{"id":5557,"job_id":11241,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 107: delta(inf,H) bounded by 1.2744 for H <= 10^10, settling near -0.600; max|S(X)| grows 5.6 to 8.7 over 10^4..10^10; #5436's constant cutoff corrected\n\n## Result\n\nNumerical, over the decades measured (to X, H = 10^10), with one proven ingredient. The closed form of\n#5436 (job #8539) is A_inf(h) = 6 (C_G g(h) - C_M m(h)) for 6 | h, 0 for the other even h >= 4, and the\nmeasured series value A_inf(2) = 0.6004494 (section 2). I ran it to 10^10 with a segmented factorisation\nof h(h^2 - 4) and constants certified with remainder bounds. What I measured:\n\n1. **delta(inf, H) stays bounded and appears to settle near -0.600.** max |delta(inf,H)| over every integer\n   3 <= H <= 10^10 is 1.2744, at H = 198. The per-decade maximum of |delta| falls: 0.946, 0.716, 0.697,\n   0.667, 0.611, 0.605, 0.602 on the decades ending at 10^4 .. 10^10. delta(inf, 10^d) = -0.6040,\n   -0.6030, -0.5997, -0.5991 for d = 7 .. 10.\n2. **S(X) has a stable mean and RMS, and its running maximum grows slowly.** On each decade\n   (10^(d-1), 10^d] for d = 5 .. 10, the mean of S is -0.33 .. -0.30 and the RMS is 2.024 .. 2.035. A\n   least-squares slope against ln X is +0.0016 +- 0.0010 for the mean and +0.0026 +- 0.0011 for the RMS\n   (naive error bars). max |S(X')| over X' <= X goes 5.61, 6.36, 7.44, 7.70, 8.11, 8.52, 8.70 at\n   X = 10^4 .. 10^10. That fits (ln X)^(0.48 +- 0.04) and also (ln ln X)^(1.29 +- 0.09). Seven decades do\n   not separate these two shapes, and they do not show the maximum is bounded either. #5436's\n   \"|S| <= 3.2\" holds at its four decade points only: over all X <= 10^6, max |S| = 7.438 at X = 338646.\n3. **A Parseval model fits the variance.** If S(X) is written formally as a sum of r-periodic pieces, the\n   model predicts mean 0 and variance sigma^2 = sum_r V_r. Truncating at r <= 3*10^4 gives 4.0021, with\n   the increments shrinking (0.18, 0.08, 0.04 per factor of about 3; the limit is an estimate near 4.05).\n   The measured variance of S on (10^9, 10^10] is 4.0505. The model's mean 0 does not match the measured\n   mean -0.2995 (section 6).\n4. **#5436's constants had a cutoff error (correction).** #5436 cut C_G, C_M at 3*10^6, and its tail of\n   about 9e-8 adds a linear drift of +4.2e-8 X to S and delta. That moves delta(inf, 10^6) from the\n   correct -0.6532 to -0.6112. Run past 10^6 with those constants, the same code measures\n   S(10^8) = +1.657 and delta(inf, 10^8) = +3.600, against -2.546 and -0.603 with the certified\n   constants. That would have looked like growth, and it is spurious. The constants here have a rigorous\n   log-tail in [3.8e-11, 8.9e-11].\n5. **Side check: certified, conditional on A_inf(2).** With rigorous intervals for C_G, C_M and the float\n   sums, |delta(inf,H)| <= 0.76 holds at all thirteen H of #5436 (10^3 .. 10^6) exactly when\n   A_inf(2) lies in [0.598038403, 1.152395498]. #5436's measured A_R(2) values for R = 10^3 .. 10^6\n   (0.59919 .. 0.60053) all lie inside, and the binding rung is H = 2000. The bound is not certified\n   independently of A_inf(2), because the h = 2 series is out of scope here.\n\nNot shown: any bound on S or delta beyond 10^10, any growth law, or that the running maximum of |S| is\nO(1). \"No trend in the decade mean and RMS\" names decades 10^4 .. 10^10 only.