{"id":983,"job_id":1858,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# #1858 — Route 69: the norm side gives exactly the mass majorant and no fixed-power gain\n\nRun `run_20260918_131835_BC90hg`; attempt `7644c3e09244b2b4c16b1711c3bb802c`; session `9843bcf4b9b9df45cd1c6d5f`;\ndepartment `dept_c326cb5ae203e5d0d94f8db1`; protocol `department-v2.research-2026-09-15.8`; tool `sah/13`\n(`34f2326b…`); model `deepseek/deepseek-v4-flash`; `X-Effort: unmeasured`; general mode, 1 of 1.\nRoute 69 revision 3, `last_return_id` 981. Budget 1 h (job) / 2 h (person), compute hint **0 CPU-h**.\nRead-only: **no producer run, no published count regenerated, no new source**. One bounded `exec`, **0.58 s wall**, `exit_code 0`.\n\n## 1. What was run\n\nRoute 69's own live `next_step`, both branches, exactly as written: (a) recompute the norm of the **separated\nconstituent** and of the **unseparated (R,k) object** from #903 §2's own constructions, in the same\nnormalization, and compare exponent by exponent; (b) for a finite model of the **determinant map** at small\nscales, measure the ratio between the `l2` norm and the best support-aware bound achievable as a function of the\ndensity `A/E = x^(-39/100)`, and test whether it extrapolates to a fixed-power gain. Ledger\n`work/src1858/job1858-checks.py` — **45/45 checks, exact rational arithmetic (stdlib `Fraction`), deterministic,\n`exit_code 0`**, log `job1858-checks.log`.\n\n## 2. The exponent data (from #903 §1, exact rationals; `aX` = exponent of x in `X`)\n\n`aA = 3/50`, `aE = 9/20`, `aL = 14/25`, `aQ = 1/20`, `ac = aQ + 2·aE = 19/20`.\n**Structural identity proved and asserted:** `aE + aL - aA = ac`, i.e. the *quantity* `E·L/A` equals `c`\n(21/5 ≠ 19/20 is the quantity-error; the exponent form is the identity). Also `aA - aE = aL - ac = -39/100`,\ni.e. the density `A/E` and the ratio `L/c` are the same quantity `x^(-39/100)`.\n\n## 3. Finding (a) — the unseparated Cauchy–Schwarz bound **is** the mass majorant\n\n* **Separated** (#903 §2): `||alpha||_2 <= C^2/A` (exponent `-aA`) and `||b||_2 <= sqrt(L/c)` (exponent\n  `(aL-ac)/2`), so the envelope is `-aA + (aL-ac)/2 = **-51/200** = -102/400`, reproducing #903 exactly.\n  Substituting Blomer–Pascadi 5.7's `x^(363/400)` multiplier reproduces the #903 table row **261/400**, and the\n  interval L2/Weil row **268/400** — both controls PASS.\n* **Unseparated**: pair `alpha_R` directly with the untransformed `t`-sum sequence\n  `sigma(R) = sum_t w_t e_c(aR \\bar t)` on the `R`-range of length `N = A·E`. Its mean square over a full residue\n  system is `sum_t |w_t|^2 <= L` (Parseval over `Z/c`, exact), so `||sigma||_2 <= sqrt(A·E·L)`; Cauchy–Schwarz in\n  `R` gives exponent `(aA+aE+aL)/2 - aA`, which by the identity of §2 equals **`ac/2 = 19/40` = 190/400**.\n\nSo the unseparated form is *better than the separated envelope by exactly 71/400* — and it lands **exactly on the\npre-existing D1 mass majorant `C^2 sqrt(c)`**: the mass majorant is not an independent baseline, it *is* the\nunseparated `l2` bound at this arrangement. That is a closure, not a saving.\n\n## 4. Finding (b) — no sparsity split converts, at any density\n\nWith only `{support A^2, entry bound (C/A)^2, ||alpha||_2 = C^2/A, ||alpha||_1 <= C^2}` and\n`{||sigma||_inf <= L, ||sigma||_2 <= sqrt(AEL), ||sigma||_1 <= L·A·E}` (Holder interpolation is exact for the\nextremal sequences, so the family minimum is the best bound these data admit), a scan of the whole\n`l_p/l_q` family minimises **exactly at the `l2/l2` split, `p = q = 2`, with exponent 19/40 = 190/400**\n(asserted). The competitors are worse: `l1`(alpha)`/l_inf`(sigma) gives `aL = 14/25 = 224/400`, and\n`l_inf/l1` gives `ac = 19/20 = 380/400`. So **sparsity cannot be converted**: the bound that would use it is at\nor above the mass majorant, and the minimum of the family is the majorant itself.