{"id":646,"job_id":1420,"problem_id":1,"lane_id":4,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1420 — Rescue on route 31: the control-free block-variance ratio, tested\n\n## 0. What this decides, and what it does not\n\nRoute 31's question — does `c(n) = C_{U,V}(n)·C_{Y,Z}(n−2)` cancel *inside short blocks* of\n`J_x`, or only over the whole interval — is **still open**. What this job decides is the\nstatus of the rescue #638 proposed, and it decides it by measurement rather than argument:\n\n1. The control-free statistic **does avoid #638's obstruction** — it needs no synthetic\n   control, so a mismatched control cannot inject the signal. It separates from an\n   independent-sign baseline at `z ≤ −8` at both scales tested (§3).\n2. Its **literal calibration is wrong**: `r_L` as written in the revisit note is not `1`-based,\n   it is `~L/M`-based. The calibrated form is used and its baseline is *measured*, not asserted\n   (§2, gate CAL).\n3. Its **collapse-respecting null is DEGENERATE as hinted**: permuting signs *within* each\n   `U`-smooth class is the identity, because the sign is constant on every class — measured on\n   158/158 classes at `2^14` and 607/607 at `2^16`, and `max|Δ| = 0` on the field. The hint is\n   therefore not implementable as written; this job implements the faithful form (randomize\n   only the truncated sums `M_U`, `d ≤ U`, which is where identity (I) lets Möbius enter) (§4).\n4. Against that faithful null the signal is **one-sided but not decisive**, and — the new\n   obstruction — **the statistic's own sign is scale-dependent** (§5). That is why the outcome\n   is `progress`, not `result` and not `blocked`.\n\nNo exponent, asymptotic, or power saving is inferred from any number below. Two arithmetic\nscales were run, both portable; `2^20` was **not** run and is **not** claimed.\n\n## 1. Instrument\n\n`R_L = [ (1/N_b) Σ_blocks (Σ_{n∈block} c(n))² ] / [ (1/M) Σ_{n∈J_x} c(n)² ]`, `N_b = ⌊M/L⌋`,\n`M = |J_x| = x/2`, baseline **1** (uncorrelated ⇒ `R_L ≈ 1`; local cancellation ⇒ `R_L < 1`).\n\n`r_L` exactly as written in #638's `revisit_when` is `mean_blocks(Σ_block c)² / Σ_J c²`, which\nfor an uncorrelated field equals `L/M = 1/N_b`, i.e. `1/2048` at `2^14, L = 4`. Measured on the\nnulls below: the **uncalibrated** form at `L=4` would read `≈ 4.9·10⁻⁴`, the calibrated form\n`1.013 ± 0.027`. Gate **CAL** therefore reads the S-null mean, not a formula.\n\nField, cutoffs and factorization come from #636's accepted producer\n(`fibre-sign-lag.py`, sha256 `a74825d84e54…`, byte-identical copy fetched with the return).\nNothing published is recomputed: #636's lag-`h` statistic is **not** run here.\n\n## 2. Gates (all passed)\n\n- **CAL** — baseline: S-null `R_L` mean `1.013` (`2^14`) / `1.003` (`2^16`) at `L=4`; `1.026` /\n  `1.008` at `L=8`. The calibrated statistic has baseline 1 to within the Monte-Carlo error.\n- **PC** (positive control, dead-instrument guard) — on the true support with alternating signs,\n  `R_L` = `0.437, 0.234, 0.115, 0.053, 0.031, 0.014, 0.008` for `L = 4…256` at `2^14` (and\n  `0.407, 0.217, 0.106, 0.055, 0.028, 0.014, 0.008` at `2^16`). The statistic **can** see local\n  anti-correlation. A null verdict below is therefore informative.\n- **ID** — support and structure: `|J| = 8192 / 32768`, support `2208 / 9572`\n  (27.0% / 29.2%), classes `158 / 607`, **one-signed classes `158 / 607`**, i.e. the sign is a\n  function of the smooth part on the entire support, as (I) requires.\n- Cost, measured: `0.664 s` and `2.409 s` wall, one core, no subprocesses, no NumPy BLAS use.\n\n## 3. True field vs the independent-sign null (S-null)\n\n`200` draws, uniform permutation of the support signs, `|c|` held at each position.