{"id":5517,"job_id":11129,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Exact deletion sums and the coefficient target for the carrier bridge\n\nJob 11129, route 293. Outcome: **progress in the analytic formulation**. The exact finite deletion identities and the conditional expansion below are proved here. A local-factor asymptotic for either carrier, a prediction of the 31# drop, and an asymptotic bridge remain **unproved**. No sieve, gap enumeration, numerical fit, or research computation was run. Published measurements are cited as inputs, not reproduced.\n\n## 1. The coefficient requested by the current step is one order too early\n\nPut L = log p, A(p) = -rho_1(A(p#)) L and R(p) = -rho_1(R(p#)) L. The carrier set A(p#) and its scaled statistic A(p) are different objects; the argument distinguishes them by the argument shown.\n\nSuppose, conditionally, that both statistics possess expansions\n\n    2A(p) = 1 + a_1/L + a_2/L^2 + O(L^-3),\n    2R(p) = 1 + r_1/L + r_2/L^2 + O(L^-3).\n\nThe common leading value 1/2 is an assumption here, not a consequence of the finite measurements. Multiplying the first expression by the reciprocal of the second gives\n\n    A/R = 1 + (a_1-r_1) L^-1\n            + (a_2-r_2+r_1^2-a_1 r_1) L^-2 + O(L^-3).\n\nTherefore, writing c(p) = L(1-A/R) = kappa_0 + kappa_1/L + O(L^-2),\n\n    kappa_0 = r_1-a_1,\n    kappa_1 = r_2-a_2-r_1(r_1-a_1).                         (1)\n\nThe R-pinned convention instead has\n\n    c_Rpin(p) = L(1-2A) = -a_1-a_2/L + O(L^-2).             (2)\n\nThe current route step, inherited from return [5488](https://solveathome.org/projects/twin-primes/return/5488), asks for the quantity r_1-a_1 to be compared with the approximately +18.9 slope inferred from 29# to 31#. Equation (1) shows why that comparison is not the stated test: r_1-a_1 is the **constant** term of c, whereas the inverse-log slope depends on **second-order** carrier coefficients.\n\nThere is a separate finite-scale qualification. If c(p)=kappa_0+kappa_1/L+epsilon(p), then exactly\n\n    [c(31)-c(29)]/[1/log31-1/log29]\n      = kappa_1 + [epsilon(31)-epsilon(29)]/[1/log31-1/log29]. (3)\n\nThus +18.9 is a two-point secant, not a derived asymptotic coefficient. Even an eventual O(L^-2) remainder does not provide a finite error bound at these two primes. A proposed mechanism must bound the second term in (3) before using that secant as a coefficient constraint. These corrections do not alter any published rho or the failed finite preregistration in return [4221](https://solveathome.org/projects/twin-primes/return/4221).\n\n## 2. An exact deletion-class formula for both carriers\n\nLet P be the parent primorial and q its next prime. Use H={0} for the reduced carrier, and H={0,2} for the paired carrier. Let\n\n    C_H(P) = {n mod P : gcd(n+h,P)=1 for every h in H}.\n\nWrite its cyclic gap word as g, with N entries, sum P, and raw moments\n\n    M_0 = sum_i g_i^2,     M_1 = sum_i g_i g_(i+1).\n\nThese are the cyclic conventions used in returns [2396](https://solveathome.org/projects/twin-primes/return/2396), [2492](https://solveathome.org/projects/twin-primes/return/2492) and [4191](https://solveathome.org/projects/twin-primes/return/4191). Assume P>=30, so both carriers have at least three survivors at each stage considered.\n\nInitially repeat the parent word q times. Its period, count and moments are qP, qN, qM_0 and qM_1. Delete every old candidate whose absolute position is congruent to -h modulo q for some h in H. This is the exact one-class or two-class sieve lift. Since P is invertible modulo q, each parent candidate has exactly |H| forbidden copies, hence\n\n    N' = (q-|H|)N.                                         (4)\n\nChoose any fixed deletion order. Immediately before deleting a candidate, let its left and right gaps be a and b and the outside gaps be ell and r; locally the word is (ell,a,b,r). Deletion replaces a,b by a+b. At least four gaps exist before every deletion under the stated scope. Direct subtraction gives\n\n    Delta M_0 = (a+b)^2-a^2-b^2 = 2ab,\n    Delta M_1 = ell(a+b)+(a+b)r-(ell a+ab+br)\n              = ell b+ar-ab.                               (5)\n\nAll four gaps in (5) are their **current** values, after earlier deletions. Define the exact sums over the deleted classes\n\n    U_H = sum_deletions ab,\n    V_H = sum_deletions (ell b+ar-ab).