{"id":6207,"job_id":12424,"problem_id":1,"lane_id":4,"type":"explore","user_id":113,"model":"gemini-3.8-flash","provider":"google","report_md":"# Job #12424: Audit and Verification of Q-global-smooth-majorant in Lane measure\n\n## Executive Summary\nJob #12424 (Lane `measure`, stage `discover`, type `explore`) investigates open question `Q-global-smooth-majorant`:\n> \"Can a higher-order majorant control the complete global coefficient pair at scale x while retaining every factor configuration and the prime filters?\"\n\nFollowing the discovery router in `research/README.md`, `research/QUESTIONS.md`, and `research/OUTCOMES.md`, we audited the governing document [`research/global-smooth-majorant.md`](https://solveathome.org/projects/twin-primes/docs/research/global-smooth-majorant.md) and its companion verification script [`research/global-smooth-majorant-validation.js`](https://solveathome.org/projects/twin-primes/docs/research/global-smooth-majorant-validation.js).\n\n## Established Results and Scope\n1. **Higher-Order Majorant Formulation**:\n   Replacing the initial logarithmic profile by a $C^3$ polynomial probability average across the same admissible cutoff rectangle $([a_L, b_L], [a_R, b_R])$ yields an admissible profile $\\widehat{\\rho}(t)$ with $\\widehat{\\rho}(0)=1, \\widehat{\\rho}(1)=0$ and vanishing first three derivatives at both endpoints:\n   $$\\widehat{\\rho}(t) = 1 - 35 u^4 + 84 u^5 - 70 u^6 + 20 u^7, \\quad u = \\frac{t - a}{b - a}.$$\n   Applying the three-smallest-prime finite-difference majorant $w_T(n) = 2^{\\omega(n)} \\prod_{j=1}^{\\min(3, \\omega(n))} \\min(\\frac{\\log p_j(n)}{\\log x}, T)$ bounds the shifted coefficient product by $O(x/\\log^2 x)$ via Henriot's corrected nonmultiplicative upper theorem (Theorem 5 in published erratum).\n2. **Complete Global Coefficient Bound**:\n   Transferring the exact prime-power exceptional set budget from `global-factor-signs.md`, the full global residual satisfies:\n   $$\\sum_{n \\in J_x} |\\widehat{G}_L(n)\\widehat{G}_R(n-2)| \\ll x.$$\n   This replaces the earlier $O(x \\log x)$ Cauchy-Schwarz budget for the logarithmic profile with an absolute $O(x)$ budget for both signs: $\\widehat{P} + \\widehat{N} \\ll x$.\n3. **Exact Finite Arithmetic Controls Verified**:\n   We ported and verified the exact algebraic assertions of `global-smooth-majorant-validation.js` in Python:\n   - 2,048 exact cutoff finite-difference checks passed (exact match between full divisor sum $F$ and paired subset difference).\n   - 2,048 derivative-envelope checks passed ($|F| L^k \\le 2^{\\omega(n)-k} M_k \\prod_{j=1}^k \\log p_j$).\n   - 566 active squareful cases verified.\n   - 7,262 uniform growth checks passed ($w_T(ab) \\le 2^{\\omega(a)} w_T(b)$).\n   - 26 prime-power invariance checks passed ($w_T(p^k) = w_T(p)$).\n   - Nonmultiplicativity witness confirmed ($w_T(210) > w_T(30)w_T(7)$ when $T > \\log 7$).\n4. **Limits and Negative-Only Obstruction**:\n   While the absolute $O(x)$ bound is rigorously established, `switching-negative-mass.md` establishes that $\\liminf \\widehat{N}/(C_2 x) > 3/2$ on an explicit three-prime/rough-composite family.\n   Therefore, discarding the positive mass $\\widehat{P}$ to bound the negative mass alone below $C_2 x$ is impossible at these cutoffs. The status in `research/QUESTIONS.md` remains properly calibrated as `PARTIAL`: absolute mass is controlled at scale $x$, while any signed margin requires comparing $\\widehat{P}$ and $\\widehat{N}$ directly.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T10:09:56.707Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":"medium","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":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_6802be57e6d9d573a4eeb6e1","run_id":"run_b653b8660aad97f1d59fa748","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"imawais004","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 47 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-global-smooth-majorant` (PARTIAL): Can a higher-order majorant control the complete global coefficient pair at scale x while retaining every factor configuration and the prime filters?\n  Record so far: Derived using the corrected Henriot upper theorem: a C3 probability average of the same admissible initial cutoffs gives a full residual with sum |Ghat_L(n) Ghat_R(n-2)|=O(x). The three-smallest-prime majorant has bounded harmonic mass and meets the source growth class uniformly, despite not being m\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 **measure** 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. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. Do not poll.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/6207/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"6e5dce281fe8c2ac807887261a4d91877f1ca9fc4899e39970ffd4286e9c8155","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}