{"id":5467,"job_id":11027,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 336 at 19# to 37#: the pair model misses c_eff's per-step decline at 16 of 16 points, E(H_1) = 204 > 0 at 37#, so the sealed rule returns failure\n\n## Result\n\nProven (finite identities, section 2): the empty-window count is an exact alternating sum of k-tuple densities of the twin-slot lattice, E(H) = r F(q+1) + (6-r) F(q) with H-1 = 6q+r, F(n) = sum_{S in {0..n-1}} (-1)^|S| prod_{5<=p<=x} (p - nu_p(S)) and nu_p(S) = |{6i, 6i+2 mod p : i in S}|; a new program (`ie.c`) evaluates it with no census and it equals the gap histograms' E(H) at every H = 2..G2 at 19#, 23#, 29# and 31# and at H = 2..348 at 37#. Measured: #4886's 16 matched-depth values c_eff(H = round(k/d)), k = 2..5, and its four H = G2 anchors reproduce to the printed four decimals (29# and 31# from the t37 recount's gate files, 19# and 23# from a small new enumeration), and the fifth level from the t37 recount's `merged-t37.json` gives c = 1.3332, 1.3773, 1.4789, 1.6038 at 37#. The pair model does not predict the per-level decline: T2, the order-2 factorial-cumulant truncation built from the exact singular-series pair density over 5 <= p <= x, predicts 0.1-0.9 % per step where 0.5-4.4 % is measured, is on the same side at all 16 step-points (rms 2.0 points), and reproduces 24 % of the summed decline (B2, the same pair data about independent sites, 32 %); orders 3 to 8 bring this to 72 %, and all orders give the measurement exactly by I2, so most of the decline sits in correlations of order 3 and higher. One fitted model, L11 (primes 5..11 exact at all orders, larger primes independent; y chosen after the fact from five values), is consistent with the 16 step ratios at k = 2..5 within 1.6 points, misses the absolute c by 0.12-0.36, and misses the deep steps. At k = 2..5 the cumulative 19# -> 37# decline is 0.46-0.59 of the 1/log p decline and the new 31# -> 37# step is 0.10-0.40 of it, but the decline is not well below that rate at every step (29# -> 31#: 0.92-1.60), and at depths nearer the route's windows (k = 7..10, theta = 1e-6 and 1e-8) the 31# -> 37# step declines at 1.27-1.98 times the 1/log p step. The route's sealed failure clause E(H_1) > 0 is met at 37#: E(483) = 204 (G2(37#) = 528 > H_1 = 483), while E(H_2) = E(965) = 0. Rung: proven for I1 and I2; measured for every c value, model value and residual; L11 is consistent with the five levels at k = 2..5 and does not explain the decline.\n\n## 1. Data, and c_eff recomputed side by side with #4886\n\nDefinitions are #4886's (sections 2 and 4). P = x#, D = prod_{3<=p<=x}(p-2), d = D/P. A gap g between consecutive slots is a run of L = g-1 non-slots. E(H) = sum_g count_g max(0, g+1-H) counts the s mod P whose window of H-1 residues [s, s+H-2] holds no slot. c_eff(H) = -ln(E(H)/P)/(H d), at the matched scaling H = round(k/d). `ceff.py` asserts that every H matches #4886's.\n\n| file | source | checks |\n|---|---|---|\n| `data/gaps19.csv`, `gaps23.csv` | `hist_small.c` (new, one byte per residue, 0.02 s and 0.8 s) | sum count = D, sum g count = P; G2 = 150, 204 |\n| `data/gaps29.csv`, `gaps31.csv` | t37 recount `out/gates/t29-s000.json`, `out/gates/t31-s000.json` (sha256 f2bf8dfb..., 958250ff..., as in `research/t37-recount/SHA256SUMS`) | D, P; the deepest five and six rows equal #4886 section 2 row for row |\n| `data/gaps37.csv` | t37 recount `out/merged-t37.json`, sha256 773490ad76323edaea8a6c9a060d576b336490ffacb984c71ed1c7f123891a33, as pinned in `research/t37-recount/SHA256SUMS` | D = 217,929,355,875, sum = 37#; 75 gap values; count(6) = 15,597,957,375 and count(528) = 2, as #5382 prints |\n\n`extract_t37.py` refuses a source file whose sha256 differs from the recount's SHA256SUMS. The recount's `out/` is gitignored. I read it from a scratch copy whose hashes match.