# natal-cap-12 — the sub-CRT overlap sign: proofs, refutation, and the priced law 2026-08-14. Companion to `natal-cap-12-overlap-sign.js` (all statements below are machine-verified there; "verify A/B/C" and the T2 line in the output are the checks). Question under attack (campaign TODO 2, from natal-cap-06 readings 3/7): is S₂ = Σ_{q x ≥ 11 are primes. The class {r ≡ c mod Q} ∩ [0,W) is {c + tQ : 0 ≤ t < L_c} with L_c = ⌈(W−c)/Q⌉; writing W = Q⌊W/Q⌋ + ρ (ρ ≥ 1 since gcd(Q,W)=1) gives L_c = ⌊W/Q⌋+1 for c < ρ and ⌊W/Q⌋ for c ≥ ρ. For each modulus m ∈ {30} ∪ {p}, r ≡ v (mod m) ⟺ t ≡ (v−c)Q̄_m (mod m); substituting the natal conditions (r ∈ {11,17} mod 30; r ∉ {0, −2} mod p) gives the displayed t-conditions. The two forbidden classes mod p are distinct (their difference is −2Q̄_p ≢ 0), and the two allowed classes mod 30 are units times units, so per period of lcm = W the t-pattern count is 2·∏(p−2) = N. ∎ The forbidden-class *gap* in t-space is −2Q̄_p, not −2: the combo windows are **dilated** natal patterns, one per pair — the rotation-and-dilation ensemble of natal-cap-03/05, anchored at a specific origin. ## L2 — Class-sum identity Σ_{c mod Q} cnt_c = N (the classes partition N_x). Hence the CRT value N/Q is the **exact average over all Q classes**, and per-pair deviations measure anchoring only, not normalization. (Verified exhaustively for the pair 13·17 at @11.) ## L3 — Fair-price identity For every class c, N·L_c/W = N/Q + N·(Q·[c<ρ] − ρ)/(QW). *Proof:* substitute L_c from L1 and W = Q⌊W/Q⌋ + ρ. ∎ By cap-05's exact ensemble-mean theorem, N·L_c/W is the average count of a window of length L_c over all rotations of the (dilated) pattern — the honest "fair price" of the window. So CRT **overprices every short class by exactly Nρ/(QW)** and underprices every long class by N(Q−ρ)/(QW). ## L4 — Dead origin c = 0 satisfies c < ρ always (ρ ≥ 1), so combo (0,0) always draws the long window; but its t = 0 position is r = 0, which is never natal (0 ≡ 0 mod 7). Additionally r = W−2 (≡ −2 mod every tile prime, ≡ 28 mod 30 — never natal) lies in the (0,0) window iff Q | W−2, i.e. q and q′ both divide W−2 (counted in the run: 3 pairs @17, 1 @23, 0 elsewhere). No other position of any combo window is dead independently of (q, q′): in mixed combos r = 0 and r = W−2 would force q′ | 2 or q′ | W, impossible. ## T1 — Structured-bias theorem Define, per combo, dev := cnt − N/Q and the decomposition dev = len + head + resid, len = N·L_c/W − N/Q (exact, L3 — deterministic), head = −(N/W)·(#forced-dead positions) (L4 — deterministic), resid = cnt − (fair price of the window's non-forced positions). **Theorem.** For every pair (q,q′) at every level x: 1. combo (0,0): len + head = −Nρ/(QW) − [Q | W−2]·N/W **< 0 always** (long-window credit N(Q−ρ)/QW minus the dead origin N/W). 2. combo (−2,−2): len = −Nρ/(QW) **< 0**, UNLESS ρ = Q−1, i.e. W ≡ −1 (mod Q), i.e. **q and q′ both divide the Euclid number W+1** (then len = +N/(QW)·(Q−ρ) = +N/(QW)). These pairs are enumerable: @17, W+1 = 510511 = 19·97·277 gives exactly the 3 pairs (19,97), (19,277), (97,277); no other level ≤ 23 has any. 3. mixed combos: len = ±, sign [c < ρ], no fixed sign; the two mixed classes sum to c ≡ −2 and behave as coin flips over pairs (measured mean 0.00–0.43 N/W). *Proof.* (1) c = 0 < ρ always, so L = ⌊W/Q⌋+1 and L3 gives len = +N(Q−ρ)/(QW); the origin subtracts N/W = NQ/(QW); sum −Nρ/(QW). (2) c = Q−2 ≥ ρ unless ρ = Q−1; ρ = Q−1 ⟺ W ≡ −1 mod Q ⟺ Q | W+1 ⟺ q,q′ | W+1 (q,q′ prime). Apply L3. (3) immediate from L3. ∎ **Corollary (the priced law).** Summing over pairs, with ρ/Q equidistributing (measured: mean 0.4965 @23), E[dev per pair] = −2·(N/W)·E[ρ/Q] + o(N/W) ≈ −N/W, i.e. S₂ ≈ ΣCRT − (#pairs)·N/W. Measured (structured part vs total deficit): −18.5/−19.0 @13, −206/−415 @17, −2444/−1864 @19, −35657/−34275 @23 — the deterministic term IS the deficit; the remainder Σresid alternates in sign (−0.6, −209, +580, +1382; z = −0.03, −2.70, +1.65, +0.90) and has no persistent sign. ## T2 — Tail lemma (the largest-Q pairs are ABOVE CRT) **Theorem.** If ⌊W/Q⌋ < nextprime(x), then cnt₍₀,₀₎ = [ qq′ ∈ N_x ] exactly. *Proof.* The (0,0) window is {tQ : 0 ≤ t ≤ ⌊W/Q⌋}. t = 0 is dead (L4). Any t ∈ [2, ⌊W/Q⌋] has a prime factor ≤ t < nextprime(x), hence a prime factor p₀ ≤ x (including 2, 3, 5); then p₀ | tQ, so tQ is ≡ 0 mod p₀ — excluded by the natal conditions (for p₀ ∈ {2,3,5} via the mod-30 classes 11, 17, which are units). Only t = 1, r = Q remains. ∎ Verified: 1,205,518 pairs @23 (0 violations; likewise at all other levels). **Consequence.