# The Minus-Half Theorem — the exact −1/2 anticorrelation constant *(2026-08-14. Companion to `natal-cap-26-minus-half.js`, which verifies every identity below — the pushforward identity and the exact −1/2 to relerr ≤ 1e−15 on the full spectrum at x = 7, 11; the deviation identity and ledger to ≤ 1e−9 for every scour prime at x = 11..19. Notation as cap-23: level x, W = x#, natal set N, N̄ = 2∏(p−2), δ = N̄/W, S(j) = Σ_{r∈N} e(jr/W); scour prime q, lA = ⌈W/q⌉, lB = ⌊(W+1)/q⌋, L = lA+lB, a = q·lA − W; R(q) = 2Cov_adj/(V_A+V_B). C(m) = natal autocorrelation (the J₅ product), g(m) = C(m)/W − δ². Pushforward variable t_j = (qj mod W)/W.)* Status of this result and of the anchored calm as a whole: [anchored-calm.md](anchored-calm.md). Cap-23 closed with one open leg: under k/q-equidistribution of the resonant spectral weight, R(q) → E[sin²θcos2θ]/E[sin²θ] = −1/2. This file proves what is provable there. The surprise: **for the flat spectral functional the −1/2 is EXACT — a two-line consequence of the mod-30 difference set — and the whole equidistribution question compresses, exactly, into two residue-class correlation sums.** --- ## Proposition 1 (Pushforward identity) For every integer h: **Σ_j |S(j)|² e(h·qj/W) = W·C(qh mod W).** *Proof.* |S(j)|² = Σ_{r,r'∈N} e(j(r−r')/W); summing e(j(r−r'+qh)/W) over all j gives W exactly when r'−r ≡ qh (mod W), else 0. ∎ So the Fourier coefficients of the pushforward of the spectral measure |S(j)|² under j ↦ t_j are the natal autocorrelations at the lags q, 2q, 3q, … — everything J₅-computable. (Verified @7, @11: h = 1..10, every scour prime, relerr ≤ 4e−16 against the full CRT spectrum.) ## Theorem 2 (Minus-half, flat form — EXACT) For every level x and every q coprime to 30 (in particular every scour prime): **Σ_{j≠0} |S(j)|² sin²(πt_j) cos(2πt_j) / Σ_{j≠0} |S(j)|² sin²(πt_j) = −1/2** exactly — no equidistribution hypothesis, no error term. *Proof.* The difference set of {11, 17} mod 30 is {0, ±6}. For a unit u mod 30 and 1 ≤ h ≤ 5, hu mod 30 never lies in {0, ±6} (check the 8 units; first hit is h = 6, when q ≡ ±1 mod 5). So C(qh mod W) = 0 for h = 1..5: by Prop. 1 the pushforward integrates every trig polynomial of degree ≤ 5 exactly as the uniform measure does. Both test functions have degree 2: numerator = −¼WN̄ + ½WC(q) − ¼WC(2q) = −¼WN̄, denominator = ½WN̄ − ½WC(q) = ½WN̄ (the j = 0 term drops itself: sin²(0) = 0). Ratio −1/2. ∎ The heuristic constant of cap-23 is not a limit — it is an identity. The natal comb's two teeth sit 6 apart in a 30-wheel; 6/30 = 1/5, and no lag qh with h ≤ 5 can come back around. All approximation lives in the difference between the flat functional and the true abutting-window kernel — and that difference is *exactly* the next proposition. ## Proposition 3 (Deviation identity — the bridge) The combined lag weight of 2Cov_adj + (V_A+V_B)/2 is ψ_d = 2w(d) + (lA−d)₊ + (lB−d)₊ (w the cap-23 trapezoid), and **ψ_d = L for every 1 ≤ d < lA; ψ_d = 2(L−d) for lA ≤ d < L.** Hence, exactly: **R(q) + 1/2 = [ (L/2)·B1 + B2 ] / (V_A+V_B)**, with - **B1 = Σ_{|d| 1/2 ⟺ Cov > 0): skeleton = 0.625 (13@11), 0.544 (107@17), 0.606 (2083@19), 0.521 (2221@19) — carried by deep (support ≥ 3) terms: {30,7} +0.314 and {30,7,11} +0.321 for 13@11; {30,7,13,17} +0.392 for 107@17; {30,7,11,13} +0.245 for 2221@19; a broad multi-T spread for 2083@19. Cap-23 found "no divisibility law" — correct: the law is multi-prime alignment of the two lag classes q·Z and a+q·Z with the J₅ spike classes, deterministic and exactly computable but not a one-liner. Near-exceptions form a continuum (q = 1621@19: dev +0.449). - **The −0.4-not-−0.5 gap is real**: mean rigid (|T| ≤ 2) skeleton ≈ 0.000 at x ≥ 13 — the systematic +0.10..0.13 offset is deep skeleton and does not trend to 0 through @19. ## Sharpest true form - **[PROVEN, all x, all q coprime to 30]** Theorem 2: the flat −1/2 is exact, with 5 harmonics of immunity (Prop. 1 + the mod-30 difference set). - **[PROVEN, all x, q]** Prop. 3: R(q) + 1/2 = two explicit residue-class correlation sums over V_A+V_B; Prop. 4: the no-30 part ≤ APB(x,q), uniformly in q, with the explicit constant A_x. - **[CERTIFIED EXACT @11–@19]** R(q) = −1/2 + G30(q) + no30(q) with G30 computed exactly for all 599 primes and |no30| ≤ 0.007 (measured; bounded a-priori with 100× slack); the non-resonance condition G30 + APB < 1/2 certifies Cov < 0 for 7/10 @11 and 4/34 @13 (vacuous at @17/@19); the 4 exceptions are exactly the deep-skeleton resonances G30 > 1/2. - **[OPEN]** an all-x bound on the skeleton itself — the aggregate calm needs only **G30_agg < 1/2** (measured +0.10..+0.21). Its deep terms are the same diffuse support-≥2 cloud as cap-02, now CONFINED to the lags ≡ 0, ±6 (mod 30) of the two classes q·Z and a+q·Z. The wall, in a smaller room. ## Where this sits This file owns the Minus-Half Theorem and the reduction that follows it: the all-x aggregate statement is exactly **G30_agg(x) < 1/2**, and everything else in the chain is proven. That is strictly smaller than what cap-23 left open, which asked for equidistribution of the whole spectral weight; the constant needs none. The reduction has since been carried further. The skeleton's 2ⁿ ledger collapses to one closed-form kernel and G30_agg < 1/2 is certified as an exact integer inequality at six levels @11 through @29 ([natal-cap-30-skeleton-bound.md](natal-cap-30-skeleton-bound.md) Theorems A and B, [natal-cap-36-skeleton-door.md](natal-cap-36-skeleton-door.md) P6), and the residual all-x statement is named the Skeleton Equidistribution Conjecture. Calibration of every part of the anchored calm is one table in [anchored-calm.md](anchored-calm.md).