Use the phase relationships.
Grouping 32 successive powers of two for every odd starting integer reduces the complete clock bound to about 2.987 × 10⁻⁸ at the base radii. Every omitted power and every odd starting integer remains in the bound.
What can an experiment
actually tell us?
A measurement rarely identifies everything. The useful question is whether it determines the answer you need—and whether that answer survives the uncertainty you allow.
PreparationHarmonic readingsArithmetic structureInfinite tailsFinite resolution
ONE COMMON TESTDo different answers
still fit the same data?
still fit the same data?
Locate a sourceRecover integers
A proved transfer framework, with independently checked computational consequences in two declared mathematical models. It is a research result, not a claim of physical calibration or external peer review.
01
START WITH THE ANSWER SET
Keep every explanation that fits.
Suppose the source is a number x between 0 and 4. An unknown offset u affects two readings. We observe 5 and 7. Each sensor error has magnitude at most ε.
EXACT ELEMENTARY MODEL
Why this example can validate itself
The offsets are unrestricted real numbers. Subtracting the two readings with a shared offset gives 2 = x + e₂ − e₁. The exact feasible interval is [2 − 2ε, 2 + 2ε] ∩ [0,4]. Every point in it has an admissible pair of errors and a common offset. Integer answers are the integers in that interval. One reading, or independent unrestricted offsets, permits every x in [0,4]. The slider uses integer hundredths, so all membership decisions are exact.
This is an answer set for the displayed observation. It does not assert a uniform noise threshold for every observation. Continuous uncertainty is reported as an interval, not silently rounded into certainty.
02
THE MATHEMATICAL CONNECTION
Transfer a guarantee through the experiment.
The central object is the set of answers consistent with the data. A valid processing step must retain the true explanation. A certificate then separates different answers or encloses the remaining uncertainty.
𝒬D(y) = { q(x) : y = FD(x,u) + e, with all variables admissible }
- Declare the questionA target label? A continuous source? An integer coefficient? State the source class and measurement norm.
- Remove shared nuisanceFor linear nuisance, project out its full admitted subspace. An affine clock preserves polynomial drift; arbitrary jitter may not.
- Charge what remainsSensor error, finite approximation, normalization, clock/model directions and complete nonlinear or infinite remainders.
- Pass the query gateSeparate every pair of different labels; bound the inverse error for a continuous answer; stay below ½ for integer rounding.
THE REFINEMENT
A disturbance has a direction, not only a size. Passing that direction through the query inverse can give a much smaller error bound than assuming it points in the experiment’s weakest direction.
Read the compact transfer theorem and proof
After a valid nuisance operation, suppose y = Aⱼu + Dⱼ(z)u + e, u ≠ 0, ‖e‖ ≤ rⱼ‖u‖, Aⱼ* Aⱼ ≥ μⱼI > 0. Keep the same joint parameter z across all readings in one explanation. Bound Fⱼ ≥ sup ‖Dⱼ(z)‖ and Vⱼ ≥ sup ‖Aⱼ†Dⱼ(z)‖.
The true reading lies in a tube of radius δⱼ = Fⱼ + rⱼ. If, for each distinct label pair,
[Aⱼ, −Aₖ]*[Aⱼ, −Aₖ] ≻ diag(cⱼI, cₖI),
δⱼ²/cⱼ + δₖ²/cₖ ≤ 1,
δⱼ²/cⱼ + δₖ²/cₖ ≤ 1,
with cⱼ > 0 and cₖ > 0, then the labels cannot share a reading. After selecting the correct label, nominal least squares satisfies
‖û − u‖ / ‖u‖ ≤ Vⱼ + rⱼ / √μⱼ.
Proof. A shared reading would imply ‖Aⱼu − Aₖv‖ ≤ δⱼ‖u‖ + δₖ‖v‖. Weighted Cauchy–Schwarz and the strict Gram inequality contradict this. For reconstruction, Aⱼ†Aⱼ = I, so û − u = Aⱼ†Dⱼ(z)u + Aⱼ†e; take norms. A linear query uses its own inverse gain. For integer recovery, a separately established coordinate error strictly below ½ makes rounding unique.
