{"id":5530,"job_id":11204,"problem_id":1,"lane_id":null,"type":"measure","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Route 356: a second implementation reproduces #5446's x = 29 rows j = 11..14 to 4.3e-9 in r; certified E[g] grid; short-window Delta 0.0059, 0.0024, 0.0053, 0.0079 confirmed, not monotone\n\nLane `route-356-second`, 2026-10-23. Every number below is measured (finite exact computations, outputs in this directory) unless it is labelled otherwise.\n\n## Result\n\nThe four shortest x = 29 true-ladder rows of #5446 (j = 11..14, L = 3,159,029 to 394,878) reproduce. A second implementation, written from the definitions before `uprofile.c` was opened, agrees with #5446's r to within 4.3e-9 on these rows, which is inside the rounding of the printed 8 decimals. Over the whole x = 29 ladder j = 0..14 the largest difference is 4.8e-9. Two algorithms inside the lane agree with each other to 7.8e-14 in r on j = 11..14: segmented marking and smallest-prime-factor factoring in double-double. A third evaluation of j = 14 in 50-digit decimal agrees to 1e-15. The model side E[g](y, L) has a rigorous enclosure, in 128-bit integer fixed point with floor and ceiling bounds, of width at most 2.2e-18 at x = 29 (j = 0..3 and 11..14) and at x = 31 (j = 6..8). #5471's E[g] values lie within 4.7e-11 of it, so the model grid is certified to 1e-9 at the named points (and well below that). With both sides from this lane, Δ(u) = ln y (r_true − E[g]) at j = 11..14 is 0.005854, 0.002403, 0.005272, 0.007942. These are #5471's values to 1e-6. **Verdict: the second of the brief's two outcomes in its stated form.** The rows reproduce to 1e-7, and Δ stays above 0.006 at the shortest window (0.0079 at u = 1.14). The short-window Δ values are therefore properties of the exact finite-y objects at these four L. They are not an artefact of #5446's code or of #5471's model code. One qualification: the four values are not a smooth rise. They go 0.0059, 0.0024, 0.0053, 0.0079 after 0.0040 at j = 10. This lane did not test whether they depend on u, or on the arithmetic of the particular L = ⌊W/2^j⌋. \"A genuine short-window u-shape\" is the brief's label for this outcome, and four non-monotone points do not establish a shape. The cheapest test of that is in the Filing note.\n\n## Definitions (what the code implements)\n\nSources for the definitions: finding 82649 on `paper/variance-note.md` §7 Corollary 3 (sah-dumps 2026-10-21 `findings.jsonl`), which gives the comb membership r mod 30 in {11, 17} and the modulus M = ∏_{p≤y} p; the #5446 report (window ladder L = ⌊W/2^j⌋, y the largest prime ≤ √W, W = x#); the #5471 report (the decoupled model). The variance note itself is not in the dumps (`document_versions.jsonl` has its metadata only) and was not opened by this lane.\n\nTrue side. n is in the comb when n mod 30 ∈ {11, 17} and p ∤ n(n+2) for 7 ≤ p ≤ y. The window start m is uniform modulo M, and N_L(m) = #{comb n ∈ [m, m+L)}. By CRT the events at different primes are independent, which gives the pair correlation J(d) = ρ₃₀(d)/30 · ∏_{7≤p≤y} ρ_p(d)/p, with ρ₃₀(d) = #{a ∈ {11,17} : a + d mod 30 ∈ {11,17}} (2 at d ≡ 0, 1 at d ≡ ±6, 0 elsewhere; the code computes it by brute force) and ρ_p(d) = p − #{0, −2, −d, −d−2 mod p} (p − 2 if p | d, p − 3 if d ≡ ±2, p − 4 otherwise). Then δ = J(0) = (2/30) ∏(p−2)/p and Var N_L = Σ_{|d|<L} (L − |d|)(J(d) − δ²). The code writes J(d)/δ² = c(d) K h(d) with c(d) = 30 ρ₃₀(d)/4, K = ∏ p(p−4)/(p−2)², h(d) = ∏_{p|d}(p−2)/(p−4) ∏_{p|d±2}(p−3)/(p−4), so Var = δL + δ²(2KA − L²) with A = Σ_{0<d<L} (L−d) c(d) h(d).