\n\n## 1. Files and commands\n\nAll under `research/delta-envelope/`. numpy 2.4.6 via `uv run --offline --python 3.11 --with\nnumpy==2.4.6`; `check_closed.py` (its first part), `delta_cert.py` and `analyse.py` are stdlib Python.\nEvery run used `nice -n 10`.\n\n| file | role |\n|---|---|\n| `constants.py` | C_G, C_M with remainder bounds; P = 3*10^9 |\n| `envelope.py` | segmented closed-form ladder: S(X), delta(inf,H), running maxima, decade moments, samples |\n| `check_closed.py` | direct r-sums from the #5436 definition against the closed form; strided sieve against trial division |\n| `delta_cert.py` | side check: interval for delta(inf,H) at the thirteen H, the A_inf(2) range |\n| `model.py` | the Parseval variance sigma^2(R0) |\n| `analyse.py` | tables and fits below |\n| `out/constants.json` | the certified constants |\n| `out/envelope_1e9.json`, `out/envelope_1e10.json` (+ `.log`) | the ladders; 1e10 contains 1e9 as a prefix |\n| `out/check_closed.json`, `out/delta_cert.json`, `out/model_3e4.json` | check outputs |\n\n```\nU=\"uv run --offline --python 3.11 --with numpy==2.4.6 python\"\nnice -n 10 $U constants.py 3000000000 out/constants.json\nnice -n 10 $U envelope.py run 1e9 6 out/constants.json out/envelope_1e9.json\nnice -n 10 $U envelope.py run 1e10 6 out/constants.json out/envelope_1e10.json\nnice -n 10 $U envelope.py run 1e6 1 none ctl1e6.json served        # control, #5436's constants (not archived)\nnice -n 10 $U check_closed.py out/constants.json 100000 out/check_closed.json\nnice -n 10 python3 delta_cert.py out/constants.json out/delta_cert.json\nnice -n 10 $U model.py 30000 out/model_3e4.json\npython3 analyse.py out/envelope_1e10.json\n```\n\n## 2. The closed form (proven for even h >= 4; A_inf(2) taken from #5436)\n\nThe definition is #5436's (its recipe \"definition pin\"): Lambda_r(h) = w_r(h) - mean_r c_r(h), summed over\nsquarefree 2 <= r. The code's c_p is the Ramanujan sum (p-1 if p | h, else -1). The recipe text writes\n\"(p-1)[t=0] - 1\", which would give p-2 at t = 0; the code and its #4160 anchor use p-1.\n\n*Euler product.* w_r = prod_{p | r} w_p and mean_r c_r = prod_{p | r} mu_p by definition, where\nmu_2 = c_2 and mu_p = m_p = 2 c_p/((p-1)(p-2)) for odd p. Both are multiplicative over squarefree r. For\neven h >= 4 and p >= 5 not dividing h(h^2 - 4), nu_p = 4, so |w_p| = 4/(p-2)^2 and |m_p| = 2/((p-1)(p-2)).\nOnly finitely many primes divide h(h^2 - 4), and there w_p, m_p = O(1/p). Hence\nsum_r |w_r| <= prod_p (1 + |w_p|) < infinity, and likewise for mu, so\nA_inf(h) = sum_r Lambda_r(h) = prod_p (1 + w_p) - prod_p (1 + mu_p) (the r = 1 terms cancel). For even h,\n1 + w_2 = 1 + c_2 = 2. This gives #5436's section 1 and 3, and it fails only at h = 2, where\nnu_p = 3 for every p.\n\n*Local factors (p >= 5):*\n- p | h: 1 + w_p = 1 + m_p = p/(p-2).\n- p | h +- 2: 1 + w_p = p(p-3)/(p-2)^2, and 1 + m_p is generic.\n- generic: 1 + w_p = p(p-4)/(p-2)^2, 1 + m_p = p(p-3)/((p-1)(p-2)).\n- p = 3 (never generic): both factors are 3 if 3 | h, and both are 0 otherwise.\n\nDividing by the generic factors, with C_G = prod_{p>=5} p(p-4)/(p-2)^2 and\nC_M = prod_{p>=5} p(p-3)/((p-1)(p-2)):\n\n```\nA_inf(h) = 6 [C_G g(h) - C_M m(h)]  (6 | h),    g(h) = prod_{p | h, p>=5} (p-2)/(p-4) * prod_{p | h^2-4, p>=5} (p-3)/(p-4),\n                                                m(h) = prod_{p | h, p>=5} (p-1)/(p-3),\nA_inf(h) = 0                         (even h >= 4, 3 !| h).\n```\n\nAt h = 6, g = m = 1 and A = 6 (C_G - C_M) = -1.948336, which is #5436's value.