Pricing a numerical coincidence a tolerance achieved

Pricing a numerical coincidence at the tolerance achieved

David Coates — Independent Researcher, Reynoldsburg, Ohio, United States ORCID 0009-0009-9192-4797 · Draft, August 2026

In memory of Shirley.

Abstract

A claim that some measured quantity "is" a simple expression in a small pool of constants is normally reported as a percentage agreement, at a tolerance the claimant chose. We replace the declared tolerance with the achieved one. Because the coverage of the target space by ε-neighbourhoods of the pool is identically the distribution function of the distance from a target to its nearest pool point, the resulting score is a probability integral transform: it is exactly Uniform(0,1) under the null, for every pool, every budget, every density, with no calibration step and no asymptotics. Three consequences follow. The score decomposes exactly into a part fixed by the framework's own choice of pool and a part contributed by the target, so forced and selective content can be separated by computation rather than by judgement. The score's growth rate under increasing precision or increasing index identifies the claim's provenance — whether it is a genuine near-miss, an exact identity, or a convergent that was always going to be close. And the score cannot be inflated by enlarging the budget: the set of targets scoring b bits has measure exactly 2⁻ᵇ whatever pool is used. We give the extraction rule, and worked examples: four in which the instrument returns zero — three of them on the author's own published work — and one in which a claim passes at 5.82 bits.

1. What is classical

Almost all of the arithmetic below is old, and the paper is only honest if that is said before anything else.

Under octave equivalence a multiplicative pool {2^a g^b} reduces to the orbit of an irrational rotation on the circle, x ↦ x + α (mod 1) with α = log₂ g. The gap structure of such an orbit is the three-gap theorem, conjectured by Steinhaus and proved by Sós, Surányi and Świerczkowski in the 1950s [Sós 1958]: at most three distinct gap lengths, the largest equal to the sum of the other two. The gap lengths and their multiplicities are known in closed form via the Ostrowski expansion of N, in terms of ‖q_n α‖ for convergent denominators q_n.

Equidistribution is Weyl (1916); unique ergodicity of the rotation is what licenses reading a covered measure as a probability.

The application to tuning is also established. Carey and Clampitt (1989) connect the three-gap theorem to well-formed scales; the pentatonic (5), diatonic (7) and chromatic (12) are convergents of log₂(3/2), and the two semitone sizes of Pythagorean tuning together with the comma are exactly the three gaps. Clader (2018) gives the expository account.

Consequently the following, which the author derived independently, are rediscoveries and are claimed as such: that the saturation staircase of the {2^a 3^b} lattice steps at convergent denominators 1, 2, 5, 12, 41, 53; that its gap values are exactly the named Pythagorean intervals (fourth, minor third, whole tone, apotome, limma, limma less one and two commas, and the comma) to 0.00000 cents; and that the largest gap equals the sum of the other two.

What follows is a statistical construction on top of that arithmetic. It is not a contribution to Diophantine approximation.

2. The substitution

Let a pool be a finite point set on the circle with cyclic gaps g₁…g_N summing to 1. An arc of half-width h about each point covers

C(h) = Σᵢ min(2h, gᵢ) (1)

exactly — a per-gap ledger, not a merge algorithm. Since a relative tolerance ε is a half-width h = log₂(1+ε), (1) is the coverage at tolerance ε.

Now let D be the distance from a target to its nearest pool point. Its distribution function is

F_D(d) = Σᵢ min(2d, gᵢ) = C(d) (2)

identically. Coverage is the CDF of the nearest-point distance. Therefore by the probability integral transform, C(D) ~ Uniform(0,1) exactly, and with

B := −log₂ C(D) (3)

we have P(B > b) = 2⁻ᵇ, for any pool, any budget, any density. No calibration, no asymptotics, no simulation.

The substitution is small and its consequence is not: a declared tolerance is a free parameter chosen before the data, whereas D is a statistic of the data.

Scoring at the achieved tolerance removes the parameter rather than policing it.

