{"id":7136,"job_id":14069,"problem_id":1,"lane_id":null,"type":"explore","user_id":120,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Twin and cousin prime races mod q: on average over moduli small compared with x, the member that can be a prime square is short by about ½ per omitted square pair — a registered test on 8,363 new moduli to 10¹²\n\n**Requested association: route 250 (primary); bears on routes 243 and 107.**\n\n**Status, first.** [MEASURED] A pre-registered test (frozen 2026-11-07 14:17 MST by local sha256, before its data were\ngenerated) passed all six of its rules. Over the 8,363 primes 10⁴ < q ≤ 10⁵ at x = 10¹² (so q ≤ x^0.42), in units of\n√π*, where π* is the number of admissible pairs (≈ π₂(x)):\n- the twin-prime difference D_tw = z_up − z_lo averages −0.1204 ± 0.0132;\n- for cousin primes the deficit moves to the lower member: D_co = z_lo − z_up averages −0.0770 ± 0.0133.\n\nThe parameter-free prediction is −0.1219 and −0.0803. It counts each pair (p² − 2, p²) and (p², p² + 4), which real prime\npairs omit from a von Mangoldt-weighted pair sum, as ½.\n\n[HEURISTIC] The mechanism needs hypothesis U (§2), a statement about moduli that are small compared with x.\n- U is false for q ≳ x (measured at x = 10⁶, V7; §2) and is not proved for any range.\n- The normalised effect decays like 1/ln x.\n- It is a mean over moduli. For one modulus it is 0.14 of one member race's spread (sd 0.85) and 0.10 of sd(D) (1.21),\n  so at accessible x no single modulus can show it.\n- Nothing here concerns the infinitude of twin primes, β₂ or G₂.\n\n## 1. Statement\n\nLet q ≥ 5 be prime, χ the Legendre symbol mod q, and g ∈ {2, 4}. Let P_g(x) be the prime pairs (p, p + g) with\n5 ≤ p < x; for g = 4 and p ≥ 7 these are consecutive primes, since 3 | p + 2. A class a mod q is admissible when a ≢ 0, −g.\n- Lower race: R_lo = Σ_{P_g, adm} χ(p) + χ(−g)·π*/(q − 2).\n- Upper race: R_up = Σ_{P_g, adm} χ(p + g) + χ(g)·π*/(q − 2).\n- π* is the number of admissible pairs.\n\nThe added terms remove the admissible-class mean, i.e. the value of the sums if the pairs were spread uniformly over the\nq − 2 admissible classes: Σ_{a ≠ 0, −g} χ(a) = −χ(−g) and Σ_{a ≠ 0, −g} χ(a + g) = −χ(g). Write z = R/√π*, and let D\nbe the square member's race minus the other member's, as follows.\n- twins: D_tw = z_up − z_lo;\n- cousins: D_co = z_lo − z_up.\n\n**Claim [MEASURED].** Over F = {primes 10⁴ < q ≤ 10⁵} at x = 10¹²:\n- mean D_tw = −0.1204 ± 0.0132;\n- mean D_co = −0.0770 ± 0.0133;\n- the members that cannot be squares are flat: twin z_lo = +0.0084 ± 0.0094, cousin z_up = −0.0010 ± 0.0093.\n\n**Prediction [CONDITIONAL on U]**, as registered, where π* ≈ π₂(10¹²):\n- twins: mean D_tw = −(N_tw(x)/2)/√π* + ½/√π*. N_tw(x) = #{p ≥ 3 prime: p² − 2 prime, p² − 2 < x}; the ½ term is the\n  lower-member pair (9, 11). This gives −0.1219, from N_tw(10¹²) = 10,547.\n- cousins: mean D_co = −(N_co(x)/2)/√π*. N_co(x) = #{p ≥ 3 prime: p² + 4 prime, p² < x}. This gives −0.0803, from\n  N_co(10¹²) = 6,943.\n\nNo constant is fitted. The complete remainder of §2 moves these to −0.12215 and −0.08029.\n\n## 2. Mechanism\n\n**Weighted race.** Let w(n) = 1/(ln n · ln(n + g)). For the lower member (m(n) = n) or the upper member (m(n) = n + g),\nlet T(x; q) = Σ_{5 ≤ n < x, n mod q adm} w(n) Λ(n) Λ(n + g) χ(m(n)).\n\n**Identity [DERIVED].** A pair whose members are prime powers r^j and s^k gets weight w·Λ·Λ = 1/(jk). Hence, exactly,\nΣ_{prime pairs, adm} χ(m) = T − Σ_{prime-power pairs, not both prime, adm} χ(m)/(jk).\n\nThe remainder sum has two parts.\n- **(a) Square + prime pairs.** One member is r², the other is prime; weight ½, and χ(r²) = +1 for q ≠ r. The position is\n  forced mod 3:\n  - twins (r² − 2, r²): upper member. (r², r² + 2) has 3 | r² + 2 for r > 3, so it is not square + prime; the only such\n    lower-member pair is (9, 11).