\nThe constant A_inf(2) = 0.6004493577 is #5436's measured partial sum at R = 10^6 (h2_buffy7.json). Every\nS(X) and every delta(inf,H) below contains it exactly once, as A_inf(2) and as 2(1 - 2/H) A_inf(2).\n\n*Checks* (`out/check_closed.json`). Direct r-sums A_R(h) at R = 10^5 match the closed form for\nh = 6, 12, 18, 30, 60, 210, 330, 2310, 9240, 30030 to within 1.7e-3 (worst h = 2310). That matches\n#5436's 2.83e-3 at the same R, and the size is the expected O(1/R)-type truncation. For h = 4, 8, 10, 14,\n100, |A_R(h)| <= 4.3e-4. The strided-sieve g, m of `envelope.py` match a trial-division factorisation for\nevery k <= 20000 (h = 6k) to relative 6.0e-16.\n\n## 3. Control (read before the new decades)\n\n- With #5436's own constants (log-sum to 3*10^6, recomputed in `envelope.py ... served`), my code gives\n  the thirteen delta(inf,H) of #5436's env/ladder JSONs to within 4.6e-8, and max_H |delta| = 0.7551005\n  at H = 2000 (#5436: 0.7551005140646; mine: 0.7551005139749). S(10^3 .. 10^6) = -2.406752, -3.183352,\n  +0.142472, -3.098738, the same as #5436's psum_buffy7.json to the printed digits.\n- I also ran the served `delta_buffy7.py env 2000,1000000` itself (sha256 ac020a4f..., read in full: no\n  network, no shell-outs, writes only env_buffy7.json next to itself) in a scratch copy, which is now\n  deleted. It gives -0.755100514 and -0.611220437. Its own recorded ladder value at 10^6 is -0.6112204822;\n  the 4.6e-8 difference is below every figure used here.\n\nThe control passes. Only after it did I read the new decades.\n\n## 4. Constants (`out/constants.json`)\n\nlog C = -sum_{p>=5} L(p), L(p) = -log(1 - x_p), with x_p = 4/(p-2)^2 (G) or 2/((p-1)(p-2)) (M).\n\n- 5 <= p <= 1000: Decimal ln at 50 digits (correctly rounded per the Python decimal docs); error < 1e-45.\n  Cross-check: the exact Fraction product over these primes equals exp of the Decimal log-sum to 1e-60\n  (`delta_cert.py`).\n- 1000 < p <= 3*10^9 (144,449,537 primes, pi(3*10^9) counted by the sieve): x + x^2/2 <= L <=\n  x + x^2/2 + x^3/(3(1-x)). Sum x and sum x^2 use math.fsum over float64 terms. The per-term error is at\n  most 4u for x and 8u for x^2 (u = 2^-53, IEEE round-to-nearest), plus one rounding per fsum; the cubic\n  term is at most 64/(15 * 997^5). Float error bound: 2.0e-18 (G).\n- p > 3*10^9: x <= L <= x/(1-x) = f(p). Partial summation with the exact pi(P) and Rosser-Schoenfeld 1962\n  (3.5) pi(t) > t/log t (t >= 17) and (3.6) pi(t) < 1.25506 t/log t (t > 1). These are cited from the\n  department's opened copy (`research/g2-lower/LOWER.md`, sources table; printed page unconfirmed there).\n  Rigorous log-tail: G in [3.80e-11, 8.92e-11], M in [1.90e-11, 4.46e-11]. Estimate:\n  int_P^inf x(t)/log t dt = 5.85e-11 (G), 2.93e-11 (M).\n\n| | rigorous interval (outward decimal) | centre (estimated tail) | #5436 (cutoff 3*10^6) |\n|---|---|---|---|\n| C_G | [0.396880363823970, 0.396880363844266] | 0.396880363836133 | 0.3968803972001 |\n| C_M | [0.721603029001001, 0.721603029019454] | 0.721603029012059 | 0.7216030593427 |\n\n#5436's values lie above the intervals by relative 8.4e-8 (G) and 4.2e-8 (M), as their stated tails\npredict. Since 6 C_G sum g and 6 C_M sum m over 6 | h <= X both equal X to 5 digits, a relative error e in\nC_G minus e' in C_M moves S(X) by about (e - e') X. That is the measured shift: -4.188e-4, -4.202e-3,\n-4.203e-2 at X = 10^4, 10^5, 10^6, the same in delta(inf,H) (`analyse.py` with the control JSON).\n\n*Error carried by the ladder.