3. Forced and selective

The target's position factorises. Let G be the gap it lands in (size-biased) and U its normalised position within that gap. Then with V = min (U, 1−U),

D = G · V, G ⊥ V, 2V ~ Uniform(0,1) exactly.

 (4)

Verified on the {2^a3^b} lattice at |b| ≤ 5: Kolmogorov–Smirnov p = 0.92 for 2V overall, and p = 0.69, 0.53, 0.048 within each of the three gap classes separately; corr(log G, log V) = −0.001.

Define B_V := −log₂(2V).

On the unsaturated branch,

B_D − B_V = −log₂(N G) (5)

exactly (verified to 3×10⁻¹⁵). The right-hand side depends on G alone — the pool's own inhomogeneity, which the framework chose. B_V depends on the target alone. Both have the same null law;

E[B_D] = E[B_V] = 1/ln 2, so

(5)

is a redistribution, not a bias. When the gaps are equal, B_D = B_V identically.

This makes the forced/selective split a computation. B_V is the headline number: it prices what the target contributed and nothing else.

4. Provenance from a slope

Score the same claim twice — at two precisions, or at two indices of a family — and take the difference. The growth rate identifies the failure mode: growth of B regime constant genuine claim; achieved ε is fixed log₂ n parabolic / repeated root; error decays as 1/n

n·log₂|λ₊/λ₋|

convergent-forced; error decays geometrically

d·log₂10 = 3.3219 per decimal digit

exact identity; the null is false

The third row is a spectral-gap law: for a linear recurrence x² = px + q, the achieved tolerance of the n-th consecutive-term ratio decays as |λ₋/λ₊|ⁿ, so

dB/dn = log₂|λ₊/λ₋| (6)

Verified to ten digits: Fibonacci 1.3884838273, Jacobsthal 1.0000000000, Pell 2.5431066063, bronze 3.4473579372, each matching log₂|λ₊/λ₋|. The sign pattern gives a second, independent read: alternation is forced for every q > 0 recurrence, since continued-fraction convergents always bracket their limit. A claim resting on "the two values straddle the limit" is restating a theorem.

The fourth row is the divergence of §9 of the companion note made quantitative: I(d) = d·log₂10 − log₂(2N/ln 2), with slope invariant across pools and only the intercept carrying N. It reproduces the two figures published there, 48.16 bits at 16 digits and 161.11 at 50.

5. What cannot inflate the score

Invert (3). The set of targets scoring at least b bits is a union of arcs of total measure exactly 2⁻ᵇ — 11 arcs at |b| ≤ 5, 25 at |b| ≤ 12, 41 at |b| ≤ 20, all of measure 0.5000, 0.2500, 0.03125 for b = 1, 2, 5. The pool decides where qualifying targets sit; never how many.

Consequently there is no inverse in the budget. Scoring π at |b| = 2, 3, 5, 8, 12, 20, 41 gives 0.59, 0.29, 2.39, 1.76, 1.21, 0.49, 0.11; e gives 1.85, 1.37, 0.72, 0.39, 1.30, 0.59, 0.07. Non-monotone in every case: enlarging the budget brings points nearer and raises coverage, and the two cancel. A budget cannot be designed to reach a chosen score.

Two things can inflate a score, and both must be declared in advance: the coefficient bound when integer-relation detection is used, and the complexity bound when the target family is generated recursively. An undeclared bound turns any family into a dense set worth nothing. This is the same failure as an undeclared exponent budget, in two further costumes.

6. The extraction rule

For each claim, in order, stopping at the first that applies:

Is the target exactly a pool member?

Integer-relation detection (PSLQ) against the declared basis with maxcoeff declared in advance; a relation counts only if the working precision exceeds three times the digits its coefficients require. Confirm by the 3.3219 bits/digit slope. → FALSE NULL.

Is it a convergent of the pool's dominant eigenvalue?

Score at two indices; slope log₂|λ₊/λ₋| → FORCED. Logarithmic growth → repeated root, also forced.