\n\nModel side, as #5471's report states it: π₂ = π₃ = 1, π₅ = 3/4, π_p = 2/(p−1) for 7 ≤ p ≤ y, independent; n is the product of the chosen primes; g(n) = {L/n}(1 − {L/n}) n/L. Then Pr[n] = C ∏_{p|n, p≥5} ρ_p with C = (1/4) ∏_{p≥7} (p−3)/(p−1), ρ₅ = 3, ρ_p = 2/(p−3). The lane re-proved E[1/n] = δ: the factor at p ≥ 7 is 1 − π_p + π_p/p = (p−2)/p, and at p = 5 it is 2/5. For n > L, g(n) = 1 − L/n. Hence E[g] = B + (1 − S₀) − L(δ − S₁) with S₀ = Σ_{n≤L} Pr[n], S₁ = Σ_{n≤L} Pr[n]/n, B = Σ_{n≤L} Pr[n] g(n).\n\n## Order of work and when the served code was opened\n\n1. Definitions read from the #5446 and #5471 reports and finding 82649. `true_ladder.c`, `dd.h`, `defcheck.py` written.\n2. Definition-level controls (these check the derivation from the definition to the formula, not only the arithmetic): `defcheck.py` enumerates every window start m mod M at y = 13 (M = 30,030) and y = 17 (M = 510,510) and computes Var N_L exactly as a Fraction. `true_ladder direct` matches it to every printed digit (10 decimals of Var) at L = W/2^j, j = 0..5 (y = 13) and j = 1..3 (y = 17). `true_ladder seg` matches to 4e-10 in Var.\n3. The four rows j = 11..14 at x = 29, by both paths (seconds).\n4. The diagonal control: x = 29, j = 0, and the rest of the ladder j = 1..10, then x = 23 j = 0. The brief asked for the diagonal control first. In fact it ran about two minutes after step 3. Steps 2 and 3 were the first controls. The order does not change any number.\n5. `model_sieve.c`, `modelcheck.py` written and run. `uprofile.c` was **first opened at 2026-10-23T08:00:58Z**, after steps 1 to 5 had produced the x = 29 rows j = 0..14 and the model rows x = 29 j = 11..14 and x = 31 j = 7, 8. `uprofile_model.c` was **first opened at 2026-10-23T08:01:28Z**, after the model code was written and run. Both files were checked against `SHA256SUMS.txt` (`44c5bcb7…`, `4ca18c61…`); neither was built or run.\n6. A final pass, `run_all.sh`, rebuilt both programs from the committed source and regenerated every `.out` file here in one sequential process (timings below).\n\nWhat reading the served code showed. `uprofile.c` implements the same formula with the same ρ₃₀, ρ_p, y and L conventions. The lane's segmented path uses the same algorithm as `uprofile.c`: per prime, the d ≡ 0, ±2 positions inside the three contributing classes mod 30, stepping p in the class index. So that path is a second piece of code, not a second algorithm. The direct path (factoring d, d ± 2 with a smallest-prime-factor table, h(d) in double-double) and the 50-digit decimal check (`truecheck_decimal.py`, J(d) as the full product with no K/h split) use different algorithms. Arithmetic also differs: `uprofile.c` uses x86 long double (64-bit mantissa) and the lane uses double-double (about 106 bits). Per the project's rule, separate code checks arithmetic. The derivation of the formula from the definition is checked by step 2 alone. The definition itself (comb {11, 17}, uniform m mod M, y, L) comes from the same sources #5446 used, so it is not independently derived. `uprofile_model.c` enumerates conductors by depth-first descent in long double with Neumaier sums. `model_sieve.c` uses a segmented sieve over m = n/6 and exact integer interval arithmetic. These are different algorithms and different arithmetic.\n\n## Results\n\n### True side, x = 29 (y = 80,429, ln y = 11.295130087)\n\nFull table with Var columns in `compare.out`; outputs `true_x29_seg_j0_7.out`, `true_x29_seg_j8_14.out`, `true_x29_direct_j11_14.out`, `truecheck_decimal_x29_j14.out`.