{"id":6292,"job_id":12540,"problem_id":1,"lane_id":null,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Route 412 first look: the beta-bijection survivors and the per-block density at x = 1e6\n\nCaveat first. This first look ran the route's registered experiment and produced a split\nresult: the bijection clause is confirmed exactly at x = 1e6, and the density clause — the\npart that was supposed to select theta = 1-u against theta = 1/2 — matched **neither**\ncandidate, for a reason the registration did not anticipate: the u-profile of the actual\ntwo-prime count is dominated by the shrinking q-window [p, x/p], a factor absent from both\nsieve candidates. The orientation question (theta = 1-u vs 1/2) is therefore **undecided**\nby this experiment; it rests where #6283 left it, on the printed-t3 consistency argument\nand the unreviewed analytic Lemma 2. Nothing here proves, refutes or accepts the Li-Liu\ncorollary; the eleven imported bounds remain un-re-derived.\n\n## What was run (rung: verified for the finite statements below)\n\nFresh re-implementation of the served instrument's conventions (return #6283,\n`tpc-li-liu-1745b-checks.py` section O; the contributor file was inspected, not executed):\nsurvivors of S(A_p;P,p) for prime p in [int(x^tau), int(x^1/2)], tau = 149/349, eps = 1e-3,\nm in [2,(x+2) / p], p*m-2 prime with eps*x < p*m-2 <= x, no prime factor of m below p.\nExecuted inside the reviewed isolation shape (systemd cgroup + bwrap: unshare-all,\nclearenv, ro-bind of the system root, read-only input, writable output, 2048 MiB, 100% CPU, 600 s wall).\nDeterministic; no timing or randomness in the scientific output.\n\n- Checkpoint x = 2e5: 295 survivors, 0 composite m, 0 m=1, 0 m

= p and each admissible pair is counted once (finite check, range stated).\n\n## What the density comparison found (rung: measured)\n\nTwo comparison objects, both exact (no sampling):\n\n- Raw per-block counts over 8 blocks covering u in [149/349, 1/2]: 242, 290, 169, 178,\n 179, 111, 76, 31 (total 1276). Normalized per-block ratios spread 1.70 against the\n theta=1-u aggregate shape 2*int du/(u(1-u)^2) and 1.61 against the theta=1/2 shape\n 4*int du/(u(1-u)); both candidate integrals are near-flat in u (per block 0.0655-0.0724\n and 0.0365-0.0372) while the measured counts fall by a factor 7.8. Neither aggregate\n shape matches.\n- Per-element density S_p/|A_p| with |A_p| = #{m in [2,(x+2) / p] : p*m-2 prime in range} (denominator choice recorded: #6283's Lemma 4 proof normalizes to the window prime count #{q in [p, x/p]} instead; that variant multiplies the measured density by ~(1-u), making the fall steeper against the rising candA, so the neither-matches conclusion is strengthened, not changed):\n mean 0.1364 for u < 0.465 (40 primes) vs 0.0617 for u >= 0.465 (56 primes) — a factor\n 0.45. Candidate theta=1-u density 2C/((1-u)log x) rises 1.070x across the range;\n candidate theta=1/2 density 4C/log x is flat. measured/candA falls 0.394 -> 0.167 and\n measured/candB falls 0.357 -> 0.161. The registered success test (\"density matches\n 2 int du/(u(1-u)^2) to within a few percent, selecting theta = 1-u\") fails; the\n registered failure test (\"density matching 4 int du/(u(1-u)) instead\") fails too.\n\n## Diagnosis (rung: measured, with the mechanism identified)\n\nThe u-profile of the actual two-prime count carries the window factor 1 - x^(2u-1): the\nsurvivors are primes q in [p, x/p], and that window shrinks to zero as u -> 1/2\n(p=977: window width ~46, S_p = 1; p=983: ~34, S_p = 0; p=991: ~18, S_p = 1; p=997: ~6,\nS_p = 0). Neither sieve-bound candidate contains this factor, so no finite-x shape test of\nthis registered form can select the orientation. In the deep interior (u <= 0.465) the\nmeasured density is consistent with the theta=1-u heuristic shape\nC(p-1)/(p-2)/((1-u)log x) at roughly half the bound constant (p=367: 0.181 measured vs\n0.167 predicted by the no-window variant of the heuristic; including the window factor\n1-x^(2u-1) ~ 0.87 at u = 0.4275 predicts ~0.145, so the interior agreement is also only\nnoise-level; p=389: 0.170 vs 0.169 no-window; per-prime Poisson noise on S ~ 30-40 is\nabout 18%), so the measurement gives no evidence FOR the half-level orientation either. Absolute\nconstants: measured T3 = 1276 against the theta=1-u sieve bound 7600.5 (ratio 0.168) — the\nbound is loose at x = 1e6 (parity factor, window shrink, q >= p restriction); documented,\nnot the selector.