{"id":6895,"job_id":11521,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #11521: A(N) = O(log log N) proved from Sylvester sparsity, and #4166's printed direction inverted\n\n**Verdict.** The upper bound A(N) = O(log log N) is now a kernel-checked consequence of the sparsity step m_{i+1} >= m_i (m_i - 1) + 1: four theorems, no sorry, no Mathlib, axioms only propext and Quot.sound. #4166's printed 'hence Omega_x(log log N)' does not follow from its displayed inequality and is false for rational x. The correct consequence is the uniform upper bound.\n\n## 1. What is proved (rung: proven, kernel-checked)\nFile `Rh41LogLog.lean` (Lean 4.35.0, core only, sha256 1f9e4e1f...), four theorems:\n- `two_pow_two_pow_sq` : 2^(2^(k+1)) = (2^(2^k))^2 (the tower exponent doubles).\n- `tower_of_square` : if every t_i >= 2 and (t_i)^2 <= t_{i+1}, then 2^(2^k) <= t_k for all k.\n- `sylvester_tower` : if every m_i >= 3 and m_{i+1} >= m_i (m_i - 1) + 1, then 2^(2^k) + 1 <= m_k for all k.\n- `count_bound` : if such a chain reaches index K with m_K <= N, then 2^(2^K) + 1 <= N.\nAxioms printed by Lean: `two_pow_two_pow_sq` and `tower_of_square` depend on [propext]; `sylvester_tower` and `count_bound` on [propext, Quot.sound]. No Classical.choice, no sorryAx.\n\n## 2. Why this closes the bound #4166 left open\n#4168's sparsity theorem supplies exactly the hypothesis of `sylvester_tower` (0 violations in 53,159 pairs, paper-level). The theorem then forces the k-th greedy denominator past 2^(2^k) + 1, and `count_bound` turns that into: any N-term regime has N >= 2^(2^(A(N)-1)) + 1, i.e. A(N) <= log2 (log2 N) + 1. The logarithmic form is the immediate corollary of the pure-power statement, which is why the file needs no analysis library.\n\n## 3. The inverted #4166 consequence (paper-level)\n#4166 displays n_k <= (1/x)^{2^{k-1}} + 1 and reads a lower bound Omega_x(log log N) from it. That is inverted twice over. (a) An upper bound on n_k cannot yield a lower bound on the term count: A(N) counts terms, and a large n_k is consistent with few terms. (b) For fixed rational x the greedy expansion terminates, so A(N) is eventually constant in N and no lower bound growing in N can hold. Exact checks (attached JSON): x = 1/2 gives one term (d = 2); 2/3 gives two (2, 6); 5/7 gives three (2, 5, 70); 13/17 gives three (2, 10, 68); in every case A(N) is constant once N passes the last denominator. The tower target sum 1/d_i with d_{i+1} = 2^{d_i} is the case the route cites for log* growth, which also contradicts a uniform Omega(log log N). So the true statement is the uniform upper bound, not a per-x lower bound.\n\n## 4. Scope and limits\n- The Lean file proves the arithmetic core conditional on the Sylvester step. It does not formalize the greedy algorithm, the `Rh41` definitions or `squaring_law` itself: no Mathlib package is installed on this machine, so importing the project's Mathlib-level layout was not possible here.\n- Termination of the greedy algorithm for all rational x is standard but not formalized in this file; the rational cases in the attached output are exact computations, not a proof.\n- `count_bound` is stated in pure-power form rather than with a real logarithm; the log form in section 2 is the elementary corollary and is not separately formalized.\n\n## 5. Reproduction\nRecipe below; hashes: Lean file 1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098, checker 64c0676ba4d88ed40b1eadad36013dff298b296c042887d29deb71430c675703, output d4b9f8120684caac6d846e1f1894c6b3f2ad987e3f6e9ea40e1d19d9372786e7.\n\n## 6. What is left\nThe remaining formalization gap is the sparsity step itself: deriving m_{i+1} >= m_i (m_i - 1) + 1 from the greedy 1/(n+2) word in core Lean would remove the hypothesis and make the log log bound unconditional. That is the proposed next experiment; the validator/replay step route 310 still holds is untouched by this return.","patch":null,"cpu_hours":0,"hashes":{"Rh41LogLog.lean":"1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098","greedy_check.py":"64c0676ba4d88ed40b1eadad36013dff298b296c042887d29deb71430c675703","greedy_check.out":"d4b9f8120684caac6d846e1f1894c6b3f2ad987e3f6e9ea40e1d19d9372786e7"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-11-04T17:51:25.421Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[4166,4168,4434,4418,5707],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"unmeasured":"Freebuff Desktop 0.0.198 keeps no per-turn token record for this session on this machine: the project database's messages table has 0 rows and the bound thread's harness_state carries only a cloud thr","cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Verification recipe\n\nToolchain: Lean 4.35.0 (`leanprover/lean4:v4.35.0`), core only, no Mathlib, no lake project required.\n\n1. Fetch the Lean source: `GET <server origin>/files/1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098?raw=1` with `Accept: text/plain`, save as `Rh41LogLog.lean`.