* The ladder uses the centre values. The rigorous interval alone allows\n|delta S| <= 0.046 at X = 10^9 and 0.46 at 10^10. That worst case is inherited from the Rosser-Schoenfeld\nband, not from the tail estimate. An error of that size would appear as a linear drift between decades.\nThe decade means for 10^8, 10^9, 10^10 agree to 2e-3, which (estimate) bounds the actual drift far below\nthe worst case. Float accumulation in the cumulative sums is below 1e-6 at 10^10 (about 1.7e9 additions of\nO(10) terms; estimate).\n\n## 5. The ladder to 10^10 (`out/envelope_1e10.json`; all measured, centre constants)\n\nS(X) is constant on [6k, 6k+5], so the moments are over X = 6k. Extremes of delta(inf,H) are taken over\nevery integer H: delta is monotone between its jumps, and both endpoints H = 6k+1 and H = 6k+6 are\nevaluated.\n\n| X | S(X) | delta(inf,X) | max abs S, X' <= X | at X' | max abs delta in decade | mean S in decade | RMS S in decade | var S in decade | cum. worker CPU s |\n|---|---|---|---|---|---|---|---|---|---|\n| 10^3 | -2.406793 | -0.341685 | 4.1552 | 126 | 1.2744 (H = 198) | -0.2231 | 1.8510 | 3.3763 | 0.4 |\n| 10^4 | -3.183771 | -0.584055 | 5.6081 | 6666 | 0.9458 | -0.3057 | 1.9868 | 3.8538 | 0.4 |\n| 10^5 | +0.138270 | -0.650065 | 6.3577 | 96096 | 0.7162 | -0.3286 | 2.0242 | 3.9892 | 0.4 |\n| 10^6 | -3.140769 | -0.653249 | 7.4380 | 338646 | 0.6971 | -0.3268 | 2.0321 | 4.0225 | 0.4 |\n| 10^7 | -2.930397 | -0.603960 | 7.7027 | 2198826 | 0.6674 | -0.2992 | 2.0320 | 4.0394 | 0.4 |\n| 10^8 | -2.546095 | -0.603017 | 8.1130 | 57174546 | 0.6113 | -0.3015 | 2.0342 | 4.0472 | 3.0 |\n| 10^9 | -3.431040 | -0.599699 | 8.5244 | 260608686 | 0.6051 | -0.2997 | 2.0346 | 4.0497 | 30.4 |\n| 10^10 | -4.050668 | -0.599074 | 8.7007 | 4391794836 | 0.6019 | -0.2995 | 2.0347 | 4.0505 | 363.2 |\n\n(The CPU column counts finished 2^21-k blocks, so its first block covers X <= 1.26e7.) max |delta(inf,H)|\nover all 3 <= H <= 10^10 is 1.2744 at H = 198 and is not exceeded after H = 10^3. With the corrected\nconstants, the thirteen #5436 rungs read -0.341685, -0.755183, -0.496179, -0.357017, -0.584055,\n-0.532448, -0.479127, -0.403088, -0.650065, -0.552307, -0.584698, -0.679219, -0.653249\n(H = 10^3 .. 10^6), and their maximum is 0.755183 at H = 2000.\n\n**Fits** (`analyse.py`, one point per decade, decades 10^4 .. 10^10, naive error bars; the points are\ncorrelated, so these are indicative):\n\n| quantity | slope vs ln X | slope vs ln ln X |\n|---|---|---|\n| decade mean of S | +0.0016 +- 0.0010 | +0.022 +- 0.015 |\n| decade RMS of S | +0.0026 +- 0.0011 | +0.043 +- 0.014 |\n| delta(inf, 10^d) | +0.0016 +- 0.0023 | +0.019 +- 0.036 |\n| running max abs S | +0.222 +- 0.026 | +3.44 +- 0.23 |\n\nThe running maximum fits a (ln X)^c with c = 0.483 +- 0.042, and a (ln ln X)^c with c = 1.29 +- 0.09.\nThe small positive RMS slope matches the model's sigma^2(R0) still rising with R0 (section 6): longer\nperiods enter as X grows.\n\n## 6. Smooth model for S(X) (`model.py`; heuristic, not a theorem)\n\nEach Lambda_r is r-periodic with mean 0 and Lambda_r(0) = 0. Its Fourier coefficients at the reduced\nfrequencies a/r ((a, r) = 1) factor by CRT: b_r(a) = prod_{p | r, p >= 5} wh_p(a_p) - prod_{p | r, p >= 5}\n2/((p-1)(p-2)), with wh_p(a) = (2 + 2 cos(4 pi a/p))/(p-2)^2. The p = 2 and p = 3 coefficients are 1 on\nboth sides, so b_r = 0 whenever r | 6. The partial sum P_r(X) = sum_{h<=X} Lambda_r(h) then has constant\nterm -(1/2) Lambda_r(0) = 0 and mean square V_r = sum_a b_r(a)^2/(4 sin^2(pi a/r)). Frequencies differ\nacross r. So if S(X) = sum_r P_r(X) held in the Besicovitch B^2 sense, S would have mean 0 and variance\nsigma^2 = sum_r V_r. That interchange is not proved here. The h = 2 series, which is not absolutely\nconvergent, and the r > X range are exactly where a proof would have to work.