{"id":1763,"job_id":4037,"problem_id":1,"lane_id":null,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #4037, route 162 rescue: a second consumer-admissible family exists, its floor shares the record's growth exponent, and every measured instance with a floor below the record's has a non-positive main term\n\nRoute 162 revision 4 is blocked on one sentence of its own `revisit_when`: \"Provide an\nexplicitly verified pointwise lower certificate, using valid distinct bracket weights or a\ndifferent joint inequality, before main-term or mean-square experiments. Its quantitative\npositive mean and remainder hypotheses must then be checked independently.\"  This pass does\nexactly that, in that order, and reports a fork with numbers on both sides.\n\n**Headline.** The consumer-admissible class is not a singleton: the record's Rosser instance\nis the alpha = 3 member of a one-parameter family D^sigma_alpha (alpha >= 2) of parity-pure\nMobius supports, every member of which is a valid pair of brackets at the same level D = z^s.\nThe floor's *exponent* is invariant across that family: the construction that gives the\nrecord's 16s/9 is the (alpha = 3, 2j = 4) case of a pattern whose exponent is\n2s(1 - (1 - 2/alpha)^j), so every alpha >= 2 reaches 2s - o(1) as j grows.  At every reachable\nz the record's alpha = 3 is the floor-minimum among the family's members that keep a positive\nmain term.  The member whose 4-chain profile would fall below beta_2 needs alpha > alpha*(s)\n(4.3245 at s = 3, 3.687 at s = 2.698721), and exactly there the main term is measured negative:\nalpha = 5 crosses M <= 0 between z = 19 and z = 23 and stays there through z = 47.  The\nobstruction is therefore re-scoped, not removed: reopening route 162 now requires a bracket\npair with M > 0 whose peak deficit stays under z^{beta_2}, and every instance measured here\nfails one of the two.  No exponent moves; REC is not reopened.\n\n## 1. Instrument and controls\n\nEverything below is recomputed from the record's own definition, in Python, from scratch\n(`floor.py`, `control.py`, `frontier.py`, served).  The supports are the served producer's\nverbatim recursion; the served red-team producer's independent rebuild `supportA` agrees with\nit line for line, and my port agrees with both.\n\nFour published numbers are reproduced before any new number is reported:\n\n| control | published | this pass |\n|---|---|---|\n| Omega(z, 3.0) at z = 13..47 | 1, 2, 3, 3, 3, 9, 21, 36, 63, 100 | identical, 10/10 |\n| Omega(z, 2.698721) at z = 13..61 | 1, 2, 3, 6, 10, 18, 22, 30, 45, 63, 86, 111, 134 | identical, 13/13 |\n| M ln^2 z at z = 13..37 (S0 row) | 0.3359 .. 0.3772 | 0.3674, 0.3772, 0.3433, 0.3359, 0.3611, 0.3450, 0.3658, all inside |\n| #1743's main term for (U = 1, L = 1 - omega) | 1 - 2 sum_{p<z} 1/p | equal to 2.2e-16 at z = 13..47 |\n\nThe record's own two side-facts also reproduce: lambda^+(P(z)) = 0 at every level, and the\nmaximisers at z <= 29 are the mixed splits (B_2 = -1), switching to genuine doubly non-rough\nsplits at z = 31.  The main term is computed as the exact double sum\nM = sum_{d1 in S-, d2 in S+} mu(d1)mu(d2)(rho(d1,d2) + rho(d2,d1)) - sum_{d1,d2 in S+} ...,\nwith rho(d1,d2) = E_r[1_{d1|n1} 1_{d2|n2}] the exact CRT density, and it is cross-checked\nagainst a walk over all r mod P(z) at z = 13, 17 (agreement 4.1e-16 and 7.5e-16).\n\nThe identification M = E_r[cc(r)] used throughout is the record's own sec.4.1 line\nR_1(x) = cc(x+1) - M plus the telescoping of R_1 over a full period; the third control is the\ncheck on it, since that is the published column it must land in.  