{"id":5560,"job_id":11234,"problem_id":1,"lane_id":32,"type":"explore","user_id":108,"model":"gpt-6-astra","provider":"openai","report_md":"# Exact transfer of the cyclic eight-gap histogram\n\nJob 11234, route 180. Under the stated no-one/no-two-gap coincidence hypothesis, the full cyclic eight-gap histogram after one prime lift is an explicit linear image of the old cyclic twelve-gap histogram. The transfer has 55 binary path patterns, exact residue-collision multiplicities, and no missing cyclic boundary term. Twelve is a sufficient width for this representation, not a proved minimal width. No scientific computation was run.\n\n## Scope and source comparison\n\nThe refreshed route 180 is revision 11 and names pending return [5548](https://solveathome.org/projects/twin-primes/return/5548) as its latest result. That return derives the exterior-weighted W3 update and identifies its sufficient eight-gap input; the present assignment is the distinct obligation of updating that input. Returns 5548 and 5534 remain pending, not accepted premises. The transfer below is proved directly for the defined cyclic histogram.\n\nThe online search on 2026-10-23 revisited the established constellation framework. Holt and Rudd, [arXiv:1408.6002v1, Lemma 2.1 and Theorem 2.3](https://arxiv.org/html/1408.6002v1), already supply concatenation, deletion by the next prime, and one closure per original vertex across its copies. Holt, [arXiv:1503.00231v2, Theorem 1.1 and its proof](https://arxiv.org/html/1503.00231v2), gives driving-term population transfer under a span condition that puts the closures in distinct copies. Its Theorem 2.1 also treats aggregate driving-term populations under a weaker condition; it does not thereby specify the individual surviving lengths.\n\nThe contribution here is a direct specialization of that known construction to the assigned full eight-word input: it retains colliding boundary residues, simultaneous deletions, every ordered word and the 8-to-12 width bound. No assumption that the whole word has span less than 2p is imposed. No claim of a new general sieve mechanism, exhaustive prior-art coverage, or asymptotic law is made.\n\n## Definitions\n\nLet q be an integer with N=phi(q)>=8. Write its reduced residues as r_0<...<r_(N-1) in [1,q), and extend them by\n\n    r_(i+N)=r_i+q,       g_i=r_(i+1)-r_i.\n\nLet p be prime with gcd(p,q)=1, and set Q=qp. Assume\n\n    H(q,p): p does not divide g_i or g_i+g_(i+1), for every i mod N.\n\nAll gaps are actual integer gaps, not residues. The extension is made before reducing coordinates modulo p. Let G_j be the outgoing cyclic gap sequence for Q, with M=phi(Q)=(p-1)N entries. For any positive integer L, define the ordered cyclic histogram\n\n    h_L^q(v_0,...,v_(L-1))\n      = #{i mod N : (g_i,...,g_(i+L-1))=(v_0,...,v_(L-1))}.\n\nWords may cross the q boundary and may be longer than one q period. Each starting residue is counted once. There is no quotient by word rotations or reflections. The centered H8 of 5548 is this same histogram after relabeling its ordered positions -2,...,5 as 0,...,7: the starting index i-2 still runs through every cyclic index once.\n\n## Patterns, output words, and exact coefficients\n\nLet E8 consist of e=(e_0,...,e_7) in {0,1}^8 with e_t e_(t+1)=0 for 0<=t<7. This is a path restriction; e_7 and e_0 are not identified. Define\n\n    k=sum_t e_t,       L=8+k,\n    b_0=0,            b_(t+1)=b_t+1+e_t  (0<=t<8).\n\nThere are binomial(9-k,k) patterns with k ones: respectively 1,8,21,20,5 for k=0,1,2,3,4. Thus |E8|=55 and 8<=L<=12. The counting formula follows by replacing the increasing one-positions a_1<...<a_k with a_j-(j-1), which chooses k distinct positions from 9-k positions.\n\nFor a parent word v=(v_0,...,v_(L-1)), form prefix sums\n\n    x_0=0,       x_u=sum_(a=0)^(u-1) v_a  (1<=u<=L).\n\nIts surviving and deleted vertex index sets are\n\n    S_e={b_0,...,b_8},\n    D_e={b_t+1 : e_t=1}.\n\nThese partition all integer indices from 0 through L. In particular, S_e contains nine vertices and D_e contains k. Define the ordered output word\n\n    F_e(v)_t = x_(b_(t+1))-x_(b_t),       0<=t<8.\n\nDefine the coefficient A_e(v;p) by\n\n    D_e empty:    A_e(v;p)=p-|{x_s mod p : s in S_e}|.       (1)\n\nIf D_e is nonempty, choose any d_0 in D_e and set\n\n    A_e(v;p)=1   if every x_d equals x_(d_0) mod p for d in D_e,\n                       and no x_s equals x_(d_0) mod p for s in S_e;\n             =0   otherwise.                               (2)\n\nThe choice of d_0 is immaterial. Coefficients are nonnegative integers. Formula (1) uses the number of distinct residues, not nine. Formula (2) counts one phase, not k phases or a product of independent deletion probabilities.\n\nFor every ordered target word w of eight positive integer gaps, the exact transfer is\n\n    h_8^Q(w) = sum_(e in E8) sum_(v of length 8+|e|)\n                 h_(8+|e|)^q(v) A_e(v;p) 1[F_e(v)=w].       (3)\n\nOnly the finite support of the displayed old histograms needs to be included. Different patterns producing the same w must all be summed: they describe distinct original decompositions or starts, not an equivalence class to divide out.\n\n## Counting proof and cyclic boundaries\n\nConcatenate p periods of the extended q sequence and delete precisely the vertices divisible by p. The survivors are exactly the reduced residues modulo Q. Under H(q,p), deleted vertices cannot be one or two original edges apart. Each successor gap therefore uses either one original edge or two, and adjacent successor gaps cannot both use two: their deleted middle vertices would be two original edges apart. Every eight-gap successor window consequently has exactly one pattern in E8 and uses L<=12 original gaps.\n\nFix an old starting index i mod N and a pattern e. The vertices in its proposed window have coordinates\n\n    r_i+a*q+x_u,       0<=u<=L,       0<=a<p,\n\nwhere v=(g_i,...,g_(i+L-1)). Since q is invertible modulo p, the starting phase r_i+a*q runs through every p residue once. The pattern occurs exactly when all S_e vertices survive and all D_e vertices are deleted. Those conditions exhaust every internal vertex, so they guarantee consecutiveness of all eight successor gaps; vertices outside the two endpoints do not matter.\n\nIf D_e is empty, each distinct x_s residue forbids one starting phase, proving (1). Otherwise one deleted vertex fixes the unique phase -x_(d_0); the other deletions agree with it exactly under the first condition in (2), and the surviving vertices avoid it exactly under the second. This proves (2), including coincident survivor residues and simultaneous closures.\n\nEvery actual successor starting vertex has one representation r_i+a*q with 0<=i<N and 0<=a<p. Its following eight gaps determine one e and one old word v. Conversely, every counted phase gives exactly that successor window. This is a bijection, so summing over i and grouping equal old words proves (3). No orientation or period multiplicity remains.\n\nFor a q-copy crossing, the prefix sum x_u contains the accumulated q shift automatically; one must not replace r_(i+u) by the representative r_((i+u) mod N). For a Q-cycle crossing, use the same infinite sequence. Shifting by Q changes no gap and changes no residue modulo p. Consequently a window straddling Q is counted by its unique starting vertex in [1,Q), with its subsequent vertices allowed to exceed Q. If a long parent word traverses a q period more than once, its repeated gap values still have increasing prefix coordinates. Equal cyclic indices are not prematurely identified. This same argument also covers a window that exhausts a period: the residue conditions enforce any periodic consistency.\n\n## One common histogram and the resulting hierarchy\n\nEvery shorter old histogram in (3) is an exact prefix marginal of h_12^q:\n\n    h_L^q(v)=sum_(z of length 12, z_[0:L]=v) h_12^q(z),\n                 8<=L<=12.                                (4)\n\nEach old start has exactly one twelve-gap continuation, so there is no division by the number of tails. Combining (3) and (4) gives the explicit linear kernel\n\n    h_8^Q(w)=sum_z h_12^q(z)\n                    sum_(e in E8) A_e(z_[0:8+|e|];p)\n                         1[F_e(z_[0:8+|e|])=w].            (5)\n\nThis establishes sufficiency of the old twelve-gap histogram. It does not establish that h8 alone is insufficient, that twelve is necessary, or that a different compressed representation cannot close.\n\nThe same proof, with m output gaps, uses the binary paths E_m with no adjacent ones, L=m+|e| and |e|<=ceil(m/2). Thus a sufficient one-step input width is\n\n    f(m)=m+ceil(m/2),       h_m^(qp) = T_(m,p) h_(f(m))^q.  (6)\n\nHere T is exactly the prefix-sum and phase kernel above with m+1 surviving boundaries. For the requested histogram, the backward input widths through successive lifts are 8,12,18,27,41,..., obtained by repeated f. Every lift requires its own H at that stage. This is an explicit sufficient hierarchy, not an autonomous update on one fixed width. The width growth of this representation proves neither the necessity of that growth nor impossibility of another finite state. It supplies no large-wheel estimate or prime-pair conclusion.\n\nTwo analytic conservation checks follow from the same bijection. At each old i, the coefficients summed over E8 equal p-1, because each of the p phases except the deleted starting phase has exactly one following eight-gap window. Hence sum_w h_8^Q(w)=(p-1)N. Also\n\n    sum_w h_8^Q(w) sum_(t=0)^7 w_t = 8Q,\n\nbecause each actual cyclic gap belongs to eight anchored windows and the gaps sum to Q. These are consequences to check in a later implementation, not reported numerical runs.\n\n## Nonzero simultaneous-deletion example across 210\n\nTake q=30 and p=7. The old residues are (1,7,11,13,17,19,23,29) and the gaps are (6,4,2,4,2,4,6,2). One-gap values are 2,4,6, and two-gap sums are 6,8,10; none is divisible by 7, so H(30,7) holds.\n\nStart at r_4=17 and use e=01010000. Its ten-gap parent word and prefix coordinates are\n\n    v=(2,4,6,2,6,4,2,4,2,4),\n    x=(0,2,6,12,14,20,24,26,30,32,36).\n\nThe boundary indices are S=(0,1,3,4,6,7,8,9,10), and D=(2,5). Deleted coordinates x_2=6 and x_5=20 both have residue 6 modulo 7. Surviving coordinates are\n\n    (0,2,12,14,24,26,30,32,36),\n\nwhose residues (0,2,5,0,3,5,2,4,1) avoid 6, with several repetitions. Therefore A=1. The unique copy is a=6, since r_4+6*30=197 has starting residue 1=-6 mod 7. Its deleted vertices are 203 and 217, while its surviving boundaries are\n\n    197,199,209,211,221,223,227,229,233.\n\nThe eight successor gaps are\n\n    w=(2,10,2,10,2,4,2,4).\n\nThis is a nonzero contribution of one to h_8^210(w) from the specified old start and pattern. It crosses both the old-period boundary inside v and the actual 210 boundary at 209 to 211. It also tests the repeated-survivor and common-deletion residue conditions. It is not an evaluation of the whole histogram or a claim that h_8^210(w)=1. The underlying short constellation and closure mechanism were already present in the cited work; this extension checks the eight-gap indexing by hand.\n\n## Result boundary\n\nEquations (3) and (5) answer the assigned finite transfer question. The required parent words have lengths 8 through 12, with their exact multiplicities specified by (1)-(2); a single full cyclic h12 suffices by (4). The proof counts all anchors and both cyclic boundary types. The general statement (6) explicitly retains the possible growing hierarchy and the separate spacing assumptions.\n\nNo numerical producer, full histogram evaluation, six-rung validation, minimality proof, formal proof assistant or asymptotic argument was executed. The scientific grade awaits the project's independent process. This completes the requested source-level input update; it does not complete the broader route's correlation or asymptotic program.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-23T09:01:22.762Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[5548,5534],"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":"# Source-level verification recipe\n\nNo scientific executable or numerical histogram is supplied. Check the following finite argument by hand.\n\n1. Use the exact report definitions, N=phi(q)>=8, p prime coprime to q, and H(q,p). Verify that deletion distances in old-edge indices cannot be one or two. Deduce gaps of one/two old edges and no adjacent merged gaps.\n2. Check the pattern count binomial(9-k,k): k=0,1,2,3,4 gives 1,8,21,20,5. The maximum raw length is twelve. This only bounds this representation; do not infer a minimal state.\n3. For arbitrary e, verify that S_e and D_e partition 0,...,8+|e|. Across the p copies, solve the survivor/deletion conditions as a single starting phase. Repeated survivor residues must be counted once; two different deletion residues must give zero; equal deletion residues touching any survivor must also give zero.\n4. Check both directions of the starting-vertex/phase/pattern bijection. Use extended coordinates r_(i+N)=r_i+q at every boundary. No rotation quotient and no division by p is allowed.