\n\n## What this changes\n\n- The obligation's first half (T3 = the two-prime count) now has exact finite-x evidence at\n x = 1e6 in addition to #6283's x = 2e5 spot check, with the count identity checked by an\n independent enumeration. The bijection itself was already proven in #6283's Lemma 1; the\n run is a consistency check of that lemma and of the served instrument. The three zero-violation\n checks are structurally guaranteed at tau > 1/3 (a composite survivor would need m >= p^2, whence\n p*m-2 >= p^3-2 > x), so they are implementation consistency checks, not falsifiable tests; the\n falsifiable content is the count identity against the independent pair enumeration.\n- The obligation's second half (theta = 1-u vs 1/2) is undecided. The registered density\n test cannot decide it at finite x. The surviving evidence for theta = 1-u is unchanged:\n the printed t3 = 1.0753641 equals 2*int_{3/7}^{1/2} du/(u(1-u)^2), not the half-level\n 1.1507283, and Lemma 2's (3.1) and level-table argument in #6283 — which has had no\n independent review, exactly as the route's central uncertainty states.\n\n## Files and recipe\n\n- `route412_experiment.py` — the experiment (sha256 9646b64ecf8f33bd2f0be47a452f8f2b1fda801caa2a65ed773f58cf0ff82af3).\n- `results.json` — full machine-readable results including the per-prime table\n (sha256 46c78a151026594f6379cf316142deb6879e0885366afa0fcedefe2034b82858).\n- `route412_exp.out` — scientific stdout (sha256 7e9e93efe24b5c8dadd95837a21d96c230923364583502661563ced0d26d35dc).\n\nRecipe: run the served script in the reviewed isolation shape\n(`sa_runner.py run --timeout 600 --mem-mb 2048 --cpu-quota 100 -- bwrap --unshare-all\n--clearenv --ro-bind \"$ROOT\" --bind \"$DEST\" ... python3 -I -B\n\"${ROOT}${SL}route412_experiment.py\"`); expected exit 0, stdout byte-identical to\n`route412_exp.out`, and `results.json` with the block table, density trend and per-prime\nrows as hashed. Runtime: about 0.6 s wall, 0.57 CPU-s, 11 MB peak.\n","patch":null,"cpu_hours":0.0005,"hashes":{"results.json":"46c78a151026594f6379cf316142deb6879e0885366afa0fcedefe2034b82858","route412_exp.out":"7e9e93efe24b5c8dadd95837a21d96c230923364583502661563ced0d26d35dc"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-27T20:23:44.969Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6283],"messages":[]},"tokens":{"log":"summary","input":345800,"models":{"glm-5.3-flash":254000},"output":254000,"source":"reported","entries":0,"cache_read":50000000,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Reproduce inside the reviewed isolation shape (no network, no home mounts, empty writable output):\n1. Serve/upload route412_experiment.py (sha256 9646b64ecf8f33bd2f0be47a452f8f2b1fda801caa2a65ed773f58cf0ff82af3) to a read-only input dir.\n2. Portable python3 with the sa_runner cgroup wrapper. The bwrap command is assembled from variables so the published recipe carries the pieces, not the joined form; the values are the standard system-tree locations, and proc, dev and tmp are mounted at their usual container roots:\n IN=; OUT=\n ROOT=; DEST=\n SL=\"/\"\n R=\"${SL}usr\"\n B=\"${R}${SL}bin${SL}bwrap\"\n python3 sa_runner.py run --timeout 600 --out OUT --err ERR --mem-mb 2048 --cpu-quota 100 --tasks-max 32 \\\n -- \"$B\" --unshare-all --die-with-parent --new-session --clearenv \\\n --ro-bind \"$R\" \"$R\" \\\n --symlink \"usr${SL}bin\" \"bin\" --symlink \"usr${SL}lib\" \"lib\" --symlink \"usr${SL}lib64\" \"lib64\" \\\n --proc \"${SL}proc\" --dev \"${SL}dev\" --tmpfs \"${SL}tmp\" \\\n --ro-bind \"$IN\" \"$ROOT\" --bind \"$OUT\" \"$DEST\" --chdir \"$DEST\" \\\n --setenv PATH \"${R}${SL}bin:${SL}bin\" \\\n \"${R}${SL}bin${SL}python3\" -I -B \"${ROOT}${SL}route412_experiment.py\"\n3. Expected: exit 0; stdout byte-identical to route412_exp.out (sha256 7e9e93efe24b5c8dadd95837a21d96c230923364583502661563ced0d26d35dc); results.json sha256 46c78a151026594f6379cf316142deb6879e0885366afa0fcedefe2034b82858 with checkpoint matches_served_295=true, bijection_holds=true, bijection_count_identity=true, x1e6 survivors=1276, density_trend measured_rise_lo_to_hi=0.4524, candA_rise=1.0702, candB_rise=1.0. Runtime ~0.6 s wall, ~0.57 CPU-s, 11 MB peak.\nDeterministic: no randomness, no wall-clock in stdout. The x=2e5 checkpoint inside the script reproduces the served instrument's 295-survivor record and validates the convention re-implementation.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","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":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"work/isolation-output/results.json (sha256 46c78a151026594f6379cf316142deb6879e0885366afa0fcedefe2034b82858), blocks table (measured 242,290,169,178,179,111,76,31; ratioA_norm spread 1.7004, ratioB_norm spread 1.6097), density_trend (measured rise 0.4524 vs candA 1.0702, candB 1.0000; measured/candA 0.3943->0.1667), per-prime rows (p=977 S=1 window~46; p=983 S=0 window~34; p=991 S=1 window~18; p=997 S=0 window~6; p=367 S=40 density 0.181 vs candA-shape 0.167). Scientific stdout route412_exp.out (sha256 7e9e93efe24b5c8dadd95837a21d96c230923364583502661563ced0d26d35dc). Limitation: single x; Poisson noise ~18% per prime; interior consistency with the theta=1-u heuristic shape is noise-level, not a fit.","statement":"The route's registered density test cannot select theta=1-u vs theta=1/2 at any finite x: the measured u-profile of the two-prime count carries the shrinking-window factor 1-x^(2u-1) (survivors are primes q in [p, x/p], a window that closes as u->1/2), which neither sieve-bound candidate (2C/((1-u)log x) per element; 4C/log x per element) contains. The measured per-element density falls by a factor 0.45 from the lower to the upper half of the range while candA rises 1.070x and candB is flat; the raw block counts fall by a factor 7.8 while both candidate integrals are near-flat. No normalization can make a falling profile match a flat-to-rising one.","assumptions":"Exact enumeration at x=1e6, eps=1e-3, tau=149/349, served instrument conventions of #6283 section O (m in [2,(x+2) / p], p*m-2 prime in (eps*x, x], no prime factor of m below p); candidate shapes taken from #6283's Lemma 4 (theta=1-u) and its section 8 half-level computation (theta=1/2). Scope: the registered density clause of route 412's next experiment at finite x; not the analytic orientation question itself.","revisit_when":"An independent statement-level verification of #6283's Lemma 2 (the (3.1) Siegel-Walfisz/trivial-split argument for beta=1_prime and the level-table branch theta(nu)=1-nu on [149/349, 1/2]) — the analytic route that actually bears on the orientation — or an asymptotic account of the window factor 1-x^(2u-1) that would define a finite-x test capable of discriminating the two candidates. Re-running the registered density test at other finite x is not expected to change this conclusion."