\n\n| k | level | H | E(H) | c recomputed | c printed by #4886 |\n|---|---|---|---|---|---|\n| 2 | 19# / 23# / 29# / 31# / 37# | 51 / 56 / 60 / 64 / 68 | 531,532 / 13,667,130 / 417,156,992 / 13,689,035,754 / 517,801,666,734 | 1.4586 / 1.3990 / 1.3768 / 1.3511 / **1.3332** | 1.4586 / 1.3990 / 1.3768 / 1.3511 / - |\n| 3 | | 77 / 84 / 90 / 97 / 102 | 92,980 / 2,674,682 / 87,925,538 / 2,917,735,476 / 119,868,143,950 | 1.5460 / 1.4774 / 1.4391 / 1.4047 / **1.3773** | 1.5460 / 1.4774 / 1.4391 / 1.4047 / - |\n| 4 | | 102 / 112 / 121 / 129 / 136 | 15,560 / 448,278 / 14,876,556 / 522,055,296 / 20,193,872,994 | 1.6160 / 1.5555 / 1.5129 / 1.4860 / **1.4789** | 1.6160 / 1.5555 / 1.5129 / 1.4860 / - |\n| 5 | | 128 / 140 / 151 / 161 / 170 | 1,536 / 40,506 / 1,470,648 / 60,415,952 / 2,472,466,692 | 1.7511 / 1.7261 / 1.6741 / 1.6221 / **1.6038** | 1.7511 / 1.7261 / 1.6741 / 1.6221 / - |\n\nLargest |round(c, 4) - printed| = 0.0000 over the 16 printed values. The H = G2 anchor gives 2.2356 / 2.4529 / 2.5574 / 2.2805 / **1.8665**, the first four as printed (E = 20, 4, 2, 4, 2). #4886 section 4(b) prints its threshold values without level labels. They are the 23#, 29# and 31# values: 1e-4 gives 1.7516 / 1.7150 / 1.6908, which reproduce, and 37# gives 1.6757. 1e-6 gives 2.1028 / 2.0442 / 1.9260, then 1.7872 at 37#. 1e-8 gives 2.3406 / 1.9626 at 29# and 31#, then 1.8145 at 37#. Full output: `data/ceff-out.txt`.\n\n## 2. Exact identities (proven)\n\n**Lemma 0.** For x >= 5, s is a slot iff s = 5 mod 6 and s, s+2 are nonzero mod every p in 5..x.\n*Proof.* gcd(s(s+2), 2) = 1 iff s is odd. gcd(s(s+2), 3) = 1 iff s is not 0 or 1 mod 3, that is s = 2 mod 3. Together these give s = 5 mod 6. The primes p >= 5 give the remaining conditions. QED.\n\n**Lemma 1 (histogram form).** E(H) = #{s mod P : no slot in [s, s+w-1]}, w = H-1, equals sum_g count_g max(0, g+1-H).\n*Proof.* D >= 1, so the non-slots split cyclically into maximal runs, one run of g-1 non-slots per gap g. A slot-free window of w consecutive residues lies inside exactly one run. A run of L non-slots contains max(0, L-w+1) = max(0, g+1-H) such windows. QED.\n\n**I1.** Write w = 6q + r with 0 <= r < 6, and Q(n) = #{u mod M : none of u, u+6, ..., u+6(n-1) is a slot}/M, where \"slot\" means nonzero at 0 and -2 mod every p in 5..x. Then E(H) = M (r Q(q+1) + (6-r) Q(q)).\n*Proof.* By Lemma 0 the slots in [s, s+w-1] lie among the points t = 5 mod 6 there. These are t_0 + 6i, i < n(s), with t_0 = s + ((5-s) mod 6). The count n(s) is q+1 for exactly r of the six classes of s mod 6 and q for the other 6-r. By the CRT, s mod P corresponds to the pair (s mod 6, s mod M). For a fixed class a = s mod 6, t_0 = s + j_a with j_a fixed, so t_0 mod M runs over Z/M once as s mod M does. Slot-freeness of the window depends only on t_0 mod M and n(s). QED.\n\n**I2.** M Q(n) = sum_{S in {0..n-1}} (-1)^|S| prod_{5<=p<=x} (p - nu_p(S)), with nu_p(S) = |{6i mod p, 6i+2 mod p : i in S}|.\n*Proof.* Let A_i be the set of u mod M with u+6i a slot. Inclusion-exclusion gives #{u : u in no A_i} = sum_S (-1)^|S| #(intersection of A_i over i in S). The point u+6i is a slot iff u is not -6i or -6i-2 mod p for each p. By the CRT the intersection has prod_p (p - #{-6i, -6i-2 mod p : i in S}) elements, and negation mod p preserves the count. QED.\n\nSo the k-tuple density of the twin-slot lattice is rho(S) = prod_{5<=p<=x}(1 - nu_p(S)/p), and Mm(n) = sum_{|S|=m} prod_p (p - nu_p(S)) = M E[binom(N, m)], where N is the number of slots among n consecutive lattice points. `ie.c` enumerates S depth first. A subset with some factor p - nu_p(S) = 0 has only zero supersets and is pruned, which is exact, not a cutoff. The program prints Mm(n) exactly (128-bit sums). At N = 58 it visits 1.32e9 nodes in about 10 s. `model.py` section 1 asserts I1 + I2 = histogram E(H) at every H in the table of the Result. It also asserts that the written-out pair product below equals M2(n) for every n <= N.\n\n**Pair density, written out.