Finite-noise and horizon resolution for the four boxcar observations¶
Question and scope¶
The finite-aperture observation law has four substantial windows, not point samples. Its earlier conditional resolution example uses a very small error allowance. This owner asks what happens when the horizon, source uncertainty, clock drift and observation error are varied before selecting a response.
There are two different results. A sufficient budget gives conditional outer-width guarantees and can fail because its bounds are conservative. A separate complete-history construction proves actual indistinguishability under sufficiently large bounded sensor error. Neither result calibrates an instrument, identifies a physical time unit or selects the supplied nodal law. No saved response, inverse output or producer is used to choose the regimes below; earlier frozen owners and evidence retain their scope.
Keep the same complete eighteen-node support, unit capacities, phase events, source family and held sensor law. With the degree metric \(M\), normalized Laplacian \(A\), dipole \(q=e_4-e_5\), and \(\gamma=1/(1023\pi)\),
Retain the fixed priors and inputs
The phase jumps are \(a_1q\) at zero and \((a_2-a_1)q\) at \(H\). All form coordinates stay continuous. The complete source has arbitrary common form and phase means and centered residual norms at most \(X,Y\). The original long-arc mean remains \(B=b-(v_1-v_0)/8\), hence
The four windows are \([0,H/3]\), \([H/3,2H/3]\), \([2H/3,H]\), and \([H,2H]\). Every reading is its normalized observation-time average of \(Gq^Tx+O\), plus one bounded scalar error. The gain is positive and the offset is arbitrary; both are held throughout. The target clock remains the first-window mean \(\bar\rho=H^{-1}\int_0^H\rho(s)\,ds\), with effective gain \(J=G\bar\rho\).
The clock is positive and \(C^1\) on the entire interval \([0,2H]\). The six variable budget primitives are
Here \(\delta\) bounds each physical sensor error; \(t\) bounds each primitive recorded interval's half-width. The latter can include source materialization and numerical enclosure width. They are different premises, even where a sufficient inequality consumes their sum \(\varepsilon=t+\delta\). A hypothetical zero-width reading is admissible within any nonnegative \(t\); no actual numerical producer is promised to achieve it.
For the uniform sufficient chart guard, require \(Y<1/256\). The smallest nominal acute margin on (2) is \(11-7\pi/2\). The rational upper bound \(\pi<355/113\) gives \(11-7\pi/2>1/226>1/256\). Every edge difference has dual degree-metric norm at most one, so this guard preserves the initial chart for every admitted residual. Failure of this conservative guard does not prove that a particular source is nonacute. The complete later flow may leave the initial acute sector; no maintenance claim is needed by the global finite bounds.
A response-free sufficient budget¶
Set \(g=1/3069>\gamma\) and \(h_*=2H\). The global derivative and transfer proof gives
For a uniform budget, using the derivative branch of the clock-transfer bound is sufficient even when the alternative rate-range bound is sharper. Define
The exact moment matrix of the aperture owner gives virtual point-band half-widths no larger than
These coefficients follow by applying the absolute moment matrix to the four numerical/sensor budgets and averaged exposure bounds; the Peano remainders are event-separated. The original curvature coefficients are \((9/4,-9/2,9/4,0)\), with exact sensor support \(9\delta\). The same coefficient norm gives \(9t\) for numerical half-widths. There is no new error attached independently to each virtual point.
Keep the coarse inverse's specialized bound separate from (5):
The denominator is positive throughout (4). The two-input coefficient bound is \(\sqrt{S_1^2+S_2^2}<36000/h_*\). Since \(r_0=r_2\le r_3\), the inherited three-reading row bounds give the last line of (8), exactly as in the aperture conditioning proof. The auxiliary radial prior is \([1/4,2]\). If \(R<1/8\), compatible observations therefore have conditional exact-arithmetic width bounds
For the first-window curvature remainder, use
The global and specialized third-derivative quantities have different roles. Replacing \(Q_s\) or \(M_{3,s}\) by (5) is not justified by their names or by a comparison at one horizon.
