# The Aggregate 30-Skeleton Bound — certified at every computed level, @11 through @29 *(2026-08-14. Companion to `natal-cap-30-skeleton-bound.js`, which verifies the collapse identity pointwise (≤ 4e−18 @11) and against cap-26's 2ⁿ-mask ledger (relerr ≤ 1.6e−10 through @17, all 164 primes; 1.5e−7 @19 over a **58-of-435 sample chosen to include every exception**, the float ledger's own precision floor — sample disclosure added 2026-08-18, the clause read "1.5e−7 @19" flat while its @11–@17 neighbour said "all 164 primes", so the reader was invited to take both as exhaustive; the 58/435 is printed at `natal-cap-30-skeleton-bound.js`:216 and at `research/wave7-logs/cap36-cap30-repro.log`:6), and certifies every inequality below in exact BigInt. Notation as cap-26: level x, W = x#, N̄ = 2∏(p−2), δ = N̄/W; scour prime q, lA = ⌈W/q⌉, lB = ⌊(W+1)/q⌋, L = lA+lB; C(m) = c₃₀(m)·P(m) the natal autocorrelation with P(m) = ∏_{7≤p≤x} c_p(m) its 30-free part; dev(q) = R(q) + 1/2; ledger subsets T ⊆ {30, 7..x}; skeleton G30(q) = Σ_{30∈T} dev_T.)* Status of these results and of the anchored calm as a whole: [anchored-calm.md](anchored-calm.md). Cap-26 reduced the anchored calm's aggregate anticorrelation to one open statement: **G30_agg(x) < 1/2**. This file (1) collapses the skeleton — a priori 2ⁿ ledger terms — to a single closed-form kernel, (2) turns the open statement into an integer inequality and certifies it exactly at every computed level, and (3) computes, honestly, why the all-x version stays open. The certified ladder now runs to @29; the sixth level is the `--at29` pass of [natal-cap-36-skeleton-door.md](natal-cap-36-skeleton-door.md), whose split-weight accumulator is what keeps the arithmetic exact at W = 6.5·10⁹. --- ## Theorem A (Skeleton Collapse — proven, all x, all q) Define **K(m) := 15·C(m) − 2·P(m)**. Then for every lag m **Σ_{T∋30} ∏_{M∈T} φ_M(m) ∏_{M∉T} m̄_M = φ₃₀(m)·∏_p f_p(m) = K(m) / (15W)**, and hence, with the cap-26 exact variance integers IVA + IVB = W²(V_A+V_B), **G30(q) = W·NUMsk(q) / (30·(IVA+IVB))**, where **NUMsk = L·(K(0) + 2Σ_{0½ | |---|---|---|---|---|---|---|---| | 11 | 10 | +0.2132 | 0.287 | +0.2133 | 0.0024 | −0.287 | 13 (0.625) | | 13 | 34 | +0.1113 | 0.389 | +0.1109 | 0.0068 | −0.389 | none | | 17 | 120 | +0.1011 | 0.399 | +0.1013 | 0.0070 | −0.399 | 107 (0.544) | | 19 | 435 | +0.1259 | 0.374 | +0.1262 | 0.0060 | −0.374 | 2083 (0.606), 2221 (0.521) | | 23 | 1739 | +0.0945 | 0.406 | +0.0948 | 0.0039 | −0.405 | 2339 (0.575) | | **29** | **7863** | **+0.1176** | **0.3824** | **+0.1180** | **0.0054** | **−0.382** | **173 (0.511)** | ΣCov_adj < 0 is also certified as an exact integer sign at every level. Rows @11–@19 reproduce cap-26 to all printed digits; @23 and @29 extend it. @23 is the first exact deviation/skeleton pass at W = 223,092,870 (17 s), with the lowest dev_agg and the largest margin on record; @29 is the `--at29` pass at W = 6,469,693,230 with 7,863 scour primes (11.3 min), which reproduces @23 as its control. The @29 max|no30| is **0.0054, at q = 1109**, the largest in the table and still four times inside the margin; the aggregate no30 there is dev_agg − G30_agg = +0.0004, in line with every level below. That cell was carried as absent until 2026-08-17, when `natal-cap-36-skeleton-door.js` gained the column and the pass was re-run: every other figure on both rows reproduced exactly, and the control row's @23 value matches the @23 row above, so the column arrived with its control already passed. Exceptions keep thinning: 1 in 1,739 at @23 and 1 in 7,863 at @29. Cap-26's "the offset does not trend to 0 through @19" softens into no trend at all — @19 and @29 are both upward moves, and the sequence has no decay law (cap-36). **Corollary (the aggregate anticorrelation is a theorem at six levels).