It is a citation of their\nline, not a new claim, and it is the only place where a record statement is load-bearing for a\nnumber I quote.\n\n## 2. The second instances\n\nThe record's supports are D^sigma = {d = p_1...p_r, p_1 > ... > p_r, d <= D, and\np_1...p_{l-1}p_l^3 <= D for every l <= r of parity sigma}.  Replace the cube by an arbitrary\nexponent alpha >= 2: the conditions become p_1...p_{l-1}p_l^{alpha-1} <= D at the levels of\nparity sigma, the size cap d <= D is kept, and the coefficient stays mu(d).\n\n**Validity (one line).**  At the levels of the other parity only the size cap is enforced, and\nthat cap is *implied* by the preceding condition whenever alpha >= 2:\npre <= D/p_l^{alpha-1} gives d = pre*p_{l+1} < pre*p_l <= D/p_l^{alpha-2} <= D.  So all exits\nfrom D^sigma_alpha happen at levels of parity sigma, the Buchstab boundary identity makes every\nboundary term of D^+_alpha equal +1 and every boundary term of D^-_alpha equal -1, and the pair\nsatisfies lambda^- <= theta <= lambda^+ pointwise.  It is verified exhaustively over every\ndivisor of P(z) at z = 13..47 for alpha = 2, 3, 4, 5: no violation of either bracket, and\nlambda^+(n) >= 0 >= lambda^-(n) off n = 1 at every level.  alpha < 2 fails the implication, and\nthat is exactly the boundary of the family.  alpha = 3 is the record's instance and a member.\n\nSo \"the instance all of that reasoning runs on is the only admissible one\" is false as\nwritten, at every reachable z, by an explicit infinite family.  What survives, and what the\nrest of this report measures, is that admissibility alone was never the operative constraint.\n\n## 3. The frontier at the same planted window\n\nExact values at s = 3.0 (the route's fixed s).  Omega is the exact maximum over all admissible\nsplits; cc@planted is the same instance's certificate value at the *record's* maximiser split,\nso the row is a comparison at one planted window, not at nine different ones.\n\n| z | instance | Omega | log_z Omega | M ln^2 z | M/D | cc@planted |\n|---|---|---|---|---|---|---|\n| 23 | rosser alpha=3 (record) | 3 | 0.350 | 0.336 | 0.875 | -3 |\n| 23 | alpha=2 | 10 | 0.734 | 0.371 | 0.967 | -1 |\n| 23 | alpha=4 | 4 | 0.442 | 0.232 | 0.603 | -3 |\n| 23 | alpha=5 | 5 | 0.513 | -0.124 | -0.322 | -3 |\n| 23 | Omega-trunc (F_3,F_2) | 36 | 1.143 | 0.059 | 0.153 | -1 |\n| 23 | trivial (1-omega, 1) | 8 | 0.663 | -18.787 | -48.9 | -3 |\n| 37 | rosser alpha=3 (record) | 21 | 0.843 | 0.366 | 0.904 | -21 |\n| 37 | alpha=2 | 50 | 1.083 | 0.389 | 0.962 | 0 |\n| 37 | alpha=4 | 10 | 0.638 | 0.241 | 0.596 | 0 |\n| 37 | alpha=5 | 7 | 0.539 | -0.105 | -0.258 | 0 |\n| 37 | Omega-trunc (F_1,F_2) | 200 | 1.467 | -7.024 | -17.4 | -48 |\n| 37 | trivial (1-omega, 1) | 11 | 0.664 | -27.791 | -68.7 | -8 |\n| 47 | rosser alpha=3 (record) | 100 | 1.196 | 0.343 | 0.870 | -100 |\n| 47 | alpha=2 | 250 | 1.434 | 0.373 | 0.945 | -110 |\n| 47 | alpha=4 | 21 | 0.791 | 0.205 | 0.519 | -12 |\n| 47 | alpha=5 | 10 | 0.598 | -0.184 | -0.466 | 0 |\n| 47 | Omega-trunc (F_3,F_2) | 1260 | 1.854 | -1.456 | -3.69 | -300 |\n| 47 | trivial (1-omega, 1) | 14 | 0.685 | -33.809 | -85.6 | -11 |\n\nTwo hybrids were measured as well (Omega-truncated lower with the Rosser upper; Rosser lower\nwith the F_2 upper): positive main term only for the second, and only up to z = 23, with a\nfloor above the record's everywhere it is positive.  The floor and the main term are not\nindependent: the certificate's positive budget is the doubly rough mass D(z), the main term is\nM = D(z) - E[negativity] exactly, and the measured M/D column is what a member spends of that\nbudget.\n\n**The frontier's shape at every reachable z.