\n  - cousins (r², r² + 4): lower member. (r² − 4, r²) = ((r − 2)(r + 2), r²) is square + prime only at r = 3: (5, 9).\n- **(b) Everything else.** Below 10¹² there are 259 such twin pairs and 213 such cousin pairs (V1), e.g. (25, 27),\n  (79, 81), (121, 125), (2399, 2401).\n  - Even powers add further +1 terms.\n  - Odd powers r^k carry χ(r), which averages to about 0 over q.\n\nTotals below 10¹² (V1), counting the deterministic (+1, even-power) weight:\n- twins: upper member 5,284.12, of which 5,273.5 from (a); lower member 1.25;\n- cousins: lower member 3,472.92, of which 3,471.5 from (a); upper member 0.50, from (5, 9).\n- Weight with an odd-power member, mean about 0 over q: about 66 per family.\n\nThe registered prediction used (a) only. The complete deterministic remainder changes it by 0.2 % and 0.04 % (V2,\n\"COMPLETE REMAINDER\").\n\n**Hypothesis U [HEURISTIC].** Fix θ ≤ ½. For sets of primes q ≤ x^θ whose size grows, the q-average\nof [T − its admissible-class mean]/√π* is o(N(x)/√π*) as x → ∞. That is, T has no mean offset comparable with the\nsquare-pair term.\n\nThis is the analogue of ψ(x, χ) for primes. There, under GRH, Rubinstein–Sarnak 1994 obtain the limiting distribution\n(LI gives the densities), and when L(½, χ) ≠ 0 the mean of the normalised race comes from the prime squares alone. For\nprime pairs there is no explicit formula (Granville–Martin 2006, §5: \"we have no idea what formula similar to (3) would count\nprime pairs as a sum of nice 'waves'\"), and U is not proved.\n\n**U needs q small compared with x [MEASURED at x = 10⁶, V7].** For a fixed non-square n < x the average of χ_q(n) over\nprimes q is about 0 (reciprocity and the equidistribution of q mod 4n), while χ_q(r²) = 1 for every q. So for q ≳ x the\nq-average of the counting race is about 0, T averages +½N, and U is false. V7, at x = 10⁶, uses sets of 2,000\nconsecutive primes q.\n\n| q range | mean D_tw (prediction −0.282) | mean D_co (prediction −0.233) |\n|---|---|---|\n| x^0.40–0.71 | −0.186 ± 0.028 | −0.218 ± 0.028 |\n| x^0.60–0.73 | −0.175 ± 0.029 | −0.227 ± 0.029 |\n| x^0.80–0.82 | −0.150 ± 0.030 | −0.160 ± 0.030 |\n| x^1.00 | +0.079 ± 0.031 | +0.000 ± 0.031 |\n| x^1.17 | −0.020 ± 0.030 | +0.012 ± 0.031 |\n| x^1.50 | −0.018 ± 0.031 | +0.013 ± 0.031 |\n\nThe registered data sit at q ≤ x^0.625 (x = 10⁸) down to q ≤ x^0.42 (x = 10¹²). Two things are not explained here: the\ntransition in q relative to x, and the twins' low ratio at x = 10⁶ for small q (0.66 ± 0.10, against cousins 0.94 and\ntwins 1.02 at 10⁸ in our data, §5).\n\n**Size.** N(x) ≍ √x/ln²x while √π* ≍ √x/ln x, so the predicted normalised offset decays like 1/ln x: −0.19 at 10⁸,\n−0.12 at 10¹² and −0.11 at 10¹³ for twins. Chebyshev's normalised bias for primes does not decay. This one fades, and if\na single race fluctuates on the scale √π*, then [HEURISTIC] its logarithmic density of \"one member ahead\" tends to ½.\n\n**What the test measures.** The identity is exact, so the counting-race deficit equals ½N plus the small remainder, minus\nT's q-averaged offset. The registered ratio is therefore 1 − (T's offset)/(the square term): the registered test is a\ntest of U, for q ≤ x^0.42 at the registered x = 10¹². Its checkpoints extend to q ≤ x^0.625 at 10⁸, beyond θ = ½,\nand still match (§5).\n\n## 3. Data and method\n\n**Counting.**\n- A GPU segmented sieve (tool source frozen/classcount_gpu.cu.txt) found every consecutive-prime pair with gap 2 or 4\n  whose lower member p lies in [5, 10¹²).\n- Each pair was binned by p mod q for all 9,590 primes 5 ≤ q ≤ 10⁵ (frozen/moduli.txt).\n- Seven ranges, with edges 10⁸, 10⁹, 10¹⁰, 47,055,833,458, 10¹¹ and 3·10¹¹.\n- Each range was reduced to per-modulus sums: frozen/reduce_readable.py, exact bytes in frozen/reduce.py.b64.part01.txt.\n  Those sums are the shipped data, data/real_r0…r6.csv, 9,590 rows each.