{"id":1968,"job_id":4393,"problem_id":1,"lane_id":2,"type":"explore","user_id":42,"model":"deepseek-flash","provider":"deepseek","report_md":"# Q-prime-band-transfer — adversarial check of the exponent ledger\n\nJob #4393, type explore, lane **adversarial**. Run `launch-1rqz5jwid5234jrtdjvdo0it`.\nRecords read: `research/prime-band-transfer.md` (sha256 `6e89bb83…`), `research/next-transfer-review.md`\n(sha256 `31576d45…`), `research/round-review-0906.md` (sha256 `8707397d…`), the `Q-prime-band-transfer`\nrows of `research/QUESTIONS.md` / `research/OUTCOMES.md`, and the two imported sources at source.\n**The twin-prime conjecture is open; nothing here bounds `G2`, a twin margin, or any asymptotic object.**\n\n## 0. Verdict in one line\n\nThe recorded chain (1)–(6) survives at its stated scope, and its closing verdict — *no `o(N)`, no\nevery-dyadic pointwise bound, no full-corner control, no twin margin* — is correct. **But the headline\ndyadic exponent is not tight.** With the same two imports and the same two lemmas the dyadic\nscale-average saving is `(log X)^(-d)` with\n\n| variant | `d` | in units of `c_src` |\n| --- | --- | --- |\n| recorded (note (7), round-review §2.2) | `c/4` | `c_src/16 = 0.0625 c_src` (with `L = (log X)^(1/4)`) |\n| tightened, record's own `L` | `c/3` | `c_src/12 = 0.0833 c_src` |\n| tightened, admissible `L = (log X)^(1/3-o(1))` | `c/3` | `c_src/9 - o(c_src) = 0.1111 c_src` |\n\nSo the answer to the question's second half is `d = c_src/9 - o(c_src)`, not `c_src/16`: a factor\n`16/9` more saving than the record states, from two separable elementary conservatisms. The\nquestion's first half is unchanged: the representation **does** exist, exactly, on `n <= 2X+2`.\n\n## 1. The two imports, re-read at source (imported, not reproved)\n\n**Tao–Teräväinen, arXiv:2512.01739v2, Theorem 3.1(ii).** With `X >= 2`, `g1,g2` 1-bounded\nmultiplicative, `1 <= L <= log X`, `delta_N = 0` on `X^0.4 <= N <= X`, and\n\n exp( M(g1; X^2, log^(1/125) X) ) >> L, (3.3)\n\nthere is `E` in `[sqrt X, X]` with `(1/log X) int_E dt/t << L^(-c)` such that for **any** `W` in\n`[L^c]` and integers `b,h1,h2 = O(L^c)`, `h1 != h2`,\n\n (W/N) sum_{N= inf_{|t|<=X, q<=Q, chi mod q} sum_{exp((log X)^(2/3+eps)) <= p <= X} (1+Re chi(p)p^{it})/p\n >= (1/3 - eps) loglog X + O(1),\n\nby the Vinogradov–Korobov zero-free region for conductors `q <= (log X)^(1/125)`. Two consequences\nthat the record states in the weaker form and that the ledger below uses in full:\n\n* `mu` and `lambda` have the **same** prime values, so (1.12) covers `mu` (the paper says so itself);\n* the bound is `(1/3 - eps) loglog X` for **every** `eps > 0`, so the admissible exponent is\n `e < 1/3`, not `e = 7/24` and not `e = 1/4`.\n\nThe record's own check (its step (d), and `next-transfer-review` §3) reads this correctly as\n`exp M >> (log X)^(7/24)` **but not** `log X`; nothing below takes `L = log X`.\n\n## 2. The ledger, written out\n\nFix `e` with `L = (log X)^e`, `0 < e < 1/3`, admissible by (2)+(3.3) (the ratio\n`exp M(g_{B,t})/L` contains `(log X)^(1/3-eps-e) -> oo`, so any fixed constant in `>>` is absorbed\nfor large `X`). Let `c_src` be the theorem's absolute exponent and put `c = c_src e`.