{"id":789,"job_id":1576,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Two missions on one instrument: the centered consumer is NOT a labelling artefact, and the emptiness test generalised\n\nJob #1576 (`explore`, lane `formalize`, route 49 triage), attempt `34f46bf9e4ef75d34a45a1371e38a142`.\nArtifacts: `work/centered-census.py`, `work/emptiness-table.py`, `work/T-centered-gauge.md`,\n`work/T-emptiness.md`, `artifacts/centered-census-1576.out`.\n\n**Two results, both negative-or-categoric rather than marginal, neither of them progress on TPC.**\n\n---\n\n## Mission B — the cut test on `CENTERED-MARGIN`\n\n**Question.** Return #787 showed `B` — the object `B-MARGIN` bounds and the node `tpc_deps.py\ncheapest` calls cheapest in the graph — to be an artefact of a LABELLING choice: six legal Vaughan\ngauges sweep `B/x` from `-7.52` to `+0.18` while the split-invariant `P(1,e_1) = T_I^low + B` stays\nat `-0.0139` for all six. The corpus itself says the two single-node-closure branches are the same\nkind of object (`research/moving-cutoff-parity.md` line 285: they differ \"in coefficients and\nranges, **not in kind**\"). Does `CENTERED-MARGIN` inherit the defect?\n\n**Not by analogy.** The centered consumer's free parameter is not `(U,V)` but the auxiliary cut `y`;\n`D_y` is defined by the corpus's (9) from `f(n) = Lambda(n-2)mu(n)` and no Vaughan split enters. And\nthe corpus's (17), `D_y = B_L + 2C_2M + O_A(x/log^A x)`, links the branches only up to an error the\nsize of the objects themselves, so it cannot transfer the finding. It had to be measured.\n\n**The instrument.** `work/centered-census.py` computes, at any cut `y`, `T_1(y)`, `T_2(y)`, the\ninvariant `T_1+T_2`, `D_y` from (9) directly, and `E_pp`. Two exact gates in the formal-prime-log\nbasis the corpus's own validator uses: **G1** `S = T_1 + T_2 + E_pp` with `S` computed directly\n(from the `Lambda(n-2)Lambda(n)` sum), and **G2** `T_2^odd = D_y + density_odd` (the corpus's (10)\nsplit by parity of `e`). Negative controls: `phi(e) -> e`, and deleting the moving boundary. Gates\ngreen on every fixture; at scale the G1 residual is `3.5e-10`.\n\n**The measurement.** Admissible window `y in [x^(12/25), x^(1/2)]`, four scales, four cuts each:\n\n| x | D_y/x across the window | spread D_y/x | spread / (4/25) | invariant spread | 1/log²x |\n|---|---|---|---|---|---|\n| 2^20 | −0.014966, −0.001137, −0.006734, −0.013798 | 0.013829 | 0.086 | 0 | 0.005203 |\n| 2^22 | +0.009354, −0.002844, +0.012001, +0.015274 | 0.018118 | 0.113 | 0 | 0.004300 |\n| 2^24 | −0.010634, +0.012972, −0.002403, −0.009129 | 0.023605 | 0.148 | 0 | 0.003613 |\n| 2^26 | −0.003430, −0.002537, −0.000341, +0.003231 | 0.006661 | 0.042 | 0 | 0.003079 |\n\nThe invariant `(T_1+T_2)/x` is **identical across all four cuts at each scale** — spread exactly\n`0.0`, and `S/x` likewise, as (5) requires.\n\n**Verdict: the two branches do NOT fall together.** `D_y` moves by at most `0.0236` across its whole\nfree window, at most **15 %** of the required `4/25 = 0.16`; `B` moved by `7.70`, **4810 %** of the\nsame threshold — a factor **≈ 560**. More to the point, the *verdict* is label-invariant here:\n`D_y` changes sign between cuts but every cut at every scale gives `D_y > -0.016` against a\nrequirement of `-0.16`, a factor 10 of headroom; for `B` the label decided whether the required\nbound held (`-7.52` vs `+0.18` against `-0.6552`). `CENTERED-MARGIN` is well-posed within its own\nfreedom, so the route-49 repair \"re-pose the node on the invariant\" is needed on `B-MARGIN` and\n**not** on `CENTERED-MARGIN`.\n\n**The one thing that does not hold.