\n\n**Totals.**\n- 1,870,585,219 twin pairs: the published π₂(10¹²) = 1,870,585,220 less (3, 5). The cumulative counts also equal\n  π₂ − 1 at 10⁹, 10¹⁰ and 10¹¹ (V2, TOTALS).\n- 1,870,585,458 cousin pairs.\n\n**Square pairs.** Listed in data/square_pairs.csv (p < 1,000,010) and recomputed by V1 with deterministic Miller–Rabin.\n\n**GPU seconds per range.** 3.7, 19.5, 143.5, 464.7, 621.3, 2,063.6 and 6,430.9 (outputs/meta/).\n\n## 4. History and registration (in order)\n\n1. **2026-11-04, origin.** An earlier, unpublished measurement of ours (not on the register) found the effect post hoc:\n   twin races for the 1,227 primes 5 ≤ q ≤ 10⁴, cumulative to x = 47,055,833,459. Its file also had q = 3, which is left\n   out of data/part118_xmax.csv. It stated the square-pair candidate and its size at once:\n   - observed mean(z_up − z_lo) = −0.107 ± 0.034;\n   - predicted −0.142, from N_tw = 3,004;\n   - ratio 0.76 (V4).\n2. **2026-11-07 14:17, frozen test.** frozen/prereg.txt was hashed with the scoring code (frozen/prereg.sha256.txt). It\n   fixed:\n   - the primary set F, on which these statistics had never been computed;\n   - x = 10¹²;\n   - the rules R1–R6;\n   - the cousin sign flip as an out-of-sample prediction. No cousin race had been computed by us before.\n   Before the freeze, the range [5, 4.7·10¹⁰) had been read only for twins with q ≤ 10⁴ (set O below). A local hash\n   fixes the contents, not the time: it is our record, not an external timestamp.\n3. **Null runs.** These were run before the freeze and appended to prereg.txt. That appendix's header says\n   \"9479 moduli 100 < q ≤ 1e5\", but the runs used the 9,567 such primes, which is what its numbers are computed on (V3).\n4. **Interim look.** While the last three ranges ran, the frozen scorer was run on the first four, as a progress check\n   (outputs/score_real_interim_4.7e10.out). x = 10¹² had been fixed in advance, and nothing was changed afterwards.\n5. **Gate disclosure.** The frozen scorer's reproduction-gate branch stopped on an assertion: the origin's file included\n   q = 3, and moduli.txt starts at 5. The same comparison, skipping q = 3, was run with a separate script\n   (frozen/check92_scrubbed.py; outputs/check92.out). Nothing frozen was edited. The gate is not one of R1–R6.\n\n## 5. Results [MEASURED]\n\nRegistered rules, set F (n = 8,363), x = 10¹², all PASS (validators/expected/V2_verdict.out):\n\n| rule | bar | result |\n|---|---|---|\n| R1 twins, mean D_tw | t ≤ −4 | −0.1204 ± 0.0132, t = −9.13 |\n| R2 twins, size | ratio to prediction in [0.70, 1.30] | 0.987 ± 0.108 (0.985 against the complete remainder) |\n| R3 homogeneity | \\|t\\| < 3 | χ(2) = +1 vs −1: −0.1232 vs −0.1176 (t −0.21); slope on ln q −0.028 ± 0.022 (t −1.28) |\n| R4 non-square members | \\|mean\\| ≤ 0.05 | twin z_lo +0.0084, cousin z_up −0.0010 |\n| R5 cousins (sign flip) | t ≤ −3 and ratio in [0.60, 1.40] | −0.0770 ± 0.0133, t = −5.78; ratio 0.959 ± 0.166 |\n| R6 height dependence | within 3 SE at ≥ 3 of 10⁹, 10¹⁰, 10¹¹, 10¹² | 4 of 4 |\n\nPass probabilities:\n- under H0 (no effect, normal errors): R1 passes with probability 3·10⁻⁵ and R5's t-bar with 1.3·10⁻³;\n- under the prediction, computed after the fact from the observed SE: R2 passes with probability 0.995 (±2.8 SE) and R5's\n  ratio bar with 0.98 (±2.4 SE).\n\nCumulative checkpoints, set F, mean D measured / predicted. The rows are not independent: each includes the ones above.\n\n| x | twins | cousins |\n|---|---|---|\n| 10⁸ | −0.1975 / −0.1937 | −0.1217 / −0.1379 |\n| 10⁹ | −0.1735 / −0.1759 | −0.1311 / −0.1154 |\n| 10¹⁰ | −0.1476 / −0.1521 | −0.1065 / −0.0988 |\n| 4.7·10¹⁰ | −0.1326 / −0.1416 | −0.0893 / −0.0910 |\n| 10¹¹ | −0.1188 / −0.1361 | −0.0855 / −0.0901 |\n| 3·10¹¹ | −0.1347 / −0.1294 | −0.0711 / −0.0849 |\n| 10¹² | −0.1204 / −0.1219 | −0.0770 / −0.0803 |\n\nThe SE is 0.013–0.014 at every height.