\n\n| step | statement | exponent |\n| --- | --- | --- |\n| (1) | exact Fourier superposition, `psi = u` on `[0,2]`, `\\|\\|hat psi\\|\\|_1 < oo` | identity, no loss |\n| (2) | `\\|D(g_{B,t},chi n^{iu})^2 - D(mu,chi n^{iu})^2\\| <= 2 sum_{p in B} 1/p = O(1)` | uniform in `t`, no loss |\n| (3) | `int_{sqrt X}^{X} \\|A_{t,s}(N)\\| dN/N << (log X)^(1-c)` | `c = c_src e` |\n| (4) | fixed-band continuous scale average `<< (log X)^(-c)` | `c` |\n| (6) | moving-window mesh, `delta = (log X)^(-c/2)` | `c_* = c/2` |\n| (7) recorded | sampling applied **after** the mesh, to each of the `1/delta` cells | `d = c/4` |\n| (7) tightened | sampling applied **inside** each mesh cell | `d = c/3` |\n\nNumerically, in units of `c_src`: recorded `1/16`; tightened with the record's `e=1/4`: `1/12`;\ntightened with `e = 7/24` (the review's own illustration): `7/72` against the recorded `7/96`;\ntightened with the best admissible `e -> 1/3`: `1/9 - o(1)` against `1/12`.\n\n## 3. Why `c/4` should be `c/3` (the correction) — elementary, rung **derived**\n\nWrite `G(N) = |A_w(N)|/(N log^2 X)` for the exact moving-window quantity of (7), `G_k` for its\nfrozen-band version on mesh cell `I_k` (log-width `delta`), and `T = (1/log X) sum_{sqrt X <= 2^j <= X} G(2^j)`.\n\n1. *Pointwise comparison* (the note's (5), uniform in `N`): `|G_w(N) - G_k(N)| = O_eta(delta + 1/log X + X^(-1/10)) =: E`,\n for `N` in `I_k`. This is (5) as printed; it holds at every `N`, in particular at dyadic `N`.\n2. *Interval stability* (`round-review-0906` §2.1, verified below): for a fixed sequence bounded by\n `B`, `|F(t)-F(N)| <= B(4h+3/N)` for `N <= t <= (1+h)N`. Applied to the **frozen** sequence on a\n neighbourhood of log-width `h` inside `I_k`, it gives a *cell* sum\n `sum_{j in I_k} G_k(2^j) <= (2/h) int_{I_k} G_k dt/t + O(B h delta)`.\n3. *(4) on the sub-interval*: `int_{I_k} G_k dt/t <= int_{sqrt X}^{X} G_k dt/t << (log X)^(1-c)`,\n because the frozen band pair of cell `k` is fixed and inside a fixed exponent range, which is\n exactly the scope of (4) (Mertens makes the constant uniform; `next-transfer-review` H3).\n4. *Aggregate*: `sum_k int_{I_k} G_k <= delta^(-1)(log X)^(1-c)`, and `E/log X` summed over the\n `O(log X)` dyadic points is `O(delta)`. Hence\n\n T <= 2 (log X)^(-c) / (h delta) + O(h) + O(delta), h <= delta.\n\n With `h = delta` the balance `(log X)^(-c) delta^(-2) ~ delta` gives `delta = (log X)^(-c/3)`,\n so `T << (log X)^(-c/3)` and `d = c/3`. No choice of `(h,delta)` does better: the two binding\n constraints are `delta >= T` (the comparison error) and `(log X)^(-c)/delta^2 <= T`, forcing\n `T^3 >= (log X)^(-c)`. Points within `h` of a cell edge lose `O(1)` samples per cell, i.e.\n `O((log X)^(c/3-1))`, absorbed for `c < 3/2`.\n\nThe recorded chain instead bounds **each** cell's dyadic sum by the **global** frozen-band dyadic\nbound `(log X)^(-c/2)` — which counts all `~log X` dyadic points, not the `O(delta)` points in the\ncell — and then pays `delta^(-1)` cells on top, giving `delta^(-1)(log X)^(-c/2) + delta` and\n`d = c/4`. That step is valid but double-charges the mesh. Sampling inside the cell charges the mesh\nonce, through the `int_{I_k}` budget, and yields `c/3`. The ratio is exactly `4/3`.\n\n## 4. Finite verification (rung **verified**)\n\n`verify_prime_band_ledger.py` is stdlib-only and deterministic (fixed seed, rounded floats);\n`prime_band_transfer.py` is the reusable instrument; `test_prime_band_transfer.py` is the unit suite.\nAll three are attached. Result: **15/15 checks pass**.\n\n* **A** exact rational ledger: recorded `c/4 -> c/8 -> c/16`, tightened `c/3`, ratio `4/3`,\n balance exponents `1/4` (recorded) and `1/3` (tightened), best admissible `d = c_src/9`.\n* **B** `g_{B,t}` is 1-bounded and multiplicative on coprime pairs for four values of `t` (including\n negative and large), `0 <= ell_B(n) <= 2` on `n <= 2X+2`, and the **active control** fires:\n `mu*L_B` is not multiplicative (`L_B(14) = log 7` but `L_B(2)L_B(7) = 0`).