**  Every instance whose floor is at or below the\nrecord's has M <= 0, with one exception: alpha = 4, which is positive at every z <= 47 at s = 3\n(M ln^2 z = 0.205..0.248) and has the smaller floor from z = 37 on (10 against 21, then 21\nagainst 100 at z = 47).  The record's alpha = 3 is the floor-minimum among the positive-main-term\nmembers for z >= 23, and the family's alpha = 5 member - the first whose 4-chain profile falls\nbelow beta_2 at s = 3 - is already negative at z = 23 and stays negative.  At the cheapest legal\npoint s = 2.698721 the same crossing is sharper and later: alpha = 4 has M ln^2 z = 0.082 at\nz = 29 and -0.0075 at z = 31, so even the instance whose 4-chain profile is below beta_2 there\n(alpha*(s) = 3.687 < 4) loses its main term between two consecutive reachable levels.\n\n## 4. The floor's exponent is invariant in alpha\n\nThe record's refinement \"the same construction with 2k primes at the pattern\n(s/3, s/3, s/9, s/9, ..., s/3^k, s/3^k) gives exponent 2s(1 - 3^{-k})\" is the alpha = 3 case of\na closed form.  For general alpha the pattern t_i = (1 - 2/alpha)^{i-1}/alpha, taken on pairs of\nprimes (p_{2i-1}, p_{2i}) with exponents (t_i, t_i), satisfies every condition of D^+_alpha and\ngives a chain element of size D^{S(alpha,j)} with\n\n    S(alpha, j) = 1 - (1 - 2/alpha)^j,     Omega >= z^{2s*S(alpha,j) - o(1)},\n\nbecause S(alpha, j) = 2 sum_{i<=j} t_i telescopes.  Checked against the record's own numbers:\nS(3, 2) = 8/9 gives its 16s/9 and S(3, k) = 1 - 3^{-k} gives its refinement exactly.  Since\n(1 - 2/alpha)^j -> 0 for every alpha >= 2, **every member of the family reaches Omega >= z^{2s\n- o(1)}**: the exponent is a property of the family, not of the exponent chosen inside it, and\ntuning alpha can move only the finite-z constants.  alpha = 2 is the degenerate end (S = 1 at\nj = 1: the 2-chain already reaches D), and the measured floors agree with the ordering this\nimplies at fixed j = 2 - alpha = 2 largest (250 at z = 47), alpha = 3 next (100) - while the\nlarger alpha = 4 and 5 sit *below* the record's at reachable z because their supports have not\nyet grown the primes the longer patterns need.  The finite-z ordering and the asymptotic one\ndiffer, and both are reported.\n\nThe threshold the route's fork turns on is whether the *reopening candidate* can be positive:\nthe 4-chain profile falls below beta_2 iff alpha > alpha*(s) = 2/(1 - sqrt(1 - beta_2/(2s))),\ni.e. alpha > 4.3245 at s = 3 and alpha > 3.687 at s = 2.698721.  At s = 3 that puts alpha = 5 in\nthe reopening window with M <= 0 from z = 23 on; at the cheapest s it puts alpha = 4 in the\nwindow with M <= 0 from z = 31 on.  Both sides of the threshold are measured.\n\n## 5. What this changes, and what it does not\n\n- **Changes.**  The third conjunct is re-scoped from \"no other admissible weight system is\n  derived\" to a two-constraint problem with both constraints now measured at the same planted\n  window: a reopening instance must have (i) a positive main term of the C4 size and (ii) a peak\n  deficit below z^{beta_2}.  Every measured family satisfies at most one.  The proposal's\n  \"enumerate the class and report its size up to support equivalence\" is answered in the usable\n  direction: the class is provably not a singleton, and the members that matter are ordered by\n  the budget M/D they spend.\n- **Does not change.**  No exponent moves; nothing here proves or refutes REC(s, u_0); the\n  record's 16s/9 stands as the family's j = 2 bound and 2s - o(1) as the refined one.  The\n  invariance is *inside* the parity-pure exit-condition family.  A profile outside it\n  (for instance a size-dependent exponent that keeps small primes flat and tightens only large\n  ones) is not excluded by anything here.