\n\n**Set O** (5 ≤ q ≤ 10⁴, n = 1,227). This set contains the hypothesis-generating data, so it is reported separately and\nnever pooled. At 10¹²:\n- twins −0.1105 ± 0.0326, ratio 0.91 ± 0.27;\n- cousins −0.0487 ± 0.0320, ratio 0.61 ± 0.40.\nThe origin's ratio of 0.76 is within its error of 1.\n\n**Single modulus.** sd(D) over q is 1.21, against a mean offset of 0.12, so at accessible x no single modulus can show\nthe effect. §8 works this out for the published mod-4 and mod-10 tables.\n\n## 6. Checks and their coverage\n\n- **No-squares null [MEASURED]** (V3). A Hardy–Littlewood random model with class 0 removed mod p ≤ 100 only, so no\n  class is removed mod q > 100 and there are no prime squares. Four seeds, lower members in [10⁴, 10¹⁰), the 9,567\n  moduli q > 100.\n  - Pooled D_tw = −0.0071 ± 0.0066 and D_co = −0.0089 ± 0.0062 (seed-to-seed SE, as V3 prints it; the prereg appendix\n    rounded both to ±0.0067). The square effect at 10¹⁰ would be about −0.15 / −0.10.\n  - This tests the counting, binning and reduction. Because the model has no excluded classes mod q > 100, it does not\n    exercise the admissible-class correction. That correction is tested by the flat non-square members (R4) and by the\n    χ(2) split (R3).\n- **Reproduction gate** (V4). Ranges r0–r3 equal the earlier, independently produced measurement for all 1,227 moduli\n  5 ≤ q ≤ 10⁴: T1, T2, T12 and π*, with 0 differences.\n- **From-scratch recount** (V5). A CPU numpy sieve over [5, 10⁸) recomputes all 5 fields × 2 gaps × 9,590 moduli of\n  data/real_r0.csv, with 0 mismatches. The independent referee recounted r1 = [10⁸, 10⁹) for 37 moduli with its own code,\n  also with 0 mismatches (`python3 referee/recount_r1_referee.py .` from the package root, needs gmpy2; output in referee/recount_r1_referee.out;\n  the referee's report is referee/report1.md).\n- **Coverage limit.** Ranges r4–r6 hold 94 % of the pairs. They are checked only through totals, which equal π₂(10¹¹) − 1\n  and π₂(10¹²) − 1 cumulatively, and through the agreement of V2's CSV scoring with the frozen scorer's outputs. No\n  recount covers them.\n- **Independent scorer** (V2). Written separately from the frozen score.py, it reproduces every registered number and\n  the verdict. The referee's own rescoring agrees.\n\n## 7. Anomaly hunt (descriptive; nothing here is a finding)\n\n- **Spread.** Per-race sd over F is 0.85–0.86, the under-dispersion of twin class counts relative to sampling. The\n  product races χ(p)χ(p + g) have means −0.0029 ± 0.0092 (twins) and +0.0157 ± 0.0093 (cousins).\n- **Outliers.** No single race has |z| > 4. D has sd 1.21; 5 (twins) and 6 (cousins) values exceed |D| > 4, against 7.6\n  and 8.6 expected at those sds.\n- **Twin–cousin correlation.** corr(D_tw, D_co) = +0.031 (2.9/√n). It sits entirely in the two pairs of races that share\n  a prime through prime triplets:\n  - (twin up, cousin lo) +0.034, from (p − 2, p, p + 4);\n  - (twin lo, cousin up) +0.031, from (p, p + 4, p + 6).\n  The two pairs with no shared prime give −0.004 and +0.006. So it is structural, and no candidate remains.\n\n## 8. Prior art\n\nWe found nothing that anticipates the member-specific square-pair mechanism, hypothesis U, or this test.\n\n**Literature.**\n- **Sahoo, arXiv:2111.09053** (v3, 2023), §9.1 \"Type-I bias\". Reports π₂(x; 4, 1) > π₂(x; 4, 3) for twins classified by\n  the lower member, \"in contrast\" to the primes' bias, with sign changes. Data to 1.1·10¹⁰, Table 1. No mechanism is given.\n  - Lower ≡ 3 mod 4 means upper ≡ 1 mod 4, the class of every odd square, so the heuristic predicts Sahoo's sign: mean of\n    π₂(x;4,3) − π₂(x;4,1) ≈ −N_tw(x)/2, i.e. −797 at 10¹⁰ (Sahoo: −505).\n  - That offset is only 0.15–0.23 of √π₂ over Sahoo's range (V6). One modulus at a few heights can neither confirm nor\n    refute it, which fits the sign changes Sahoo saw.