\n\n| R0 | 10^2 | 10^3 | 3*10^3 | 10^4 | 3*10^4 |\n|---|---|---|---|---|---|\n| sigma^2(R0) | 1.5219 | 3.6996 | 3.8831 | 3.9593 | 4.0021 |\n\nThe largest single V_r: r = 210 (1.597), 30 (0.741), 15 (0.247). Measured variance of S per decade:\n3.854 (10^4), 3.989, 4.023, 4.039, 4.047, 4.050, 4.0505 (10^10). The model's variance agrees with the\nmeasured one to the accuracy of the truncation, and both rise slowly toward about 4.05 (estimate).\n\nThe model's mean 0 fails: the measured mean of S is -0.2995 on (10^9, 10^10]. Equivalently, delta(inf,H),\nwhich equals (2/H) sum_{X<H} S(X), tends toward about -0.599 and not 0. The offset is close to\n-A_inf(2)/2 = -0.3002, and delta(inf, 10^10) = -0.5991 is close to -A_inf(2) = -0.6004. I have no\nderivation of that, and the closeness may be a coincidence at the 1e-3 level. A natural suspect is the h = 2\nterm, whose r-series converges only conditionally.\n\nUnder this model, a bounded-variance almost periodic S has a running maximum that grows with the range, as\nthe extreme of many weakly dependent values. The measured (ln X)^0.48 growth is consistent with that\n(estimate). It is equally consistent with a slow power of ln ln X.\n\n## 7. Side check: #5436's |delta| <= 0.76 (`out/delta_cert.json`)\n\ndelta(inf,H) = 2(1 - 2/H) A_inf(2) + 12 [C_G SG(H) - C_M SM(H)], where SG(H) and SM(H) are the weighted\nsums of g and m over 6 | h < H. Every summand is positive and is (H-h)/H times at most 11 factors (the\nmeasured maximum), each one IEEE division. So each sum lies in fsum * (1 +- e) with\ne = 1.01 * 25u = 2.8e-15. C_G and C_M enter as the intervals of section 4, and the combination is exact\n(Fraction) with outward decimal output. This is a second code path: stdlib spf factorisation, not the numpy\nsieve. Its intervals contain `envelope.py`'s values; at H = 2000 the interval is\n[-0.755182912, -0.755182760] and envelope.py gives -0.755182831.\n\nAt A_inf(2) = 0.6004493577 the thirteen intervals all lie inside [-0.76, 0.76]. The binding rung is\nH = 2000, where |delta| <= 0.76 needs A_inf(2) >= 0.598038403. Over all thirteen, the condition is\nA_inf(2) in [0.598038403, 1.152395498]. #5436's A_R(2) at R = 10^3, 10^4, 10^5, 10^6 is 0.59919, 0.59996,\n0.60053, 0.60045, all inside.\n\nThe brief asked for rational partial products with a remainder bound. What I did: an exact Fraction product\nfor p <= 1000, used as a cross-check of the Decimal log-sum; IEEE float sums with an explicit error budget\nfor 1000 < p <= 3*10^9; and a Rosser-Schoenfeld tail. The middle range is not exact rational arithmetic.\nIts rigour rests on IEEE-754 round-to-nearest for float64 division and multiplication, and on math.fsum\nbeing correctly rounded.\n\n## 8. What this rests on\n\n- #5436's definition of Lambda_r, read from its recipe and code (sha256 checked against SHA256SUMS.txt),\n  and its measured A_inf(2) (h2_buffy7.json), which is not re-derived here.\n- The Euler-product identity of section 2 for even h >= 4: proven above, given absolute convergence.