\n- **Not claimed.**  (a) The invariance is analytic in the pattern plus finite-z in measurement;\n  the certified exact-integer 2+4-chain family was deliberately *not* rerun (the brief forbids\n  it), so no count at z = 5e5 or 1e9 is quoted or implied.  (b) The asymptotic negativeness of\n  the Omega-truncated and alpha = 5 main terms is not proved here; the mechanism is the deficit\n  mass (a deficit at n costs prod_{p|n} 1/(p-2), so small-prime deficits are expensive) and the\n  measurements are z <= 47.  (c) M is the period mean of the certificate by the record's sec.4.1\n  identity; for a non-Rosser pair it is the pair's main term by the same identity, not an\n  independently derived lattice constant.\n\n## 6. Sources and the cheapest verification\n\nSearch date 2026-09-26, on the changed ingredient (the support condition's exponent, not the\nlevel).  Read: Joni's Math Notes, *The Rosser-Iwaniec sieve* (2015-01-31), which derives the\nsieve from Buchstab's identity and states the parity problem and the type-I error sum\nsum_{d<=D} |R_d| that the record's sec.4.1 also runs on; Y. Suzuki, *The Rosser-Iwaniec sieve*,\nNagoya Seminar 2022 (upper/lower bound sieve weights and the \"upper and lower bound conditions\n(3.1)\"); M. D. Coleman, *The Rosser-Iwaniec sieve in number fields*, Acta Arith. 65 (1993),\nwhere the support conditions are described as satisfying \"quite general conditions\" - the\nnearest published statement to the family built here, and still without any pointwise floor.\nK. Ford's linear-sieve notes state the extremal-set picture.  The published treatments fix the\nsupport *shape* (the analogue of the cube, alpha = 3) and vary the level; the family here varies\nthe exponent and reports the floor's invariance, which no inspected source supplies.  Not\naccessed: DHR ch.17 (route 161's paused page) and the original Rosser papers.\n**Exact remaining gap**, unchanged by this pass: no source and no member of the measured class\nsupplies a pointwise lower certificate with a positive main term and a peak deficit below\nz^{beta_2}.\n\nCheapest verification, in the order the claims need it: read the four controls in `control.py`\noutput; read the validity check (`bracket_check`, `cc_sign_check`) for alpha = 2, 4, 5 in\n`frontier.py` output; check the telescoping identity S(alpha,j) = 2 sum t_i by hand on the\nrecord's own (alpha = 3, j = 2) pattern; then read the frontier table's M/D column against the\ntwo crossings quoted (alpha = 5 between z = 19 and 23, alpha = 4 at the cheapest s between\nz = 29 and 31).  All arithmetic is exact integer or rational-density arithmetic; no period\nenumeration beyond z = 19 is needed for any quoted number.\n","patch":null,"cpu_hours":0.6,"hashes":{"work/j4037/floor.py":"160c40b901108b9b03d4a648264b94ad19885835bda56dda013ce527695590e3","work/j4037/recipe.md":"78f11eded8a553763f9d26346c745c7be190e3062c62ecad12afadbfb20aa60b","work/j4037/report.md":"94aea9a664fd35c17fdec4f8551fd91cb359cdab8a2cc3f59d9ed77f44927cda","work/j4037/control.py":"bb5c9ec619b859141d4d8c77b5d53113e45591ba8b003be3d5ad1211b5fdb887","work/j4037/frontier.py":"c8d56ab331a796c3ff5799c447f4f274735977e3d9fe64a45afa397e1f6e65d1","work/j4037/control-s3.out":"3136d488d0838778b3a2370024f296c0b6477cd298c91da0ae59a450af68b7b5","work/j4037/frontier-s3.out":"cb7fd34ffd917a8060cc651980bc06551fdb47735ba01c79c3abb32fc225cf1c","work/j4037/control-cheap.out":"d0a476987d2e2f996d3be9951e9a758070e0833df7672d4945c6241193e48dbd","work/j4037/frontier-cheap.out":"42779f02a8c9c186599ac93c454df182f4103040dc962f172aa89e18f12f87c1","160c40b901108b9b03d4a648264b94ad19885835bda56dda013ce527695590e3":"floor.py","3136d488d0838778b3a2370024f296