Ø Frozen Predictions · the document of record
Reproduction: verify_between_family.py · Status: a result, not a ruling. The status of KS-30.4 is the author's to set; this document reports what the search found.
RIGIDITY.md (2 August 2026) establishes that 21 cannot move within the corpus's family of formulas, and states its own limit in §4(a):
Rigidity is not uniqueness. This shows that within the corpus's own family the integer cannot move. It says nothing about whether a completely different family of formulas, built from different primitives, would also work. That is KS-30.4's between-family limb, and it is open and unclosed.
That is the single most damaging open item in the corpus's defence against the numerology charge, because it is the one an outside reader will raise first. This is an attempt to settle it by enumeration rather than by assurance.
The space the sceptic names — "a handful of small integers and π, and a free choice of where to put them" — made concrete:
| primitives | the integers 1..30, and π |
| operations | + − × ÷ |
| complexity | the number of primitive tokens; competitors get five, the corpus's own count |
| target | m_p/m_e = 1836 + α·A, so A = 20.9217554797 lands on CODATA 2022 |
| tolerance 1 | 1.333 × 10⁻³ — the corpus's own order-α miss, so a competitor must be at least as good |
| tolerance 2 | 4.385 × 10⁻⁶ — the CODATA measurement bar itself, eleven-digit agreement |
The corpus's α-coefficient is 21·(1 − 1/(84π)) = 20.9204225285: five leaves — 21, 1, 1, 84, π — and 1.333 × 10⁻³ from the target. That miss is the 5.3 ppb (304σ) order-α residual the corpus publishes and then closes with its α² term.
2,971 distinct formulas of five leaves or fewer reach the ratio at least as accurately as the corpus's own. 15 of them reach eleven-digit agreement at order α, which the corpus's formula does not — it needs its α² term to get there. The five closest:
20.9217554461 err −3.4e−08 22 + 10/(19 − 9π) 20.9217557252 err +2.5e−07 20 + 21/(22 + 18/23) 20.9217557627 err +2.8e−07 24 + ((14/17)/13 − π) 20.9217565580 err +1.1e−06 21 − 19/(20(9 + π)) 20.9217567635 err +1.3e−06 21 + 28/(π − 19·19)
The proton ratio's precision, on its own, proves nothing. Any presentation of this corpus that leads with the eleven digits is misleading, and should stop. RIGIDITY.md §4(c) said this in words — "the digit count proves less than it looks… the defence does not rest on the proton's digit count and should never be presented as if it does" — and this is the number behind the words: about three thousand ways to do as well, and fifteen ways to do better, inside a grammar no larger than the corpus's own.
The corpus does not claim the ratio alone. It claims one integer carries three unrelated observables at once: the mass scaffold 1836 = 4N² + 3N + 3² (which is simply 1836 written in base N), the gravitational constant as α^N · F · ℏc/m_e² with F a small structural factor, and the visible fraction as 1/N. N sits in an exponent in G, so one step costs a factor of 137. Every integer from 14 to 28:
| N | F that G demands | small? | 1/N | visible error | 1836 in base N |
|---|---|---|---|---|---|
| 19 | 7.0 × 10⁻⁵ | no | 5.263% | +7.74% | 5, 1, 12 |
| 20 | 0.00955 | no | 5.000% | +2.35% | 4, 11, 16 |
| 21 | 1.309 | yes | 4.762% | −2.52% | 4, 3, 9 |
| 22 | 179.4 | no | 4.545% | −6.95% | 3, 17, 10 |
| 23 | 2.5 × 10⁴ | no | 4.348% | −11.00% | 3, 10, 19 |
Requiring F ∈ [0.5, 3] and the visible fraction within 3%, exactly one integer survives: 21, and the F it demands is 1.3092 against the corpus's 1 + 1/π = 1.3183 — the 0.69% that AP28 published and never repaired, and that AP44 later realised at −0.036%.
None of Leg A's ~3,000 formulas has a second life. They reach one number and stop. That asymmetry — thousands of ways to hit one observable, one way to hit three — is the whole of the corpus's structural claim, and it is now enumerated rather than asserted.
Settled against the corpus. The proton ratio's precision has no evidential weight on its own. The corpus should never present it as though it had.
Settled for the corpus. Within the shared-integer family, N = 21 is alone in carrying all three observables across the searched range.
Still open. Whether some other primitive set — one not of the form "a single integer in three places" — also carries three observables at once. This search does not reach that question: it enumerates one coefficient, not three formulas jointly. Settling it needs a grammar over the whole triple, and that is a larger piece of work.
KS-30.4's between-family limb is therefore narrowed, not closed. It should stay LIVE. The one-observable reading of it is answered — and answered against the corpus. The cross-observable reading survives the strongest search that has been run at it so far.
Every number above regenerates from CODATA 2022 by verify_between_family.py. Copyleft 2026. Don't be a cunt. Be kind.
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