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: lab/base3/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), lab/base3/sieves.py (float sums, lattice sums, gcd sieves) and lab/base3/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^(2pii/3)`, hexagonal lattice, norm `a^2 - a*b + b^2`	definitional; norm multiplicativity asserted in code	`N(zw) = 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/(3sqrt(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

"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.

The zeta(3) analogy was softened from "has no pi^3 form" to "has no known pi^3 form".
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.

Their sample measurements (0.6111 and 0.7764) were replaced with my own: 0.608042 at N = 3000,

and 0.77809 at 4e5 pairs.

The general "base picks the field" corollary is marked as a pattern rather than a theorem, with

the 5-fold crystallographic obstruction noted.