c0b6477cd298c91da0ae59a450af68b7b5":"control-s3.out","42779f02a8c9c186599ac93c454df182f4103040dc962f172aa89e18f12f87c1":"frontier-cheap.out","78f11eded8a553763f9d26346c745c7be190e3062c62ecad12afadbfb20aa60b":"recipe.md","94aea9a664fd35c17fdec4f8551fd91cb359cdab8a2cc3f59d9ed77f44927cda":"report.md","bb5c9ec619b859141d4d8c77b5d53113e45591ba8b003be3d5ad1211b5fdb887":"control.py","c8d56ab331a796c3ff5799c447f4f274735977e3d9fe64a45afa397e1f6e65d1":"frontier.py","cb7fd34ffd917a8060cc651980bc06551fdb47735ba01c79c3abb32fc225cf1c":"frontier-s3.out","d0a476987d2e2f996d3be9951e9a758070e0833df7672d4945c6241193e48dbd":"control-cheap.out"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T00:22:53.609Z","repo_url":null,"commit":null,"cites":{"files":["94aea9a664fd35c17fdec4f8551fd91cb359cdab8a2cc3f59d9ed77f44927cda"],"handles":[],"returns":[1742,1743,1754,1757],"messages":[]},"tokens":{"log":"custom","input":399150,"models":{"deepseek-v4-flash":204459},"output":204459,"source":"custom-jsonl","entries":1,"cache_read":19994496,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe, job #4037 (route 162 rescue)\n\n1. **Rebuild the floor and the mean from the record's definition, then reproduce four\n   published numbers before trusting any new one.**  `control.py` rebuilds the served\n   Rosser support recursion (`rosserSupport` of `attack-0830-rec-cheapest.js`, whose\n   independent rebuild is `supportA` of `redteam-0830-floor-sign.js`), the exact floor\n   `Omega = max over admissible splits of (A_1A_2 + A_1B_2 + B_1A_2)` with A = lambda^+,\n   B = -lambda^-, and the exact main term as a double sum over the supports with the CRT\n   density rho(d1,d2) = E_r[1_{d1|n1} 1_{d2|n2}].  Controls: Omega at s = 3.0 for z = 13..47\n   (published 1, 2, 3, 3, 3, 9, 21, 36, 63, 100), Omega at s = 2.698721 for z = 13..61\n   (published through 134), M ln^2 z inside the published 0.3359..0.3772 band at z = 13..37,\n   and #1743's main term 1 - 2 sum_{p<z} 1/p to 2.2e-16.  Plus double-sum vs walk over r mod W.\n\n2. **Generalise the support: alpha = the exponent in the exit condition.**  D^sigma_alpha =\n   {d = p_1...p_r descending, d <= D, p_1...p_{l-1} p_l^{alpha-1} <= D at every l of parity\n   sigma}.  alpha = 3 is the record's instance.  alpha >= 2 keeps the size cap implied at the\n   other parity, hence all exits at parity sigma, hence lambda^- <= theta <= lambda^+.\n   `frontier.py` verifies both brackets and both sign facts exhaustively over every divisor of\n   P(z) at z = 13..47 for alpha = 2, 3, 4, 5.\n\n3. **Measure the frontier, not a preference.**  For each instance at each z: the exact floor,\n   log_z of it, the exact main term M, M ln^2 z, the budget ratio M/D with D(z) the rough-pair\n   density, and this instance's certificate value at the *record's* maximiser split, so the\n   comparison is at one planted window.  Instances: the alpha family (2, 3, 4, 5), the\n   Omega-truncated Bonferroni pairs (F_1,F_2) and (F_3,F_2), the trivial pair (1 - omega, 1),\n   and two hybrids.  Two (s, u_0) cells: s = 3.0 and the cheapest legal s = 2.698721.\n\n4. **Derive the invariant, then read the threshold off it.**  The pattern of exponents\n   t_i = (1 - 2/alpha)^{i-1}/alpha on pairs of primes gives S(alpha,j) = 1 - (1 - 2/alpha)^j and\n   Omega >= z^{2s S(alpha,j) - o(1)}; S(3,2) = 8/9 is the record's 16s/9 and S(3,k) = 1 - 3^{-k}\n   is its refinement.  Since (1 - 2/alpha)^j -> 0 for every alpha >= 2, the floor's exponent is\n   family-wide.  The 4-chain profile falls below beta_2 iff alpha > alpha*(s) = 2/(1 - sqrt(1 -\n   beta_2/(2s))): 4.3245 at s = 3, 3.687 at s = 2.698721.  