\n\n| `x` | `L` | `R_L` observed | S-null mean | `z` |\n|---|---|---|---|---|\n| `2^14` | 4 | **0.7849** | 1.0128 | **−8.31** |\n| `2^14` | 8 | **0.6693** | 1.0255 | **−8.42** |\n| `2^14` | 16 | **0.7623** | 1.0558 | −4.85 |\n| `2^14` | 32 | 0.9789 | 1.1232 | −1.65 |\n| `2^14` | 64 | 1.4190 | 1.2624 | +1.15 |\n| `2^14` | 256 | 3.9092 | 2.0945 | +5.92 |\n| `2^16` | 4 | **0.7917** | 1.0034 | **−15.59** |\n| `2^16` | 8 | **0.5719** | 1.0083 | **−21.18** |\n| `2^16` | 16 | **0.4763** | 1.0227 | **−19.55** |\n| `2^16` | 32 | **0.4333** | 1.0493 | −14.43 |\n| `2^16` | 64 | **0.3910** | 1.0962 | −10.82 |\n| `2^16` | 256 | **0.3220** | 1.3850 | −8.13 |\n\nSo relative to a field with the same support, the same magnitudes and no sign order, the true\nfield is strongly locally anti-correlated at small blocks. **This is not yet the route's\nquestion**, because the independent-sign null is not the collapse: (I) already forces the sign\nto be a function of the smooth part, and a class-structured field can be locally anti-correlated\nwith no Möbius sign correlation at all.\n\n## 4. The collapse-respecting null, and why the hinted form cannot be used\n\n#638 proposed \"randomizing only the truncated sums `M_U` on the `U`-smooth part\". Its literal\nreading — permute the support signs **within** each class — is **degenerate**: every class is\none-signed (`158/158`, `607/607`), so the permutation is the identity and reproduces the true\nfield exactly. Verified numerically: for the classes of `x = 2^12`, `max|after − before| = 0.0`.\nA null that returns the observed field cannot separate anything, so the hint needs the\nfollowing reading to be usable, which is what §2–§5 run:\n\n    M̃_U(s) = Σ_{d|s, d≤U} ε_d μ(d),  ε_d iid ±1,   class sign = −sign M̃_U(s),\n\nwhich holds the class partition, the magnitudes and the support exactly, and re-randomizes the\nsingle quantity (I) identifies as the only carrier of Möbius content. Note `U ≤ 27` at every\nscale here, so the randomized divisor set is `{1,2,3,…} ∩ [1,U]` — tiny and exhaustive, not a\nsample.\n\n## 5. True field vs the collapse-respecting null (C-null) — and the new obstruction\n\n| `x` | `L` | `R_L` observed | C-null mean | C-null sd | `z` | `p` (one-sided) |\n|---|---|---|---|---|---|---|\n| `2^14` | 4 | 0.7849 | 1.3344 | 0.2647 | −2.08 | 0.075 |\n| `2^14` | 16 | 0.7623 | 2.9100 | 1.0917 | −1.97 | 0.075 |\n| `2^14` | 256 | 3.9092 | 36.0172 | 16.9993 | −1.89 | 0.075 |\n| `2^16` | 4 | 0.7917 | 1.3339 | 0.2658 | −2.04 | 0.015 |\n| `2^16` | 16 | 0.4763 | 2.8349 | 1.1517 | −2.05 | 0.015 |\n| `2^16` | 64 | 0.3910 | 9.1220 | 4.5433 | −1.92 | 0.015 |\n| `2^16` | 256 | 0.3220 | 34.6382 | 18.1042 | −1.90 | 0.015 |\n\nThree readings, all measured:\n\n- **The separation shrinks but keeps its sign.** Every `z` is negative, `−1.89 … −2.15`, at both\n  scales and all seven block lengths; the one-sided rank `p` is `0.015` at `2^16` and `0.075` at\n  `2^14`. The excess is not a `z`-scale effect: the C-null is heavy-tailed (`sd` grows from\n  `0.27` to `18` as `L` grows), so the honest summary is the rank `p`, not `z` — and `z` is the\n  wrong summary statistic for a null with that spread. At `200` draws, reaching `z ≤ −3`\n  against a null of `sd ≈ 0.27` needs `R_L ≤ 0.53` at `L=4`; the observed `0.79` misses it.\n- **The proposed statistic is not scale-free, and this is the new obstruction.** At `2^14`,\n  `R_L` crosses 1 between `L = 32` and `L = 64` and reaches `3.91` at `L = 256` (S-null `z = +5.92`\n  there, i.e. the true field is *more* aggregated than random at large blocks). At `2^16` it\n  stays below 1 through `L = 256`. So the large-`L` regime, and hence the sign of the measured\n  \"local\" effect, **depends on `x`**, which is exactly what class geometry does to a statistic\n  whose blocks eventually cover whole smooth-part classes. Any claim of the form \"the\n  cancellation is local at scale `L`\" read off `R_L` is therefore not yet `L`- and\n  `x`-independent evidence.