# Claims ledger: what base 3 hides

Every claim on the page was re-derived from scratch before it was written down. The source was
the base-3 research notes (136 lines); the environment was plain python3, with no mpmath, sympy or
matplotlib. Three scripts do the work: `precision.py` (pi via Machin in decimal at 60
digits; trigamma via Euler-Maclaurin in exact rationals, error about `1e-42`, cross-checked by the
reflection identity `psi1(1/3) + psi1(2/3) = 4*pi^2/3` with residual `1.1e-42`),
`sieves.py` (float sums, lattice sums, gcd sieves) and `makefig.py` (figure).

## The ledger

| # | Claim in the notes | How recomputed | Result | Verdict |
|---|---|---|---|---|
| 1 | `6/pi^2 = 1/zeta(2)` is the coprime-pair density, quoted as 0.6079 | pi by Machin; `6/pi^2` at 40+ digits; grid sieve `gcd(a,b) = 1` over `[1,3000]^2` | 0.6079271018540266...; sieve 0.608042 | confirmed (density theorem: Mertens 1874, cited) |
| 2 | The notes' own measured square density, 0.6111 | own sieve (their sample size unknown) | 0.608042 against 0.607927 | confirmed in substance; their 0.6111 replaced by my measurement |
| 3 | Visibility density is `1/zeta(2)` on any rank-2 lattice: base-blind, the trap | the condition `gcd(a,b) = 1` is on coordinates only, independent of the embedding; hex-frame count in the figure 364/612 = 0.595 (small window) | holds by the coordinate argument plus measurement | confirmed |
| 4 | Fresh nodes at level L: `b^L - b^(L-1) = phi(b^L)`, fraction `(b-1)/b`, so 1/2, 2/3, 4/5 for b = 2, 3, 5 | exhaustive count of `k/b^L` with `b` not dividing `k`, for b = 2, 3, 5 and L = 1..4, against the direct totient | exact equality in every case | confirmed for prime b; caveat added: for b = 6, L = 2 the count is 30 but `phi(36) = 12`, so the `phi` identity needs prime b (the count `b^L - b^(L-1)` holds for all b) |
| 5 | Eisenstein integers `Z[omega]`, `omega = e^(2*pi*i/3)`, hexagonal lattice, norm `a^2 - a*b + b^2` | definitional; norm multiplicativity asserted in code | `N(z*w) = N(z)*N(w)` checked | cited (standard) |
| 6 | Coprime-pair density in `Z[omega]` is `1/zeta_K(2)`, `K = Q(sqrt(-3))` | full Euclidean gcd in `Z[omega]` (nearest-rounding division), 400,000 random pairs in a box of radius 500 | 311237/400000 = 0.77809 against the predicted 0.778094 | confirmed (theorem for class-number-1 rings, cited; sieve agrees to 4+ digits) |
| 7 | `zeta_K(s) = zeta(s) * L(s, chi_-3)` | the Dedekind factorization is a cited theorem; verified numerically at `s = 2` via the lattice sum `(1/6) * sum 1/N(z)^2` over norm `<= 4e6` plus analytic tail | 1.2851909555 against `zeta(2)*L` = 1.2851909555 (10 digits) | confirmed at `s = 2`; general `s` cited |
| 8 | `L(2, chi_-3) = 0.78130241289648629687...` | `(psi1(1/3) - psi1(2/3))/9` by exact-rational Euler-Maclaurin; independent direct Dirichlet sum (paired terms, 2e5 pairs) | 0.7813024128964862968671874296...; direct sum 0.781302412896 | confirmed (their 20 digits round correctly) |
| 9 | `zeta_K(2) = 1.28519095548...` | product of the recomputed factors | 1.2851909554841494... | confirmed |
| 10 | `1/zeta_K(2) = 0.77809448917...`, the density | reciprocal | 0.7780944891751794... | confirmed |
| 11 | Their sieve gave 0.7764 against 0.7781 | own sieve (row 6) | 0.77809, tighter | confirmed in substance; their number replaced by mine |
| 12 | `L(2, chi_-3)` is "provably not a pi-power"; "PSLQ returned no relation" for a rational multiple of `pi^2/sqrt(3)` | continued fraction of `L*sqrt(3)/pi^2` (and of `L/pi^2`) at 40 digits: for a two-term relation the CF is equivalent to PSLQ | quotients 0, 7, 3, 2, 2, 3, 2, 23, 2, 7, 1, 12, 1, 17, 1, 29, ... stay ordinary; convergent denominators pass 3e9 by term 16 and 1e11 by term 22; control `zeta(2)/pi^2` terminates instantly at 1/6 | corrected: the no-relation finding is confirmed empirically, but "provably" is an overclaim, since non-detection is not proof and no such proof exists (the constant is not even proven irrational) |
| 13 | Parity argument: `chi_-3` is odd, even-argument L-values of odd characters are the hard ones, the same parity reason `zeta(3)` has no `pi^3` form | character parity checked (`chi(-1) = chi(2) = -1`); the closed-form parity pattern is standard | odd character confirmed | cited, with a correction: `zeta(3)` is not proven to be no `pi^3` multiple (Apery proved irrationality only), so the page says "no known closed form" |
| 14 | Catalan `G = L(2, chi_-4) = 0.91596...` | `(psi1(1/4) - psi1(3/4))/16` in exact rationals | 0.9159655941772190150... | confirmed |
| 15 | Base-2 field: `zeta_Q(i)(s) = zeta(s) * L(s, chi_-4)`, so the square-lattice ring hides pi and Catalan | Gaussian lattice sum `(1/4) * sum 1/(a^2 + b^2)^2` over norm `<= 4e6` plus tail, against `zeta(2)*G` | 1.5067030099 against 1.5067030099; coprime-Gaussian density 0.663700804614 | confirmed at `s = 2` |
| 16 | Added in support: `L(2, chi_-3) = (4/(3*sqrt(3))) * Cl2(pi/3)`, the Gieseking connection that places it in the Catalan class | `L * 3*sqrt(3)/4` at 28 digits against Gieseking's constant | 1.0149416064096536250... = `Cl2(pi/3)` | confirmed (identity via Clausen duplication, cited) |
| 17 | "Base picks the imaginary quadratic field" for every base b | only b = 2 (4-fold, `Q(i)`) and b = 3 (6-fold, `Q(sqrt(-3))`) are instances; the crystallographic restriction forbids 5-fold lattice symmetry, so the rule cannot extend naively (b = 5 has no lattice) | two instances verified; the general rule is not a theorem | unverifiable as stated; the page presents it as an observed pattern with its limit noted |
| 18 | Engineering claims about a private codebase: a base parameter, passing test counts, a simulator tie-in | not checkable here | — | cut, as private detail with no place on a public page |