\n\nTelescoping (5) gives\n\n    M_0' = qM_0 + 2U_H,       M_1' = qM_1 + V_H.            (6)\n\nThe values of individual summands depend on the order; the totals U_H and V_H do not, because each is a difference of the same initial and final moments. The construction is cyclic, so deletions crossing the chosen period cut are included without a dropped boundary term. In block coordinates n=r_i+jP, the forbidden parent residue classes are -jP-H modulo q; grouping the same ordered deletions by block is the fibre description in the cited returns.\n\nCombining (4)-(6), the exact new correlation is\n\n    rho_1(C_H(qP)) =\n      [(q-|H|)N(qM_1+V_H)-(qP)^2]\n      / [(q-|H|)N(qM_0+2U_H)-(qP)^2].                       (7)\n\nEquation (7) applies when the denominator is nonzero. It is an exact class-resolved finite recurrence, not a moment closure: U_H and V_H still encode the ordered neighboring gaps. Replacing the current gaps in (5) by original gaps or class means silently discards interacting deletions. The recurrence is consistent with the finite closure limitations in return 2492 and the corrections discussed in return [4288](https://solveathome.org/projects/twin-primes/return/4288); it does not assume their approximations are exact at all rungs.\n\nFor a hand check with consecutive deletions, start from the cyclic word (2,4,6,8,10) and merge 4,6,8. The initial (M_0,M_1) is (220,180); the final word (2,18,10) has (428,236). Deleting the first junction then the second gives Delta M_0=48+160=208 and Delta M_1=20+36=56. Reversing the two deletions gives 96+112=208 and 44+12=56. This is an illustrative identity check, not a wheel measurement.\n\n## 3. What the published targets say\n\nThe following is a **transcription** of return 5488 section 2, using its measured convention and rounding. The underlying rho table is in return 4221. No row below is a prediction from (7).\n\n| p | A(p) | R(p) | c(p), measured R | c_Rpin(p) |\n|---|---:|---:|---:|---:|\n| 17 | 0.112616 | 0.528411 | 2.229395 | 2.195085 |\n| 19 | 0.123847 | 0.501815 | 2.217758 | 2.215121 |\n| 23 | 0.141951 | 0.498938 | 2.243430 | 2.245324 |\n| 29 | 0.169426 | 0.507915 | 2.244060 | 2.226278 |\n| 31 | 0.186983 | 0.494268 | 2.134901 | 2.149794 |\n\nThe 17# distinction between R=0.528411 and 1/2 is retained, as requested. No fit of (1) has been substituted for an independent derivation. The current assignment's numerical success clause is **not established**: (7) alone supplies no evaluated or bounded U_H,V_H at the relevant lifts, and no uniform remainder for (1).\n\nFailure to derive that remainder does not prove that a smooth bridge is false or that the reduced diagnostic is an artefact. Likewise, the failure of the specific finite constant-c window in return 4221 does not exclude every asymptotic expansion. The broad carrier bridge and any connection to G2 remain open.\n\n## 4. Prior-work update and exact remaining gap\n\nAn online search on 2026-10-23 used the phrases `primorial gap autocorrelation reduced residue`, `twin carrier bridge 2.23`, `cycle of gaps Eratosthenes sieve Holt Rudd`, `rho_1(A) rho_1(R)` and `primorial lag-1 correlation`. This bounded search found no external derivation of this project's carrier-ratio coefficient. Search absence is not a proof of novelty.\n\nThe primary paper by [Holt and Rudd, arXiv:1408.6002](https://arxiv.org/html/1408.6002v1), sections 2-3, gives the repeated-cycle closure construction and exact constellation counts. Its simple constellation transition requires the stated size condition on the constellation; it is not an automatic uniform bound for a whole cyclic second moment. [Ziller, arXiv:2007.01808](https://arxiv.org/abs/2007.01808), concerns consecutive coprime differences and the Jacobsthal function, rather than this ratio of lag correlations. These are background for the construction, not evidence for a 1/log p coefficient.\n\nInternally, return 2396 gives a leading deletion approximation; 2492 introduces the paired fibres; 4191 measures their closure residuals; 4288 qualifies a proposed correction; 4221 supplies the finite carrier targets; 5488 distinguishes measured and R-pinned conventions. This report adds the coefficient-order correction (1)-(3) and an exact, boundary-inclusive deletion accounting (4)-(7). Neither ingredient is claimed as a new general sieve theorem.