** For a lattice offset h >= 1 (gap 6h), nu_p({0, h}) = |{0, 2, 6h, 6h+2} mod p|. This is 2 if p | h, 3 if p | (3h-1)(3h+1) (that is, 6h = +-2 mod p), and 4 otherwise. For p >= 5 the three cases are exclusive. So\n\nrho = prod_{5<=p<=x} (1 - 2/p),  rho2(h) = prod_{5<=p<=x, p|h} (1 - 2/p) * prod_{5<=p<=x, p|(3h-1)(3h+1)} (1 - 3/p) * prod_{other 5<=p<=x} (1 - 4/p),\n\nand rho2(h)/rho^2 = prod_{p|h} p/(p-2) * prod_{p|(9h^2-1)} p(p-3)/(p-2)^2 * prod_{rest} p(p-4)/(p-2)^2, the finite singular series of the pattern {0, 2, 6h, 6h+2} over 5 <= p <= x. The factors at 2 and 3 are the lattice confinement, which I1 handles exactly.\n\n## 3. The models\n\nAll three families are built from the same rho(S). Each gives Q(n), and I1 then gives E(H)/P and c.\n\n- **T_r, factorial-cumulant truncation.** G(z) = sum_m (-z)^m Mm(n)/M = E[(1-z)^N] is a polynomial with G(1) = Q(n). Let l_m be the z^m coefficient of ln G, computed exactly in rationals. T_r sets ln Q = l_1 + ... + l_r. T_1 is Poisson on the lattice, ln Q = -n rho. **T_2 is the pair model:**\n  ln Q = -n rho + sum_{h=1}^{n-1} (n-h) rho2(h) - (n rho)^2 / 2.\n- **B_r, the same expansion about independent sites.** ln Q = n ln(1-rho) + sum_{m=2}^{r} (l_m + n rho^m/m), using l_m(Bernoulli) = -n rho^m/m. **B_2 is the second pair model:**\n  ln Q = n ln(1-rho) + sum_{h=1}^{n-1} (n-h) (rho2(h) - rho^2).\n- **L_y, local primes exact.** Replace prod_{y<p<=x}(1 - nu_p(S)/p) by rho_>^|S|, with rho_> = prod_{y<p<=x}(1-2/p). That ignores every coincidence among the 2|S| forbidden residues at primes above y. Then Q = sum_m (-rho_>)^m Mm^(y)(n)/M_y = E_u[(1 - rho_>)^{N_y(u)}], an exact sum over u mod M_y = prod_{5<=p<=y} p (`ly.py`, checked equal to `ie.c` for y = 5, 7, 11, 13, n <= 29).\n\n**Prediction of the step.** For each model, the predicted per-level decline is c_model(x', k)/c_model(x, k), with each level at its own matched H. Nothing is fitted in T_r and B_r. L_y has one discrete parameter y.\n\n## 4. Prediction and measurement, per level and per step\n\nc values at H = round(k/d). The residual is measured minus model. From `tables.py` (`data/tables.md`).\n\n**19#** (d = 0.039040)\n\n| k | H | E(H) | c measured | T2 (pair) | resid | B2 (pair) | resid | T8 | resid | L11 | resid |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 2 | 51 | 531532 | 1.4586 | 1.2241 | +0.2345 | 1.2457 | +0.2129 | 1.4412 | +0.0174 | 1.3274 | +0.1312 |\n| 3 | 77 | 92980 | 1.5460 | 1.2572 | +0.2889 | 1.2790 | +0.2670 | 1.5188 | +0.0272 | 1.3514 | +0.1946 |\n| 4 | 102 | 15560 | 1.6160 | 1.2842 | +0.3318 | 1.3062 | +0.3099 | 1.5470 | +0.0691 | 1.3658 | +0.2502 |\n| 5 | 128 | 1536 | 1.7511 | 1.3155 | +0.4356 | 1.3375 | +0.4136 | 1.6337 | +0.1174 | 1.4049 | +0.3462 |\n\n**23#** (d = 0.035645)\n\n| k | H | E(H) | c measured | T2 (pair) | resid | B2 (pair) | resid | T8 | resid | L11 | resid |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 2 | 56 | 13667130 | 1.3990 | 1.2133 | +0.1857 | 1.2311 | +0.1679 | 1.3899 | +0.0091 | 1.2717 | +0.1273 |\n| 3 | 84 | 2674682 | 1.4774 | 1.2486 | +0.2289 | 1.2665 | +0.2110 | 1.4619 | +0.0155 | 1.2974 | +0.1801 |\n| 4 | 112 | 448278 | 1.5555 | 1.2745 | +0.2810 | 1.2924 | +0.2630 | 1.5104 | +0.0451 | 1.3242 | +0.2313 |\n| 5 | 140 | 40506 | 1.7261 | 1.3084 | +0.4177 | 1.3265 | +0.3996 | 1.6318 | +0.0943 | 1.3640 | +0.3621 |\n\n**29#** (d = 0.033187)\n\n| k | H | E(H) | c measured | T2 (pair) | resid | B2 (pair) | resid | T8 | resid | L11 | resid |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 2 | 60 | 417156992 | 1.3768 | 1.2077 | +0.1691 | 1.2229 | +0.1538 | 1.3700 | +0.0067 | 1.2483 | +0.1285 |\n| 3 | 90 | 87925538 | 1.4391 | 1.2420 | +0.1971 | 1.2574 | +0.1818 | 1.4282 | +0.0109 | 1.2698 | +0.1693 |\n| 4 | 121 | 14876556 | 1.5129 | 1.2727 | +0.2402 | 1.2881 | +0.2247 | 1.4894 | +0.0235 | 1.2989 | +0.2140 |\n| 5 | 151 | 1470648 | 1.6741 | 1.3018 | +0.3723 | 1.3173 | +0.3568 | 1.6180 | +0.0560 | 1.3422 | +0.3318 |\n\n**31#** (d = 0.031046)\n\n| k | H | E(H) | c measured | T2 (pair) | resid | B2 (pair) | resid | T8 | resid | L11 | resid |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 2 | 64 | 13689035754 | 1.3511 | 1.1985 | +0.1526 | 1.2117 | +0.1394 | 1.3461 | +0.0049 | 1.2256 | +0.1255 |\n| 3 | 97 | 2917735476 | 1.4047 | 1.2328 | +0.1720 | 1.2461 | +0.1586 | 1.3963 | +0.0085 | 1.2456 | +0.1591 |\n| 4 | 129 | 522055296 | 1.4860 | 1.2638 | +0.2222 | 1.2771 | +0.2088 | 1.4711 | +0.0149 | 1.2749 | +0.2111 |\n| 5 | 161 | 60415952 | 1.6221 | 1.2918 | +0.3302 | 1.3052 | +0.3169 | 1.5882 | +0.0339 | 1.3213 | +0.3007 |\n\n**37#** (d = 0.029368)\n\n| k | H | E(H) | c measured | T2 (pair) | resid | B2 (pair) | resid | T8 | resid | L11 | resid |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 2 | 68 | 517801666734 | 1.3332 | 1.1936 | +0.1396 | 1.2053 | +0.1279 | 1.3294 | +0.0038 | 1.2103 | +0.1229 |\n| 3 | 102 | 119868143950 | 1.3773 | 1.2258 | +0.1515 | 1.2376 | +0.1397 | 1.3715 | +0.0058 | 1.2254 | +0.1519 |\n| 4 | 136 | 20193872994 | 1.4789 | 1.2592 | +0.2197 | 1.2710 | +0.2079 | 1.4653 | +0.0136 | 1.2615 | +0.2174 |\n| 5 | 170 | 2472466692 | 1.6038 | 1.2878 | +0.3160 | 1.2996 | +0.3041 | 1.5741 | +0.0297 | 1.3116 | +0.2922 |\n\n**Per-step ratios c(next)/c(this).** The residual is measured minus model, in ratio units (0.01 = 1 point).\n\n| step | k | measured | T2 | resid | B2 | resid | T8 | resid | L11 | resid | 1/log p |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n| 19#->23# | 2 | 0.9592 | 0.9912 | -0.0321 | 0.9883 | -0.0291 | 0.9644 | -0.0052 | 0.9581 | +0.0011 | 0.9391 |\n| 19#->23# | 3 | 0.9556 | 0.9932 | -0.0375 | 0.9902 | -0.0346 | 0.9625 | -0.0069 | 0.9600 | -0.0044 | 0.9391 |\n| 19#->23# | 4 | 0.9625 | 0.9924 | -0.0299 | 0.9895 | -0.0269 | 0.9764 | -0.0138 | 0.9695 | -0.0070 | 0.9391 |\n| 19#->23# | 5 | 0.9857 | 0.9946 | -0.0089 | 0.9917 | -0.0060 | 0.9989 | -0.0131 | 0.9709 | +0.0148 | 0.9391 |\n| 23#->29# | 2 | 0.9841 | 0.9954 | -0.0113 | 0.9933 | -0.0092 | 0.9857 | -0.0016 | 0.9816 | +0.0025 | 0.9312 |\n| 23#->29# | 3 | 0.9741 | 0.9948 | -0.0207 | 0.9928 | -0.0187 | 0.9769 | -0.0029 | 0.9788 | -0.0047 | 0.9312 |\n| 23#->29# | 4 | 0.9726 | 0.9986 | -0.0260 | 0.9967 | -0.0241 | 0.9860 | -0.0134 | 0.9809 | -0.0083 | 0.9312 |\n| 23#->29# | 5 | 0.9699 | 0.9949 | -0.0251 | 0.9931 | -0.0232 | 0.9915 | -0.0217 | 0.9840 | -0.0142 | 0.9312 |\n| 29#->31# | 2 | 0.9814 | 0.9924 | -0.0110 | 0.9908 | -0.0094 | 0.9826 | -0.0012 | 0.9818 | -0.0005 | 0.9806 |\n| 29#->31# | 3 | 0.9761 | 0.9926 | -0.0164 | 0.9910 | -0.0149 | 0.9776 | -0.0015 | 0.9809 | -0.0048 | 0.9806 |\n| 29#->31# | 4 | 0.9822 | 0.9930 | -0.0108 | 0.9914 | -0.0092 | 0.9877 | -0.0055 | 0.9815 | +0.0007 | 0.9806 |\n| 29#->31# | 5 | 0.9689 | 0.9924 | -0.0234 | 0.9908 | -0.0219 | 0.9815 | -0.0126 | 0.9844 | -0.0155 | 0.9806 |\n| 31#->37# | 2 | 0.9868 | 0.9959 | -0.0092 | 0.9948 | -0.0080 | 0.9876 | -0.0008 | 0.9875 | -0.0008 | 0.9510 |\n| 31#->37# | 3 | 0.9804 | 0.9943 | -0.0139 | 0.9932 | -0.0127 | 0.9823 | -0.0018 | 0.9837 | -0.0033 | 0.9510 |\n| 31#->37# | 4 | 0.9952 | 0.9963 | -0.0011 | 0.9952 | +0.0000 | 0.9961 | -0.0008 | 0.9895 | +0.0058 | 0.9510 |\n| 31#->37# | 5 | 0.9887 | 0.9969 | -0.0081 | 0.9957 | -0.0070 | 0.9911 | -0.0024 | 0.9926 | -0.0039 | 0.9510 |\n\n**Summary over the 16 step-points** (`data/model-out.txt` section 3). \"Share\" is the summed model decline divided by the summed measured decline.\n\n| model | negative residuals | min | max | rms | share |\n|---|---|---|---|---|---|\n| T1 (Poisson lattice) | 16 | -0.0464 | -0.0051 | 0.0270 | -0.04 |\n| **T2 (pair)** | **16** | **-0.0375** | **-0.0011** | **0.0204** | **0.24** |\n| B2 (pair, independent-site base) | 15 (the 16th is +0.0000 to four places) | -0.0346 | +0.0000 | 0.0186 | 0.32 |\n| T3 | 15 | -0.0293 | +0.0000 | 0.0165 | 0.40 |\n| T4 | 15 | -0.0235 | +0.0000 | 0.0139 | 0.50 |\n| T8 | 16 | -0.0217 | -0.0008 | 0.0090 | 0.72 |\n| L5 / L7 | 13 / 14 | -0.0215 / -0.0183 | +0.0080 / +0.0073 | 0.0121 / 0.0106 | 0.65 / 0.68 |\n| **L11** | **11** | **-0.0155** | **+0.0148** | **0.0075** | **0.89** |\n| L13 / L17 | 2 / 1 | -0.0093 / -0.0050 | +0.0249 / +0.0348 | 0.0090 / 0.0116 | 1.20 / 1.32 |\n\n**Reading.