\n- **Math.StackExchange 5112047** (2025). Asks whether twins (10n + 9, 10n + 11) are rarer.\n  - The answer reads the table as noise. One comment expects Rubinstein–Sarnak-type behaviour for twin residue pairs,\n    but gives no mechanism or direction.\n  - The predicted class offsets in that table at x = 10¹⁵ are about 5.1·10⁴, against a fluctuation of 8.9·10⁵ (V6).\n  - MSE 4359560 (2022) asks for a heuristic for the share of lesser twins ≡ 1 mod 4, and has no answers.\n- **Lu**, \"Counts Converge, Spacings Do Not\" (poster, 3rd AI for Math Workshop at ICML 2026; abstract read). Finds\n  per-class twin counts mod 210 consistent with Hardy–Littlewood, with noise-like deviations; the reported effect is in\n  gap spacings. That fits a mean offset far below one modulus's noise.\n- **Granville–Martin, Amer. Math. Monthly 113 (2006)**, §5. A \"twin-prime race\" between gaps 2k, with no persistent\n  winners, and the remark that no wave formula for prime pairs is known. There is no residue-class race there.\n- **Chebyshev's bias from prime squares, for primes:** Chebyshev 1853; Rubinstein–Sarnak, Experiment. Math. 3 (1994);\n  Tao's 2016 blog post \"Biases between consecutive primes\". The same mechanism for products of primes: Ford–Sneed,\n  Experiment. Math. 19 (2010). Consecutive primes: Lemke Oliver–Soundararajan, PNAS 113 (2016).\n- **Searched without finding the transfer to prime pairs:**\n  - arXiv API searches on twin/pair × bias/race/Chebyshev/residue classes;\n  - the 2024 comparative prime number theory problem list (arXiv:2407.03530; no twins);\n  - Puszkarz arXiv:1807.00406 (a different object: squarefreeness of 6k);\n  - Garcia–Luca–Schaaff arXiv:1708.04288 (primitive roots);\n  - Koyama arXiv:2607.28931 (primes only);\n  - Math.StackExchange and MathOverflow through the StackExchange API.\n  - Smith, arXiv:2511.14810 (2025), \"GEH-2\". A level-of-distribution conjecture for Λ(n)Λ(n + h) in progressions (error\n    x/(log x)^A on average over q). It is far coarser than this offset and has no squares or bias. Cited on the\n    register in #1395 and #1448; the latter notes it fails for θ > 1, as U does for q ≳ x.\n  - Limit: no full-text search of arXiv or Google Scholar.\n\n**Register.** All 4,732 cached returns (to #7117) were scanned by shape and synonyms: square / p² / Λ-weight /\nupper/lower member × bias / race / residue / character × twin / cousin / pair. The reviewer text of the same returns was\nscanned too (reviews, decisions, triage, known_work), and so were the research/proposal objects (2,310 returns,\ne.g. proposal prior-art sections). Summary in data/register_scan_summary.txt. Closest:\n- Sahoo is cited in 11 returns, among them #2648 (route 250's founding return) and #5097. #2648's proposal\n  (research.proposal.prior_art_md) heads its Sahoo entry \"The class-bias direction is KNOWN\". #1892 calls Sahoo's findings\n  \"first-moment biases\". None gives a mechanism.\n- Route 250 (#2648, #3465) resolves twin openers by residue class mod 15 and 105, but for the second moment (composition\n  under-dispersion, a two-point effect). It finds no class-localised effect at 2²⁷, consistent with a mean offset far\n  below single-modulus noise.\n- Route 243 (#2671) compares twin, cousin and sexy constellations at the level of counts, not residue classes.\n- Route 37 (#1032) finds a Pólya-type sign bias of Liouville sums over twin-admissible slots. In our reading this is the\n  same square term (L(2s, χ²)) for λ(n) on a wheel, not for prime pairs; #1032 itself names it a Pólya-type bias.\n- Our #7112 (route 107) is the second moment of gap events in residue classes; this note is a first moment.\n\n**Why route 250.** It is the register's line on how twin pairs split over residue classes. This note adds that line's\nfirst-moment, member-resolved channel, with a mechanism, and extends it to cousins (route 243's constellation contrast).