2b. For a physical relation, is its form fixed by dimensions?

Buckingham π on the dimension matrix; a single free dimensionless group means the form carries no information and only the prefactor is selective.

Is an uncertainty stated?

 If not, record NO ERROR BAR and stop.

A claim without a stated uncertainty is not in scoreable form. Otherwise score at ε_eff = max(ε_achieved, relative standard error) and report B_D, B_V and Δ, with B_V as the headline.

Step 3 is not a formality. In the audit below it disposes of half the numerical claims examined, and it is the single most common defect.

7. Worked examples

7.1 A convergent in disguise

Claim: the B-DNA pitch-to-groove ratio 34/21 = 1.619 approximates φ to 0.06% (author's own, zenodo.20573572).

The agreement is arithmetically correct: 0.06265%. But 34/21 = F₉/F₈, and by Hurwitz's theorem φ is the worst-approximable irrational, with |φ − p/q| → 1/(√5 q²) optimally. At q = 21 the predicted relative error is 0.06267%. Observed over predicted: 0.9995. The number was fixed by continued fraction theory before any molecule was measured. Verdict FORCED, 0 bits.

Two independent confirmations. First, the same paper's L₈/F₈ = 47/21 ≈ √5 at 0.09066% matches the Binet closed form 2ψ⁸/(φ⁸−ψ⁸) to six digits. Second, the input cannot support the precision: major-groove width is quoted to ±1–2 Å, and 34/22 (the standard value) gives 4.49% rather than 0.063%. A single ångström swings the error seventyfold. Using consistent modern values — 35.7 Å pitch, 22 Å groove — gives 0.29%, five times weaker, and needs no Fibonacci framing.

That is the defensible version of the observation.

7.2 A prediction, declared and falsified

The same corpus reports peaks in exoplanet period ratios at the Jacobsthal.convergents 21/11 and 43/21. Since |J_{n+1}/J_n − 2| = 3/(2ⁿ − (−1)ⁿ) exactly, and the framework's own declared resolution floor is σ = φ⁻⁹ = 0.013156, exactly four convergents are resolvable. The third, 85/43 = 1.976744, sits at 0.0233 — comfortably above the floor — and had never been looked for.

A protocol was written and hashed (sha256 f99e81af…1108) before the NASA

Exoplanet Archive was queried: band [1.85, 2.15], nine bandwidths fixed in advance, a result counting only if it held at a majority of them, and the failure condition stated. T1 failed: 0 of 9 bandwidths. The published peaks reproduce at tight bandwidth (216 ratios in band, matching the published count), but the next convergent is absent. Recorded as declared: the convergent-peak hypothesis is wrong as stated. A second test nominally passed and is not claimed — its hits came from two peaks merging at wide bandwidth, satisfying the letter of the pre-registered rule and failing its intent.

Two independent arguments agree. Mean-motion resonance strength falls steeply with order, and 21/11 and 43/21 are 10th and 22nd order — libration widths far below anything observable in 216 systems. And straddling is forced: consecutive convergents always bracket their limit, so "two peaks either side of 2:1" carries zero selective bits and matches ordinary resonance physics equally well.

7.3 A published framework: what survives and what does not A recently deposited framework builds, from a signed "antimatter" variant of Pascal's triangle, a chain of claims ending in a quantum error-correcting code said to surpass the Bravyi–Poulin–Terhal bound and an associative memory of countably infinite capacity with O(1) retrieval. Eight claims are checkable.

Three hold and five fail, and the split falls in one place. Verified (3/8). The row sums of the signed triangle are exactly 2 — twenty rows out of twenty, no exceptions. The sum of the positive interior entries of row n is exactly the n-th term of A000975 — twenty out of twenty, cleanly indexed a(0) through a(18), not approximately but as the sequence itself. And the claimed identity 2ⁿ + L(n) = L(n+2) holds for n = 1…14.