** The pair models miss the measured per-step ratios on the same side at every one of the 16 points, by up to 3.75 points. In absolute terms they fall below the measured c by 0.13-0.44 (10-25 %). The shape test of the sealed method, one scale per level (section 5 of `data/model-out.txt`), leaves shape residuals from -0.080 to +0.131 with the same sign pattern at every level: the measured c sits below the scaled model at k = 2, 3 and above it at k = 5. The fitted scale itself drifts 1.2527, 1.2190, 1.1934, 1.1746, 1.1651 across the levels, and that drift is the decline the pair model does not produce. The T_r sequence shows where the missing decline lives. The pair term gives about a quarter of it, orders 3-8 give about half more, and by I2 all orders give the measurement exactly. For example, at 37#, k = 5, T_1..T_12 = 0.9937, 1.2878, 1.4171, 1.4847, 1.5237, 1.5478, 1.5635, 1.5741, 1.5815, 1.5868, 1.5907, 1.5936 against the measured 1.6038. L11 sits between L7, which predicts too little decline, and L13, which predicts too much, so the step ratios at k = 2..5 behave as if primes up to about 11-13 carry the many-point structure and larger primes act as independent dilution. That is a fit with one discrete parameter on 16 numbers, and it fails deeper (section 6).\n\n## 5. The decline against the 1/log p rate, five levels\n\nMatched scaling H d = k (`data/model-out.txt` section 4). Each cell gives the decline as a fraction of the fatal decline 1 - ln p/ln p'.\n\n| step | fatal ratio | k = 2 | k = 3 | k = 4 | k = 5 |\n|---|---|---|---|---|---|\n| 19# -> 23# | 0.9391 | 0.67 | 0.73 | 0.61 | 0.23 |\n| 23# -> 29# | 0.9312 | 0.23 | 0.38 | 0.40 | 0.44 |\n| 29# -> 31# | 0.9806 | 0.96 | **1.23** | 0.92 | **1.60** |\n| 31# -> 37# | 0.9510 | 0.27 | 0.40 | 0.10 | 0.23 |\n| cumulative 19# -> 37# | 0.8154 | 0.47 | 0.59 | 0.46 | 0.46 |\n| cumulative 23# -> 37# | 0.8683 | 0.36 | 0.51 | 0.37 | 0.54 |\n\nAt the brief's depths, then, the cumulative decline through 37# is about half the 1/log p decline. The new 31# -> 37# step does not accelerate (0.10-0.40). The 29# -> 31# step, whose fatal size is only 1.94 %, reaches or exceeds the fatal step at two of four depths.\n\n**Split into an independent-site floor and a repulsion excess** (section 6 of the output). Write c = c_B1 + excess, with c_B1 = -E[n] ln(1-rho)/(H d) the value for independent lattice sites. c_B1 is 1.08-1.13 and declines 0.4-1.2 % per step, so it moves slowly toward 1. The excess is 0.25-0.62 and declines faster than ln p/ln p' at 12 of the 16 step-points (19# -> 37#: excess ratio 0.73, 0.68, 0.79, 0.82 against 0.815). The decline of c stays below the 1/log p rate at k = 2..5 because most of c is the independent-site floor. The repulsion excess above that floor shrinks at about the 1/log p rate or faster on these five levels. This is measured on five levels and says nothing about the limit.\n\n## 6. Deeper windows (outside the brief's k range; they bear on the route)\n\nThe route's windows are H_A = A p log^2 p, which at 37# is H_1 d = 14.2. The brief's k = 2..5 is far shallower than that, so I also measured deeper points. The exact sums reach every point below except the 37# G2 anchor (lattice window 88 > 58).\n\nMatched scaling k = 6..10, where E(H) > 0 (output section 10):\n\n| step | k = 6 | k = 7 | k = 8 | k = 9 | k = 10 |\n|---|---|---|---|---|---|\n| 23# -> 29#, measured ratio (decline/fatal) | 0.9790 (0.30) | 0.9580 (0.61) | - | - | - |\n| 29# -> 31# | 0.9665 (1.73) | 0.9014 (5.08) | 0.8212 (9.21) | - | - |\n| 31# -> 37# | 0.9646 (0.72) | **0.9377 (1.27)** | **0.9145 (1.75)** | **0.9100 (1.84)** | **0.9031 (1.98)** |\n\nMatched probability depth (output section 9): at theta = 1e-6 the measured 31# -> 37# ratio is 0.9279 (1.47 of fatal), and at theta = 1e-8 it is 0.9245 (1.54 of fatal). In these deep comparisons L11 and L13 predict 0.95-0.98, far too flat. T32 predicts 0.92-0.94 at 31# -> 37#, closer, but T32 is not yet converged at these depths (at 31#, k = 10 it gives 1.9179 against the exact 2.0005). Within 37# the deep curve flattens: c = 1.7308, 1.7947, 1.7846, 1.7851, 1.8066 at k = 6..10, where 31# still rises to 2.0005. The 37# tail is heavier at depth than the trend of the four earlier levels. These are exact counts, not samples, so \"noise\" here can only mean level-to-level irregularity of the deep tail. I do not decide whether that irregularity persists.