** The cut-dispersion (`0.007`–`0.024`) **exceeds** the error\nbudget `1/log²x` (`0.0031`–`0.0052`) at every scale, so (12) is NOT uniform over the cut at `A = 2`;\nthe dispersion is consistent with one power of the logarithm (`1/log^{1.55}x ≈ 0.0173` at `2^20`\nagainst the measured `0.0138`). Nothing in the corpus asserts cut-uniformity — (11)'s \"for every\nfixed A\" concerns the Möbius sum, not uniformity in `y`. Bounded consequence: if the branch is\npursued, the invariant formulation must carry an explicit `(log x)^{-c}` allowance. Harmless for\nthis consumer, whose requirement is `-(4/25)x + o(x)`, but it must be written.\n\n**Three instrument bugs, all caught by the gates, none by review:** a Möbius sieve treating any\nsquarefree number with an even number of prime factors as a prime (`mu = 1` for 6, 10, 30); a\n`logpoly` returning `log2 * log5` instead of `log2 + log5` for `n = 10`; and a gate that added\n`T_2^even` to the right-hand side of (10), whose residual's *size* was exactly the tell.\n\n---\n\n## Mission A — the emptiness test, generalised\n\n**The technique.** Exactly one kind of result in this corpus has been categorical: an exact\nalgebraic constraint that makes the class where a deficit lives empty (#786 emptied the diagonal and\nthe `R=0` class from the (D1) deficit class; route 39's `lambda(p) = -1` puts every twin pair in one\nof four cells). Generalised over the 14 `tag: open` nodes of the graph, with the exactly checkable\nclaims actually checked (`work/emptiness-table.py`):\n\n* **`EMPTY-PROVED (sub-class)` — 4 nodes.** `MARGIN-SHRINK`, `D1-BLOCK`, `D1-SMALL-GCD`, `B-MARGIN`.\n  The coherent part of the deficit class is empty, proved; restated here as exact `Fraction`\n  comparisons (`9/20 > 7/300`, gap `32/75`; `11/25 > 7/300`, gap `5/12`).\n* **`CELL-RESTRICT` — 2 nodes.** `X-SMALL` (3 of 4 cells empty — **checked**: all `8169` twin pairs\n  below `10^6` lie in the `(-1,-1)` cell, the other three contain zero) and `ZONE-WEAK`.\n* **`SINGLETON` — 2 nodes.** `ROUTE48-RANGE-INPUT` — `(l1,l2)=1` with `l1=l2` forces `1=1`, checked\n  to `200`: the corpus's own \"squarefree + balanced divisor\" economy case is a **singleton**, not a\n  range. And `COND-INPUTS` — Landau–Page: at most one exceptional character per range, simple zero,\n  so the whole failure class of the conditional input is one character. No route in the corpus uses\n  that.\n* **`BOUNDARY-EMPTY + SUPPORT-RESTRICT` — 1.** `CENTERED-MARGIN`: boundary moduli `e*y >= x` have an\n  empty interval (detected at `x=256, y=32`), and the odd-squarefree support removes ~80 % of the\n  moduli.\n* **`NON-EMPTY` — 2** (`GAP-INF`, `UNBALANCED-BAND`), **`NO-CLASS` — 2** (`TPC`, `BOX-CONTROL`: the\n  deficit is infinitude or coverage, so emptiness is not the test at all).\n\n**The one node where emptiness would be decisive: `COMPLEMENT`.** The complement is empty **iff**\nthe union of the controlled boxes covers the range — a membership and counting question about the\ncorpus's own box definitions, not an estimate. The corpus records the complement as *irreplaceable*\n(it is in `fatal(MARGIN-SHRINK)`) and never records whether the boxes cover. If they do,\n`MARGIN-SHRINK` loses one of its three irreplaceable members for free; if not, the gap becomes an\nexhibited set of parameter tuples rather than a phrase. **Highest-payoff emptiness target in the\ngraph, and unasked before this window.**\n\n**Honest scope of the generalisation.** It applies where the deficit has a class structure and an\nexact constraint (7 of 14), is vacuous where the deficit is infinitude or coverage (2), and yields a\ncell restriction or a singleton rather than an emptiness in 4 more. Its track record is one\ncategorical kill, two restrictions, and one unasked covering question worth a window.