\n\n| j | L | r, this lane | r, #5446 (8 dp) | lane − #5446 |\n|---|---|---|---|---|\n| 0 | 6,469,693,230 | 0.377418019306 | 0.37741802 | −6.9e-10 |\n| 11 | 3,159,029 | 0.650027129585 | 0.65002713 | −4.1e-10 |\n| 12 | 1,579,514 | 0.675024805688 | 0.67502481 | −4.3e-9 |\n| 13 | 789,757 | 0.700734949166 | 0.70073495 | −8.3e-10 |\n| 14 | 394,878 | 0.726022500765 | 0.72602250 | +7.6e-10 |\n\n- Diagonal control (j = 0): Var = 5,307,862.62359, against #5446's 5,307,862.6235 and the variance note's 5,307,862.63 ± 1.4 (as #5446 quotes it). x = 23 diagonal: Var = 243,740.37291, against 243,740.3729 (#5446) and 243,740.37 ± 3.1e-3 (note, as quoted).\n- j = 1..10 also agree with #5446 to within 4.8e-9 in r. The largest Var difference is at j = 1: 2,813,444.69596 against 2,813,444.6958, a difference of 1.6e-4, or 6e-11 relative. The lane's segmented path holds h(d) in double, and a crude a priori worst-case bound for its rounding at L ≈ 3e9 to 6e9 is of order 1 in Var (estimate, not a certificate). The difference is far below that bound and far below 1e-7 in r. The double-double direct path cannot run at j = 1 because its factor table would need L words. So this lane reports j = 0..10 to about 1e-9 in r, not to the last printed Var digit.\n- j = 11..14: direct − seg ≤ 7.8e-14 in r; the decimal check at j = 14 gives r = 0.726022500764775112 against the direct path's 0.726022500764775.\n\n### Model side: certified enclosures\n\nOutputs `model_x29_j0.out`, `model_x29_j1_3.out`, `model_x29_j11_14.out`, `model_x31_j6_8.out`. Each line gives E[g] lower and upper bounds, both rigorous: integer fixed point at scale 2^118, every term rounded down for the lower bound and down + 1 for the upper bound. δ and C use lo − ⌈2lo/q⌉ and hi − ⌊2hi/q⌋ per factor.\n\n| x | j | L | E[g] (enclosure width) | #5471 − lane |\n|---|---|---|---|---|\n| 29 | 1 | 3,234,846,615 | 0.399818325591469 (6.1e-19) | −2.3e-11 |\n| 29 | 2 | 1,617,423,307 | 0.423119843339822 (1.9e-19) | −1.2e-11 |\n| 29 | 3 | 808,711,653 | 0.446978680652715 (7.1e-20) | −5.7e-12 |\n| 31 | 6 | 3,133,757,658 | 0.512027981376567 (7.8e-19) | +1.1e-11 |\n| 31 | 7 | 1,566,878,829 | 0.533851207690880 (2.2e-19) | +6.1e-12 |\n| 31 | 8 | 783,439,414 | 0.555856699335469 (7.6e-20) | +2.5e-12 |\n\nAlso run: x = 29 j = 0 (0.377144439037523, #5471 0.377144438991, difference 4.7e-11) and j = 11..14 (#5471 within 4.7e-13). #5471's values sit outside the lane's 1e-18 enclosures by 1e-13 to 5e-11. That is the size of the float floor #5471 itself names, and it is far inside 1e-9.\n\nModel-side controls: the conductor count at x = 13 is 548 and at x = 29 j = 0 is 107,819,335, both as #5471 reports. The bands at x = 13 (P_> 0.350725586, R 0.066478692, B 0.012344728, E[g] 0.296591623) match the note's values as #5471 quotes them to every printed digit. At x = 23 the lane gets P_> 0.3944369261, R 0.0372348166, B 0.0066403128, E[g] 0.3638424223, against the quoted 0.394436926, 0.037234816, 0.006640313, 0.363842423: R and E[g] differ in the ninth decimal (5.8e-10 and 7.7e-10). That is the size of the float floor #5471 attributes to the note's table. The lane's values are enclosures of width 3e-20. At y = 29 (x = 17, j = 0..3), `modelcheck.py` sums over all 256 prime subsets in exact Fractions. It uses no band split and no E[1/n] identity. Its 24-digit values lie inside the C enclosures at all four windows (`modelcheck_y29.out`, `model_y29.out`).\n\nOne loose end, quoted and not opened by this lane. #5471 says the variance note's \"BigInt-exact witness\" for x = 29 j = 0 is 0.377144535, which is 9.6e-8 above this lane's certified 0.377144439038. This lane did not open that witness and does not know its convention. It does not touch route 356, because Δ at u = 2 uses the same E[g] on both readings.