The third passes as arithmetic and fails as a discovery. The paper states the defining recurrence ℓ_n + ℓ_{n−1} = 2ⁿ − 1 on the same page; subtracting consecutive instances gives a(n+1) − a(n−1) = 2ⁿ, which is the claimed theorem.

Under §4, that is a claim restating its own premise: zero selective content.

Refuted (5/8).

The uniqueness theorem is false for the closed form its own proof uses. The proof defines H(n,k) = round(φ^{2(n−|k|)}/√5), which depends only on n − |k|.

Hence C(3,0) = C(4,1) = C(5,2) = C(6,3) = C(7,4) = (8,8): the map is constant on every diagonal. Injectivity is asserted from a formula that manifestly lacks it.

That closed form does not generate the paper's own tables. Row 4 is printed as (1, 0, 2, −2, 1); the formula returns (0, 1, 3, 8, 21, 8, 3, 1, 0). The proof and the data describe different objects.

The code comparison is inverted. For [[F_{2n+1}, 1, F_{n+1}]], d/√N = 0.852, 0.851, 0.851 at N = 233, 1597, 10946 — the same O(√N) scaling as the surface code, whose ratio is exactly 1, with a worse constant. And kd² = 169 against N = 233 satisfies the BPT bound comfortably rather than surpassing it.

The memory cannot work, and the paper measures this. Addressing runs through SHA3-512, which destroys locality by construction: flipping one bit of a test object moves the hash by 37% of its value. Nearest-address recovery from a corrupted cue therefore returns an arbitrary object. The document's own benchmark reports 2% success (1/50); a simulated run in the same document

reports 99.90%.

How far the rule needed to go. Across four documents from one external programme, exactly one claim reached the scoring step at all. The others halted at step 1 (a constant defined as (3−φ)/2 rather than measured), at step 2 (a six-fold binary cascade on 64 codons, forced because 64 = 2⁶ and true of any 64 objects), or at step 3 (no uncertainty stated anywhere in forty-five pages).

That is the rule's practical value: the bit count is rarely needed, because three cheap checks dispose of most claims before any coverage is computed.

Where the split falls. Everything that survives is a property of the table— an exact combinatorial object with exact structure. Everything that fails is built on top of it: the closed form, the code parameters, the retrieval scheme.

The object is sound; the superstructure is not attached to it. That is the common shape of the failures in §7.1–7.3 and it is what the extraction rule is built to locate: not "is this wrong" but "which layer stopped being load-bearing".

A note on the author's own position. The present author is a co-author with one author of that framework on an unrelated paper, and has a further claim scored in §7.4 arising from the same collaboration. Five of the withdrawn claims in the corpus were withdrawn during that collaboration for reasons unrelated to their merit. All of this is disclosed here rather than in a footnote, because a paper about how claims are made and unmade should record the pressures that unmade some of its own.

7.4 A claim that passes

Not everything returns zero, and the instrument is worth nothing if it cannot show this.

The P1/P2 partition of the twenty amino acids — P1 = {A,N,D,Q,E,G,H,S,T} — gives a residue-weighted composition of 0.498178, a deviation of 0.1822 percentage points from balance, measured over 208,897,834 residues. Enumerating all C(20,9) = 167,960 nine/eleven partitions, a partition drawn at random lands this close with probability 0.0177, so a pre-specified partition scores 5.82 bits.

Everything turns on provenance, and here it is documented: the partition is fixed by the PI/10 classification, which predates the composition analysis by decades. It was not selected for balance. Had it been, the correction would be total — P(at least one of 167,960 this close) = 1.000000, zero bits.

The claim survives, and one adjacent claim does not. The partition ranks 2,970 of 167,960 on balance alone, and the best available (A,N,D,E,H,L,K,M,T) is 0.000080 pp — 2,278 times closer. So the partition is balanced, worth 5.8 bits as a pre-specified test, and is not the optimum on that criterion. A companion paper claims global optimality on a joint criterion including a weighted-gap invariant; that joint claim is untested here and this result does not bear on it.