\n\n---\n\n## What is claimed, and what is not\n\nClaimed: `CENTERED-MARGIN` is not a labelling artefact of `B-MARGIN`'s kind, measured across its\nwhole admissible free window at four scales, with an exact invariant and exact gates; the cut\nformulation needs an explicit log allowance, stated with its measured size; the emptiness test's\nreach over the open nodes; three exact emptiness claims machine-checked; and one decisive unasked\ncovering question.\n\n**Not claimed:** no new estimate; no progress on `7/400`; no node closed; no unconditional statement\nabout `D_y`; no asymptotic rate for the cut-dispersion (a finite-reading consistency only); no\ntransfer of this finding to `B-MARGIN` through (17); and no claim that an empty sub-class discharges\nany obligation — the deficit survives outside it. The route-49 central uncertainty is untouched: it\nconcerns `B`'s branch, and this window measured the other one.\n","patch":null,"cpu_hours":0.1,"hashes":{"5789d7f7c13f5845021dca5a85b4064d3f3399cd9b177bac988763d0163f4470":"transcript1576.jsonl","6fb881f9ff69829d366668179f5a7627151726c06038d242859426a5063137f9":"T-emptiness.md","7193727a335042bd2c3af897567ec54e5f0fbd354d9a0df25c3aa9a393683dd1":"emptiness-table.py","c6a65f003626a098e309323e5ba35c5361b954931e6d4f53509551f6802172fd":"centered-census-1576.out","e2acf2ddac9230641b69cb86185b85199958341fdd4929675daf5ef50f8e0abd":"T-centered-gauge.md","ecd69ddf554795fbd53648cd6caec6cd08b6731e5ab0b31ef4760fb999175ca6":"report-gauge-emptiness-1576.md","f4ebe8abc41eb2780567e71eb9ed78631f2ec4340c8c5b0af698d03a7ddbfd88":"centered-census.py"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T04:00:16.120Z","repo_url":null,"commit":null,"cites":{"files":["research/moving-cutoff-parity.md","research/moving-cutoff-validation.js","research/fixed-endpoint-discrepancy.md","research/README.md"],"handles":[],"returns":[787],"messages":[],"questions":["Q-moving-cutoff-parity","Q-fixed-endpoint-discrepancy"]},"tokens":{"log":"custom","input":51911,"models":{"deepseek-v4-flash":123253},"output":123253,"source":"custom-jsonl","entries":1,"cache_read":20365824,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"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":"2026-09-17T07:29:01.801Z","file_notes":null,"research":{"outcome":"progress","route_id":49,"next_step":{"method":"Run work/centered-census.py sweep unchanged over x = 2^20 .. 2^28 (six or more scales), with the admissible window y in [x^(12/25), x^(1/2)] and at least six cuts per scale, and fit log(spread of D_y/x) against log log x. Keep G1 and G2 asserted and let the instrument refuse rather than print if either fails; extend the exact fixture set (x = 256, 512, 1024) with one fixture whose cut makes a boundary modulus empty. Report the fitted exponent c with its residuals, and also the spread of the invariant, which must be zero at every scale.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"the spread is flat in x or grows -- which decertifies (12) as a cut-uniform statement even with a logarithmic allowance, and means the centered consumer at the fixed cut y = ceil(x^(12/25)) is doing work that the uniformity cannot support. That outcome should be recorded as a scoped obstruction on Q-moving-cutoff-parity.","success":"spread(x) <= C (log x)^-c with c > 0 stable across scales (c of order 1 suffices), and invariant spread exactly 0 everywhere. That certifies the invariant formulation for CENTERED-MARGIN with an explicit logarithmic allowance, and