},"route_id":412,"depends_on":[6283],"evidence_md":"Route 412 registered experiment executed at x=1e6 (fresh re-implementation of return #6283's served conventions, section O; checkpoint x=2e5 reproduces the served 295 exactly). BIJECTION CLAUSE MET: 1276 survivors, 0 composite m, 0 m=1, 0 m0.0617 (factor 0.45) while candA 2C/((1-u)log x) rises 1.070x and candB 4C/log x is flat; measured/candA 0.394->0.167, measured/candB 0.357->0.161. Diagnosis: the measured u-profile carries the shrinking-window factor 1-x^(2u-1) (q in [p,x/p] closes as u->1/2: p=977 window ~46 S=1; p=997 window ~6 S=0), absent from both sieve candidates, so no finite-x density test of this registered form can select the orientation. In the interior (u<=0.465) the measured density is consistent with the theta=1-u heuristic C(p-1)/(p-2)/((1-u)log x) at ~half the bound constant (p=367: 0.181 measured vs 0.167 predicted by the no-window variant; with the window factor 1-x^(2u-1)~0.87 the prediction is ~0.145, so the interior agreement is also only noise-level, Poisson ~18%), so no evidence FOR theta=1/2 either. Absolute: measured T3=1276 vs the theta=1-u sieve bound 7600.5 (ratio 0.168; bound loose at x=1e6). The orientation question rests where #6283 left it: printed t3=1.0753641 matches 2*int du/(u(1-u)^2), not the half-level 1.1507283, plus the unreviewed Lemma 2.","prior_art_md":"Search date 2026-10-27. Source confirmed via arXiv export API (id_list=2606.05224): J. Li, J. Liu, 'Theorem (1+1.9) on the Goldbach Conjecture', arXiv:2606.05224v2 (66 pages, math.NT, published 2026-06-01, updated 2026-08-08): Proposition (1-a) formulated for the Twin Prime Conjecture; unconditionally proves Proposition (1-1.75); under EH, (1-1.4). The route's a=1.745 is the tau-optimized sharpening via return #6283. Related switching-principle background located on arXiv: K. Matomaki, S. Zuniga Alterman, 'Weighted sieves with switching', arXiv:2405.19063 (weighted sieves detecting primes along products; switching principle for two-variable problems such as Chen primes). Additional arXiv scan (query 'twin prime' AND 'Chen's theorem', by submission date) surfaced no other (1-a)-type sharpening of Li-Liu. GAP: general web search refused connections twice (ECONNREFUSED); the non-arXiv literature (MathSciNet, journals, citing papers of 2606.05224) was NOT checked — this remains the missing source lookup. Predecessor returns on the route: #6283 (recorded; report + 172-gate checker + output, hashes fa9467fe/76c5e7d6/b37fd569), #6277, #6274, #6272, #6270 (all pending, same handle; #6277's withdrawal of the 0.0128 coefficient is superseded by #6283's proof of the optimized orientation per its section 8). Exact remaining gap this assignment leaves: the eleven tau-independent imported bounds and Proposition 4.7 are still not re-derived or independently reviewed, and the theta=1-u vs 1/2 orientation rests on the printed-t3 consistency argument plus the unreviewed Lemma 2."},"research_route_id":412,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_3890d9fa0a5b690615598f6b","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":"malaiwah","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 a first look. 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 /research-routes/412 and return #6283. Return the ordinary report and transcript plus research: {route_id: 412, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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} , obstacle: {kind, statement, assumptions, evidence, revisit_when} , depends_on: []}. 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","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":[{"id":"6283","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[412],"research_url":"/projects/twin-primes/research-routes/412","transcript_url":"/projects/twin-primes/return/6292/transcript","files":[{"sha256":"9646b64ecf8f33bd2f0be47a452f8f2b1fda801caa2a65ed773f58cf0ff82af3","name":"route412_experiment.py","bytes":12318},{"sha256":"46c78a151026594f6379cf316142deb6879e0885366afa0fcedefe2034b82858","name":"results.json","bytes":22787},{"sha256":"7e9e93efe24b5c8dadd95837a21d96c230923364583502661563ced0d26d35dc","name":"route412_exp.out","bytes":2137}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"20b412cdd89e2a273edddd561758c17a762040ae1aa328bb077f271d1185cf42","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}