A second pre-registered test in the same corpus reports 11/12 correct against a declared threshold, p = 13/4096 = 0.003174, 8.30 bits. The arithmetic is exact, conditional on a fair-coin per-group null. Inverting for the null shows the result falls to p = 0.05 at a per-group baseline of 0.661 and to a coin flip at0.864, so the baseline is the quantity to check; verification awaits the deposited script.

7.5 What the audit does and does not demonstrate

The corpus of twenty closed claims was scored under the rule of §6 and every pre-declared expectation was met. That statement is much weaker than it sounds and the weakness must be stated first.

Nine rows are theorems or constructions: they could not have returned anything but unscoreable, so predicting that outcome is not a prediction. Two more state no uncertainty and cannot be scored at all — a gap in the record, not a test. Rows genuinely at risk: nine. Five of those are claims the author withdrew during a collaboration for reasons unrelated to their merit and had never scored; two of the five (√3 = e^(ln3)/2, true of every positive real, and the Kerr identification, refuted by det H = a²/M² ≠ 1) were fixed by mathematics regardless. So the instrument had a genuine opportunity to disagree on roughly seven rows, and did not.

A second correction. Multiplicity of log₂(20) = 4.32 bits is charged to the corpus. Applying it consistently to the two passing claims gives 1.50 bits for the partition and 3.98 bits for the split test. The split test has a defence — it was registered as a single test with its own declared threshold, not drawn from twenty — but the partition has none, and the uncorrected 5.82 should not be quoted without the corrected 1.50 beside it.

A third. Coverage numbers here use the octave lattice at |b| ≤ 5. Three rows — a modulation depth, a base-composition peak, a Voronoi scatter — have no octave structure, and §8 states that C is then a measure rather than a probability.

Those three are reported as computed and should be read as measures. Scoring them properly requires a second declared pool, which this paper does not supply.

7.6 Where the corrections bind

Three corrections apply to any score: saturation, the error-bar floor, and multiplicity. All three are measured in bits, so they are directly comparable, and the crossover is a single inequality. claim

ε SE floor cost mult cost

binds on

34/21 ≈ φ 0.063% ~7% 6.80 4.32 saturation — 0 bits

OR ≈

√2 0.126% 0.45% 1.83 4.32 multiplicity  Koide's relation

9.2×10⁻⁶ 1.0×10⁻⁵ 0.14 4.32

multiplicity

μ = 1/4, Chargaff — none ——

NO ERROR BAR

Multiplicity costs log₂N uniformly and can be applied once at the end; the error-bar floor costs log₂(SE/ε) and must be computed per claim. The floor dominates exactly when SE/ε > N.

Two consequences worth stating. Koide's relation agrees to 9.2×10⁻⁶ while the τ mass permits only 1.0×10⁻⁵, so the floor applies — but it costs 0.14 bits, and what actually reduces the claim is the 4.32-bit multiplicity. The famous precision was never the binding constraint. And for 34/21 the groove-width uncertainty pushes ε_eff to 7%, past ε*(5) = 6.07%: the claim is annihilated by measurement error alone, independently of the Hurwitz argument of §7.1.

Half the corpus reaches none of these tests, because no uncertainty is stated. That, not the mathematics, is the binding constraint on the audit.

7.7 The instrument on itself

During development the reference implementation set working precision at module import, silently overriding the caller. A protocol declaring 80 digits executed at 60, and integer-relation detection matched a planted target it should have rejected. Removing that assignment then froze log(2) at the 15-digit default, and the exact lattice member 432 = 2⁴3³ — which must diverge — scored 48 bits. Both defects were found by positive controls and neither by reading the code. Precision is now set and derived constants recomputed together, and
asserted at run time.

The lesson generalises: an exact identity scoring high because a constant was
quietly imprecise is precisely the failure this instrument exists to detect, and
it nearly shipped inside the instrument.
8. Limits 

It prices the pool, not the truth. A high score means the match was unlikely
under the null. π ≈ 6φ²/5 agrees to 1.5×10⁻⁵ and is refuted outright by
Lindemann; Benford's law scores nothing and is a theorem. The probability reading needs a compact group. Unique ergodicity of the rotation on ℝ/ℤ, with Lebesgue as Haar measure, is what makes C a probability.