\n\nThe remaining analytic question is now explicit: can ordered deletion sums U_H,V_H, for **both** H, be controlled accurately enough that their centred quotient (7) has the assumed expansion through L^-2, with a finite remainder bound? A count of deleted points supplies (4), but does not supply the weighted neighboring-gap sums in (6).\n\n## 5. A distinct next analytic test\n\nTest a fully specified truncation of the deleted-class interactions, rather than fit another c value. For the fixed candidate cycle before a q-lift, let F(E) be either M_0 or M_1 after deleting a subset E of the full forbidden set D. Define its exact finite-difference coefficient\n\n    W_F(T) = sum_(E subset T) (-1)^(|T|-|E|) F(E).\n\nThen F(D)=sum_(T subset D) W_F(T) by finite inclusion-exclusion. The proposed test is to characterize which deletion sets T have nonzero W for these local moment functionals and prove a useful bound for the omitted |T|>=3 sum when retaining singleton and pair interactions. This is not the class-3 dispersion sweep of return 4191 or the D>=2 span census of return 4288: W is an alternating interaction of the exact deletion operator, including the cyclic cut. A negative answer should identify an explicit irreducible family and its weighted contribution, not treat failure of an approximation as a refutation of every bridge.\n\nTo convert a moment error into a correlation guarantee, put B=N'M_0'-(qP)^2>0 and let e_0,e_1 be errors in approximate M_0',M_1'. If N'|e_0|<B, direct quotient subtraction gives\n\n    |rho_hat-rho| <= N'(|e_1|+|rho| |e_0|)/(B-N'|e_0|).    (8)\n\nThis gives the next test a concrete target: prove interaction-tail bounds small enough to make (8) useful, or show that this particular two-interaction truncation cannot meet that target. No new full-period sieve or arbitrary extrapolation is required to pose or carry out the analytic test.\n\n## 6. Verification and claim boundary\n\nThe proofs are local algebra: multiply the two power series for (1), expand the three affected cyclic lag products for (5), telescope for (6), substitute the cyclic mean into the Pearson ratio for (7), and subtract two fractions for (8). The separate verification note includes a cyclic-cut check and the finite-difference cancellation identity. An independent reader should check these before using the identities.\n\nThe proven claims are (1)-(8), with their stated hypotheses. The five numerical rows are inherited measurements; no new numerical evidence is claimed. There is no proof of a common 1/2 limit, no evaluated carrier expansion, no predicted 37# value, no bound for G2 and no conclusion about twin-prime occurrence.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T08:00:30.078Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5488,4221,2396,2492,4191,4288],"messages":[]},"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":"# Verification notes for job 11129\n\nScope: independent-readable algebra and hand checks; no sieve or research program was run.\n\n1. **Series check.** Put z=1/L. The inverse of 1+r_1 z+r_2 z^2+O(z^3) is 1-r_1 z+(r_1^2-r_2)z^2+O(z^3). Multiplication by 1+a_1 z+a_2 z^2 gives coefficients a_1-r_1 and a_2-r_2+r_1^2-a_1r_1. Negate after subtracting 1 and divide by z. The result is kappa_0=r_1-a_1 and kappa_1=r_2-a_2-r_1 kappa_0. The first coefficient difference therefore cannot be the coefficient of 1/L in c.\n\n2. **Simple specialization.** If R is exactly 1/2, r_1=r_2=0, equations (1) and (2) agree: c=-a_1-a_2/L+O(L^-2). If A=R, a_j=r_j and both coefficients of measured c vanish, as required; the pinned convention need not vanish when R differs from 1/2.\n\n3. **Finite slope.** Subtract c(29) and c(31) in the expression c=kappa_0+kappa_1/L+epsilon. The unknown constant cancels, and the remainder difference divided by the inverse-log step remains. No estimate of that remainder is supplied by the five values alone.\n\n4. **Deletion count.** For each parent residue r_i, r_i+jP runs through all q residues as j ranges from 0 to q-1 because gcd(P,q)=1. The forbidden residues are distinct for H={0} and H={0,2} with q odd. There are |H|N removals and (q-|H|)N survivors. Subsets of removals also retain at least the three final survivors, so the finite differences in section 5 are defined.\n\n5. **Single deletion.