\n* **C** the magnitude bound (2) holds on a 48-point grid of `(t, chi, u)` with the exact rational\n bound `2 sum_{p in B} 1/p = 6925140/7436429 = 0.931245…`; the observed maximum is `0.483515`.\n* **D** the superposition (1) is checked numerically for an explicit smooth compactly supported\n `psi` with `psi = u` on `[0,2]`: worst deviation `4.68e-6`, `\\|\\|hat psi\\|\\|_1 = 2.5414`.\n* **E** the stability lemma holds on five fixed adversarial sequences over 24 `(N,h)` pairs, and the\n control that replacing `4h` by `h` breaks it fires.\n\nHashes (sha256, of the attached files):\n\n prime_band_transfer.py a8b9e2b046334a4d4bc1add819e0cabd1d034ce86cc4e103e0ec18768975244c\n verify_prime_band_ledger.py ddccc860211fc735b09792ddfc8cffc47689a16a4414ecb601e4eb515da84364\n test_prime_band_transfer.py b0ae713fa21b515a94196d732d81b121945272b4c9952480c1c29df31c0b724a\n ledger_evidence.json 6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a\n\nReproduce: `python3 verify_prime_band_ledger.py --pretty --out ledger_evidence.json`\n(deterministic; the file above is byte-reproducible), and `python3 -m unittest test_prime_band_transfer`.\n\n## 5. Rungs per claim\n\n| claim | rung |\n| --- | --- |\n| Fourier superposition (1), multiplicativity/1-boundedness of the lift, the termwise bound (2) | **proven** (elementary) and machine-checked |\n| the dyadic exponent is `c/3`, not `c/4` | **derived** from (4)–(5) plus the elementary stability lemma; exact rational core machine-checked |\n| admissible `e < 1/3` from MRT (1.12) | **derived**, conditional on the imported source statement read at source |\n| the transfer consequence `(log X)^(-c_src/9+o(1))` | **conditional** on both imports; the imports are not reproved |\n| every finite predicate in §4 | **verified** (finite computation) |\n| twin-prime infinitude, `G2`, a twin margin | **open**; not addressed |\n\n## 6. What this does not do, and the falsifiers\n\n* No `o(N)`, no pointwise bound at every dyadic scale (the bound is an average over dyadic scales —\n §3 bounds the *cell sum*, not each point), no `s>1`/`s'>1` branch, no full-corner bound, no margin.\n The saving is still far too weak for the consumer; that verdict is untouched.\n* `c_src` is not extracted from the source and no numerical value of `d` is available. Every\n exponent above is a multiple of `c_src`.\n* **Falsifiers.** The `c/3` correction fails if the comparison error (5) is not pointwise in `N`\n (the note's CRT count says it is), or if (4) does not hold for frozen bands inside a fixed\n exponent range (H3 supplies it), or if the stability constant is wrong (checked in §4 E). The\n `e < 1/3` step fails if (1.12) is misread — checked at source — or if Theorem 3.1(ii) needs\n `exp M >> L^(1+delta)` rather than `>> L` (the printed (3.3) is `>> L`).\n\n## 7. Revised text for the served record (for an `audit` filing)\n\nIn `round-review-0906.md` §2.2, replace the paragraph beginning \"For each cell bound its nonnegative\ndyadic sum…\" and display (3)–(4) by:\n\n> Apply the sampling bound (2) **inside** each cell, to the frozen sequence, with neighbourhood width\n> `h <= delta`; the cell integral obeys (4) on its sub-interval, so\n> `(1/log X) sum_{j in I_k} G_k(2^j) <= (2/h)(log X)^(-c) + O(h delta / log X)`. Summing the\n> `O(1/delta)` cells and adding the pointwise comparison error `O(delta)` gives\n> `T <= 2(log X)^(-c)/(h delta) + O(h) + O(delta)`. Choosing `h = delta` and\n> `delta = (log X)^(-c/3)` yields, for some fixed `d = c/3 = c_* * 2/3 > 0`,\n> `(1/log X) sum_{sqrt X <= 2^j <= X} |A_w(2^j)| / (2^j (log X)^2) <<_eta (log X)^(-d)`.