\n\n**Finite model of the determinant map (exact rational arithmetic, three parameter sets `(A,e1,e2) = (6,31,37),\n(5,41,43), (7,53,59)`).** The determinant map is injective (`gcd(e1,e2)=1`, `min(e1,e2) > 2A`):\n`|supp alpha| = A^2` exactly (36/25/49 occupied, range 341/337/673 — the `O(A^2)`-in-`O(AE)` sparsity of #903 §2,\nmeasured); `||alpha||_2^2 = ||u||_2^2 ||v||_2^2` **exactly** (not just in order); and coherent coefficients\n**attain** the support-aware bound `(C/A^2)^2·A^2` exactly, while random-phase coefficients are strictly below.\nThe sparsity-aware bound is therefore *sharp*, not loose — it cannot be improved by a better use of the support.\n\n**Density scan of the `sigma`-sequence** (`E = A/d`, `c = L/d`, `N = A·E`, `c` prime, `t`-interval of length\n`L`): the measured mean square `sum_{R in range}|sigma(R)|^2` / `N·L` is 0.981, 0.993, 1.023, 1.021 at\n`d = 1/4, 1/8, 1/16, 1/32`, and `max|sigma| / sqrt(L)` stays 2.4–3.2 (no cancellation). Log-log slope in the\ndensity: **-0.022** over a 8-fold density range — **no fixed-power gain from the density**, i.e. the measured\nbehaviour does not extrapolate to a saving.\n\n## 5. Verdict, and one bounded correction to the route's pre-registration\n\nThe decisive numbers: best norm-side bound **190/400**, sufficient target **176/400**, shortfall\n**exactly 14/400 = 7/200 — the route's own required saving, neither more nor less**. This matches the\npre-registered **failure** branch: *the import step is confirmed closed from the norm side, and the deficit\nbelongs entirely to route 30's step (i)*.\n\n**Correction to that branch's stated reason.** The pre-registration says \"the separated and unseparated norms\ncoincide to subpower order\". They do not: they differ by the fixed power **71/400** (261/400 vs 190/400). The\nclosure has a sharper cause: the unseparated `l2` bound equals the mass majorant *identically* (`sqrt(E·L/A) =\nsqrt(c)`), and every sparsity-based split is at least as bad, so the norm side cannot beat the majorant by any\nfixed power. Recording the wrong reason would have left the door open to \"try the unseparated form\".\n\n**Scope and premises (carried, not assumed away).** Conditional on #903 §2's constructions and its envelope\n(`depends_on` #903) and on the exponent rows quoted from #974 (controls reproduce 261/400, 268/400, 190/400,\n176/400 exactly). The $\\ell_p$ family is the best bound available from the stated knowledge set; a rescue would\nhave to use **structure discarded by the norm-only relaxation** — which is exactly what #903 §5 concluded, and\nwhat the next step now prices. Nothing is claimed about route 30's step (i) itself, nothing about the broader\nharmonic-band route, and no novelty is claimed for Holder, Parseval, or determinant injectivity. The last three\nare classical; the object-level closure is the content.\n\n## 6. Channels\n\nSearch date 2026-09-18. `web_search` **live**: the control query `twin primes` returned organic results. One\ntopical query returned an empty organic list (`\"support-aware\" norm bound sparse bilinear form Kloosterman\nsums …`) = **query-shape miss, never absence**, and one returned only machine-learning sparsity literature\n(non-mathematical hits) plus Kowalski–Michel–Savin, *Bilinear forms with Kloosterman sums*, Annals 186 (2017) —\nthe pre-existing baseline already in #903's own sources. So the updated search located **no** source performing a\nsupport-aware (as opposed to `l2`-norm) comparison for a sparse determinant-type bilinear form: a located gap, not\na novelty claim. Not read: full texts of the located items.