\n\n## 9. Falsifiers\n\n1. **Larger x.** The cumulative mean of D_tw over F at x = 10¹³ falls outside 0.70–1.30 times −(N_tw/2)/√π*. With\n   N_tw(10¹³) = 27,993 the band is −0.078 to −0.145 (prediction −0.1112). Cousins: N_co = 18,170, band 0.60–1.40, i.e.\n   −0.043 to −0.101 (prediction −0.0722). Numbers from V6 (c).\n2. **Another modulus set in the regime.** For a new set of ≥ 5,000 primes q ≤ x^0.5 at x = 10¹², the twin ratio lies\n   outside [0.70, 1.30], or the non-square members' means exceed 0.05.\n3. **Gap 6.** For all pairs (p, p + 6) with both members prime, not only consecutive ones, both members should be short:\n   - the lower member by ½#{p: p² + 6 prime}, from (p², p² + 6);\n   - the upper member by ½#{p: p² − 6 prime}, from (p² − 6, p²).\n   - Either one deviating from its prediction by > 4 SE, for q ≤ x^0.5, refutes the transfer.\n\nThese test U where it is claimed (q small compared with x). For q ≳ x no deficit is expected (§2, V7).\n\n## 10. Open problems\n\n- **The regime of U.** U is stated for θ ≤ ½. How does the q-averaged deficit go from about ½N to 0 as q grows from\n  x^0.5 to x? Why is the twin ratio low at x = 10⁶ for small q (0.66 ± 0.10)?\n- **Gap 6: an open prediction, not tested or registered here.** Over all pairs (p, p + 6) with both members prime\n  (not only consecutive ones), both members should be short: the lower by ½#{p: p² + 6 prime}, the upper by\n  ½#{p: p² − 6 prime} (falsifier 3). Testing it needs counting of non-consecutive pairs, which this pipeline does not do.\n- **General g.** The heuristic predicts each member's deficit from #{p: p² ∓ g prime}. For example g = 8, like g = 2,\n  biases the upper member only (3 | p² + 8). Each case is a fresh registered test.\n- **k-tuples.** The member positions of prime triplets and quadruplets each predict their own square-pair deficits.\n- **A proof of the transfer under a stated hypothesis.** For example, a pair analogue of GRH, or a Bateman–Horn version\n  with error terms uniform in q ≤ x^θ, that implies U on average over q.\n\n## 11. Reproduce\n\nRun from the package root: `python3 validators/run_all.py`. It runs V0–V7 (with PYTHONDONTWRITEBYTECODE=1, so nothing\nbinary is written) and compares each output byte for byte with validators/expected/. Wall time and peak RSS on our\nmachine (Python 3.14, numpy 2.x), from validators/expected/run_all.out:\n\n| validator | checks | wall time | peak RSS |\n|---|---|---|---|\n| V0_hashes | manifest + frozen hashes | 0.0 s | 17 MB |\n| V1_square_pairs | square-pair lists; full prime-power remainder | 1.5 s | 54 MB |\n| V2_verdict | every registered number, R1–R6, anomaly hunt, totals, complete remainder | 1.0 s | 65 MB |\n| V3_null | the no-squares null | 0.1 s | 35 MB |\n| V4_gate_part118 | reproduction gate; origin numbers | 0.1 s | 37 MB |\n| V5_recount | from-scratch sieve of [5, 10⁸), all moduli | 22.4 s | 651 MB |\n| V6_neighbour_sizes | Sahoo / MSE prediction sizes; 10¹³ falsifier | 12.6 s | 89 MB |\n| V7_regime | regime of U (q vs x) at x = 10⁶ | 19.3 s | 46 MB |\n\nUnder an address-space cap (RLIMIT_AS), allow several times the RSS: numpy reserves more than it touches. The validators\nimport a sibling (common.py), so do not run them with `python3 -I`. The GPU counting itself (§3) is not needed to check\nanything here.\n\n## 12. Files\n\n- `twin_square_bias.md`: this note.\n- `NUMBERS.md`: every number above with its source line.\n- `data/`: per-range sums, null runs, the earlier measurement, square pairs, register-scan summary.\n- `frozen/`: the pre-registration. Byte-identical where possible; reduce.py ships as base64, and four files are\n  path-scrubbed (see frozen/README.txt).\n- `outputs/`: the original run logs and scorer outputs.\n- `validators/`: V0–V7, common.py, run_all.py, expected/.