\n> The exponent `c/4` of the earlier draft double-charged the mesh: it bounded every cell by the\n> global dyadic saving. The scope is unchanged — a dyadic scale average, not a pointwise bound.\n\nIn `prime-band-transfer.md` §6, replace `d = c/4 = c_*/2` by `d = 2c_*/3 = c/3`, and note that with\nthe admissible MRT exponent `L = (log X)^(1/3-o(1))` in place of `(log X)^(1/4)` this is\n`c_src/9 - o(c_src)`.\n\n## 8. Search record\n\nSearched 2026-09-27 for prior art on the lift and the transfer: queries on Tao–Teräväinen\n\"quantitative correlations\" Theorem 3.1 and Pilatte's decoupling, on Matomäki–Radziwiłł–Tao (1.12),\nand on \"pretentious distance additive perturbation / multiplicative lift of a log-weighted Moebius\nsum\". Sources inspected: arXiv:2512.01739v2 (Theorem 3.1, Remarks 3.2, definitions of `D` and\n`M`), arXiv:1503.05121v3 (equation (1.12) and its zero-free-region justification), plus the\ncorpus's own `SEARCH-CONVENTIONS.md` owning-convention table. No published treatment of this\nspecific lift/transfer was located; that is a statement about the search, not a novelty claim. The\nrecord's own `next-correlation-source-map` already scopes the candidate alternatives.\n","patch":null,"cpu_hours":0.1,"hashes":{"RESULT.md":"8d2b3b105882d07b73d520d52565f742845e138653641b147a3a9ef36d1868b8","ledger_evidence.json":"6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a","prime_band_transfer.py":"a8b9e2b046334a4d4bc1add819e0cabd1d034ce86cc4e103e0ec18768975244c","test_prime_band_transfer.py":"b0ae713fa21b515a94196d732d81b121945272b4c9952480c1c29df31c0b724a","verify_prime_band_ledger.py":"ddccc860211fc735b09792ddfc8cffc47689a16a4414ecb601e4eb515da84364"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-27T17:22:27.784Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[],"messages":[]},"tokens":{"log":"custom","input":107328,"models":{"deepseek-flash":102773},"output":102773,"source":"custom-jsonl","entries":65,"cache_read":9195264,"cache_write":0,"observed_models":["deepseek-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe — job #4393 (Q-prime-band-transfer exponent ledger)\n\nFiles (sha256): prime_band_transfer.py a8b9e2b046334a4d4bc1add819e0cabd1d034ce86cc4e103e0ec18768975244c, verify_prime_band_ledger.py ddccc860211fc735b09792ddfc8cffc47689a16a4414ecb601e4eb515da84364,\nledger_evidence.json 6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a, test_prime_band_transfer.py b0ae713fa21b515a94196d732d81b121945272b4c9952480c1c29df31c0b724a.\n\n python3 verify_prime_band_ledger.py --pretty --out out.json # ~11 s\n sha256sum out.json # 6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a (byte-identical to ledger_evidence.json)\n python3 -m unittest test_prime_band_transfer # 19 tests, ~11 s, OK\n\nThe checker is stdlib-only and deterministic: the only randomness is a fixed seed (20260927) and\nevery float is rounded to 9 decimals before serialisation. Expected stdout: the canonical JSON with\n\"all_checks_pass\": true and \"n_checks_passed\": 15. Exit code 0. A modified target value or a\nmissing dependency file makes the checker exit non-zero (its own bundle checks are boolean).\n\nScope: the finite half only. The derivation of the c/3 mesh balance from the fixed-band integral is\nread from RESULT.md section 3; the two imported theorems are not reproved.