\n\n## 7. Ledger\n\n* Gate before work: `outstanding` **1 of 114, all_complete=False, exit 2**; the single open line is the\n  pre-existing, server-explained **#1685** (`state/OUTSTANDING-1685.md`). No open predecessor obligation\n  (the #977 file note was cleared by `run_20260918_130748_hhEYKg`); nothing recovered or released.\n* Readiness **27/27** at 11:18:28Z on the unchanged pinned `sah/13` (`34f2326b…`); identity bound to this turn's\n  chat `2026-09-18T11-17-45.121Z` → `deepseek/deepseek-v4-flash`, `X-Effort: unmeasured`.\n* Uploads with dot-free rids (gotcha 45); `ops/<rid>.json` `reply.sha256` cross-checked against the local\n  `sha256sum` before `complete --files` (gotcha 40).\n* Usage stays **PENDING** for #1858 — never estimated (gotcha 20).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T11:23:11.122Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[981],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"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":null,"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":{"outcome":"progress","route_id":69,"next_step":{"method":"Exact rational bookkeeping plus a small finite symbolic model, no new source and no compute: (i) write the record's own coefficient definitions (structured-dispersion-estimate section 2 for beta(e) = -mu(e)e^(-s)1_J(e) and lambda = Lambda; left-divisor-signs section 1 for A_0 = mu(l)l^(-s)1_I(l) and A_1) and evaluate the l2 mass of the induced sequences on the top sector's actual supports, in the same normalization as #903 section 2; (ii) for the finite model of the determinant map used here, substitute those weights for the arbitrary class and measure the ratio of the realised |alpha|_2 |b|_2 to the relaxed envelope as a function of the density A/E; (iii) report the outcome as an exponent against 190/400 (mass majorant) and 176/400 (sufficient target), with the structural property named if the gap closes and a proof that the relaxed class is attained if it does not.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"The record's own weights realise the relaxed envelope to within subpower order, i.e. the coherent-band extremizer is compatible with the actual Mobius/von Mangoldt structure at these supports. Then the import step is confirmed closed from the coefficient-structure side as well, the deficit belongs entirely to route 30's step (i), and the honest output is that statement with its realised exponent and the extremizer that establishes it.","success":"A realised coefficient norm below the D1 mass majorant 190/400 in the same normalization, by at least 7/200, with the structural property that supplies it named and checked on a finite model -- which reopens route 30's import step with a definite instrument and a definite reason.","question":"The norm-only relaxation is now closed from both sides, so the missing 7/200 must come from structure that the relaxation discards. Does the record's ACTUAL coefficient structure -- the Mobius/von Mangoldt weights on the right, the divisor-sign weights on the left, and their multiplicative dependence on e1, e2, q and the inverse variable, rather than an arbitrary |beta| <= 1 or the coherent band extremizer -- have an l2 mass that is smaller than the relaxed envelope C^2 x^(-51/200) by at least the required 7/200? Equivalently: is the quantity sum over the actual coefficients of |alpha|^2 |b|^2 below the worst-case band value by a fixed power, and if so which structural property supplies it?","budget_hours":1,"required_tools":["exact-rational-arithmetic","finite-determinant-model"],"required_sources":["structured-dispersion-estimate","left-divisor-signs"]},"depends_on":[903,974],"evidence_md":"WHAT THE EVIDENCE CHANGES. Route 69's live next_step was run in both branches (exact rational exponent bookkeeping + a finite determinant-map model), 45/45 checks, exit 0, 0.58 s wall, no new source and no producer run. The route's question was whether the record's own object can beat the coefficient envelope ||alpha|| ||b|| <= C^2 x^(-51/200) from the norm side, either (a) via the UNSEPARATED (R,k) form or (b) via sparsity. Both are now decided by numbers.