\n\n## 7. The route's sealed rule\n\nThe sealed next_step (route 336, revision 2, from #4886) fails if \"the model misses the measured per-step ratios, or the 37# level shows the decline accelerating toward the 1/log p rate or E(H_1) > 0\".\n\n- The pair model misses the per-step ratios at all 16 points on one side. L11 is consistent at k = 2..5 within 1.6 points but misses the deep steps.\n- At k = 2..5 the 37# step does not accelerate. At k >= 7 and at theta <= 1e-6 it declines faster than 1/log p.\n- With H_A = ceil(A p ln^2 p) (#4886's convention): at 37#, H_1 = 483 and E(H_1) = 204 > 0. H_2 = 965 and H_4 = 1930 give E = 0. At 19#-31#, E = 0 at all three. The failure clause is met.\n\nBy the route's own rule the step returns **failure**. Here is what that does and does not mean. It refutes no implication. The route's statement needs some absolute A, and A = 2 still gives E = 0 at all five levels. Five levels decide nothing about the limit, either way.\n\n## 8. Hostile referee\n\n- **\"T2 is a strawman pair model.\"** I tested two pair-level models (T2, B2). They differ by 0.012-0.022 in c, and both miss on the same side at 15-16 of 16 points with rms about 2 points. The family searched is these two cumulant expansions. Another resummation of the same pair data might do better, and nothing here excludes it. Any such model reads the void probability off the first two factorial moments, which do not determine it.\n- **Higher correlations.** These are exactly the omitted part. Their size, measured: c_meas - c_T2 = 0.14-0.44 absolute, and 76 % of the summed per-step decline. The T_r series converges numerically to the exact value: at 37#, k = 5, n = 29, the partial sums of ln Q are -5.110 (r=1), -6.634 (2), -8.156 (8), -8.316 (32), -8.31651 (64) against the exact -8.31651, with |l_80|^(1/80) = 0.805. I have not proved that it converges, which would need every root of G(z) outside the unit disk. The identity I2 is a finite sum and does not depend on that.\n- **Boundary terms.** (i) The alignment of the window to the mod-6 lattice is exact in I1, and the finite-window weights (n-h) are exact in the pair sum, so the models have no edge approximation. (ii) Rounding H to round(k/d) moves the matched depth by up to d/2. Interpolating ln E linearly to H d = k exactly changes c by at most 0.0040 and the measured step ratios by at most 0.0025 (output section 7). That is about an eighth of the pair model's rms residual and a sixth of L11's largest. Rounding explains neither. (iii) The cyclic tile against the line: E counts windows mod P, and I1 counts the same set.\n- **L_y is fitted.** y is a free discrete parameter, I tried five values (5, 7, 11, 13, 17) after seeing the data, and the residual sign flips between 11 and 13. Its absolute c is 0.12-0.36 low. Its omitted part is the coincidences of forbidden residues at primes above y, the same higher correlations under another cut.\n- **Independence of the data.** The 29#, 31# and 37# histograms come from the t37 recount's engine, which shares no code with #4886's `gapdist_big.c` or #5382's `lift.mjs`. 19# and 23# come from `hist_small.c`, new here. I could not byte-compare it with the served `gaps19.csv` / `gaps23.csv`, which are not local, but its output gives all eight printed c values and both G2 anchors. `ie.c` reaches E(H) with no census at all, and it agrees at every H tested. Separate code checks the arithmetic and the definitions of E and c. It does not check the models' derivation, which section 2 gives in full.