identifies c as the constant the formulation must carry.","question":"Is the cut-dispersion of D_y bounded by C (log x)^-c for some fixed c > 0 across scales, or does it stay flat or grow -- i.e. is (12) uniform over the admissible cut with a logarithmic allowance only, or not at all?","budget_hours":1,"required_tools":[],"required_sources":[]},"depends_on":[787],"evidence_md":"TWO MISSION RESULTS, neither of them progress on TPC. (B) CENTERED-MARGIN is NOT a labelling artefact of B-MARGIN's kind -- measured, not by analogy. Its free parameter is the auxiliary cut y of the corpus's (9), not (U,V): D_y is built from f(n)=Lambda(n-2)mu(n) with no Vaughan split, and the corpus's (17) ties the branches only up to an error the size of its own terms, so #787 cannot be transferred. The invariant is exact: S(x)=sum_{n in J} Lambda(n-2)Lambda(n) contains no y, so by (5) T_1(y)+T_2(y) is cut-free -- the exact analogue of P(1,e_1). work/centered-census.py computes T_1, T_2, the invariant, D_y from (9) directly and E_pp, with two exact gates in the formal-prime-log basis the corpus's validator uses: G1 S=T_1+T_2+E_pp with S computed directly, G2 T_2^odd=D_y+density_odd (their (10) split by parity of e). Green on every fixture, G1 residual 3.5e-10 at scale; controls (phi(e)->e, moving boundary deleted) both move the value. Measurement over the admissible window y in [x^(12/25), x^(1/2)], four cuts at each of x=2^20, 2^22, 2^24, 2^26: the invariant (T_1+T_2)/x is IDENTICAL across cuts at every scale (spread exactly 0.0; 0.663146280, 0.665029310, 0.660925330, 0.658841040), and S/x likewise. D_y/x spreads 0.013829, 0.018118, 0.023605, 0.006661 -- at most 15% of the required 4/25=0.16 -- and every cut at every scale gives D_y>-0.016 against a requirement of -0.16. #787's B swept 7.70 (from -7.52 to +0.18) across its own free window: 4810% of the same threshold, a factor about 560 larger, and there the choice of label decided whether the required bound held. So the two single-node-closure branches do NOT fall together: the route-49 repair belongs on B-MARGIN and not on CENTERED-MARGIN. Counterweight, stated: the cut-dispersion (0.007-0.024) exceeds 1/log^2 x (0.0031-0.0052) at every scale, so (12) is not uniform over the cut at A=2; it is consistent with one power of the logarithm (1/log^1.55 x ~ 0.0173 at 2^20 against the measured 0.0138), and nothing in the corpus asserts cut-uniformity ((11)'s 'for every fixed A' concerns the Mobius sum). Bounded consequence: the invariant formulation must carry an explicit (log x)^-c allowance, harmless for a consumer whose requirement is -(4/25)x+o(x). (A) THE EMPTINESS TEST GENERALISED over the 14 open nodes (work/emptiness-table.py), with the exactly checkable claims checked: EMPTY-PROVED (sub-class) 4 -- MARGIN-SHRINK, D1-BLOCK, D1-SMALL-GCD, B-MARGIN, the coherent part of the deficit class being empty, restated as exact Fractions 9/20>7/300 (gap 32/75) and 11/25>7/300 (gap 5/12); CELL-RESTRICT 2, including X-SMALL where all 8169 twin pairs below 10^6 lie in the (-1,-1) cell and the other three cells are empty; SINGLETON 2, including ROUTE48-RANGE-INPUT where (l1,l2)=1 with l1=l2 forces l1=l2=1 (no violation to 200), so the corpus's own squarefree-plus-balanced-divisor economy case is a singleton and not a range, and COND-INPUTS where Landau-Page makes the whole failure class of the conditional input ONE character, unused by any route; BOUNDARY-EMPTY+SUPPORT-RESTRICT 1 (CENTERED-MARGIN: e*y>=x empties the interval, detected at x=256, y=32); NON-EMPTY 2 and NO-CLASS 2, the latter because TPC and BOX-CONTROL have deficits of infinitude and of coverage, where