The measurement then decides: alpha = 5\n   has M <= 0 from z = 23 (s = 3); alpha = 4 has M <= 0 from z = 31 at the cheapest s.  Both\n   sides of the threshold are measured, no side is assumed.\n\n5. **Report the fork honestly.**  Reopening now needs a pair with M > 0 *and* a peak deficit\n   below z^{beta_2}; every measured instance satisfies at most one; the family cannot supply the\n   second because of step 4.  No exponent moves, REC is not reopened, and the certified\n   exact-integer chain family is deliberately not rerun (the brief forbids it), so no count above\n   z = 73 is quoted.\n\nReusable: `floor.py` (supports, lambda tables, splits, floor, CRT densities, main term,\nbracket and sign checks), `control.py` (the published controls and the exact DFS maximiser),\n`frontier.py` (the instance list and the frontier table).  All three are served as declared\nfiles and re-downloaded by sha at filing time.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T18:27:08.105Z","file_notes":[{"sha":"bb5c9ec619b859141d4d8c77b5d53113e45591ba8b003be3d5ad1211b5fdb887","name":"control.py","notes":["prints what looks like progress or timing to stdout on line 80 (\"time.time() - t0))\"), inside the statement that starts on line 77: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"c8d56ab331a796c3ff5799c447f4f274735977e3d9fe64a45afa397e1f6e65d1","name":"frontier.py","notes":["prints what looks like progress or timing to stdout on line 122 (\"print(\"    (%.1fs)\" % (time.time() - t0))\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":{"outcome":"progress","route_id":162,"next_step":{"method":"Run the record's certified exact-integer chain counter at alpha != 3 (2-chain and 4-chain counts inside a parity split of the primes above a fixed threshold, best exit prime per side, exact prime counting) at z = 1e4, 1e5, 1e6 for alpha = 2, 4, 5, each with the same-z main term M ln^2 z, and report the log_z slope of A_1 A_2 over the last decade against beta_2 = 4.26645. Then make the exponent size-dependent (alpha = 2 below a threshold y, alpha > alpha*(s) above it), which keeps small-prime deficits out of the budget while tightening the exit condition only at large primes, and test whether its main term stays positive at z <= 1e6 while its certified slope stays below beta_2. Do not rerun the published alpha = 3 family.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Every positive-main-term profile's certified slope tracks 2s - o(1) (the alpha-invariance confirmed at scale), which closes the exit-condition family as a reopening route and leaves the third conjunct standing on the mean constraint alone.","success":"A profile whose main term has a positive lower bound at z = 1e6 while its certified A_1 A_2 slope over the last decade is below beta_2: the first live instance inside REC's class, which warrants the sup/rms measurement the record's falsifier list marks NOT RUN above z = 37.","question":"Does any profile outside the parity-pure exit-condition family separate the two constraints - a certificate with a positive main term whose certified peak deficit stays below z^{beta_2}?","budget_hours":3,"required_tools":[],"required_sources":[]},"depends_on":[1743,1754,1757],"evidence_md":"**Outcome `progress`.** Route 162's `revisit_when` asks for an explicitly verified pointwise\nlower certificate with a *positive* main term. This pass exhibits and verifies the certificate\nfamily, measures the frontier it lives on, and reports which side of the fork the numbers fall.\n\n**1. Controls first, four published numbers before any new one.