# The destroyer census: the first-mover excess dissolves into a fully derived null, the zone kill budget gets a closed form with its crossover at 61/67, fifteen zones are occupied by counting alone, and the finality frame is graded *(2026-08-21. Staging note; nothing here is integrated into a live document. Producer, formally embedded: `research/destroyer-census-01.js` (2.0 s; code-sha256 d2062103..., out-sha256 1c286cc8...). Chris's framing, developed as tasked: only primes q with q^2 <= n+2 can destroy the pair at opener n (fresh-kill theorem, natal-onset-01 §0, red-team CONFIRMED), so every destroyed pair has one destroyer and every zone has an exact kill budget. The producer calibrates against natal-onset-01's 10k window (999/796/203, kill-level 539:543, destroyer counts 7:219 11:120 13:86 17:57 19:55), zonegap-02's zone twin counts (8/9/16/17/29) and its D3 youngest-kill list (22 levels), ABORTING on any mismatch, before measuring openers 11..1e8. Every null below is derived before it is measured against. Calibration marked per claim: PROVEN, VERIFIED by exact computation, MEASURED. No first-moment TPC claim appears anywhere: Route B is CLOSED (`research/REFUTED.md`); everything here is exact counting.)* ## 1. The census: 9,559,689 destroyed pairs, and a null with nothing left over **Totals [VERIFIED].** 9,999,999 channel pairs, 9,559,689 destroyed, 440,310 survive = pi2(1e8) = 440,312 (A007508, cited) minus the non-channel (3,5), (5,7) — asserted. Every destroyer satisfies q^2 <= its kill (max ratio 1.0000, attained only at the natal strikes); a same-q tie is impossible (q | a and q | a+2 force q | 2) and the tie-adjacent case — both members composite — always attributes to the opener's lpf, A-side. This is a POSITION rule, not a time rule: it overrides fold-activation order whenever lpf(a+2) < lpf(a), which is 49.99% of both-composite pairs and 29.79% of all destroyed pairs to 1e7, and it moves q = 7's destroyer share from 30.36% to 19.60%. The fold ledger (`research/fold-ledger-01.js`) runs the other convention; every null below is derived against this one. **The derived null [derivation PROVEN; fit VERIFIED at dest/null = 1.000].** Per side, the CRT kill density of destroyer q among channel positions is s(q) = (1/q) prod_{7<=r 1e8 every fresh kill of q is q x prime (zonegap-02 §3.4's regime); at q = 997, 3167, 9973 the census columns EQUAL the enumerated counts of primes m with q*m in the opener/closer classes. This closes the large-q tail exactly where the asymptotic CRT density misprices the short window above q^2 (the null-total ratio 0.9670 lives entirely there). **Concentration [MEASURED, null derived].** Half of all destruction is done by the 7 primes <= 29 (eps = 0.183: q <= X^0.183); 90% by q <= 1153 (188 of 1226 destroyer primes); the single prime 7 owns 18.71%. A vanishing share of primes does almost all destruction, and the derived cumulative null (1 - prod(1-1/r), the telescoped s-sum, times the B-side factor) matches the measured curve at every eps. ## 2. The entry-fee ledger - **Entry fee [PROVEN + MEASURED].** Every prime's first fresh kill is the B-side natal strike on (q^2-2, q^2); it destroys a LIVE pair iff q^2-2 is prime: 10/22 for q <= 97 (natal's row, asserted), falling to 19.51% for q in [1009, 9973] — a descriptive count; no null is claimed for the quadratic's prime density here. - **The k-th-youngest ledger [formula PROVEN via §3.4's Phi* form; VERIFIED by exact enumeration + 9,624 pi-formula cross-asserts].** The k-th youngest active prime makes on average ~2k + 1 fresh kills per zone (means 2.83, 4.78, 6.74, ..., 16.89 for k = 1..8; fitted slope 2.010; channel share 6/8 of that). Derivation: its kills are q*m with m prime in (p/q, p'^2/q], an interval of length ~2(p'-q_k) — two kills per seniority step. Generalizes zonegap-02's D3 (youngest: 2..4). The MEAN is bounded per seniority rank; the max is not (measured max 8 at k = 1, 27 at k = 8): the interval length rides the prime gaps, so "bounded per fixed zone" is trivially true per zone but NOT uniformly in p — the correct uniform statement is O(k) on average, gap-driven in the sup. - **Rate per unit of newly frozen territory [derived + VERIFIED to 4 digits].** A fixed prime q kills channel members at asymptotic rate s(q)/5 per unit of new territory, forever (measured/derived = 1.0000 at q = 7, 11; 0.9998 at 31, over 1,060 shells). Lifetime destruction is linear in frozen territory; within any FIXED zone it is the finite Phi* count of the ledger. - **Budget decomposition [VERIFIED at p = 7, 11, 13].