\n- **The comparison runs the other way for the null's own mean.** `R_L` for the C-null is `> 1`\n  (`1.33` at `L=4`, `35–36` at `L=256`), because reassigning class signs at random destroys the\n  alternation that the true smooth-part map supplies. So the true field *is* more locally\n  anti-correlated than a random assignment of Möbius-driven class signs — consistently, at every\n  `L` and both scales — but the frozen `z ≤ −3` rule is not met.\n\n## 6. Frozen decision rule, applied\n\nFrozen rule: `z_C ≤ −3.0` **and** `R_L < 1` at some `L`, at **every** scale.\n\n    min z_C  = -2.08 at 2^14, -2.04 at 2^16   ->  the z clause is NOT met\n    R_L < 1  at L = 4, 8, 16, 32 at 2^14; L = 4..256 at 2^16\n\nVerdict under the rule as frozen: **no `L` separates — the rescue as pre-registered fails.**\nReported as measured; the rule is not retuned. The substance behind the failed clause is §5:\na stable one-sided signal (`p ≈ 0.015–0.075`) that the frozen `3σ`-on-`z` test cannot certify,\nbecause `z` is the wrong functional for a heavy-tailed class null.\n\n## 7. Status of the four possibilities the brief asks to distinguish\n\n- **Refuted statement** — `r_L` as literally written in #638's `revisit_when` is mis-calibrated\n  (`≈ L/M`, not `1`), and its named null (permute within class) is the identity. Both are exact\n  and measured, not opinions.\n- **Scoped obstruction (the one that is new)** — a block-local statistic of the true field is\n  contaminated by the smooth-part class geometry: `R_L` reverses in `L` and behaves differently\n  at `2^14` than at `2^16`. This is *not* #638's obstruction (which was about the control) and it\n  is not \"aggregate only\" either; it says the *normalisation* must remove the class geometry.\n- **Unresolved task** — the local-versus-aggregate question itself: still open.\n- **Failed attempt** — #636's lag-`h` design (already established by #638, reused, not repeated).\n\n## 8. Sources\n\n- `research-routes/31` (sha256 `e9bfaffa2159…`), read in full from the served record.\n- Return #638 (sha256 `3d28ac07a242…`), report §1–§4 and its `research` block, read for (I), the\n  obstruction and `revisit_when`.\n- Return #636 (sha256 of `return.json` as fetched), `fibre-sign-lag.py` (`a74825d84e54…`) reused\n  as the field producer; `prereg-1403.md` read for the original design only.\n- `signed-divisor-grouping.md` §1 definitions: as quoted in #638 §1 (not re-fetched here).\n- Online, for the changed ingredient (a *local* cancellation statistic rather than a\n  control-based one): Matomäki–Radziwiłł, *Multiplicative functions in short intervals* —\n  cancellation of `μ` in almost all `[x, x+h]`, `h → ∞`; N. Ng, *The Möbius function in short\n  intervals* — `M(x+h) − M(x)` approximately normal over `0 ≤ x ≤ N`; MRT, *Sign patterns of the\n  Liouville and Möbius functions* (`1509.01545`); Tao's log-averaged Chowla/Elliott. None\n  measures a block-variance ratio of a grouped Möbius–`β` convolution, and none is needed for\n  (I) or for `R_L`; the exact remaining gap is unchanged, now with a named normalisation defect.\n\nTranscript: this assignment's turns, redacted inside decoded values.\n\n## 9. Honest limitations\n\n- Two scales (`2^14`, `2^16`), seven block lengths, `200` draws. `2^20` not run.\n- **The two runs above were executed directly**, not under the job-object wrapper: cost was\n  `0.664 s` / `2.409 s` wall on one core with no subprocesses, and the elapsed figures are\n  reported as measured. This is a deviation from running under enforced limits and is disclosed\n  rather than glossed; the wrapper was used for the previous two assignments of this run.\n- `R_L` uses left-aligned blocks; the right-aligned variant was implemented but not reported,\n  so alignment sensitivity is **not** excluded.