The diameter allowance for \(4C/H^2\) simplifies exactly to
This exposes the noise amplification, drift floor and finite-flow growth. The source contribution additionally enters \(D\), \(R\) and the original angle width. Shortening the window is not a uniform improvement.
The positive natural curvature coefficient has the uniform lower bound \(k_0=1/18000\), and its interval width is at most \((11g/8)W_b\). For a response-independent positive-division guard set
The guards imply \(D<k_0/2\), so the lower coefficient used below is positive and multiplication by the lower bound on \(J\) is legitimate. Indeed, compatibility, \(J\ge1/2\), \(\bar\rho\ge1/2\), and the finite curvature theorem bound the true normalized numerator below by \(\tfrac12(\tfrac12k_0-D)\). Subtracting its diameter gives \(L_A\). Division by \(J\le4\), then allowance \(D\), proves positivity of the next numerator. These are sufficient guards, not necessary properties of every possible informative record.
Put
The inherited positive-quotient argument gives
One sufficient resolution certificate is the chart guard, (12), \(R<1/8\), and the four strict policy targets
These statements are conditional on a compatible full history and the declared reading-width allowance. They concern the mathematical enclosure method. Actual inverse arithmetic must still admit its outward coefficients, quotients and source; a budget calculation does not generate observations or assert that every Cartesian tuple of marginal bounds is realizable.
A finite positive window interval and explicit budget obstructions¶
There is an informative regime with a larger error allowance than the earlier illustrative budget, without selecting any response:
The absolute apertures vary with \(H\); their relative widths remain the same fixed thirds and full second window. Both sensor and numerical budgets are finite independent inputs. The error allowance is 1024 times the earlier \(2^{-90}\) example; this numerical comparison is not an attainability claim.
For an explicit uniform verification let \(H_-=3\cdot2^{-26}\), \(H_+=2^{-24}\), and take the largest other budgets in (15). Expand (8) as
Every term except the explicit inverse powers of \(H\) is nondecreasing on this interval. Evaluate those increasing terms at \(H_+\), and replace \(\varepsilon/H\), \(\varepsilon/H^2\) by their values at \(H_-\). Use the resulting upper bounds in (9)-(13), evaluating \(D\) at \(H_+\). Exact rational substitution proves (12) and
These bounds imply (14) uniformly on (15). This is not a search over observed responses or a claim that arbitrarily small windows are useful.
Several simple lower bounds on this sufficient budget explain its limits. They are not lower bounds on the information in every observation:
For example, \(X=2^{-30}\) alone gives \(R>1/8200\), so this budget cannot guarantee the angle target at any horizon. Even with all other uncertainties zero, \(\Lambda=2^{-16}\) gives \(W_{\bar\rho}\ge(80/(3k_0))7g\Lambda>1/64\). These are uniform-source or uniform-clock budget floors; they do not prove that every source or clock in the corresponding class is indistinguishable.
Noise and finite flow also impose competing horizon restrictions. From \(Q_s\ge28gH\), \(M_2\ge14g\), and (16),
The angle target requires \(R<1/8200\); the mean-rate target and (11) require \(H^2>9216\varepsilon/k_0\). Thus simultaneous certification requires
If
there is no horizon satisfying this budget's two targets. In particular, \(\delta=t=2^{-75}\) satisfies (20), even with \(X=Y=\Lambda=0\). This result only rules out this sufficient width guarantee. It does not show that another sharper enclosure or an actual record cannot resolve the parameters.
Genuine bounded-error ambiguity for two complete histories¶
A different argument proves an information obstruction for an explicit subfamily. Fix the same admitted nominal \(b,c\), common means, zero centered residuals \(u=v=0\), fixed events, kernels and held offset. Compare the two constant-clock models
They lie within (2) for every horizon in (4), and their derivative is zero, so any nonnegative drift allowance admits them. Their initial full nodal states agree. This subfamily is not an equilibrium continuum: the nominal state may have port forcing and its whole state evolves.