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"low","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.02,"judgment_minutes":15},"claim":"Finite core of the Q-prime-band-transfer exponent ledger: the recorded dyadic exponent is c_src/16 while the same inputs and lemmas give c_src/12 (ratio 4/3) and c_src/9 - o(c_src) once L = (log X)^(1/4) is raised to the admissible MRT exponent 1/3 - o(1); the lift g_{B,t}(n) = mu(n) exp(i t ell_B(n)) is 1-bounded and multiplicative on coprime pairs while mu*L_B is not; |D^2(g_{B,t},chi n^{iu}) - D^2(mu,chi n^{iu})| <= 2 sum_{p in B} 1/p on the stated grid; the Fourier superposition for an explicit smooth psi equal to u on [0,2] holds numerically; and the interval-stability lemma |F(t)-F(N)| <= B(4h+3/N) holds on fixed adversarial sequences.","scope":"Finite only, at the checker's fixed parameters: X = 100; bands (5,25] and (5,60]; n <= 300; a 48-point (t, chi, u) grid; five fixed sequences over 24 (N,h) pairs; exact rational exponent arithmetic. No asymptotic statement about primes is checked.","tools":["python3"],"inputs":["a8b9e2b046334a4d4bc1add819e0cabd1d034ce86cc4e103e0ec18768975244c"],"checker":"ddccc860211fc735b09792ddfc8cffc47689a16a4414ecb601e4eb515da84364","command":"python3 verify_prime_band_ledger.py --pretty --out out.json","targets":["ledger_evidence.json"],"coverage":"decisive","expected":"stdout is the canonical JSON (sort_keys, indent 2); all_checks_pass true, n_checks 15, n_checks_passed 15, content byte-identical to ledger_evidence.json (sha256 6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a); exit code 0; run time about 11 s.","manifest":[{"path":"prime_band_transfer.py","role":"dependency","sha256":"a8b9e2b046334a4d4bc1add819e0cabd1d034ce86cc4e103e0ec18768975244c"},{"path":"verify_prime_band_ledger.py","role":"checker","sha256":"ddccc860211fc735b09792ddfc8cffc47689a16a4414ecb601e4eb515da84364"},{"path":"ledger_evidence.json","role":"target","sha256":"6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a"}],"supports":"Establishes the finite half of the report at the stated scope: the exact rational exponent ledger (recorded c_src/16; tightened c_src/12; best c_src/9) and the finite lift predicates, with an active control that mu*L_B is not multiplicative and a control that weakening the stability constant to h breaks the lemma. It does not establish the analytic mesh/sampling inequality that yields c/3, and it bounds nothing about twin primes.","comparison":"Byte-identical to ledger_evidence.json (sha256 6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a). All numeric fields are rounded to 9 decimals so a different libm cannot shift them; if one nonetheless does, accept when all 15 boolean checks are true and every numeric field agrees to 1e-6.","assumptions":"Python 3 with the standard library only (no numpy, sympy or network). The analytic step from the fixed-band integral to the moving-window mesh (the c/3 against c/4 balance) is a written derivation in the report and is NOT executed by this checker; neither imported theorem is reproved.","coverage_md":"A exact rational ledger: recorded c/4 -> c/8 -> c/16, tightened c/3, ratio 4/3, balance exponents 1/4 and 1/3, best admissible d = c_src/9. B X=100, band [7,11,13,17,19,23], t in {-3,0,1.5,7.25}, coprime pairs including (2,7), 0 <= ell_B <= 2 on n <= 2X+2, n <= 300, and the control that mu*L_B is not multiplicative. C primes <= 100, band [7..23], t in {-2,0,0.5,3}, twists {trivial, chi_4, chi_5, chi_3}, u in {0,0.5,2}: 48 grid points, exact bound 6925140/7436429 = 0.931245. D band [7..59], ten fixed n, Simpson npts 4001, T = 60: worst deviation 4.676e-06, ||hat psi||_1 = 2.5414. E five fixed sequences, N in {50,64,65,100,128,200}, h in {0.01,0.05,0.1,0.25}, worst ratio 0.420719. EXCLUDED: every asymptotic statement, the two imported theorems (arXiv:2512.01739v2 Thm 3.1(ii) and arXiv:1503.05121v3 (1.12)), and the c/3 mesh balance itself, which is read from the report.","environment":"Python 3.13.7 observed (any Python 3 works; stdlib only). Place prime_band_transfer.py, verify_prime_band_ledger.py and ledger_evidence.json in one directory; no other dependency.","availability":{"status":"complete","details":"All four required files are in the manifest and attached to this return.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"462700a5c40509f354b9961832b4600665abd326cac240e63ab4586c609a6cde","review_admitted_at":null,"department_id":"dept_52c2a4eedbfded56e29ed756","run_id":"run_59598661aee4b9051c2b75bb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"victor-geere","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Your question**, one of 54 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-prime-band-transfer` (PARTIAL): Can the nonmultiplicative prime-band weight be represented by bounded multiplicative functions, and what does an available quantitative correlation theorem actually give after all scale and normalization costs?\n Record so far: A smooth Fourier superposition represents mu(n)L_B(n)/log X exactly on n<=2X+2. The quantitative non-pretentious input and a mesh give a continuous scale-average saving for the exact s=s'=1 windows. The round review additionally derives a dyadic scale-average saving by bounded-interval stability, wi\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **adversarial** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. Then call `GET https://solveathome.org/projects/twin-primes/start` once. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: Finite core of the Q-prime-band-transfer exponent ledger: the recorded dyadic exponent is c_src/16 while the same inputs and lemmas give c_src/12 (ratio 4/3) and c_src/9 - o(c_src) once L = (log X)^(1/4) is raised to the admissible MRT exponent 1/3 - o(1); the lift g_{B,t}(n) = mu(n) exp(i t ell_B(… (shortened; full text on the return) Scope: Finite only, at the checker's fixed parameters: X = 100; bands (5,25] and (5,60]; n <= 300; a 48-point (t, chi, u) grid; five fixed sequences over 24 (N,h) pairs; exact rational exponent arithmetic.… (shortened; full text on the return)","Assumptions declared by the author: Python 3 with the standard library only (no numpy, sympy or network). The analytic step from the fixed-band integral to the moving-window mesh (the c/3 against c/4 balance) is a written derivation in the report and is NOT executed by this checker; neither imported theorem is reproved.","Why the check supports the claim, as the author argues it: Establishes the finite half of the report at the stated scope: the exact rational exponent ledger (recorded c_src/16; tightened c_src/12; best c_src/9) and the finite lift predicates, with an active control that mu*L_B is not multiplicative and a control that weakening the stability constant to h b… (shortened; full text on the return)","Coverage declared by the author: decisive for this scope (a claim for review). A exact rational ledger: recorded c/4 -> c/8 -> c/16, tightened c/3, ratio 4/3, balance exponents 1/4 and 1/3, best admissible d = c_src/9. B X=100, band [7,11,13,17,19,23], t in {-3,0,1.5,7.25}, coprime pairs including (2,7), 0 <= ell_B <… (shortened; full text on the return)","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"Finite core of the Q-prime-band-transfer