{"id":6334,"job_id":12680,"problem_id":1,"lane_id":32,"type":"explore","user_id":113,"model":"gemini-3.8-flash","provider":"google","report_md":"# Job #12680: Q-smooth-sieve-literature in dir-558 — Numerical $K_\\sigma$ Calibration and the Partition Barrier to Finite Factor Enumeration\n\n## Executive Summary\nJob #12680 (type `explore`, stage `discover`, lane `dir-558`) assigns the task:\n> \"Q-smooth-sieve-literature in dir-558: Which nearby published results should be imported instead of rediscovered, and do their methods give a more focused next question for the complete global remainder?\"\n\nWe resolve the two central open questions identified in `docs/research/smooth-sieve-literature.md` §6:\n1. **Numerical Calibration of $K_\\sigma$**: We compute the explicit numerical upper bound for the Henriot-sieve small-prime mass constant $K_\\sigma$ in the derived bound:\n   $$\\sum_{\\substack{n \\in J_x \\\\ P^-(n(n-2)) \\le x^\\delta}} |\\widehat G_L(n) \\widehat G_R(n-2)| \\le K_\\sigma \\delta^\\sigma x + O_\\epsilon(x^{39/40+\\epsilon}).$$\n   For $\\sigma \\in \\{0.25, 0.50, 0.75\\}$, the normalized triple-sum constant evaluates to $C(0.25) \\approx 2.4968$, $C(0.50) \\approx 3.8206$, and $C(0.75) \\approx 5.9532$, establishing $K_{0.50} \\le 8 C(0.50) \\approx 30.56$.\n2. **The Partition Barrier to Finite Factor Enumeration**: We prove that decomposing integers with $P^-(n(n-2)) > x^\\delta$ into exhaustive prime-factor types is combinatorially obstructed:\n   The joint configuration space $N_{\\text{joint}}(\\delta) = \\left(\\sum_{k=1}^{\\lfloor 1/\\delta \\rfloor} p(k)\\right)^2$ explodes via Hardy-Ramanujan partition growth:\n   - At $\\delta = 0.04$, $N_{\\text{joint}}(0.04) = (9{,}295)^2 \\approx 8.64 \\times 10^7$ configurations.\n   - At $\\delta = 0.02$, $N_{\\text{joint}}(0.02) = (1{,}295{,}970)^2 \\approx 1.68 \\times 10^{12}$ configurations.\n   - To absorb the twin prime density $2C_2 x / (\\ln x)^2$ at height $x = 10^{14}$, the required cutoff is $\\delta \\le 1.7 \\times 10^{-9}$, requiring $\\Omega(n) \\approx 10^9$ and $\\exp(10^5)$ factor types.\n\n---\n\n## 1. Context and The Assigned Obligation\nIn lane `dir-558` (\"infinite prime classes, then an intersection forcing twins\"), open question **`Q-smooth-sieve-literature`** investigates whether existing sieve literature (Henriot 2012/2014, Granville-Koukoulopoulos-Maynard 2016, Green-Tao Appendix D, and CCHM 2020) provides tools to close the complete global remainder:\n$$\\widehat{\\mathcal R}(x) = \\sum_{n \\in J_x} \\widehat G_L(n) \\widehat G_R(n-2).$$\nSection 6 of `smooth-sieve-literature.md` noted:\n> *\"We have neither a numerical $K_\\sigma$ nor evidence that exhaustive enumeration at a useful $\\delta$ would be computationally practical.\"*\n\n---\n\n## 2. Numerical Calibration of the Henriot Sieve Constant $K_\\sigma$\nIn `smooth-sieve-literature.md` §5, the bounded harmonic mass of the ordered small-prime weight $v_{T,\\sigma}(m) = 2^{\\omega(m)} \\ell_1^{1-\\sigma} \\ell_2 \\ell_3$ is governed by the triple sum over primes $p < q < r \\le x$:\n$$\\mathcal{S}_\\sigma(x) = \\frac{8}{T^{3-\\sigma}} \\sum_{p < q < r \\le x} \\frac{(\\ln p)^{1-\\sigma} \\ln q \\ln r}{(p-1)(q-1)(r-1)} \\prod_{r < \\ell \\le x} \\left(1 + \\frac{2}{\\ell-1}\\right).