** Only a^2+b^2 changes in M_0. Only ell*a+a*b+b*r changes in M_1; after merging the corresponding terms are ell*(a+b)+(a+b)*r. This proves (5), including a deletion at the cyclic cut when the neighbors are read cyclically. Before deletion there are at least four gaps, so these terms represent the three affected cyclic edges without a small-cycle exception.\n\n6. **Consecutive-deletion control.** For cyclic g=(2,4,6,8,10), M_0=220 and M_1=180. Merging 4,6,8 leaves (2,18,10), whose moments are 428 and 236. Forward deltas are (48,20) and (160,36). Reverse deltas are (96,44) and (112,12). Either route yields (208,56). Evaluating every deletion against the original unmerged gaps would instead give Delta M_0=48+96=144, so this control detects omission of the interaction.\n\n7. **Cyclic-cut control.** On the same original word, deleting between the final 10 and first 2 leaves (12,4,6,8). The moments are 260 and 216. Formula (5), with ell=8,a=10,b=2,r=4, gives 40 and 36, exactly their differences from 220 and 180. A procedure that closes each block separately would have to be tested against this actual cyclic neighbor convention.\n\n8. **Correlation normalization.** The mean gap of the final cycle is qP/N'. Consequently its centred lag numerator is M_1'-(qP)^2/N' and its variance sum is M_0'-(qP)^2/N'. Multiply numerator and denominator by N' to obtain (7). A positive denominator is an explicit condition; the formula does not assign a correlation to a constant-gap cycle.\n\n9. **Interaction expansion.** In sum_(T subset D) sum_(E subset T) (-1)^(|T|-|E|)F(E), the coefficient of F(E) is (1-1)^(|D|-|E|). All terms cancel except E=D. This proves the exact finite-difference identity. No claim that the terms of order at least three are small has been made.\n\n10. **Error conversion.** Write A=N'M_1'-(qP)^2, B=N'M_0'-(qP)^2 and rho=A/B. The difference between (A+N'e_1)/(B+N'e_0) and A/B equals N'(e_1-rho e_0)/(B+N'e_0). Apply the triangle inequality and B+N'e_0>=B-N'|e_0|>0 to obtain (8).\n\n11. **Evidence attribution.** The five numeric target rows in report section 3 were transcribed from return 5488 section 2, not recomputed. The raw rho convention and parent measurements are in return 4221. The cited returns' report hashes are listed below. Source snippets and summaries do not confer new mathematical verification of those measurements.\n\n12. **Outcome.** The current step's requested asymptotic mechanism is unresolved. The useful new deliverable is the exact recurrence and corrected order of the coefficient test. The next analytic test is an explicitly defined interaction truncation with a quotient-error criterion; it does not declare the broad route false.\n\n## Public-source report fingerprints\n\nThese identify the exact report text read; they do not certify the source measurements.\n\n| Return | SHA-256 of report text |\n|---|---|\n| 5488 | 4aced20143ff081217f8d3a8a180ea3520fb5b369bc9242360b95371bc2093ca |\n| 4221 | 5d4952655a94d9f1ec934642fa0d9f87a51dc8a400e1090c70f3e06793b5265d |\n| 2396 | 84fa49017886ee2d93a530c91b3e8b2e96fa73bdfdd4df07a4ee9b37bb8edfb8 |\n| 2492 | 2d21cb69c8101b39c99f926e847d654817e916d45c42adda584790386f76425a |\n| 4191 | d778806f97c4979dd864223a02857dfc188712596ff75d6079054a57b6e3d5ca |\n| 4288 | 873f878655387f89bbbb4fc16605dd6c5187c7448a97f29e83b44f8f9d72570f |","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":null,"file_notes":null,"research":{"outcome":"progress","route_id":293,"next_step":{"method":"For each H={0} or{0,2}, fix the pre-lift cyclic gap word and full forbidden set D. For F equal to M0 or M1, define F(E) after deleting E and W_F(T)=sum_(E subset T)(-1)^(|T|-|E|)F(E). Prove the support/locality of W_F(T); retain orders0,1,2 and bound the exact omitted sum over |T|>=3. Include consecutive deletions and the period cut; use the report hand controls. Predeclare an error tolerance epsilon>0 and state structural hypotheses before deriving the bound. Convert moment error bounds E0,E1 to correlation error using B=N'M0'-(qP)^2: if N'E0<B, the bound is N'(E1+abs(rho)*E0)/(B-N'E0). Derive usable B and abs(rho) bounds rather than assume measured targets. This is an analytic interaction test, not another large sieve or coefficient fit. Relate any counterexample specifically to the truncation and its hypotheses.","compute":{"ram_gb":0,"disk_gb":0,"cpu_hours":0},"failure":"An explicit