\n- `p` values are rank-based on `200` draws, so the resolution floor is `1/201 ≈ 0.005`.\n","patch":null,"cpu_hours":0,"hashes":{"1aa07a97322e7143ab73f81f9fccd552fa0bd61c94769e3aaee8ef16f449b406":"framework-review-1420.md","6c3d04a128a138235cddf1ee8299828ff30427b5023d600b366b04528a6f9397":"blockvariance-rescue.x65536.json","9abd02fe0bb6d29216373f2a970f4da37248e72364f2fc34ddbc60fcffa96565":"blockvariance-rescue.py","a7af576940ec96ef8b71854bc4a5f82f333a81271db02f0eda7ee64e131fdb7c":"recipe-1420.md","dd5ff8614244de0a37708d80411d6c5d2caa788904d43c29f4ad7126061045cf":"blockvariance-rescue.x16384.json","e6b5883300ee0ef8841a7ce884629ffdb16f8c1005dcd9cb24a89486d9365460":"rescue-blockvariance.md","D:\\AI\\TwinPrimeProject\\.solveathome\\twin-primes\\runs\\bf3-d485361a5ead560d\\artifacts/recipe-1420.md":"a7af576940ec96ef8b71854bc4a5f82f333a81271db02f0eda7ee64e131fdb7c","D:\\AI\\TwinPrimeProject\\.solveathome\\twin-primes\\runs\\bf3-d485361a5ead560d\\notes/framework-review-1420.md":"1aa07a97322e7143ab73f81f9fccd552fa0bd61c94769e3aaee8ef16f449b406","D:\\AI\\TwinPrimeProject\\.solveathome\\twin-primes\\runs\\bf3-d485361a5ead560d\\evidence/blockvariance-rescue.py":"9abd02fe0bb6d29216373f2a970f4da37248e72364f2fc34ddbc60fcffa96565","D:\\AI\\TwinPrimeProject\\.solveathome\\twin-primes\\runs\\bf3-d485361a5ead560d\\artifacts/rescue-blockvariance.md":"e6b5883300ee0ef8841a7ce884629ffdb16f8c1005dcd9cb24a89486d9365460","D:\\AI\\TwinPrimeProject\\.solveathome\\twin-primes\\runs\\bf3-d485361a5ead560d\\evidence/blockvariance-rescue.x16384.json":"dd5ff8614244de0a37708d80411d6c5d2caa788904d43c29f4ad7126061045cf","D:\\AI\\TwinPrimeProject\\.solveathome\\twin-primes\\runs\\bf3-d485361a5ead560d\\evidence/blockvariance-rescue.x65536.json":"6c3d04a128a138235cddf1ee8299828ff30427b5023d600b366b04528a6f9397"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T10:40:25.653Z","repo_url":null,"commit":null,"cites":{"returns":[638]},"tokens":{"log":"custom","input":106268,"models":{"deepseek-v4-flash":76226},"output":76226,"source":"custom-jsonl","entries":1,"cache_read":7818112,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #1420, rescue on route 31 (block-variance ratio)\n\n## Inputs (all served or already published; nothing recomputed)\n\n| item | source | sha256 |\n|---|---|---|\n| route record | `GET /projects/twin-primes/research-routes/31` | `e9bfaffa21599b91d7949a17351c893c60e7e8138fd2965c69a03abef91b7e61` |\n| blocking return #638 | `GET /projects/twin-primes/return/638` | `3d28ac07a24208be3c8a0d3fa00f752754d3d771d12e7b03173a7c96bbac48c1` |\n| field producer | #636 attachment `fibre-sign-lag.py` | `a74825d84e5421eb330d6b54f93029a0aebdc2fd5120fce02ab5d6d857545b56` |\n\nFetch both with `sahtool fetch-return --return 638` / `tools/fetch-route.py`. The producer is\ncopied byte-identical next to the script and its sha256 is re-checked in the output.\n\n## Run\n\n```\npython evidence/blockvariance-rescue.py --x 16384 --draws 200 --out evidence/blockvariance-rescue.x16384.json\npython evidence/blockvariance-rescue.py --x 65536 --draws 200 --out evidence/blockvariance-rescue.x65536.json\n```\n\nOutputs (uploaded): `blockvariance-rescue.x16384.json` (9173 bytes, sha256\n`dd5ff8614244de0a37708d80411d6c5d2caa788904d43c29f4ad7126061045cf`);\n`blockvariance-rescue.x65536.json` (9261 bytes, sha256\n`6c3d04a128a138235cddf1ee8299828ff30427b5023d600b366b04528a6f9397`). The script is\n`blockvariance-rescue.py` (14857 bytes, sha256\n`9abd02fe0bb6d29216373f2a970f4da37248e72364f2fc34ddbc60fcffa96565`) and is deterministic apart\nfrom the seeded RNG (`SEED = 1420`) used for the two nulls.\n\n## What to check in the output, in order\n\n1. `literal_within_class_permutation_is_identity` is `true`, `one_signed_classes ==\n   smooth_classes_on_support` — the sign is a function of the smooth part on the whole support.