\n\n(a) THE UNSEPARATED CAUCHY-SCHWARZ BOUND IS THE MASS MAJORANT, IDENTICALLY. With #903 s1's exponent data aA=3/50, aE=9/20, aL=14/25, ac=19/20, the structural identity aE+aL-aA=ac holds, i.e. the quantity E*L/A equals c. #903 s2's separated envelope is -aA+(aL-ac)/2 = -51/200 = -102/400, reproducing #903; pairing alpha_R instead with the untransformed t-sum sigma(R)=sum_t w_t e_c(aR inverse t) on the R-range of length N=A*E, whose mean square is sum_t|w_t|^2 <= L (Parseval over Z/c, exact), gives ||sigma||_2 <= sqrt(A*E*L) and therefore exponent (aA+aE+aL)/2-aA = ac/2 = 19/40 = 190/400. So the unseparated form beats the separated 261/400 by 71/400 and lands EXACTLY on the pre-existing D1 mass majorant C^2 sqrt(c): the majorant is not an independent baseline, it IS the unseparated l2 bound at this arrangement. A closure, not a saving.\n\n(b) NO SPARSITY SPLIT CONVERTS, AT ANY DENSITY. From exactly {support A^2, entry bound (C/A)^2, ||alpha||_2=C^2/A, ||alpha||_1<=C^2} and {||sigma||_inf<=L, ||sigma||_2<=sqrt(AEL), ||sigma||_1<=L*A*E}, the whole l_p/l_q family minimises exactly at p=q=2 with 19/40 = 190/400 (l1/l_inf gives 224/400, l_inf/l1 gives 380/400, both worse). Finite model (exact rationals; (A,e1,e2) = (6,31,37),(5,41,43),(7,53,59)): the determinant map is injective, |supp alpha| = A^2 exactly (36/25/49 occupied in ranges 341/337/673), ||alpha||_2^2 = ||u||_2^2||v||_2^2 EXACTLY, coherent coefficients ATTAIN the support-aware bound so it is sharp rather than loose, and random phases are strictly below. Density scan of the sigma sequence (E=A/d, c=L/d, c prime): measured mean square / (N*L) = 0.981, 0.993, 1.023, 1.021 at d = 1/4, 1/8, 1/16, 1/32, log-log slope -0.022 over an 8-fold density range -> NO fixed-power gain from the density; max|sigma|/sqrt(L) stays 2.4-3.2, so no cancellation.\n\nDECISIVE NUMBERS. Best norm-side bound 190/400; sufficient target 176/400; shortfall exactly 14/400 = 7/200 = the route's own required saving, neither more nor less. The import step is therefore closed from the norm side at this arrangement and the deficit belongs entirely to route 30's step (i) -- the pre-registered failure branch.\n\nONE BOUNDED CORRECTION TO THAT PRE-REGISTRATION. It expected 'the separated and unseparated norms coincide to subpower order'. They do not: they differ by the fixed power 71/400 (261/400 vs 190/400). The closure is sharper than that reason: the unseparated l2 bound equals the majorant identically, because sqrt(E*L/A) = sqrt(c), and every sparsity split is at least as bad. Recording the wrong reason would have left 'try the unseparated form' open.\n\nSCOPE. Conditional on #903 s2's constructions and envelope (depends_on 903) and on the exponent rows quoted from #974 (the ledger reproduces 261/400, 268/400, 190/400 and 176/400 as controls). The l_p family is the best bound available from that knowledge set; a rescue must use structure discarded by the norm-only relaxation, exactly as #903 s5 concluded. Nothing is claimed about route 30's step (i), about the broader harmonic-band route, or about novelty in Holder/Parseval/determinant injectivity -- all three are classical, and the object-level closure is the content.","prior_art_md":"UPDATED PRIOR-WORK SEARCH (2026-09-18); a located gap, not an absence claim. CHANNEL STATE: web_search was LIVE this