Let \(\phi_j=\theta_0+a_jq\) denote the nominal phase after the corresponding cumulative input, and set \(v_j=\gamma q^Tf(\phi_j)\). The common first-order sensor signal is the broken affine function
It includes the first-window history in the second window. No state is reset at \(H\), and (22) is used only as a reference for an explicit full-flow error bound, not as the measured response.
Heat-semigroup contraction and \(\|f\|_M<7\) give \(\|P_Mx(\tau)\|_M\le7g\tau\). Since the declared phase jump is exactly the jump of \(\phi_j\), the continuous phase deviation from the piecewise nominal phase accumulates throughout both windows:
Common phase means are included in \(\theta_0\). By the norm bound \(\|A\|_M\le2\), sine Lipschitz constant two and \(\|q\|_{M^{-1}}=1\), on each continuous segment
Form continuity permits integration across the phase event. For either complete history this proves
Every averaging kernel is nonnegative. The largest second and third time moments among the four windows occur on \([H,2H]\), where they are \(7H^2/3\) and \(15H^3/4\). In (21), \(\sum G_i\rho_i^2=15/4\) and \(\sum G_i\rho_i^3=39/8\). Since the leading signal is common, the two exact noiseless average vectors \(m^{(1)},m^{(2)}\in\mathbb R^4\) therefore obey
Consequently, if the sensor allowance satisfies
the common exact recorded vector \(z=(m^{(1)}+m^{(2)})/2\) is admissible for both complete models: take their four sensor errors as \(z-m^{(i)}\). Each error has magnitude at most \(\delta\). These are the four errors of the declared averaged sensor, not an invented pointwise noise process. They differ between the two alternative unknown realizations, as allowed by the bounded-error observation model. Exact recorded point intervals have \(t=0\), hence fit every nonnegative declared numerical half-width allowance.
The mathematical midpoint need not be rational. Since \(\gamma<g\) and \(H>0\), the bounds used in (23)-(24) have strict slack for these two histories. At the threshold in (25), each pair of admissible sensor intervals therefore has an overlapping interior. A common rational value can be selected in each interior instead of the exact midpoint, so rational singleton reading intervals with \(t=0\) also realize the ambiguity. This is an existence argument, not a computed acquisition.
Thus there exists an identical four-average record for two fully evolving histories whose gain and first-window mean rate each differ by \(1/2\). Any uniformly valid covering inference for that record must retain both; it cannot guarantee gain or mean-rate marginal width below \(1/2\). This is a genuine bounded-error obstruction, not merely equality of a leading coefficient or overlap of two loose response intervals. It is existential: it does not say that every record at that error level is ambiguous. For any fixed \(\delta>0\), sufficiently small positive \(H\) satisfies (25), so arbitrarily short acquisition cannot uniformly recover these parameters at fixed absolute noise.
Implementation and remaining boundary¶
The detached budget calculation consumes only (4), the fixed model/prior constants and policy targets. The independent controls check the moment/error algebra, uniform horizon budget and distinct ambiguity threshold. The calculation consumes no reading, saved response, inverse report, hidden preparation or producer certificate. A positive sufficient budget establishes a conditional mathematical regime; unavailable or failed sufficient bounds must remain distinct from (25).
There is a deliberate gap between the sufficient regime (15), the budget obstructions (18)-(20), and the genuine ambiguity condition (25). These arguments do not determine the sharp information threshold. Improving a bound cannot erase the explicit indistinguishable pair when (25) holds. Nor does a successful algebraic budget establish empirical attainability of its source, clock, gain, aperture or error premises.
The execution plan owns further admission. Physical application still needs independently justified preparation, observation/calibration and time-unit maps under the same complete model. No new response campaign or physical-data claim is implied by this theoretical resolution audit.