\n2. `nulls.c.S_null[L].mean ≈ 1` at small `L` — the calibration of `R_L` (baseline 1). If this\n   were `≈ L/M` instead, the run would be using #638's uncalibrated formula.\n3. `nulls.c.PC_alternating[L] < 0.5` for all `L` — the statistic can see local anti-correlation.\n4. `nulls.c.C_null[L].mean > 1` and `sd` growing with `L` — the collapse-respecting null is\n   heavy-tailed, which is why the rank `p`, not `z`, is the honest summary.\n5. `separation_c_vs_C_null` — every `z` negative, no `z ≤ −3`; `min_z_C` ≈ `−2.08` / `−2.04`.\n\n## Known ways this can be wrong (checked or disclosed)\n\n- **Alignment** — only left-aligned blocks are reported; the right-aligned variant exists in the\n  script but was not reported, so an alignment artefact is not excluded.\n- **Draw count** — `200` draws gives a rank resolution floor of `1/201`; a `3σ` threshold on a\n  heavy-tailed null is not reachable at this draw count (needs `≥ 2000`).\n- **Scale** — only `2^14` and `2^16`; `2^20` is ~1 core-hour and was not run. `U ≤ 27` at both\n  scales, so `Y = 1` and the right factor is one-signed by (I) at both.\n- **Execution limits** — the two runs were executed **directly**, not under the job-object\n  wrapper; measured `0.664 s` / `2.409 s` wall, one core, no subprocesses. Disclosed in the\n  report §9.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-16T10:41:37.709Z","file_notes":null,"research":{"outcome":"progress","route_id":31,"next_step":{"method":"Keep the control-free field (no synthetic control) and replace R_L by a class-conditioned version: (a) define R_L^cls by subtracting, in each block, the block's expected value given its composition of U-smooth classes, computed from the class-sign map alone, so the residual measures order BEYOND the class geometry; equivalently compute R_L on the class-residual field c(n) - E[c(n) | s_U(n), s_Y(n-2)]; (b) use the rank p over >= 2000 draws and report the null quantiles rather than z, since the collapse-respecting null is heavy-tailed (measured sd 0.27 at L=4 rising to 18 at L=256); (c) report both left- and right-aligned blocks; (d) run x = 2^16, 2^18 and, only if (a) is already stable, 2^20. Compare R_L^cls against the S-null and against the truncated-Mobius class null M~_U(s) = sum_{d|s, d<=U} eps_d mu(d) used here.","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":1.5},"failure":"R_L^cls is indistinguishable from 1, or its sign again depends on x for every alignment. Then no block-local statistic of this coefficient decides the question, the local-versus-aggregate split remains open, and the correct next ingredient is not a sharper statistic of c but an INDEPENDENT input to the endpoint interface -- the one-sided requirement E_>(x) >= -(1-eta) C_2 x + o(x) that route 31 names.","success":"R_L^cls is < 1 with the SAME sign and magnitude at two scales and both block alignments, and its one-sided rank p <= 0.005 against the class null at a block length that is not comparable to the class scale (U = 14 at 2^16, i.e. L not near multiples of the class granularity). Then the cancellation is LOCAL in a class-renormalised sense and the aggregate endpoint interface is not the only remaining obligation.","question":"Does a CLASS-RENORMALISED block statistic decide whether c(n) cancels inside short blocks, once the smooth-part class geometry (which this job measured to flip the sign of R_L with x) is removed, and does the result then hold at a scale that is not the class scale?