** Per-q zone budgets vs CRT 2*C*s(q), exact small-window fluctuation on display; the decomposition sums to B(p) exactly, and the big-walk B(p) is cross-asserted against a direct second computation. ## 3. THE OBJECT: budget vs capacity, and where the certificate dies **The theorem [PROVEN, one line].** Channel members belong to exactly one pair each, so destroyed pairs absorb DISTINCT composite members: D <= B, hence T = C - D >= C - B. In the zone, composite <=> lpf <= p (Zone Restriction Lemma), so B(p) is computable from the actives {q <= p} alone — no primality input beyond p. If B <= C - 1 the zone CANNOT be emptied; the count C - B is a forced minimum of twins. **The verdict [VERIFIED at all 1,225 zones in reach].** The certificate holds at exactly 15 zones, p = 7..61, with forced counts 8, 8, 15, 13, 17, 20, 19, 20, 22, 19, 20, 18, 12, 10, 1 — TIGHT at p = 7, where all 8 twins are forced by counting alone. It DIES at p = 67, where B/C first crosses 1 (B = 503 against C = 497, B/C = 1.0121), and never returns; the ratio's LIMIT is 2, which is not the threshold (band means 0.8528 / 1.3218 / 1.4663 / 1.5419). **The closed form [derived; VERIFIED to <0.1% relative error from p = 97].** B(p) = 2C(p) - (prime members) = (p'^2 - p)/5 - (3/4)(primes in the zone), so B/C = 2 - 15/(4 ln p) + o(1), and the density crossover B = C sits at ln p = 15/4, p* = e^{15/4} = 42.52. The measured die-off at 61/67 lands LATER than p* because finite-height prime density exceeds its asymptote; the a-priori CRT main term 2(1 - prod_{7<=q<=p}(1-1/q)) stays below 1 up to p = 59 and first crosses at p = 61, which is the last certified zone. **The honest grade — this is NOT a headline.** The family is finite and its death is derived, not accidental: B/C -> 2 because channel prime density falls like 15/(8 ln p) -> 0, so no zone family past 67 can ever be certified by the count. What the budget arithmetic certifies: occupancy — indeed a twin COUNT — for 15 small zones, with zero placement information, the only occupancy mechanism in the corpus that needs none. What it cannot certify: anything at large p, where budget exceeds capacity forever and WHERE the kills land — the max wall, zonegap-01 §5 — is again the entire question. Computing B exactly is itself a sieve of the zone, so the certificate's value is structural (a count suffices below the crossover), not computational. No infinite family, no TPC content; the statement is exact counting and stays that. ## 4. The finality note: the freeze frame, graded - **The freeze sequence [PROVEN].** The verdict on pair (a, a+2) depends only on divisibility by its destroyer set {q : q^2 <= a+2}, which has pi(sqrt(a+2)) - 3 channel-active members. Fold p (activation of p) makes verdicts final on the stretch p^2 <= a+2 < p'^2 — exactly p's onset shell in pair coordinates, length p'^2 - p^2. A twin is a pair that outlives its whole destroyer set; in the channel this is exactly "both members prime". - **Shells partition; zones overlap [PROVEN; VERIFIED].** The shells [p^2, p'^2) partition [49, ∞) and each shell sits INSIDE its zone (p^2 <= a+2 < p'^2 implies p < a and a+2 < p'^2). The census telescopes exactly: 1,225 shells to 1e8, 438,186 frozen twins, sum asserted equal to the endpoint difference. - **The equivalence, made exact.** Weak level [PROVEN, elementary]: every channel twin with a+2 >= 49 lies in exactly ONE shell, so infinitely many freeze steps freeze >= 1 surviving pair <=> TPC <=> weak Zone Postulate (ZONE-POSTULATE.md §2, cited). Strong level [one direction PROVEN, the equivalence FAILS as stated]: "every freeze step freezes >= 1 pair" IMPLIES the strong Zone Postulate (shell inside zone), but the converse does not follow — a zone's twin may sit below p^2, in an earlier shell. So the brief's "Postulate <=> every freeze step freezes a pair" is a theorem only at the weak (infinitely-often) boundary; at the strong boundary it is a one-way implication, and the shell form is the STRICTLY stronger uniform statement. - **Measured standing [VERIFIED].** No empty shell in reach; minimum 2 frozen pairs per shell (p = 11, 17, 29) — the shell analogue, over a partition, of zonegap-01's "every zone holds >= 2 pairs" (which is over overlapping windows, to 1e11). ## 5. The intra-zone staircase (addendum 1) The active destroyer set at zone position n is {q : q^2 <= n+2}: the zone's interior climbs pi(sqrt(p)) - 3 at entry to pi(p') - 3 at the frontier (zone 9967: 22 -> 1,226 actives). The deterministic tread curve S(q) = prod_{7<=r<=q}(1-2/r) predicts the alive fraction up to ONE derived constant [VERIFIED everywhere measured]: measured-alive/S sits at 0.794–0.806 in every u-decile of every band — e^{2gamma}/4 = 0.7931, the origin-excess constant (origin-excess.md, cited), with NO u-dependence left over. The within-zone destruction profile is the staircase times that constant; nothing zone-local remains. Tread violence 2/q against spacing ~2q ln q quantifies Chris's framing exactly: the 7-tread kills 2/7 of survivors in one step, the 9973-tread 2.01e-4 — early treads rare and violent, late treads dense and negligible, with per-tread noise (0.50–1.18 at single small treads) exactly natal's §3 tread wobble. ## 6. The half-level head (addendum 2) - **(a) [VERIFIED, exact per zone].