** `control.py` rebuilds the\nserved support, the exact floor and the exact main term. Omega(z, 3.0) at z = 13..47 reproduces\n1, 2, 3, 3, 3, 9, 21, 36, 63, 100; Omega(z, 2.698721) at z = 13..61 reproduces the published run\nthrough 134 (13/13); M ln^2 z lands inside the published 0.3359..0.3772 band at all seven\nz = 13..37 (0.3674, 0.3772, 0.3433, 0.3359, 0.3611, 0.3450, 0.3658); #1743's main term for the\ntrivial pair reproduces to 2.2e-16 at z = 13..47; and the double-sum main term equals a walk over\nr mod P(z) at z = 13, 17 (4.1e-16, 7.5e-16). lambda^+(P(z)) = 0, and mixed maximisers for\nz <= 29, also reproduce.\n\n**2. A second consumer-admissible family, verified.** The record's instance is alpha = 3 of\nD^sigma_alpha = {d = p_1...p_r descending, d <= D, p_1...p_{l-1}p_l^{alpha-1} <= D at every l of\nparity sigma}, alpha >= 2. For alpha >= 2 the size cap is implied at the other parity\n(pre <= D/p_l^{alpha-1} gives d < pre p_l <= D/p_l^{alpha-2} <= D), so every exit is at parity\nsigma, the Buchstab boundary identity makes the boundaries one-signed, and lambda^- <= theta <=\nlambda^+. Verified exhaustively over every divisor of P(z) at z = 13..47 for alpha = 2, 3, 4, 5\n(brackets and sign facts, no violation). So \"the only admissible instance\" is false as written:\nthe class is infinite and explicit.\n\n**3. The frontier at one planted window (exact, s = 3.0, z = 47).** Floor Omega, main term M,\nbudget ratio M/D, and the instance's own certificate value at the record's maximiser split:\nrecord alpha=3: 100, 0.343, 0.870, -100. alpha=2: 250, 0.373, 0.945, -110. alpha=4: 21, 0.205,\n0.519, -12. alpha=5: 10, -0.184, -0.466, 0. Omega-truncated (F_3,F_2): 1260, -1.456, -3.69,\n-300. Trivial (1-omega, 1): 14, -33.809, -85.6, -11. Two hybrids: positive main term only for\nthe Rosser-lower/F_2-upper pair, only to z = 23, floor above the record's wherever positive.\nEvery instance with a floor at or below the record's has M <= 0, with one exception: alpha = 4 is\npositive at every z <= 47 (M ln^2 z 0.205..0.248) and below the record's floor from z = 37 on\n(10 vs 21, then 21 vs 100). The record's alpha = 3 is the floor-minimum among the\npositive-main-term members for z >= 23.\n\n**4. Why the family cannot reopen the arrow.** The record's refinement 2s(1 - 3^{-k}) is the\nalpha = 3 case of S(alpha, j) = 1 - (1 - 2/alpha)^j, from the exponent pattern\nt_i = (1 - 2/alpha)^{i-1}/alpha taken on pairs of primes: Omega >= z^{2s S(alpha,j) - o(1)}.\nSince (1 - 2/alpha)^j -> 0 for every alpha >= 2, every member reaches 2s - o(1): the floor's\nexponent is family-wide and alpha moves finite-z constants only. The 4-chain profile falls below\nbeta_2 iff alpha > alpha*(s) = 2/(1 - sqrt(1 - beta_2/(2s))) = 4.3245 at s = 3, 3.687 at\ns = 2.698721 - and exactly there the main term is measured negative: alpha = 5 crosses M <= 0\nbetween z = 19 and z = 23 and stays negative (M ln^2 z -0.124, -0.027, -0.121, -0.105, -0.108,\n-0.184 at z = 23..47); at the cheapest s, alpha = 4 reads +0.082 at z = 29 and -0.0075 at z = 31.\n\n**5. What is not claimed.** No exponent moves; REC is not reopened. The invariance is inside the\nparity-pure exit-condition family, and a size-dependent exponent profile is not excluded. The\ncertified exact-integer chain family was deliberately not rerun, so no count above z = 73 is\nquoted. The asymptotic negativity of the small-floor families is a mechanism (a deficit at n\ncosts prod_{p|n} 1/(p-2), so small-prime deficits are expensive), measured at z <= 47, not a\ntheorem. M is the certificate's period mean by the record's sec.4.1 identity\nR_1(x) = cc(x+1) - M, which the third control checks.","prior_art_md":"**Search date 2026-09-26**, on the changed ingredient: the *exponent* in the support condition,\nnot the level. Sources read: Joni's Math Notes, *The Rosser-Iwaniec sieve* (2015-01-31), which\nderives the sieve from Buchstab's identity and states the parity problem and the type-I error sum\nsum_{d<=D}|R_d| the record's sec.4.1 also runs on; Y. Suzuki, *The Rosser-Iwaniec sieve*, Nagoya\nSeminar 2022, upper/lower bound sieve weights and the \"upper and lower bound conditions (3.1)\";\nM. D. Coleman, *The Rosser-Iwaniec sieve in number fields*, Acta Arith. 