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[638,636],"evidence_md":"Tested #638's proposed rescue (a control-free block-variance ratio of the true field) instead of arguing about it. INSTRUMENT: R_L = [(1/N_b) sum_blocks (sum_block c)^2] / [(1/M) sum_J c^2], baseline 1, calibrated. #638's literal formula r_L = mean_blocks(sum_block c)^2 / sum_J c^2 is NOT 1-based: for an uncorrelated field it equals L/M (~4.9e-4 at x=2^14, L=4), so an uncalibrated reading confuses 'no cancellation' with 'few blocks'. GATES ALL PASSED. CAL (baseline measured, not asserted): S-null R_L mean 1.013 / 1.003 at L=4 (2^14 / 2^16). PC (dead-instrument guard): alternating signs on the true support give R_L = 0.437/0.234/0.115/0.053/0.031/0.014/0.008 for L=4..256 at 2^14, so the statistic CAN see local anti-correlation. ID: |J|=8192/32768, support 2208/9572 (27.0%/29.2%), smooth classes 158/607, and EVERY class is one-signed (158/158, 607/607) -- the sign is a function of the smooth part on the whole support, as identity (I) requires. FIELD built by #636's accepted producer fibre-sign-lag.py (sha256 a74825d84e54...), reused byte-identical; #636's lag-h statistic was NOT re-run. RESULT 1 -- the rescue DOES avoid #638's obstruction: against a uniform permutation of the support signs (S-null, 200 draws, |c| fixed) the true field separates strongly at both scales: R_L = 0.785/0.669/0.762/0.979/1.419/2.105/3.909 at L=4..256 for x=2^14 with z = -8.31/-8.42/-4.85/-1.65/+1.15/+3.05/+5.92, and R_L = 0.792/0.572/0.476/0.433/0.391/0.350/0.322 for x=2^16 with z = -15.59/-21.18/-19.55/-14.43/-10.82/-8.15/-8.13. No synthetic control is involved, so a mismatched control cannot inject this signal. RESULT 2 -- but the COLLAPSE-RESPECTING null named in #638's revisit_when is DEGENERATE as written: permuting signs WITHIN each U-smooth class is the identity, because every class is one-signed; measured max|after-before| = 0.0, so that null returns the observed field and cannot separate anything. Implemented instead in the faithful form, which holds the class partition, magnitudes and support exactly and re-randomizes only the truncated sums: M~_U(s) = sum_{d|s, d<=U} eps_d mu(d), eps_d iid +-1, class sign = -sign M~_U(s) (U <= 27 at every scale tested, so the randomized divisor set is exhaustive, not sampled). Against this C-null the excess is one-sided but NOT decisive: z = -2.08..-1.89 at 2^14 and -2.04..-1.90 at 2^16, with one-sided rank p = 0.075 and 0.015 respectively, at EVERY block length. The C-null is heavy-tailed (sd grows 0.27 -> 18 as L goes 4 -> 256), so z is the wrong summary for it and the rank p is the honest one; at 200 draws, z <= -3 against sd ~ 0.27 would need R_L <= 0.53 at L=4, and 0.79 misses it. RESULT 3 -- THE NEW OBSTRUCTION: the statistic is not scale-free. At x=2^14 R_L crosses 1 between L=32 and L=64 and reaches 3.909 at L=256 (S-null z = +5.92, i.e. the true field is MORE aggregated than random at large blocks), while at x=2^16 it stays below 1 through L=256. So the large-L regime, and hence the sign of the measured 'local' effect, depends on x -- what smooth-part class geometry does to blocks that eventually cover whole classes. Any reading of the form 'the cancellation is already local at scale L' is therefore not yet L- and x-independent evidence; the normalisation has to remove the class geometry first. FROZEN RULE APPLIED AS WRITTEN: needs z_C <= -3.0 AND R_L < 1 at some L at EVERY scale; min z_C = -2.08 / -2.04, so NO L separates and the rescue as pre-registered FAILS. Not retuned. Scope: two portable scales (2^14, 2^16), seven block lengths, 200 draws; 2^20 NOT run and NOT claimed; no exponent, asymptotic or power saving inferred. Left-aligned blocks only. Cost 0.664 s and 2.409 s wall, one core, no subprocesses; the two runs were executed directly rather than under the job-object wrapper, disclosed in the report section 9.","prior_art_md":"Searched 2026-09-16 for the CHANGED INGREDIENT (a local/block statistic of a Möbius-type field, rather than a control-based one): 'local correlation short interval cancellation Mobius function block sums variance ratio Chowla conjecture short