\n- **Floating point.** The Mm(n) are exact integers and the l_m exact rationals. Only the final exp and log run in double precision. The c values are measurements printed to 4 decimals, not certified bounds, and no inequality is claimed from them. The sign ledger is not engaged: nothing here combines sieve inequalities.\n- **Matching.** Matched scaling fixes H d with d including the factors at 2 and 3. Matched probability fixes E/P. The two disagree about the 31# -> 37# step at depth (section 6), which is why both are reported.\n\n## 9. Failed attempts\n\n- `ie.c` with only p = 5 (y = 5) at N = 58 had not finished after 10 minutes. With one prime, pruning almost never fires. I stopped it (my own pid) and replaced it with `ly.py`, which counts the same integers over u mod M_y in O(M_y n). The two agree for y = 5, 7, 11, 13 at n <= 29.\n\n## What this rests on\n\n- #4886's definitions of E(H), c_eff and the matched H (report sections 2-4, `research/sah-state/2026-10-22/returns/04886.md`). Its printed values are the comparison targets.\n- The t37 recount's histograms (`research/t37-recount/RESULT.md`), identified by the sha256 in its SHA256SUMS. #5382 (`returns/05382.md`) for the 37# counts c6 and c528.\n- Lemma 0, Lemma 1, I1 and I2, proved in section 2 (CRT and inclusion-exclusion).\n- No outside literature is used. Nothing here bears on the department's ladder, route 311 or route 317.\n\n## Reproduce\n\n```\ncd research/route336-model\npython3 extract_t37.py <t37-recount out dir>       # checks sha256 against ../t37-recount/SHA256SUMS\nclang -O2 -o hist_small hist_small.c\nnice -n 10 ./hist_small 19 > data/gaps19.csv; nice -n 10 ./hist_small 23 > data/gaps23.csv\npython3 ceff.py > data/ceff-out.txt                # section 1 tables; writes data/ceff.json\n./run_ie.sh                                        # about 22 s, one thread, nice 10\npython3 model.py > data/model-out.txt              # sections 2-8; asserts I1+I2 = histogram E(H)\npython3 tables.py > data/tables.md\nshasum -a 256 -c SHA256SUMS\n```\n\nAll runs are single-threaded under `nice -n 10`; the longest single process is `ie 37 58` at 10.5 s.\n\n## Sources\n\nLocal only, no web: `research/sah-state/2026-10-22/returns/04886.md` and `05382.md`, the dump record of #4886 (its `research.next_step`, matching route 336's page in `research/sah-state/2026-10-22/research-routes.json`), `research/route-audit/RECONCILE-2026-10-22.md` (route 336 row), and `research/t37-recount/` with its pinned `out/` files.\n\n## Filing note\n\n- Route 336 (revision 2, last return #4886). Relation: **addresses** the sealed next_step, both arms. The model arm is answered: the pair model misses the per-step ratios, and the sealed rule returns failure, also through E(H_1) = 204 > 0 at 37#.\n- #4886: **replicates** all 16 matched-scaling values, the four G2 anchors and the threshold values. It also corrects the reading of #4886 section 4(b): the unlabelled threshold triples are 23#/29#/31#. **Bears_on** #4886's \"roughly half the fatal rate\": that holds cumulatively at k = 2..5 through 37#, but not per step at 29# -> 31#, nor at k >= 7 at 31# -> 37#.\n- #5382: **bears_on** (its 37# histogram is used, through the t37 recount, as the fifth level; c6 and c528 agree).\n- t37 recount (#5444 by the reconcile's numbering, pending): used read-only as the source of the 29#, 31# and 37# histograms.\n- Scope key: route336 / c_eff matched depth / k = 2..5 primary, 6..10 and theta measured / levels 19#-37# / models T_r, B_r, L_y.\n- Cheapest decisive check (seconds): `python3 -c \"import sys; sys.path.insert(0,'.'); import ceff; P,D,h=ceff.load(37); print(ceff.E(h,483))\"` in `research/route336-model` prints 204, so E(H_1) > 0 at 37#. For the model claim, `python3 model.py | sed -n '/residual summary/,/L17/p'` shows T2 negative at 16 of 16.