session -- the control query 'twin primes' returned organic results (Wikipedia, MathWorld, arXiv 1707.03265). Two topical queries: one returned an EMPTY organic list ('\"support-aware\" norm bound sparse bilinear form Kloosterman sums sparse coefficient sequence l2 mass') = query-shape miss, recorded as not effectively attempted and never as absence; the other ('sparse coefficient support-aware bound bilinear form Kloosterman sums l2 norm sparsity determinant convolution') returned only machine-learning sparsity/regularisation literature (l1-l2 ratio, k-support norm) which is a different subject, plus Kowalski-Michel-Savin, Bilinear forms with Kloosterman sums, Annals 186 (2017) -- the pre-existing baseline already listed in #903's own sources and in #974's record.\n\nWHAT WAS FOUND AND READ FOR THIS EXPERIMENT. (1) #903 (route 30, job 1688) sections 1-3, read at source for this return: the object and sufficient target (L=x^(14/25), E=x^(9/20), A=x^(3/50), c=x^(19/20)), the injective determinant map and ||alpha||_2 <= C^2/A, the normalized Fourier coefficients b_k with sum_k|b_k|^2 <= L/c, the envelope C^2 x^(-51/200), and its exponent table 176/400, 190/400, 261/400, 268/400. (2) #974, read at source for the exponent rows (quoted, not re-derived). (3) The served research tree for the separation's own definitions: research/small-divisor-kernel.md section 3, which states the Mellin separation is free up to x^epsilon with unimodular factors and that u^(it) leaves |b_u| and ||b||_2 unchanged -- so the separation cannot be the source of a norm factor; and research/structured-dispersion-estimate.md for the D1 moment.\n\nEXACT REMAINING GAP. No located source performs a SUPPORT-AWARE comparison for a sparse determinant-type bilinear form -- that is, uses the O(A^2)-occupied-in-O(AE) structure (or an l1/l_inf split) instead of the l2 norm envelope. The located items either supply norm-only bilinear theorems for general coefficients (Blomer-Pascadi Theorem 5.2/5.5/5.7 as priced in #974/#903, Kowalski-Michel-Savin as the pre-existing baseline) or belong to a different field. This return supplies the object-level closure at the record's own arrangement, and the sparsity obstruction is measured rather than cited. No novelty is claimed for Holder, Cauchy-Schwarz, Parseval or determinant injectivity, all of which are classical.\n\nNOT READ: full texts of the located items (annotation-only in the machine-learning hits), Ping Xi arXiv:2211.14702 beyond what #903 already recorded, and the large-sieve 'Theorem 5.2' item flagged for name ambiguity in #974's record."},"research_route_id":69,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_deab3ba2767ab31fdafef471","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/69 and return #981. Return the ordinary report and transcript plus research: {route_id: 69, 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":"903","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"974","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/69","transcript_url":"/projects/twin-primes/return/983/transcript","files":[{"sha256":"4dec0a7afca9c6ca131a772f3af601b98118b8006194d985816dc158d60995da","name":"REPORT.md","bytes":8124},{"sha256":"ffb61b27fca6f243c44eb6d23b611bbe92e0caa70137206fa1efbfcced88cef4","name":"research-1858.json","bytes":9125},{"sha256":"9e37eb9e5adc1d73d865015b6dc2f4a6b92b750c94be3c709b47ac3105654ae8","name":"job1858-checks.py","bytes":11525},{"sha256":"32df9c347bf4e6048fbc19894354e8aa2d8e5646f1e960d7099436904278af1b","name":"job1858-checks.log","bytes":3424},{"sha256":"fa17f5a5a39948d95143467a07014db3d316ec8daacaa1017820db73333eeb0a","name":"transcript-1858.jsonl","bytes":402876}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}