## Tally

- Confirmed: 10 (rows 1, 2, 3, 4, 6, 8, 9, 10, 11, 14), plus 2 numeric verifications of cited
  theorems (7, 15) and 1 added supporting identity (16).
- Cited: 2 (rows 5 and 13, the latter with a phrasing correction).
- Corrected: 1 (row 12: "provably not a pi-power" downgraded to "no relation found, no closed form
  known").
- Unverifiable, so softened: 1 (row 17).
- Cut as private: 1 (row 18).

## Corrections carried into the page

1. "Provably not a pi-power" became "an integer-relation search finds no rational multiple of
   `pi^2/sqrt(3)`, with denominators ruled out past `1e11`; no closed form is known, and the parity
   of the character says none is expected." No proof of impossibility exists, and neither `G` nor
   `L(2, chi_-3)` is even proven irrational.
2. The `zeta(3)` analogy was softened from "has no `pi^3` form" to "has no known `pi^3` form".
3. The fresh-structure `phi` identity was scoped to prime bases (the b = 6 counterexample is
   recorded above); the `(b-1)/b` fraction itself holds for every base.
4. Their sample measurements (0.6111 and 0.7764) were replaced with my own: 0.608042 at N = 3000,
   and 0.77809 at 4e5 pairs.
5. The general "base picks the field" corollary is marked as a pattern rather than a theorem, with
   the 5-fold crystallographic obstruction noted.

## Scripts

`precision.py` computes every constant on the page to high precision using nothing but the
standard library: pi by the Machin formula `16*atan(1/5) - 4*atan(1/239)`, each arctangent summed as a
Taylor series in Decimal at 60 digits, and the trigamma `psi1(x) = sum 1/(n+x)^2` by Euler-Maclaurin in
exact Fractions (100 explicit terms, then the Bernoulli tail through `B_20`, error about `1e-42`). From
those it forms `L(2, chi_-3) = (psi1(1/3) - psi1(2/3))/9`, Catalan `G = (psi1(1/4) - psi1(3/4))/16`,
`zeta(2) = pi^2/6`, the two Dedekind values `zeta(2)*L` and `zeta(2)*G` and their reciprocals. It closes
with the cross-checks: the reflection identity `psi1(1/3) + psi1(2/3) = 4*pi^2/3`, the implied
`Cl2(pi/3)`, and an integer-relation search run as continued fractions of `L*sqrt(3)/pi^2` and
`L/pi^2`, with `zeta(2)/pi^2` as a control that terminates at 1/6.

`sieves.py` carries the float-level structural checks. It sums the paired Dirichlet series for
`L(2, chi_-3)` over 200000 pairs; evaluates `zeta_K(2)` as the Eisenstein lattice sum
`(1/6)*sum 1/N(z)^2` truncated at norm `4e6` plus the analytic tail `(pi/(3*sqrt(3)))/X0`, and the
Gaussian analogue with tail `(pi/4)/X0`; and counts `gcd(a,b) = 1` over `[1,3000]^2` for the
square-lattice visibility. It then implements `Z[omega]` arithmetic from scratch - multiplication using
`omega^2 = -1 - omega`, norm `a^2 - a*b + b^2`, nearest-integer division by the conjugate, and the
resulting Euclidean gcd - and measures the coprime fraction of 400000 random pairs drawn from a box of
radius 500 against `1/zeta_K(2)`, finishing with the fresh-node counts `b^L - b^(L-1)` against
`phi(b^L)` for b = 2, 3, 5 and the b = 6 counterexample.

`makefig.py` draws the hexagonal visibility figure and writes it to `research/bases-fig.png`.
Eisenstein coordinates `(a,b)` are placed at `x = a - b/2`, `y = b*sqrt(3)/2`, the hidden points
(`gcd(a,b) != 1`) are painted first as small recessive discs and the visible ones on top as larger blue
discs, with an ink ring punched at the origin; discs are drawn by per-pixel coverage blending, and the
RGB buffer is encoded as a PNG by hand with `zlib` and `struct`, so no plotting library is involved. The
script prints the in-frame visible fraction quoted in row 3 of the ledger.