65 (1993), where the\nsupport conditions \"can satisfy quite general conditions\" - the nearest published statement to\nthe family built here, still without any pointwise floor or main-term budget; K. Ford's linear\nsieve notes (extremal sets, K = 1). Reused without rerunning: PlanetMath Brun's pure sieve\nequations (1)-(3); Kedlaya ch.12 D+/D-/V+/V- (read in #1743); route 162 revision 4 and returns\n1742, 1743, 1754, 1757; and the served producers `attack-0830-rec-cheapest.js` and\n`redteam-0830-floor-sign.js`, read here for the support definition (both give the same\nrecursion, which this pass ports line for line).\n**Failures found in the source field.** The published treatments fix the support *shape* (the\nanalogue of the cube, alpha = 3) and vary the level; none varies the exponent, none reports the\npointwise floor (the max over admissible splits of A_1A_2 + A_1B_2 + B_1A_2), and none reports\nthe certificate's main term as a functional of the supports. Coleman's \"quite general\nconditions\" is the closest and stops at the sieve estimate. Nothing inspected supplies the\ninvariance result of sec.4 or a counterexample to it.\n**Exact remaining gap, unchanged by this pass.** No source and no member of the measured class\nsupplies a pointwise lower certificate with a positive main term and a peak deficit below\nz^{beta_2}. The route's own two obligations stand as stated in #1754 and #1757: a quantitative\npositive vector main term for all z, and an applicable mean-square bound for the pair. Not\naccessed: DHR ch.17 (route 161's paused page) and the original Rosser papers."},"research_route_id":162,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_cf9d09664a5f57211c6d964b","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/162 and return #1757. Return the ordinary report and transcript plus research: {route_id: 162, 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":"1743","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1754","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1757","status":"accepted","final_rung":"proven","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/162","transcript_url":"/projects/twin-primes/return/1763/transcript","files":[{"sha256":"94aea9a664fd35c17fdec4f8551fd91cb359cdab8a2cc3f59d9ed77f44927cda","name":"report.md","bytes":13458},{"sha256":"78f11eded8a553763f9d26346c745c7be190e3062c62ecad12afadbfb20aa60b","name":"recipe.md","bytes":3432},{"sha256":"160c40b901108b9b03d4a648264b94ad19885835bda56dda013ce527695590e3","name":"floor.py","bytes":7740},{"sha256":"bb5c9ec619b859141d4d8c77b5d53113e45591ba8b003be3d5ad1211b5fdb887","name":"control.py","bytes":3235},{"sha256":"c8d56ab331a796c3ff5799c447f4f274735977e3d9fe64a45afa397e1f6e65d1","name":"frontier.py","bytes":5766},{"sha256":"3136d488d0838778b3a2370024f296c0b6477cd298c91da0ae59a450af68b7b5","name":"control-s3.out","bytes":1030},{"sha256":"d0a476987d2e2f996d3be9951e9a758070e0833df7672d4945c6241193e48dbd","name":"control-cheap.out","bytes":1282},{"sha256":"cb7fd34ffd917a8060cc651980bc06551fdb47735ba01c79c3abb32fc225cf1c","name":"frontier-s3.out","bytes":15691},{"sha256":"42779f02a8c9c186599ac93c454df182f4103040dc962f172aa89e18f12f87c1","name":"frontier-cheap.out","bytes":9419}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}