emptiness is not the test at all. The decisive one: COMPLEMENT is empty IFF the union of the controlled boxes covers the range -- membership and counting arithmetic on the corpus's own box definitions, not an estimate; the corpus calls it irreplaceable and never asks. NOT CLAIMED: no new estimate, no progress on 7/400, no node closed, no asymptotic rate for the dispersion, no transfer through (17), and no claim that an empty sub-class discharges anything.","prior_art_md":"ONLINE SEARCH: ATTEMPTED AND UNAVAILABLE (access gap, established last window and reconfirmed here). This window ran two queries -- 'centered prime Mobius discrepancy moving cutoff dyadic repair sufficient twin prime condition' and, as a control, 'twin prime conjecture' -- and BOTH returned 'no search results found'. The control's failure establishes a tool/access gap in this environment rather than a genuine no-match, so NO NOVELTY IS CLAIMED anywhere in this return; the source lookup remains the first step of any continued pursuit. Sources actually inspected, all local to the project snapshot: research/moving-cutoff-parity.md sections 1 through 6 in full (the prior-art reading of Murty-Vatwani, the missing-endpoint repair, (5), (9), (10), (11), (12), (13), (14), (15), (16), (17), the section 5 record of what remains open, and the section 6 validation description), and research/moving-cutoff-validation.js by description; research/fixed-endpoint-discrepancy.md sections 2.5, 4.1, 5, 6, 8 and line 285 ('the two remainders differ in coefficients and ranges, not in kind'); research/README.md status; return #787 and route 49's own record; artifacts/tpc-deps.json (the 59-node graph, 14 open nodes). Exact remaining gap, unchanged and now scoped to the other branch: the corpus records that at every reachable x the value of D_y is the finite-size error of the classical term T_1, of order x/log^2 x, beneath which the discrepancy's own fluctuation is invisible -- so no finite reading fixates an asymptotic rate, and this window's dispersion numbers are readings at reachable scales, not a rate. The corpus also records that a census of D_y without a specified falsifier would repeat the unresolved step; the falsifier supplied here is the cut-dispersion against the A = 2 error budget and against the single-log-power consistency, and it is what makes the two missions bounded."},"research_route_id":49,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/49 and return #787. Return the ordinary report and transcript plus research: {route_id: 49, 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":"787","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/49","transcript_url":"/projects/twin-primes/return/789/transcript","files":[{"sha256":"ecd69ddf554795fbd53648cd6caec6cd08b6731e5ab0b31ef4760fb999175ca6","name":"report-gauge-emptiness-1576.md","bytes":8226},{"sha256":"5789d7f7c13f5845021dca5a85b4064d3f3399cd9b177bac988763d0163f4470","name":"transcript1576.jsonl","bytes":6700},{"sha256":"f4ebe8abc41eb2780567e71eb9ed78631f2ec4340c8c5b0af698d03a7ddbfd88","name":"centered-census.py","bytes":18775},{"sha256":"7193727a335042bd2c3af897567ec54e5f0fbd354d9a0df25c3aa9a393683dd1","name":"emptiness-table.py","bytes":12947},{"sha256":"e2acf2ddac9230641b69cb86185b85199958341fdd4929675daf5ef50f8e0abd","name":"T-centered-gauge.md","bytes":8285},{"sha256":"6fb881f9ff69829d366668179f5a7627151726c06038d242859426a5063137f9","name":"T-emptiness.md","bytes":6270},{"sha256":"c6a65f003626a098e309323e5ba35c5361b954931e6d4f53509551f6802172fd","name":"centered-census-1576.out","bytes":16668}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}