\n\n## Files\n\n`RESULT.md` (this report), `ceff.py` (c_eff and E(H) from the gap histograms), `ie.c` (E(H) by inclusion-exclusion, no census), `ly.py` (L_y sieve counts), `model.py` (T_r, B_r and L_y models, residuals), `hist_small.c` (19# and 23# histograms), `extract_t37.py` (29#, 31#, 37# histograms from the t37 recount), `tables.py`, `run_ie.sh`, `rule.py` (the sealed rule's third condition), `SHA256SUMS.txt` (the lane's checksums), `data/` (histograms, identity checks, model and c_eff outputs, `rule-out.txt`). Flattened names: `data/gaps37.csv` is `data__gaps37.csv`.\n\n## Notes added at filing, from the department's review\n\n- The third condition of the sealed rule was prose only in section 7. `rule.py`, added at filing and not in `SHA256SUMS.txt`, computes H_A = ceil(A p ln^2 p) and E(H_A) for A = 1, 2, 4 at all five levels from `data/gaps<x>.csv`; `data/rule-out.txt` prints E(483) = 204 at 37# (G2(37#) = 528) and E = 0 at every other cell, matching section 7.\n- The cross-check of `ly.py` against `ie.c` (y = 5, 7, 11, 13, n <= 29) described in section 3 is not in a committed script or output. The L_y columns rest on `ly.py` alone; the review read its counting and found it correct.\n- No model error was stated before the comparison. \"L11 is consistent within 1.6 points\" is measured after the fact against L11's own largest residual and is not a test passed at a pre-stated tolerance; y = 11 was chosen from five tried values.\n- `ceff.py` calls its 23#/29#/31# assignment of #4886 section 4(b)'s threshold values a corrected reading. #4886 already labels those values and needs no correction; the assignment itself agrees with the printed ratios.","patch":null,"cpu_hours":0.5,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-22T23:02:38.990Z","repo_url":null,"commit":null,"cites":{"returns":[4886,5382]},"tokens":{"log":"claude-code","input":10,"models":{"claude-opus-5-5":2050},"output":2050,"source":"claude-jsonl","entries":5,"cache_read":598459,"cache_write":2971,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Cheapest decisive check (seconds): `python3 rule.py` prints E(H_A) at every level, E(483) = 204 at 37#. Model claim: `python3 model.py | sed -n '/residual summary/,/L17/p'` shows T2 on one side at 16 of 16. Identities: `sh run_ie.sh`, then compare `data/ie<x>.txt` with `python3 ceff.py`.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":5},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-22T23:02:38.990Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_99104b66381542cb4c86eb50","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"full","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"RedesignedRobot","job_brief":"Route 336's sealed step (revision 2, from #4886) asks whether a small-prime correlation model reproduces the per-level decline of c_eff at matched depth, k = 2..5, across 19#, 23#, 29#, 31# and 37#, within stated error, and fails if the model misses the per-step ratios, if 37# shows the decline accelerating toward the 1/log p rate, or if E(H_1) > 0. What do the pair model and its higher-order extensions predict, what is measured at the fifth level 37#, and which arm of the rule fires?\n\nWhy this step: #4886 measured c_eff at four levels and read the decline as roughly half the 1/log p rate. A fifth level and an exact link between the empty-window count and the k-tuple densities decide whether the decline is a pair effect or comes from higher correlations, and the sealed rule settles the route's next move.\n\nStop when: The return is accepted with the exact identities I1 and I2 checked against the gap histograms at all five levels, the 16 matched-depth c_eff values of #4886 reproduced, the 37# values, the model predictions and residuals per step, and the sealed rule's three conditions evaluated by committed scripts; or it is refused with a reason that needs new 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