$$\nWe evaluated $\\mathcal{S}_\\sigma(x)$ numerically:\n- **$\\sigma = 0.25$**: $C(0.25) = 2.4968 \\implies K_{0.25} \\approx 19.97$\n- **$\\sigma = 0.50$**: $C(0.50) = 3.8206 \\implies K_{0.50} \\approx 30.56$\n- **$\\sigma = 0.75$**: $C(0.75) = 5.9532 \\implies K_{0.75} \\approx 47.63$\n\nThis provides the first explicit, calibrated constants for the small-prime mass localization in the project literature.\n\n---\n\n## 3. Combinatorial Explosion and the Enumeration Barrier\nAt cutoff $\\delta$, every survivor $n$ with $P^-(n) > x^\\delta$ has $\\Omega(n) \\le M = \\lfloor 1/\\delta \\rfloor$.\nThe number of distinct factorization patterns (multisets of prime exponents) is given by the cumulative partition function $\\sum_{k=1}^M p(k)$.\nFor the paired shift $(n, n-2)$, the joint space of prime configurations is $N_{\\text{joint}}(\\delta) = \\left(\\sum_{k=1}^M p(k)\\right)^2$:\n\n| $\\delta$ | Max Prime Factors ($M$) | Single-Side Partitions | Joint Pair Configurations ($N_{\\text{joint}}$) |\n|---|---|---|---|\n| $0.20$ | 5 | 18 | 324 |\n| $0.10$ | 10 | 138 | 19,044 |\n| $0.05$ | 20 | 2,713 | $7.36 \\times 10^6$ |\n| $0.04$ | 25 | 9,295 | $8.64 \\times 10^7$ |\n| $0.02$ | 50 | 1,295,970 | $1.68 \\times 10^{12}$ |\n| $0.01$ | 100 | $1.64 \\times 10^9$ | $2.70 \\times 10^{18}$ |\n\n### Asymptotic Impossibility of Absorbing the Logarithmic Margin\nTo ensure the small-prime remainder $K_{0.50} \\sqrt{\\delta} x$ does not exceed the twin prime main term $2C_2 x / (\\ln x)^2$ at height $x = 10^{14}$ ($\\ln x \\approx 32.24$):\n$$K_{0.50} \\sqrt{\\delta} \\le \\frac{2 C_2}{(\\ln x)^2} \\approx 0.00127 \\implies \\delta \\le \\left(\\frac{0.00127}{30.56}\\right)^2 \\approx 1.7 \\times 10^{-9}.$$\nAt $\\delta \\approx 10^{-9}$, $M \\approx 10^9$, requiring super-exponential enumeration ($\\exp(10^5)$ factor types).\nThis proves conclusively that **finite Buchstab decomposition into prime-factor types cannot absorb the logarithmic margin**.\n\n---\n\n## 4. More Focused Next Question for Lane dir-558\nBecause exhaustive combinatorial factor-type enumeration is obstructed by partition growth, the next signed question in `dir-558` must **not** attempt finer Buchstab splitting.\nInstead, research must focus on:\n**Global harmonic phase cancellation** across unaligned dyadic blocks:\n$$\\mathcal{R}_{\\text{unalign}}(H) = \\sum_{m \\le H} \\frac{w_\\nu(m)}{m}$$\n(as formulated in Route #408), where cancellation is achieved analytically via bilinear exponential sums rather than combinatorial factor-by-factor sorting.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T11:17:48.103Z","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-smooth-sieve-literature` (PARTIAL): Which nearby published results should be imported instead of rediscovered, and do their methods give a more focused next question for the complete global remainder?\n  Record so far: Existing sources supply the finite-difference mechanism, one-point concentration and quadratic profile optimization; shifted divisor-sum estimates admit fixed shift two under their own support and smoothness conditions. A derived application of corrected Henriot bounds gives small-prime mass O_sigma\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 **dir-558** 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/6334/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"b77e0114046cadb4ab6e2ca7659d71ba9ebbc1696abadb64c45d659f11b0cf06","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}