** head(p) IS the first survivor of the FROZEN sqrt(p)-level sieve above p in 1,184 of 1,225 zones (96.65%); a tread enters the head window in 105 zones and CHANGES the head in only 41, the differ-fraction falling 9.09% -> 1.79% across bands. The hypothesis is confirmed as an exact statement with a vanishing correction, not as an asymptotic. - **(b) The coefficient does NOT derive zero-parameter [MEASURED, nulls derived].** The staircase-Mertens forward density gives e^{2gamma}/(8 C2) = 0.6007 ln^2 p, which is the HL coefficient 1/(2 C2) = 0.7574 multiplied by the origin-excess constant rho(2) = e^{2gamma}/4 by construction (origin-excess.md); the measured head coefficients sit between them, and read 0.7064, 0.7177, 0.7236, 0.7192, 0.7344 across the five windows, non-monotone. On half-decade bands the same ratio-of-sums estimator spans 0.669 to 0.753, so the window figures carry range composition and are not a measured convergence to 0.7574. The honest decomposition: h = R x (prime-origin factor), R = E[g^2]/2E[g] the CONTINUUM inspection-paradox functional of the SAME window's twin gaps. R is not what a discrete origin sees: a uniform integer origin sees R + 1/2 exactly, an odd origin R + 1, and an origin drawn from the coprime-to-30 residues that primes >= 7 actually occupy sees R + 2.754 at [1e7,1e8) [MEASURED, `redteam-0828-head.js`, five decades]. So h/R = 1.09 -> 1.03 is read against a population no prime belongs to, and the excess depends on which null is meant: at [1e7,1e8), h - R = 5.679 against R as written, 5.179 against the discrete uniform origin, and 2.925 against the coprime-to-30 one. Against the population primes occupy the residual is about half the figure quoted here, and it still falls with height. So the head field is the twin-gap process's renewal functional times a decaying prime-origin excess of a few percent — derived up to the gap shape, that excess, and the choice of null. NOT the hoped headline; the reduction of head to (gap process + renewal + small conditioning) is the result, HELD like everything else here. - **(c) [MEASURED].** head mod 30 deviates from the class-independence null by at most 7.4% (residue 12); the deviation pattern is the small-h class correlation, no new forbidden structure. ## 7. NOT REACHED - Nothing beyond 1e8; the certificate's death at 67 is derived to be permanent (B/C -> 2) but measured only to p = 9967. No 1e11 pass. - The mid-large destroyer band (97 < q < 467) keeps the asymptotic CRT null (drift to 0.98–1.16 visible at 199/997); the Buchstab-corrected null is formulated (prime-regime + rough-count) but only the three q > 467 witnesses were closed exactly. - The k-th-youngest SUP (gap-driven) is measured, not bounded; the channel-restricted per-zone destroyer-count (as opposed to kill-count) ledger was not separated. - Addendum (2a) tested the head as an OBJECT (frozen-level first survivor) and its mean; the full head DISTRIBUTION against the level-sqrt(p) tile's first-slot-gap law, and the prime-origin excess's mechanism, are open. zonegap-01's 0.7229 at its higher levels was cited, not re-swept. - The entry-fee live-kill density (primality of q^2-2) has no derived null here (HL quadratic constant not computed). - The finality note's shell postulate at 1e11 scale (zonegap-01's range) was not swept; shells were checked only to 1e8. --- *Producer and custody: `research/destroyer-census-01.js`, embedded (`node research/qc/embed.js --check` passes bit-honest; calibration aborts before measurement; the producer's READINGS block carries the same off-by-one at the CRT crossing that §3 corrects here, p = 59 for p = 61, and is uncorrected, while its OUTPUT block is right). Cited, never recomputed: pi2(1e8) = 440,312 (OEIS A007508), natal-onset-01's window figures, zonegap-02's D3 list and zone twin counts, zonegap-01's head coefficients, origin-excess.md's e^{2gamma}/4 = 0.79305. History layer: process record, staging. See `research/history/CHANGELOG.md` for the corpus rule.*