# The capture identity: the transplanted cap family collapses exactly to the rough-pair census, floor_K = T − X(K), and the depth law gets the zero-parameter candidate y* = h^{1/(2e^γ)} *(2026-08-22. Staging note; nothing here is integrated into a live document. HELD for the end-of-day adversarial roundup. Producer, formally embedded: `research/attack-quadpoint-03.js` (2.2 s; out-sha256 196602b1...; a forced re-embed stamp from adding one printed sanity figure before publication — 0 of 51 figures changed — is disclosed in its tail; `--check` passes bit-honest). Calibration marked per claim: PROVEN, VERIFIED by exact computation, MEASURED, HEURISTIC, OPEN. This note is Z2's first deliverable: the analytic race's object simplified before the race is run.)* ## 1. The capture identity [PROVEN, elementary; VERIFIED at every depth of all 1,227 anchors] **Lemma.** Fix anchor Q, the half-open stretch S_Q = [Q², Q′²), capacity C counting the channel pairs (a, a+2) with BOTH members in S_Q, twins T, actives p₁ < p₂ < ... = 7, 11, 13, ..., and the pool convention of `attack-quadpoint-01/02` (prime r at depth K reads the first min(K, idx(r)) actives below r). Then for every K > Σ_r capU_K(r) = (C − T) + X(K), equivalently **floor_K = T − X(K)**, where X(K) = #{pairs with both members composite and min(lpf(a), lpf(a+2)) > p_K}; X(0) = all both-composite pairs (CC). The window convention is load-bearing, not cosmetic. Every composite member of such a pair satisfies lpf(v) ≤ √v < Q′, hence lpf(v) ≤ Q, which is step (i)'s finality. Finality fails at v = Q′², and (Q′² − 2, Q′²) is a channel pair at 1227 of 1227 anchors, so the loose convention a + 2 ≤ hi admits a pair whose composite member is a candidate of no r: floor_K moves by +1 and T − X(K) does not, breaking the identity by exactly one at every anchor tested [MEASURED, `redteam-0828-quadpoint.js` SEC C; 12 of 12]. `attack-quadpoint-03.js` implements the strict convention on both sides and is consistent throughout. *Proof.* (i) *Candidates are composite members, bijectively.* A candidate of r is v = r·m in the window with lpf(m) ≥ r; since v is coprime to 30, lpf(m) ≥ r holds iff r = lpf(v). Every composite member has lpf ∈ [7, Q] (finality: a composite in the window has its least factor among the actives) and is therefore a candidate of exactly one r; a prime member is a candidate of none (m ≥ 2 forces v composite; m = 1 puts v = r below the window). (ii) *The depth condition reads the partner's lpf.* v survives r's depth-K conditions iff its partner is divisible by none of the first min(K, idx(r)) actives, i.e. iff lpf(partner) > p_{min(K, idx(r))}. (iii) *Classify pairs.* A composite member with a prime partner always survives (a prime partner's lpf exceeds every active). In a both-composite pair with lpfs r₁ < r₂ (equality is impossible): the r₁-member always survives — its partner's lpf is r₂ > r₁, clearing r₁'s pool, which never reaches past r₁, at any K. The r₂-member survives iff r₁ > p_K: if idx(r₂) ≤ K its pool is all actives below r₂, which include r₁, killing it; if idx(r₂) > K the test is r₁ > p_K — and r₁ > p_K itself forces idx(r₂) > K, so the two cases are consistent. (iv) *Sum.* (C − T − CC) singles + CC always-surviving first members + X(K) surviving second members = (C − T) + X(K). ∎ Producer verification: the caps recomputed on the v2 code path equal T − X(K) at every K of every one of the 1,227 anchors to Q = 10007 (uncapped below 1499, K ≤ 64 above); the v2-cited K* values, the B8 band mean, and the maximum (46 at Q = 9281) all reproduce through the identity path. ## 2. What collapses [consequences, each now one line] - **K*(Q) is a crossing point, on the set where T ≥ 1:** the least K with X(K) ≤ T − 1 — the depth cost is where the rough-pair census falls below the twin count. If T = 0 there is no such K, since X ≥ 0, so the statement presupposes T ≥ 1 and is written on that set (TODO Z2's 2026-08-27 quantifier). Measured: 0 of 1,227 anchors have T = 0, so the band data lives entirely on {T ≥ 1} and cannot test the quantifier [MEASURED, `redteam-0828-quadpoint.js` SEC G]. - **The full-depth bijection** (v1's assert): X vanishes once p_K passes the largest composite lpf — both-composite pairs cannot be rough beyond Q in a finality window. - **The K = 0 stretch-grid death:** certificate ⟺ CC ≤ T − 1, i.e. both-composite pairs FEWER than twins. Verified exact: the 8-anchor list {7, 11, 13, 19, 23, 31, 37, 43} is precisely the CC < T list. - **The whole cap machinery needs no caps:** the certificate at (Q, y) is the statement "X(y) < T in S_Q" — twins outnumber the pairs that are composite-on-both-sides yet rough beyond y. The transplant instrument was measuring a classical object in disguise. ## 3. The depth law [HEURISTIC derivation; MEASURED-consistent, not confirmed] Main terms, densities relative to channel positions at height h = Q²: rough-beyond-y density δ_R(y) ~ (15/4e^γ)/ln y (Mertens over 7..y); prime density δ_P ~ (15/4)/ln h. The pair correlations carry the SAME local product κ = ∏_{p≥7} p(p−2)/(p−1)² = 0.9389 on both sides — X/C ≈ κ(δ_R − δ_P)² and T/C ≈ κδ_P² — so κ cancels EXACTLY and the certificate condition X < T reduces to > δ_R(y) < 2δ_P ⟺ ln y > ln h · (15/4e^γ)/(2·15/4) ⟺ > **y* = h^{1/(2e^γ)}**, exponent 1/(2e^γ) = 0.280730. That exponent is the `u → ∞` form of the crossing, not its limit. Writing δ_R(y) as the bare Mertens product sets Buchstab's ω(u) to its limit e^{−γ}, while the crossing sits at finite u. Done exactly, the condition is ω(ln h/ln y) < ∏_{y