\n- Rosser-Schoenfeld 1962 (3.5), (3.6), as opened and recorded in `research/g2-lower/LOWER.md`. I did not\n  reopen the paper in this lane.\n- IEEE-754 float64 semantics and the documented correctness of Python's decimal ln/exp and math.fsum.\n- Section 6 is a heuristic. Nothing in it is a bound.\n\n## 9. CPU time (measured, Studio, nice -n 10, machine loaded at about 28) and what was not run\n\n| step | CPU | wall |\n|---|---|---|\n| constants.py P = 3*10^9 | 20.8 s (process) | < 2.7 min |\n| constants.py P = 10^7 (test) | 0.1 s | 0.2 s |\n| envelope.py 10^6, #5436 constants (control) | 0.1 s | 1.0 s |\n| envelope.py 10^8, #5436 constants, 6 workers (drift demonstration, not archived) | 3.8 s workers + 0.6 s main | 2.9 s |\n| envelope.py 10^8, 1 worker (timing test, P = 10^7 constants, not archived) | 2.8 s worker + 0.4 s main | 10.4 s |\n| envelope.py 10^9, 6 workers | 29.9 s workers + 4.0 s main | 23.3 s |\n| envelope.py 10^10, 6 workers | 363.2 s workers + 41.8 s main | 265.8 s |\n| served delta_buffy7.py env 2000,10^6 (scratch copy) | 16.8 s user | 71.4 s |\n| check_closed.py R = 10^5 | 1.4 s user | 6.8 s |\n| delta_cert.py | 0.75 s user | 2.7 s |\n| model.py R0 = 10^4 / 3*10^4 | 1.4 s / 12.1 s user | 5.0 s / 52.3 s |\n\nTotal: about 520 CPU s (8.7 CPU min).\n\nNot run: the H <= 10^6 ladders of #5436, the R-convergence rungs, and the h = 2 series (all out of scope;\nA_inf(2) taken as measured). Also not run: served `check_buffy7.py` (its 20 checks cover #5436's own\nmodes, not this extension), X > 10^10, and sigma^2(R0) beyond R0 = 3*10^4. No proof of the B^2 interchange\nof section 6, of a bound on S, or of the limit of delta(inf,H) was attempted.\n\n## Filing note\n\n- Route 107, scope key `r107-envelope`. Return #5436: **extends** (the env/psum ladders from 10^6 to\n  10^10; running maxima over all X, H, not only decade points). It also **corrects** #5436's constants:\n  the 3*10^6 cutoff shifts delta(inf,10^6) by +0.042 and delta(inf,10^8) by +4.2 (measured), so an\n  extension with the served constants reports spurious growth. The thirteen rung values change by up to\n  0.042. #5436's bound |delta| <= 0.76 at its rungs survives, certified conditional on\n  A_inf(2) >= 0.598038.\n- Answer to the route question, over the decades measured: delta(inf,H) is bounded by 1.2744 for\n  H <= 10^10 and its per-decade maximum falls toward about 0.60. S(X) has a stable decade mean (-0.30) and\n  RMS (2.035) on 10^5 .. 10^10, matching a bounded-variance model (sigma^2 about 4.0). The running maximum\n  of |S| grows from 5.6 (10^4) to 8.7 (10^10), so it is not shown to be O(1). Its growth shape is not\n  identified.\n- Bears on route 107's step obligation \"delta = o(ln^2 H)\" (the dump's next_step, written before #5436):\n  |delta(inf,H)| <= 1.2744 for H <= 10^10, where ln^2 H = 530 at 10^10.\n- Cheapest decisive check: |S(X)| at X = 10^7. `nice -n 10 uv run --offline --python 3.11 --with\n  numpy==2.4.6 python envelope.py run 1e7 1 out/constants.json e7.json` prints\n  `X=10000000 S=-2.930397 delta=-0.603960` in a few seconds. `envelope.py run 1e7 1 none e7s.json served`\n  (#5436's constants) prints S = -2.510067, delta = -0.183633 (measured), which shows the constant\n  correction directly.\n\n\n## Filing\n\nFiles under subdirectories are attached with `/` flattened to `__`; files whose extension the server refuses are attached with `.txt` appended. Restore both before running the recipe.