness for these fixed averages. It would not calibrate a physical sensor, infer the instantaneous clock, justify the supplied support/law/source, or establish a physical constituent.
Retained first finite-aperture evaluation¶
The first reserved evaluation returned
certified_reserved_aperture_inference: all 139 fixed conditions
passed. These comprise twenty-five reference-history conditions, fifty-six
for each actual clock, the shared source admission and the paired curvature
ordering. The protocol,
source archive,
attempt,
response
and manifest
retain the first outcome. The prospective statements above remain unchanged.
Exactly three forward calls completed all twelve prescribed thirty-eight-coordinate steps. Every step retained a strict Picard tube, the complete clock-scaled nodal law, cumulative observation-time integral and its initial-coordinate cancellation. The recorded bands satisfied the fixed \(\delta/16\) half-width requirement; their maximum half-width was below \(0.000256347661\delta\). This is a bound for the selected source enclosure and arithmetic, not measured sensor performance.
Both eleven-primitive inverse packets excluded the hidden preparation, sensor realization, clock profile, exposures and forward certificates. Their necessary marginals cover the original nominal angle, \(B_{\rm initial}=23/16-5/2^{65}\), its initial source box, \(J_1=63/40\), \(G=7/5\) and mean rate \(9/8\). The following decimal widths are rounded upwards from the saved exact rational bounds; the decisions use the frozen rational thresholds.
| Clock history | Original-angle width | Effective-gain width | Gain width | First-window mean-rate width |
|---|---|---|---|---|
| Positive slope | 0.000058682913 |
0.000217661061 |
0.000876129137 |
0.000548560067 |
| Negative slope | 0.000058682913 |
0.000217661061 |
0.000876129015 |
0.000548560125 |
| Frozen strict threshold | 1/1024 |
1/2048 |
1/16 |
1/64 |
All four actual/reference average differences have their predicted strict signs and lie wholly inside the independently rebuilt clock-transfer allowances. Their outward-rounded intervals in units of \(\delta\) are:
| Clock history | First third | Middle third | Last third | Second full window |
|---|---|---|---|---|
| Positive slope | [2.769323,2.770350] |
[5.143615,5.144642] |
[2.769323,2.770350] |
[-83.575660,-83.575317] |
| Negative slope | [-2.770350,-2.769323] |
[-5.144385,-5.143359] |
[-2.770350,-2.769323] |
[83.575231,83.575574] |
The retained composed curvature intervals satisfy \(C_+<C_-\). At \(H\), all thirty-six actual/reference nodal boxes intersect before and after the common phase event, consistently with exact exposure equality. Clock rate and accumulated observation are excluded from that equality check. All thirty-eight coordinates remain carried within each individual history.
The false gain and clock priors leave the coarse inference available but are excluded by curvature. The false angle prior is excluded; equal amplitudes yield rank-based abstention. Both prescribed adjacent average contrasts exclude the complete phase-blind alternative with its source and sensor allowances. These results do not establish joint realizability of every marginal tuple or identify a unique physical law.
The committed runtime base is 035965355a8066622eebabd6fe08eedee9c7aef5,
without runtime overlays. The thirty-file archive has SHA-256
b8b48ae072411a871063c1db22f22e7e5ed851b4641b9b44dd1a14ddbad1c222.
Its prospective theorem contains 34,149 bytes, with SHA-256
e670a301fc13117f12b86e5dab18d3174174b3f84ba98b40d1c5d55a48096b47.
Those bytes remain this owner's unchanged prefix, modulo checkout newline
conversion. Hashes associate retained bytes and declared execution; they
do not independently authenticate chronology or physical acquisition.
The read-only audit
rebuilds retained arithmetic and associations without another forward or
inverse evaluation.
The result supports conditional finite-aperture inference under the fixed complete model and exceptionally small supplied error/drift budgets. It does not establish an attainable instrument precision, laboratory time unit, independent measurement map, source formation or physical constituent identity. Those missing bridges remain substantive research obligations; the execution plan owns the next admission.