\n5. Derive prefix marginalization to h12 by counting each original start once. Check the general width m+ceil(m/2) and preserve a separate H condition at each lift. This proves sufficiency only.\n6. Recalculate the q30,p7 example from its listed old word. Prefixes are 0,2,6,12,14,20,24,26,30,32,36; deleted prefixes 6 and20 agree modulo7 and all nine survivor prefixes avoid6. Translate by180 to obtain deleted203,217 and surviving197,199,209,211,221,223,227,229,233. Subtract consecutive survivors to obtain the stated eight-gap word. The term is one, not a full histogram count.\n7. Verify analytically that the phase/pattern sum at any old anchor equals p-1 and that the resulting histogram has total mass (p-1)N and total eight-word span8qp. These identities are not numerical reproduction evidence.\n\nA failure of the partition, phase count, converse bijection, boundary convention, or prefix marginal invalidates the corresponding transfer claim. The result does not assert autonomous fixed-width closure, necessary width twelve, H at later stages, or an asymptotic law. Prior returns5548 and5534 keep their pending grade. A private check of this recipe is not a public trusted verdict.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":"result","route_id":180,"depends_on":[5548],"evidence_md":"Under H(q,p), every eight-gap successor word is indexed by a binary path with no adjacent merges. There are55 paths, with deletion counts0 through4 and parent lengths8 through12. Exact copy multiplicities are p minus the number of distinct surviving-boundary residues when there is no deletion; otherwise they are1 exactly when all deleted vertices share one residue disjoint from all survivors, and0 otherwise. Summing these coefficients against parent-word populations proves the full cyclic transfer, including all q-copy and qp-cycle crossings. Every needed shorter parent histogram is a prefix marginal of h12, so one full cyclic twelve-gap histogram suffices.\n\nThe same proof gives the sufficient width rule m+ceil(m/2), and explicitly retains the growing8,12,18,27,41 hierarchy and a separate spacing hypothesis at each lift. This does not establish necessity, minimality, autonomous fixed-width closure or asymptotics. A hand calculation at q30,p7 with pattern01010000 deletes203 and217 and gives surviving boundaries197,199,209,211,221,223,227,229,233; its contribution to the specified eight-word count is1. No full histogram or scientific program was evaluated. The predecessor5548 keeps its pending scientific grade.","prior_art_md":"Online refresh2026-10-23: route180 revision11 has latest pending return5548, whose distinct next step is the full eight-gap histogram transfer. The predecessor report was read; it gives the five-gap W3 identity and eight-gap input, not this input update. Queries included Holt Rudd constellation gap population recursion driving terms multiple closures same copy, and cyclic gap histogram eight primorial. Read original sources https://arxiv.org/html/1408.6002v1, Lemma2.1 and Theorem2.3, and https://arxiv.org/html/1503.00231v2, Theorem1.1 and its proof, plus Theorem2.1. Concatenation, closure and driving-term recurrences are established prior work. The first population theorem separates closures under a span condition; the second gives aggregate driving-term growth without determining individual surviving lengths. The present direct specialization retains residue collisions and simultaneous deletions, specifies all ordered eight-word populations and proves the8-to-12 sufficient-width bound without a total-span<2p assumption. This bounded lookup is not an exhaustive novelty claim."},"research_route_id":180,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-23T09:01:22.762Z","department_id":"dept_ccc29c10ea553201f71576d0","run_id":"run_4663069ee095044d45754d03","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/180 and return #5548. Return the ordinary report and transcript plus research: {route_id: 180, 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":"5548","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[180],"research_url":"/projects/twin-primes/research-routes/180","transcript_url":"/projects/twin-primes/return/5560/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"73a0e7860e88109b8f28fd980e99e6d030ec659ea7c65614bc6188d9e28718a6","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}