Thin-torus two-birth limits and source modes (stacked on #710; do not merge)

[LightChainr]

Draft. Do not merge. Stacked on #710 (which is stacked on #708). Needs those two JSON certificates; blob hashes match. No new MC, GPU, or p_c.

What

On the (w,0)×(0,m) torus, F is the mixture of birth times T1,T2, not the rank law at fixed p. Elementary barriers:

P0(p) ≤ (1-p^w)^m,   P2(p) ≤ [1-(1-p)^w]^m

so for each fixed w and p∈(0,1), as m→∞: r→1, M→0, F→1/2 pointwise, while

L(T) ⇒ ½ δ_0 + ½ δ_1.

Lower quantiles → 0, upper → 1; the median may still → q_w. Inverting a limiting flat CDF does not recover the finite median. Same split if w=o(log m) (e.g. w=j, m=⌈e^{j²}⌉). This does not refute #613 (ℓ_N/log N→∞); here that ratio → 0.

Joint limit for each fixed integer w≥2:

(m^{1/w} T1, m^{1/w}(1-T2)) ⇒ (A_w, B_w) independent,
P(A_w>x)=e^{-x^w},  P(B_w>y)=e^{-c_w y^w},
c_w = [z^0](z^{-1}+1+z)^w  (= 3,7,19 for w=2,3,4).

Finite-sample covariance of births is nonzero (exact 2×3: 1123/58800). Cylinder crossing balances two rare sectors: Pr(r=2|r≠1) is a sigmoid, unconditional F still → 1/2. Median slope Q'(1/2) ∼ 2/(m h'(q) λ_*^m): exponential bias vs exponential ill-conditioning.

Independent Bernoulli snapshots at one p are the wrong estimator for the root (at q4, m=128, E≈8.49e-10). Not a lower bound on Newman–Ziff / exact transfer.

Source: Z=1+M sinh s + E(cosh s-1). Generic orders M:16, E:28, tilted numerator 28, Z(e^s=2):29. A fixed source tilt shifts the root by 1/m; untitled matching-root bias is O(λ_*^m). Source reflection can be restored trivially by rank-1 concentration.

Width-4 quartiles from small matrix powers (not a million-row MC): m=32 IQR 0.421; m=10^6 IQR 0.957, median limit still Jacobsen n=4 q4.

Independent Grok check

• c_w = #{words in {-1,0,1}^w summing to 0} = [z^w](1+z+z^2)^w = 3,7,19,51,141,393.
• Barrier implications on exhaustive 2×3, 3×2, 4×2: full row ⇒ r≠0, empty row ⇒ r≠2, zero failures. P0,P2 sit under the stated elementary bounds at p=1/2.
• Certificate / definition JSON SHA-1 match Width-four exact rank closure: 509 futures and 15-mode scalar (do not merge) #708/All-p width-four spectrum and diamond 3L−1 onset (stacked on #708; do not merge) #710.
• 12 tests passed after apply on d543ba1. Q_025(m=32)=0.36373322… matches the report.
• Z=1+M sinh s+E(cosh s-1) is the mgf of a {-1,0,1}-valued X.

Full-repo CI not run. Do not re-dispatch the same #613/#276/#582/#337 tasks.

Out of this PR

#613 stays as stated. Next new theorem would be a sharp geometric condition for a threshold step to p_c, not more thin-torus lengths. Cross-width inference must control (i) rare-sector balance, (ii) two-birth law, (iii) source-visible modes — one object does not proxy the other two.

[LightChainr]

Do not merge. Stacked on #710/#708. Independent: c_w=3,7,19; exhaustive 2×3/3×2/4×2 barriers (full row ⇒ r≠0, empty row ⇒ r≠2); P0,P2 under the elementary bounds at p=1/2; certificate SHA-1 match; 12 tests on d543ba1. Full-repo CI not run.

[LightChainr]

Compatible, not a rewrite. Draft #718 proves the matching root on the same super-thin sequence still → p_c while this PR’s L(T) splits to the endpoints. Keep both. Do not merge either.