\n\n### Δ(u) at x = 29, both sides from this lane\n\n| j | u | r_true | E[g] | Δ = ln y (r_true − E[g]) |\n|---|---|---|---|---|\n| 0 | 2.000012 | 0.377418019306 | 0.377144439038 | 0.003090 |\n| 1 | 1.938645 | 0.400102564771 | 0.399818325591 | 0.003211 |\n| 2 | 1.877278 | 0.423416136237 | 0.423119843340 | 0.003347 |\n| 3 | 1.815911 | 0.447303599755 | 0.446978680653 | 0.003670 |\n| 11 | 1.324976 | 0.650027129585 | 0.649508819797 | 0.005854 |\n| 12 | 1.263609 | 0.675024805688 | 0.674812025470 | 0.002403 |\n| 13 | 1.202242 | 0.700734949166 | 0.700268207824 | 0.005272 |\n| 14 | 1.140875 | 0.726022500765 | 0.725319364354 | 0.007942 |\n\nEvery value equals #5471's Δ to 1e-6. For j = 4..10 this lane has r_true but did not compute E[g]. #5471's E[g] at j = 10 with this lane's r gives Δ = 0.003985 (one model value quoted, not certified here).\n\n## Reading, and what it does not say\n\n- #5471 read the short-window wobble as a probable artefact of the quoted r_true rows. That reading is refuted at these four L. The rows reproduce to 1e-9, so the wobble belongs to the exact true variance at these window lengths, and the model values beneath it are smooth and certified.\n- What is not shown: whether Δ at u < 1.4 depends on u at all. The four points are not monotone (0.0059, 0.0024, 0.0053, 0.0079), and nothing here separates a u-dependence from a dependence on the arithmetic of the specific L. For example, L mod 30 is 29, 14, 7, 18 at j = 11..14, and both the comb and the model's conductors are tied to 30 | 6·5. That hypothesis was not tested. Δ was not computed at any window outside the named ladder, so \"no smooth u-shape\" and \"a u-shape\" are both unestablished.\n- Nothing here bears on u ≥ 1.4 beyond j = 0..3, where Δ is 0.0031 to 0.0037 as #5471 had it.\n\n## CPU time (measured, `timings.txt`)\n\nOne process at a time, `nice -n 10`, on the shared Studio. \"user\" is CPU time and \"real\" is wall time under load.\n\n| step | user s | real s |\n|---|---|---|\n| definition controls y = 13, 17 (`defcheck.py`, both C paths) | 0.44 | 1.6 |\n| true x = 23 diagonal | 0.20 | 0.64 |\n| true x = 29 j = 0..7 (seg) | 13.02 | 53.9 |\n| true x = 29 j = 8..14 (seg) | 0.05 | 0.09 |\n| true x = 29 j = 11..14 (direct, double-double) | 0.09 | 0.26 |\n| decimal check x = 29 j = 14 | 0.26 | 1.03 |\n| model exact-subset check y = 29 and its C run | 0.07 | 0.26 |\n| model x = 13, x = 23 controls | 0.35 | 1.34 |\n| model x = 29 j = 11..14 | 0.01 | 0.03 |\n| model x = 29 j = 1..3 | 9.74 | 40.5 |\n| model x = 31 j = 6..8 | 11.88 | 54.6 |\n| model x = 29 j = 0 | 16.31 | 63.5 |\n| **total, final pass** | **52.4** | **217.8** |\n\nEarlier development runs of the same steps (before the final pass) cost about the same again, under 2 CPU-minutes. Total lane compute is under 0.05 CPU-hours (measured). The brief estimated 2 hours.\n\n## Not run\n\n- x = 31 true rows: none (out of scope; the x = 31 j ≤ 6 single-instrument caveat from review 1901 stands).\n- Model E[g] at x = 29 j = 4..10 and x = 31 j = 0..5, 9..12: not computed (not named in the brief).\n- True rows by the direct double-double path for j ≤ 10: not run (its factor table needs about 4L bytes).\n- Δ at windows off the ladder (L ± small offsets): not run, because the brief puts new ladder points out of scope. This is the next check.\n- `check_uprofile.py`, `uprofile.c`, `uprofile_model.c`: read for comparison only; not built or run.\n\n## Files\n\n- `true_ladder.c`, `dd.h`: true side (modes `seg`, `direct`).\n- `model_sieve.c`: model side with rigorous enclosures.\n- `defcheck.py`, `modelcheck.py`, `truecheck_decimal.py`: brute-force and exact checks (stdlib Python, run with `-I`).\n- `compare.py`, `compare.out`: the comparison tables against #5446 and #5471.\n- `run_all.sh`, `timings.txt`, `*.out`: the archived final pass.