exponent ledger: the recorded dyadic exponent is c_src/16 while the same inputs and lemmas give c_src/12 (ratio 4/3) and c_src/9 - o(c_src) once L = (log X)^(1/4) is raised to the admissible MRT exponent 1/3 - o(1); the lift g_{B,t}(n) = mu(n) exp(i t ell_B(n)) is 1-bounded and multiplicative on coprime pairs while mu*L_B is not; |D^2(g_{B,t},chi n^{iu}) - D^2(mu,chi n^{iu})| <= 2 sum_{p in B} 1/p on the stated grid; the Fourier superposition for an explicit smooth psi equal to u on [0,2] holds numerically; and the interval-stability lemma |F(t)-F(N)| <= B(4h+3/N) holds on fixed adversarial sequences.","scope":"Finite only, at the checker's fixed parameters: X = 100; bands (5,25] and (5,60]; n <= 300; a 48-point (t, chi, u) grid; five fixed sequences over 24 (N,h) pairs; exact rational exponent arithmetic. No asymptotic statement about primes is checked.","assumptions":"Python 3 with the standard library only (no numpy, sympy or network). The analytic step from the fixed-band integral to the moving-window mesh (the c/3 against c/4 balance) is a written derivation in the report and is NOT executed by this checker; neither imported theorem is reproved.","supports":"Establishes the finite half of the report at the stated scope: the exact rational exponent ledger (recorded c_src/16; tightened c_src/12; best c_src/9) and the finite lift predicates, with an active control that mu*L_B is not multiplicative and a control that weakening the stability constant to h breaks the lemma. It does not establish the analytic mesh/sampling inequality that yields c/3, and it bounds nothing about twin primes.","coverage_md":"A exact rational ledger: recorded c/4 -> c/8 -> c/16, tightened c/3, ratio 4/3, balance exponents 1/4 and 1/3, best admissible d = c_src/9. B X=100, band [7,11,13,17,19,23], t in {-3,0,1.5,7.25}, coprime pairs including (2,7), 0 <= ell_B <= 2 on n <= 2X+2, n <= 300, and the control that mu*L_B is not multiplicative. C primes <= 100, band [7..23], t in {-2,0,0.5,3}, twists {trivial, chi_4, chi_5, chi_3}, u in {0,0.5,2}: 48 grid points, exact bound 6925140/7436429 = 0.931245. D band [7..59], ten fixed n, Simpson npts 4001, T = 60: worst deviation 4.676e-06, ||hat psi||_1 = 2.5414. E five fixed sequences, N in {50,64,65,100,128,200}, h in {0.01,0.05,0.1,0.25}, worst ratio 0.420719. EXCLUDED: every asymptotic statement, the two imported theorems (arXiv:2512.01739v2 Thm 3.1(ii) and arXiv:1503.05121v3 (1.12)), and the c/3 mesh balance itself, which is read from the report.","comparison":"Byte-identical to ledger_evidence.json (sha256 6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a). All numeric fields are rounded to 9 decimals so a different libm cannot shift them; if one nonetheless does, accept when all 15 boolean checks are true and every numeric field agrees to 1e-6."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/1968/transcript","files":[{"sha256":"a8b9e2b046334a4d4bc1add819e0cabd1d034ce86cc4e103e0ec18768975244c","name":"prime_band_transfer.py","bytes":11348},{"sha256":"ddccc860211fc735b09792ddfc8cffc47689a16a4414ecb601e4eb515da84364","name":"verify_prime_band_ledger.py","bytes":9033},{"sha256":"b0ae713fa21b515a94196d732d81b121945272b4c9952480c1c29df31c0b724a","name":"test_prime_band_transfer.py","bytes":8334},{"sha256":"6e913353ef8fa5a7daa5edd793309fc6eecbaf2ede7ebc7c8e0344eb4cf1fc2a","name":"ledger_evidence.json","bytes":3214},{"sha256":"8d2b3b105882d07b73d520d52565f742845e138653641b147a3a9ef36d1868b8","name":"RESULT.md","bytes":12522}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}