This is a theorem about the group, not an assumption about physics — and it
fails exactly where the target space is not a compact group orbit. For a length
ratio or a base composition, C should be read as what it literally is.

The second multiplicity can be bounded but not applied, because the number
of quantities scanned before a match was reported is almost never recorded.
One dimension only. Everything here rests on the three-gap theorem, which
fails on a torus. Multi-generator pools have no canonical continued-fraction
algorithm (Jacobi–Perron), no three-gap bound, and no closed-form coverage.
That is where the method runs out, and it is the honest direction for extension.
Validation is thin on the positive side. One claim has passed and returned
a non-zero score (§7.4, 5.82 bits); a second reports 8.30 bits and awaits
re-verification against its deposited script. Everything else scored to date has
returned zero, FORCED, or NO ERROR BAR. The controls that establish sensitivity are otherwise synthetic: claims planted at 0.5–12 bits are recovered to 2×10⁻⁴⁵ bits, and the null survives the full chain (observed P(B > b) = 0.498, 0.252, 0.115, 0.062, 0.030 against 2⁻ᵇ; PSLQ false positives 0/600).

No claim has yet passed this instrument and then been confirmed true by independent means. The 5.82-bit result says a partition is unusually balanced given its provenance; it does not say why.

One pattern, offered as an observation rather than a result. Across everything scored here, the claims that survive are the ones that were pre-specified — the PI/10 partition, the pre-registered split test — and the claims that fail are the ones formed after the numbers were in view. The division is not between true and false claims but between claims made before looking and claims made after. If that holds on the remaining corpus, it is a finding about method rather than about arithmetic, and the remedy is procedural.
9. Prior work by the author

The grading of claims by declared evidentiary status (the Claim Meter, and
the honesty invariant convention of recording a claim's status alongside it)
was deposited by the author before the analyses reported here and is cited as
prior work rather than introduced.
The forced/selective correction of §3 is the exact form of an argument the
author made empirically in May 2026: that uniform null sampling tests the wrong hypothesis when an integer set is forced by historical theorems, demonstrated on F = {3, 7, 13, 21, 43, 71}, where a uniform null gives p ∈ [10⁻¹⁵, 5×10⁻⁴] and an alternative-recurrence null gives 7/64 ≈ 0.11. The saturation argument of §5 likewise appears in the author's March 2026 exoplanet note, which observes that the fractional symbolic ratio error test yields p < 10⁻⁷ for Fibonacci, Mersenne, Pell, Lucas and Padovan alike, and therefore distinguishes nothing. Both predate this instrument and neither is claimed as new here.

10. AI disclosure

The author used Claude (Anthropic) for symbolic verification, numerical
checking, literature search, and manuscript preparation. All mathematical content, results, and the decision as to what is claimed and what is deferred were determined by the author. Two defects in the reference implementation described in §7.7 were found by the automated controls; roughly ten further errors arising during the analysis were caught by testing rather than by reading, and none by inspection of the code. The verification in this paper rests on the deposited suites, not on any single reading.

11. Reproducibility

coates_pit.py with two independent suites: verify.py, which reproduces
Tables 1–3 of the companion note exactly (Table 3 by closed form, bisection
retired) and confirms the three-distance bound for B = 1…60; and
positive_controls.py, which plants and recovers claims of known content.
Both report ALL PASS. Code Apache-2.0; protocol documents CC-BY-4.0.
References Carey, N. and Clampitt, D. (1989). Aspects of well-formed scales. Music Theory Spectrum 11(2), 187–206.
Clader, E. (2018). Why twelve tones? The mathematics of musical tuning.
Mathematical Intelligencer 40(3), 32–36.
Diaconis, P. and Mosteller, F. (1989).