** For these pairs CRT predicts N/Q ≤ (⌊W/Q⌋+1)·N/W ≲ x·N/W ≪ 1 while the hit rate is P(qq′ ∈ N_x) — a product of two units mod 30 lands in {11,17} at rate ≈ 1/4 (vs 1/15 for a random position) and avoids −2 mod p at rate (p−2)/(p−1) (vs (p−2)/p), giving the unit-product heuristic P(qq′ natal) ≈ (1/4)·∏_{7≤p≤x} (p−2)/(p−1), measured 0.2222 / 0.1690 / 0.1618 / 0.1523 / 0.1453 at x = 11..23 vs theory 0.1875 / 0.1719 / 0.1611 / 0.1522 / 0.1453 (4-digit agreement once the scour is large enough for unit-equidistribution). Since 0.145 ≫ CRT ≈ 0.037 for L̄ = 1, **the deepest-tail pairs run systematically ABOVE CRT** (+9.1 N/W per pair @23). This is a *proven* mechanism producing positive deviations — the one-sided conjecture was structurally false, not unlucky. ## The refutation record - **Per pair:** 53.3 / 45.3 / 41.5 / 41.7 / 40.4 % of pairs sit above CRT at x = 11 / 13 / 17 / 19 / 23. Exhibits: (19,31) @11 dev +1.39; (19,71) @13 +2.06; (23,191) @17 +4.48; (37,307) @19 +12.10; (43,89) @23 +27.90 (cnt 5569 vs CRT 5541). - **Per combo:** even (0,0) alone is above CRT for 21–23% of pairs, e.g. (29,47) @23: cnt 3904 vs 3889.5. The negative structured bias −Nρ/QW < resid fluctuation scale per pair, so no per-pair sign survives. - **Aggregate:** S₂ = 30 > ΣCRT = 24.7 at @11 (ratio 1.216). The universal aggregate inequality is false. For x ≥ 13 the aggregate deficit is real but is exactly the T1 structured term (97%, 50%, 131%, 104% at 13/17/19/23), with the residual alternating sign — so any "S₂ ≤ ΣCRT for x ≥ 13" claim would hinge on Σresid never exceeding the structured −#pairs·N/W, which is a discrepancy bound, not a sign law (and at @19, @23 Σresid was positive: +580, +1382 — within 1.7σ, but positive). ## What survives (the useful form) **Priced pair law (proven core + measured remainder).** S₂ = ΣCRT − Σ_pairs [ 2Nρ/(QW) + mixedLen(q,q′) + corrections ] + Σresid where every bracketed term is computable in O(log Q) per pair (ρ = W mod Q; the mixed-combo classes via one modular inverse; corrections: +N/W for each Euclid pair q,q′ | W+1, −N/W for each pair with Q | W−2), and resid is the anchored-window discrepancy of a dilated natal pattern with measured per-pair rms 0.57 → 1.24 counts (O(1), level after level — attack-1's O(1) window law transported to dilated windows) and |Σresid| ≤ 2.7σ, sign-alternating. **What a Bonferroni certificate needs next** is not a sign theorem but a certified per-pair cap |resid(q,q′)| ≤ c(x) — exactly the campaign's open lead 3 (the discrepancy lemma), now reduced to a concrete family: initial segments of coprime-to-W combs (T2 shows the head behavior is Legendre/Buchstab oscillation: below fair for 2 ≤ L̄ < nextprime(x), above at L̄ = 1, oscillating beyond — mean ~0; the (0,0) head t < 30 pays −139,799 @23 and the bulk refunds +140,148). **The anchored calm, pair component: explained.** The sub-CRT pair/triple overlaps of cap-06 were the dead origin plus short-window arithmetic — deterministic bookkeeping, not a statistical quietness. That is one of the four original sightings dissolved; the calm's remaining sub-claims and their calibrations are in [anchored-calm.md](anchored-calm.md). ## Caveats, honestly - z-scores treat pairs as independent; pairs sharing a prime are correlated — read σ's as indicative, not exact. - The unit-product value (1/4)∏(p−2)/(p−1) assumes scour primes equidistribute over units mod 30 and over nonzero classes mod p; it is a heuristic with 4-digit empirical agreement at @19/@23, not a theorem (finite Chebyshev-type biases are visible at @11: 0.222 vs 0.188). - Triples (cap-06's 7–15% deficits) inherit the same machinery — three anchored classes mod qq′q″, origin dead in the (0,0,0) combo, all short/long terms from L3 — but were not re-measured here; the pair verdict makes the sign expectation clear (structured, not one-sided). ## Byproducts - New census numbers: Natal@5 survivors (= twins in (√W, W), by the empty-window crystallization argument, sieve-verified at 11–17 in cap-06): @19 = 38,380; @23 = 597,475. cap-08's refined @19 ladder (≥ 1877) certifies 4.9% of truth. - Euclid-number divisors of W+1 appear as a genuine (tiny) arithmetic exception class in the overlap ledger: q ∈ {59} @13, {19, 97, 277} @17, {347} @19, {317} @23 get the long −2-window in singles; pairs within one W+1 exist only at @17 (three of them).