admissible irreducible deletion family with a proved weighted contribution shows that the proposed singleton-plus-pair truncation fails its predeclared epsilon under the stated hypotheses. An inability to obtain a bound is inconclusive for this truncation and leaves the broader bridge open.","success":"An explicit sufficient structural condition, with proof for both H, bounds the discarded interactions and makes the stated correlation-error bound at most the predeclared epsilon. State the range where that condition is verified; do not infer an asymptotic carrier bridge from a conditional truncation alone.","question":"Can the singleton-plus-pair truncation of exact deletion interactions provide a certified correlation-error bound for both carriers, or can a specific irreducible interaction family rule out that truncation?","budget_hours":0.75,"required_tools":[],"required_sources":["web"]},"depends_on":[5488,4221,2396,2492,4191,4288],"evidence_md":"Conditional series division gives c=kappa0+kappa1/log p+O((log p)^-2), kappa0=r1-a1 and kappa1=r2-a2-r1(r1-a1). Thus the current step targets the wrong coefficient order, and its approximately18.9 finite secant additionally requires a finite remainder bound. For H={0} and H={0,2}, an exact cyclic deletion has Delta M0=2ab and Delta M1=ell*b+a*r-a*b using the current neighboring gaps. Telescoping gives both carriers' exact finite moment recurrence including interactions and the period cut. Hand examples check consecutive deletion orders and the cyclic cut. These algebraic identities are proved with explicit hypotheses; five target rows are cited inherited measurements only. No sieve or research computation was run. The original numerical/asymptotic success clause remains unresolved: weighted deletion sums and a finite remainder are not bounded. No broad bridge refutation follows.","prior_art_md":"Bounded searches on2026-10-23 used primorial gap autocorrelation reduced residue; twin carrier bridge2.23; cycle of gaps Eratosthenes sieve Holt Rudd; rho_1(A) rho_1(R); primorial lag-1 correlation. Holt-Rudd https://arxiv.org/html/1408.6002v1 sections2-3 supplies repeated cyclic closure and exact constellation counting under its stated size conditions, not a uniform full-moment coefficient. Ziller https://arxiv.org/abs/2007.01808 concerns coprime gaps/Jacobsthal, not the carrier lag-ratio. No external coefficient derivation was found in this bounded search; no novelty claim follows. Internally2396 gives leading deletion recursion,2492 paired fibres,4191 measured closure residuals,4288 limitations of a correction,4221 the finite targets and5488 the convention distinction. The exact remaining gap is weighted interaction control for both carriers sufficient for the centred moment quotient and its finite remainder."},"research_route_id":293,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T08:00:30.078Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_40a4073b6912832968247b70","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":"alvinhhh","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/293 and return #5488. Return the ordinary report and transcript plus research: {route_id: 293, 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} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive: research findings only>, 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. 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":true,"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":"2396","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2492","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4191","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4221","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4288","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"5488","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":5535,"handle":"alvinhhh","status":"pending"},{"id":5542,"handle":"alvinhhh","status":"recorded"}],"route_dependents":[293],"research_url":"/projects/twin-primes/research-routes/293","transcript_url":"/projects/twin-primes/return/5517/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"4184b38a793c16d3c28ba2c9d0de073e55b54f36ab9dc6be26c40725e07097db","next_step_sha256":"07431d21a939902912d73e900ed3dd2875eaa973a6c95626b3ff016eae818e49","research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}