For linear uncertainty on a box, norm convexity reduces the worst case to its vertices. Nonlinear terms still need a complete remainder. Competing explanations may choose different unknown parameters; sharing within an explanation does not force sharing across hypotheses.
Full proof and hypotheses ↗03
PATH A / SPATIAL SENSING
Six readings. Twenty-one possible targets.
A diffusing two-mode source starts at one of 21 neighboring locations. Selected spatial averages must identify its location and recover its two source coefficients. The new question: can the six-row bank tolerate both an uncertain clock and an uncertain spatial potential?
Controls are percentages of this experiment’s certified limits.
VERIFIED CONTRACT
The browser checks containment in a published uncertainty box. It does not rerun the interval matrix proof or assert failure outside that box.
What exactly does the spatial guarantee cover?
The grid has 1001 × 1001 cells with reflecting boundaries. The generator is H(g) = I ⊗ Lₓ + Lᵧ ⊗ diag(1 + gx), centered at g = 4/5. Possible targets are rows 490–510. The nonzero source lies in two specified orthonormal cosine modes. Both nominal and actual time lie in [1,2]. The norm is source-relative Euclidean sensor error and L² source error.
Every selected readout is a spatial average, not an arbitrary point detector. Sensor count saves acquisition only if those readouts are available. The six-row clock and potential radii were fixed before testing. Every target and every time in the interval is covered by exact polynomial and interval certificates; finite sample illustrations alone would not prove this.
Model, complete-time proof and raw-data replay ↗04
PATH B / ARITHMETIC SENSING
Recover integers through an infinite tail.
Use 8,900 complex readings to recover the 49 unknown integers a(2), …, a(50). Higher coefficients, polynomial drift, bounded oscillatory drift and an affine clock all affect the same data. Rounding is justified only when the complete coordinate error is strictly below ½.
Base radii: fractional scale 10⁻¹⁴; additive offset 10⁻¹¹ in the acquisition time coordinate.
VERIFIED CONTRACT
Complete error bound
The stacked bars are rounded component estimates. The certified status refers to independently checked exact inequalities at the saved contract, inherited at smaller clock radii.
Degree-eleven bound misses the target.
Its verified bound exceeds ½ for the fixed amplitude 4,000 and frequency 3 contract. Degree twelve remains the least passing degree among 6, 8, 10, 11 and 12. This is a comparison of sufficient certificates, not a proof that every degree-eleven method must fail.
Read the arithmetic model and full budget
F(t) = Σ a(n)n⁻²⁻ⁱᵗ with integer 0 ≤ a(n) ≤ d₁₄(n); a(1) = 1 is known. Samples are tⱼ = (2j + 1 − 8900)/10. Weight mass is one. This divisor envelope is a declared source class; not every sequence in it is asserted to come from a number field.
The nonpolynomial family has the selected amplitude limit (4,000 or 500) and frequency at most 3 in x = t/890, with arbitrary phase. A separate mismatch allowance is 10⁻⁶; sensor error is 4 × 10⁻⁵ in the weighted norm. A shared affine clock preserves the polynomial nuisance space. Arbitrary per-reading jitter would change that premise.
The coordinate error is a complex-modulus bound; integer rounding applies to the real part. Complete arithmetic tails, drift approximation, sensor noise, model mismatch, clock distortion, digital centering and a verified actual-matrix residual all enter the recovery error. Future data require their own residual check. Family membership requires a justified model or calibration; a small solver residual cannot establish it.
Complete bounds, degree costs and reading reconstruction ↗05
THE RESEARCH BEHIND THE TRANSFER
Five paths, one question.
The two detailed applications grew out of five investigations. Their common language is useful because it preserves their different source classes, uncertainty sets and questions.
Follow the progression through nine research packages
↗
SHARE / VERIFY / BUILD ON IT
Every claim has a route back.
Start with the public guide, inspect the mathematical report, or replay the complete evidence. The standalone lab works from a local file and makes no background network requests.
Copy this scenario JSON
A scenario file contains only the displayed control values. A URL fragment carries the same state without uploading it. File-based links work on the same folder location; share the downloadable lab with the scenario file for offline use.