# Finding: KS-30.4's between-family limb, enumerated — 4 September 2026

**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.

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## 1. What was open

`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.

## 2. What was searched, fixed before running

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.

## 3. Leg A — the proton ratio alone. **The limb fires.**

**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.

## 4. Leg B — one integer, three observables. **The limb does not fire.**

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.

## 5. What is settled, and what is not

**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.

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*Every number above regenerates from CODATA 2022 by `verify_between_family.py`.*
*Copyleft 2026. Don't be a cunt. Be kind.*