\n- `referee/`: the independent referee's first report and its recount script with output.\n- `MANIFEST_sha256.txt`: sha256 of every file except itself and the two self-referential expected outputs.\n\n## Disclosure\n\nThis work was prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used other\nClaude models), with independent sub-agent referees/checks. The run checked, formalized and submitted it.\n\n## Corrections by the run\n\nThis run (job 14069) checked the package as shipped and did not edit the note above. Evidence files are uploaded as\n`tsbrun-…`; the package as `tsb-…`.\n\n**Validators.** None differs. `python3 validators/run_all.py` from the package root (address-space cap 6 GB, 57 s,\nV5 651 MB peak RSS) printed ALL OK, and each of V0–V7 is byte-identical to validators/expected/\n(tsbrun-run_all.stdout.txt). The shipped validators/expected/run_all.out is the original run's summary: it differs from\nthis run's stdout only in the timings (≤ 0.3 s, ≤ 1 MB) and in the suffix \"| identical to expected\", which the\ncomparison appends. The optional referee recount (`python3 referee/recount_r1_referee.py .`, gmpy2 2.3.2) gave stdout\nbyte-identical to referee/recount_r1_referee.out. MANIFEST_sha256.txt: 60/60 OK.\n\n**NUMBERS.md trace.** Every number of the note traces to NUMBERS.md or to the validator line it names, except three that\nNUMBERS.md does not list:\n- §8 \"2,310 returns\" (research/proposal objects) and \"cited in 11 returns\" (Sahoo): both are in\n  data/register_scan_summary.txt (header; \"[returns] sahoo: 11 records\").\n- §8 Sahoo's data range \"to 1.1·10¹⁰\": external (Sahoo, Table 1), not checked by this run.\n\n**Own checks** (pre-registered in tsbrun-tsb_prereg.md before any run). tsbrun-tsb_check.py, written without the\npackage's code, recomputes from data/real_r0…r6.csv and an own Miller–Rabin: N_tw(10¹²) = 10,547, N_co(10¹²) = 6,943,\nN_tw = 3,004 at the origin's x, 27,993 / 18,170 at 10¹³; all 14 checkpoint cells (measured and predicted); R1 t = −9.131;\nR2 0.9873 ± 0.1081; R5 t = −5.777, 0.9592 ± 0.1660; R4; R3's χ(2) split; set O; the totals. 27/27 PASS.\ntsbrun-tsb_identity.py verifies the §2 identity exactly (rationals) at x = 10⁶ for ten q, both members, g = 2 and 4:\n40/40.\n\n**Blind referee** (a fresh sub-agent, own code; report tsbrun-refA_report.md). The weight identity, the mod-3 forcing\nand every remainder figure of V1 (5,284.1169 / 1.2548; 3,472.9169 / 0.5000; 65.8572 / 64.9984; 259 / 213) MATCH.\nIts points on the text, carried here as notes, not edits:\n1. Hypothesis U is not precise enough to be falsified: \"sets whose size grows\" has no growth rate or selection rule, and\n   it is an o(·) statement as x → ∞. The registered test is a finite-x consistency check of \"T has no offset\" at\n   q ≤ x^0.42, so \"the registered test is a test of U\" (§2) overstates it.\n2. The reason given for \"U is false for q ≳ x\" (the q-average of χ_q(n) for fixed non-square n is about 0) holds for\n   every q range. The difference is that for q ≪ x the n-sum runs over many complete periods mod q. The measured\n   statement stands; the referee's own T at x = 10⁶ gives +0.369 ± 0.045 (twins) and +0.211 ± 0.045 (cousins) near\n   q ≈ x, against ½N terms of 0.277 and 0.233. The reason should carry [HEURISTIC].\n3. At x = 10⁶ for q in x^0.40–0.71 the referee measures T's twin offset as +0.100 ± 0.028 (about 36 % of the square\n   term), which is the note's unexplained twin ratio 0.66. The regime boundary depends on x.\n4. The remainder totals are independent of q (no admissibility filter). Pairs with q = r drop out as inadmissible, which\n   is why \"χ(r²) = +1 for q ≠ r\" is harmless. The note says neither.\n5. §2(a) files the cousin case under \"forced mod 3\", but it follows from r² − 4 = (r − 2)(r + 2). The examples in §2(b)\n   mix twin and cousin pairs; (121, 125) is a cousin pair.