** With cap-26 (−1/2 exact, the deviation identity, the no-30 bound), R_agg = −1/2 + G30_agg + no30_agg < 0 at every computed level, every ingredient exact or exactly certified. The all-x statement is not covered by this: it is the Skeleton Equidistribution Conjecture, and it is OPEN. ## Proposition C (The uniformity blockers — computed, not conjectured) Three candidate all-x bounds, evaluated exactly: 1. **Branch-Abel** (the P4 machinery on the 30∈T half): |G30(q)| ≤ Ask·(L/4 + lB/2)/Vg, Ask = s₃₀·∏_p(m̄_p+s_p). Aggregate value 18.2 → 52.5 → 158 → 521 → 1593 at @11..@23 (truth: ~0.1); certifies **zero** primes at any level. The per-branch triangle bound is Θ(1/Vg) with an L-free constant, so the window length is not the obstruction. It is vacuous for two other reasons (cap-36 §"The per-branch bound"): the constant grows like 8^{π(x)}, since the sum over branches is dominated by the deepest one where M_T is W itself; and the denominator shrinks, the aggregate calm ratio ΣV/Σ(Lδ) reading 0.2349, 0.1149, 0.0505, 0.0198, 0.0080 at @11 through @23, a clean loss of 0.29 per unit ln W. 2. **Spike-product candidate** ∏_p(1 + 2/(p−2)) − 1: 0.711 @11, 1.805 @23, divergent in x (Mertens). Dead on arrival. 3. **Exact positive mass** (kernel K⁺, BigInt): 24.5 @11 → 7.3e6 @23. The skeleton's smallness is ~100% cancellation along the two incomplete lag classes; no positivity/majorization route exists. **The blocker, named.** Even the shallowest term T = {30} alone (kernel φ₃₀·∏m̄_p, period 30) Abel-bounds to ≈ 0.12·(L/2)·∏m̄_p/Vg ≫ 1/2 for x ≥ 17. Its true aggregate smallness — and hence any all-x proof — requires the window-length residues ⌈W/q⌉ mod 30 and class phases a mod 30 to equidistribute over the scour primes q: an analytic prime-equidistribution statement, provably outside the branch algebra that proved cap-26 P4. The wall's smallest room has an arithmetic door, and the door is prime equidistribution. ## Proposition D (Exception criterion — exact) q is a skeleton resonance iff **W·NUMsk(q) > 15·(IVA+IVB)** — one integer comparison per prime. It reproduces all four cap-26 exceptions to the printed digit (13@11: 0.625; 107@17: 0.544; 2083@19: 0.606; 2221@19: 0.521) and finds @23's single one, q = 2339 (0.575) — the only Cov_adj > 0 prime among 1739 — and @29's, q = 173 (0.511), the only one among 7,863. Six known counterexamples in 10,201 scour primes: this criterion is what makes the Uniform-in-q Anticorrelation refutation (cap-23) exact rather than statistical. The criterion is exact but not shallow: rigid (|T| ≤ 2) skeleton averages ≈ 0 (cap-26), so no congruence one-liner in q decides it; near-resonances form the expected continuum (@23: q = 107 at 0.483). Resonance = multi-prime alignment of the classes qZ, a+qZ with the J₅ spikes, now computable as two AP-restricted sums of one kernel. ## What this file owns, and what it does not Two proven objects and one open one: the **Skeleton Collapse Theorem** (Theorem A, all x and all q), the **Aggregate 30-Skeleton Bound (@11..@29)** (Theorem B, certified as an exact integer inequality at six levels, margins 0.287 to 0.406), and the exception criterion (Prop D). What is open is the all-x form, whose blocking term Prop C identifies as prime-equidistribution of window residues rather than branch algebra; cap-36 goes through that door, names the statement the **Skeleton Equidistribution Conjecture**, and finds it is not the blocker. The calibration of every part of the anchored calm, including the parts this file does not touch, is one table in [anchored-calm.md](anchored-calm.md). ## Next The T = {30} aggregate term: ⌈W/q⌉ mod 30 equidistribution over scour primes may be attackable via Vinogradov-type bounds — the first branch with an actual analytic handle. Cap-36 measures that door open at M = 30, 210 and 2310 with square-root cancellation, and then shows that between 91% and 111% of the skeleton's mass sits in branches whose modulus exceeds the window length, where the same question is posed at moduli of order W. The honest next move is stated there rather than here.