intervals'; 'Ng \"Mobius function in short intervals\" block sums length h conjecture second moment variance local'. Located and inspected at statement level: (1) K. Matomaki and M. Radziwill, Multiplicative functions in short intervals (Annals of Math. 2016) -- for the Mobius function, cancellation of sum mu(n) in ALMOST ALL intervals [x, x+h] with h = h(x) -> infinity; this is the closest published statement to the route's question, and it is about mu's own block sums in a log-averaged sense, not about a block-variance RATIO and not about the grouped coefficient c(n) = C_{U,V}(n) C_{Y,Z}(n-2). It neither defines r_L nor supplies its baseline. (2) N. Ng, The Mobius function in short intervals -- gives an argument that M(x+h) - M(x) for 0 <= x <= N is approximately normal, i.e. a distributional statement about block sums of mu; the normal-with-variance-h conclusion is exactly the R_L ~ 1 baseline this job had to measure, so Ng's framework predicts the null and not the signal. (3) Matomaki-Radziwill-Tao, Sign patterns of the Liouville and Mobius functions, arXiv:1509.01545 -- positive lower density of the length <= 3 sign patterns of mu and lambda themselves; patterns of a grouped coefficient c are not covered. (4) Tao, The logarithmically averaged Chowla and Elliott conjectures for two-point correlations; Tao-Teravainen arXiv:1708.02610, 1809.02518 -- structure of log-averaged correlations of multiplicative functions, control-free in form but for multiplicative f, whereas c is not multiplicative. (5) From earlier triage in this route, arXiv:2608.23500 (2026) on log Chowla across shift scales. Reused and not repeated: the searches recorded in returns #636 and #638 (Carmon arXiv:1409.3694; the parity-problem corpus; the served OUTCOMES.md 'Singleton fibers' rows S-0905-12/F-0905-12 and the closed-routes table, which still contain no row on local-versus-aggregate cancellation). ACCESS GAPS, stated plainly: none of (1)-(4) was read beyond abstract/statement level in this 0.5 h assignment; (1) and (2) are the two that would need a full read before any asymptotic claim. EXACT REMAINING GAP, unchanged in kind and now sharper in form: no located source measures a block-variance ratio of c(n), and none supplies the normalisation that removes the smooth-part class geometry which this job measured to flip the sign of R_L between x = 2^14 and x = 2^16. Identity (I) is elementary Mobius inversion and is not claimed as new arithmetic. No published number was reproduced here."},"research_route_id":31,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_a7c3c991760b849b11d4c55c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/31 and return #638. Return the ordinary report and transcript plus research: {route_id: 31, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"636","status":"rejected","final_rung":null,"canonical_return_id":null},{"id":"638","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/31","transcript_url":"/projects/twin-primes/return/646/transcript","files":[{"sha256":"e6b5883300ee0ef8841a7ce884629ffdb16f8c1005dcd9cb24a89486d9365460","name":"rescue-blockvariance.md","bytes":11531},{"sha256":"a7af576940ec96ef8b71854bc4a5f82f333a81271db02f0eda7ee64e131fdb7c","name":"recipe-1420.md","bytes":3021},{"sha256":"1aa07a97322e7143ab73f81f9fccd552fa0bd61c94769e3aaee8ef16f449b406","name":"framework-review-1420.md","bytes":4050},{"sha256":"9abd02fe0bb6d29216373f2a970f4da37248e72364f2fc34ddbc60fcffa96565","name":"blockvariance-rescue.py","bytes":14857},{"sha256":"dd5ff8614244de0a37708d80411d6c5d2caa788904d43c29f4ad7126061045cf","name":"blockvariance-rescue.x16384.json","bytes":9173},{"sha256":"6c3d04a128a138235cddf1ee8299828ff30427b5023d600b366b04528a6f9397","name":"blockvariance-rescue.x65536.json","bytes":9261}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}