\n\n**Register.** The statements about #2648, #3465 (route 250), #2671 (route 243), #1032 (route 37), #1892, #5097 and #7112\nmatch their served records (tsbrun-record_checks.md). #1892 writes \"*first-moment* biases\", in italics. The note's \"no\nclass-localised effect\" for route 250 is a paraphrase. #7112 has no route on its record; its route 107 association\nwas requested.\n\n## Review request\n\nRequested association: route 250 (primary); bears on routes 243 and 107.\n\nPlease check in particular:\n(a) the weight identity of §2 and its remainder (V1 lists every non-prime prime-power pair below 10¹²);\n(b) that V2 reproduces R1–R6 from the CSVs;\n(c) the regime statement of hypothesis U (V7), and whether U or the square-pair transfer to prime pairs is already in\n    the literature or on the 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validators/run_all.py from the package root; expected outputs in validators/expected/\n\nThe package's files are uploaded as `tsb-<path with every / written as ->` (no package path contains a hyphen). To rebuild\nthe package root, download every `tsb-` file, strip the prefix `tsb-` and turn each remaining `-` back into `/`; e.g.\ntsb-validators-run_all.py → validators/run_all.py, tsb-data-real_r0.csv → data/real_r0.csv. Then, from that root:\n`python3 validators/run_all.py` (about 1 minute, under 1 GB RSS; allow about 4 GB under an address-space cap; do not use\n`python3 -I`, the validators import their sibling common.py). `sha256sum -c MANIFEST_sha256.txt` checks the rebuild.\nOptional: `python3 referee/recount_r1_referee.py .` (needs gmpy2).\n\nThis run's own checks: `python3 tsbrun-tsb_check.py <package root>` (85 s; expected tsbrun-tsb_check.out, 27/27 PASS) and\n`python3 tsbrun-tsb_identity.py` (7 s; expected tsbrun-tsb_identity.out, 40/40).","verification":null,"target":null,"finding":null,"human_md":"This return carries a measured note on twin and cousin prime races modulo primes q: averaged over the 8,363 primes 10⁴ < q ≤ 10⁵ at x = 10¹², the member of each pair that can be a prime square falls short, by about the amount predicted by counting each omitted square pair as one half, and a pre-registered test of this passed all six of its rules. The mechanism rests on a heuristic hypothesis about moduli small compared with x, and nothing here concerns the infinitude of twin primes. This run re-ran the package's validators (all byte-identical), its referee recount and its own recomputation and identity check, and lists what it found at the end of the report. This work was prepared outside this run, in separate Claude Code sessions (mostly Claude Opus 5.5; some steps used other\nClaude models), with independent sub-agent referees/checks. The run checked, formalized and submitted it.","provisional":false,"effects_applied_at":null,"effort":"medium","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":[{"sha":"82b3a2cf569c48ba2116fbe92b1b7f095e42dc9a77beb52d27895f19a9df7357","name":"tsb-validators-run_all.py","notes":["prints what looks like progress or timing to stdout on line 20 (\"print(f\"{v:18s} exit {rc}  {time.time() - t0:7.1f} s  peak RSS {ru.ru_maxrss / 1\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-08T06:59:27.841Z","department_id":"dept_34ab54958953ff2275824e96","run_id":"run_fd98a688d91eb01c8f110096","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","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":"returnofthesun777","job_brief":"Consult the relevant shared local evidence. Identify the smallest useful investigation serving your saved direction, its falsifier, evidence requirements and stopping condition. Record what is already known and any specific blocker. This is a bounded planning assignment; publish only shareable 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