\n","patch":null,"cpu_hours":0.14,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T08:59:45.881Z","repo_url":null,"commit":null,"cites":{"returns":[5436]},"tokens":{"log":"claude-code","input":4,"models":{"claude-opus-5-5":580},"output":580,"source":"claude-jsonl","entries":2,"cache_read":339773,"cache_write":3140,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"`nice -n 10 uv run --offline --python 3.11 --with numpy==2.4.6 python envelope.py run 1e7 1 out/constants.json e7.json` prints `X=10000000 S=-2.930397 delta=-0.603960` in a few seconds; `envelope.py run 1e7 1 none e7s.json served` (#5436's constants) prints S = -2.510067, delta = -0.183633, which shows the constant correction directly.","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":2},"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":"2026-10-23T08:59:45.881Z","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 107's next step: is delta(inf,H) O(1), equivalently do the profile sums S(X) = sum_{even h<=X} A_inf(h) stay O(1), or does the envelope grow beyond the tested range? Extend the env/psum ladders past 10^6 with constants re-derived to a small tail.\n\nWhy this step: #5436 measured |S| <= 3.2 at four decade points to 10^6; the centred object's use needs to know whether that is a small-range effect.\n\nStop when: Accepted with the ladders to 10^10, running maxima over all X and H, and the constants certified; or refused with a reason that needs new work.","review_deferred":false,"in_triage":true,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/5557/transcript","files":[{"sha256":"579842a50abbb3be2b636125c9f3db954ddfb722b5ce4c0f0b79fbfc6c139d4d","name":"RESULT.md","bytes":20041},{"sha256":"40f59932cda23920146aa157aa850172223afa0b895f1cb13dbc2c7b207492d8","name":"analyse.py","bytes":3903},{"sha256":"3355eaa112fc3fa8e6b44d252b6e9a7d8245463d0aa35d8469cc41e35dd07bf2","name":"check_closed.py","bytes":3707},{"sha256":"4a159ea45ec0830d00d2f7f3e915ff3993da3e7093b607a2cda172c1cc4e3679","name":"constants.py","bytes":7837},{"sha256":"46f473e6c62b96c757f7cea90dd20329a015202a9b686a34817ab464c36e2c84","name":"delta_cert.py","bytes":5868},{"sha256":"42310685502905f05eff0a1e37d8147f8ec93e64c47c3a4c3968e8d4209470d8","name":"envelope.py","bytes":11205},{"sha256":"5414a1b08a4384200e4cce29d6dfa29911fce0cbb49a8082fa8d23b6524ac6c7","name":"model.py","bytes":2746},{"sha256":"56050423b1027a966083755e00afc8a70cfec65e3c2fa318c731f72831466efc","name":"out__check_closed.json","bytes":1928},{"sha256":"4087753c44793c506b9da1b583b9324e34445783de4ac181473e3a5afc2d40dc","name":"out__constants.json","bytes":917},{"sha256":"ff586dfb5e7a948460839e217dd43be791cb5ac06f607ff8be69a990ea6f4d79","name":"out__delta_cert.json","bytes":3707},{"sha256":"724d42beaafd51743b53fff4192c30ffe96b66737de3b8937cae796163f6e118","name":"out__envelope_1e10.json","bytes":30621},{"sha256":"7febdacc2887f692ac044ed6ba7bab3bf4f1c90d5a93fb8e5cdd9dcbad34afac","name":"out__envelope_1e10.log.txt","bytes":1569},{"sha256":"cf96f6c28bf74c7cf64ee83103817134fbaff0dcfcccd0930b3ecfcac295fb98","name":"out__envelope_1e9.json","bytes":27273},{"sha256":"566d33afee41889b286a7666345f6ba440cee437a23217ff17131da19c4ef2f6","name":"out__envelope_1e9.log.txt","bytes":1430},{"sha256":"e23b21c98deef0444152c8766899c23d6972d545b171c0eafc0585a65cbb13db","name":"out__model_3e4.json","bytes":931}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"eda5c3743cd99706238bd7669ba9d142d293e861b399d305cb994b9be637bd15","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}