\n\nExact command: `sh research/uprofile-second/run_all.sh <build-dir>` (Apple clang 21.0.0, `-O2`, arm64 macOS, Python 3 stdlib), then `python3 -I research/uprofile-second/compare.py research/sah-state/2026-10-23/attachments/05446 research/sah-state/2026-10-23/attachments/05471 > research/uprofile-second/compare.out`.\n\n## What this rests on\n\n- The definitions above, taken from finding 82649 and the #5446/#5471 reports (the variance note itself was not opened, so any difference between its Theorem 2 and the formula here would not be visible to this lane). The formula's derivation from the definition is checked by full enumeration at y = 13 and 17.\n- The model as #5471's report states it (varE-limit-theorem.md §1 not opened by this lane).\n- Served files checked against `SHA256SUMS.txt`: #5446 `uprofile.c` 44c5bcb7…, `uprofile_x29.out` 5c7e0c98…, `check_uprofile.py` 4426ee6d…, `recipe.md` 0b0c60de…; #5471 `uprofile_model.c` 4ca18c61…, plus `model_x29.json` and `model_x31.json` (read by `compare.py`).\n\n## Filing note\n\n- Route: **356**. Scope key: **`r356-uprofile-second`**.\n- Links: **#5471 extends**. This lane supplies the second true-side implementation and the certified model grid that #5471's next step asks for, and settles #5471's reading of the j = 11..14 wobble: it is not an artefact of the quoted rows. **#5446 checks**: x = 29 j = 0..14 reproduced to within 4.8e-9 in r.\n- Outcome of the brief's either/or: rows reproduced to 1e-7; Δ(u = 1.14) = 0.0079 ≥ 0.006, so the second branch holds as worded. The values are not monotone, and their u-dependence is not established (see \"Reading\").\n- Cheapest decisive check: the **j = 14 row at x = 29**. Run `true_ladder direct 29 14 14` (0.01 s CPU) and expect r = 0.726022500764775 (#5446: 0.72602250). Then `model_sieve 29 14 14` (0.01 s) gives E[g] in [0.725319364354171791759642, 0.725319364354171791772345], so Δ = 11.295130087 × 0.000703136411 = 0.007942.\n- Suggested next route step (not run): compute Δ at L = 394,878 + k for k in −30..30 at x = 29. That is 61 windows of a few milliseconds each on both sides. If Δ swings by about 0.003 with k, the short-window values reflect the arithmetic of L, not a u-shape.\n\n","patch":null,"cpu_hours":0.05,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-23T08:12:39.987Z","repo_url":null,"commit":null,"cites":{"returns":[5471,5446]},"tokens":{"log":"claude-code","input":2,"models":{"claude-opus-5-5":172},"output":172,"source":"claude-jsonl","entries":1,"cache_read":193574,"cache_write":2424,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Cheapest decisive check, j = 14 at x = 29: `true_ladder direct 29 14 14` (0.01 s) gives r = 0.726022500764775 (#5446: 0.72602250); `model_sieve 29 14 14` gives E[g] in [0.725319364354171791759642, 0.725319364354171791772345], so Delta = 11.295130087 x 0.000703136411 = 0.007942. Full pass: run_all.sh, 52.4 CPU s.","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":1},"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":"2026-10-23T08:12:39.987Z","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 356's next step (#5471): reproduce the four shortest x = 29 true-ladder rows (j = 11..14) with a second implementation and a certified model grid, and decide whether the short-window Delta values are an artefact of the quoted rows or properties of the finite-y objects.\n\nWhy this step: #5471's short-window Delta values rested on #5446's single implementation and an uncertified model grid.\n\nStop when: The return is accepted with the rows reproduced by code written from the definitions, the model side enclosed rigorously, and Delta recomputed from both sides; 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