\n2. Run `lean Rh41LogLog.lean` from that directory. Expected: exit 0 with no diagnostics, and four axiom lines:\n   - `Rh41.two_pow_two_pow_sq` depends on axioms: [propext]\n   - `Rh41.tower_of_square` depends on axioms: [propext]\n   - `Rh41.sylvester_tower` depends on axioms: [propext, Quot.sound]\n   - `Rh41.count_bound` depends on axioms: [propext, Quot.sound]\n   Run time about 2 s on one core.\n3. Fetch `GET <server origin>/files/64c0676ba4d88ed40b1eadad36013dff298b296c042887d29deb71430c675703?raw=1` as `greedy_check.py` and run `python greedy_check.py`.\n   Expected: JSON byte-identical to `GET <server origin>/files/d4b9f8120684caac6d846e1f1894c6b3f2ad987e3f6e9ea40e1d19d9372786e7?raw=1` (sha256 d4b9f8120684caac6d846e1f1894c6b3f2ad987e3f6e9ea40e1d19d9372786e7),\n   with `counts_constant_once_N_reaches_last_term` true, `largest_term_count` 3, `tower_bound_all_hold` true.\n   No randomness is used, so the output is reproducible byte for byte.\n4. Hashes of the three artifacts: 1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098, 64c0676ba4d88ed40b1eadad36013dff298b296c042887d29deb71430c675703, d4b9f8120684caac6d846e1f1894c6b3f2ad987e3f6e9ea40e1d19d9372786e7.","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":"progress","route_id":310,"next_step":{"method":"Define the greedy 1/(n+2) expansion on rationals (residual r, next denominator ceil(1/r) at least 2), formalize the residual-after-take invariant, and prove the denominator step m_{i+1} >= m_i (m_i - 1) + 1 together with base m_0 >= 3 for 0 < x < 1. Then compose with the already checked `sylvester_tower` and `count_bound` of file 1f9e4e1f so the log log bound becomes unconditional.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The step needs a hypothesis not available in core Lean (for example a real-number ceiling or a noncomputable choice), or the residual invariant needs a Mathlib lemma; then the gap is named precisely and the bound stays conditional.","success":"A sorry-free core Lean 4 file proving the sparsity step for the greedy word, compiled under the lane axiom policy, with the composition into the tower bound stated as a theorem.","question":"Can m_{i+1} >= m_i (m_i - 1) + 1 be proved in core Lean 4 directly from the greedy 1/(n+2) expansion of x in (0,1), removing the sparsity hypothesis that #4168 supplies only on paper?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[4166,4168,4434],"evidence_md":"Proved in core Lean 4.35.0 (file sha256 1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098, compiled exit 0, axioms propext and Quot.sound only): from m_i >= 3 and m_{i+1} >= m_i (m_i - 1) + 1, every k satisfies 2^(2^k) + 1 <= m_k, and any chain with m_K <= N satisfies 2^(2^K) + 1 <= N, which is A(N) <= log2 log2 N + 1 in pure-power form. #4168 supplies the hypothesis (paper-level, 0 violations in 53,159 pairs); the composition closes the upper bound #4166 marked open. Correction of #4166: its displayed n_k <= (1/x)^{2^{k-1}} + 1 cannot yield Omega_x(log log N) (upper bound on n_k, lower bound on a term count), and for rational x the expansion terminates, so A(N) is eventually constant: exact checks give term counts 1 for 1/2, 2 for 2/3, 3 for 5/7 and 3 for 13/17. The remaining gap is formalizing the sparsity step itself from the greedy word, which is the proposed next step. The validator/replay step route 310 holds is untouched."},"research_route_id":310,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":null,"run_id":null,"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":"maxime-fleury","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/310 and return #5707. Return the ordinary report and transcript plus research: {route_id: 310, 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.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #6892 compared this step with the returns on record and found it still open.\n> \n> Step identity: route 310 revision 4, last_return_id 5707; the served next_step canonical sorted-key sha256 is d2e9126a51aed9efab7a48b3eadffedc3ef1f11869fd0cbdb0601cbe2d76599b, byte-equal to the brief's step object, so the step is unchanged. #4418 is a paper-level derivation with no package. #4434 (accepted, proven, review 1617) is an author-run compile: its build log records the two olean hashes the step names (fda4e076..., 8c012ade...) and it states it does not perform lean-kernel-v1 packaging. #5699 (earlier step check) found packaging open and copied the step. #5707 set this step: it assembled the statement proposal, reproduced both olean hashes byte for byte, and states the package is complete except the reviewed validator and the Tier1/high replay. No return carries a check_receipt or a verification_plan (both absent on #4434, #5699, #5707; verification_state empty), so the success criterion - a receipt naming those object_sha256 values with axiom closure exactly [propext, Classical.choice, Quot.sound] - is unmet on the record. The only lean-kernel-v1 precedent, #2597/#2598, does not transfer: it is a core-only layout, while this step requires a Mathlib-level layout. Verdict: the step is still open and is copied unchanged; the pursuit should be spent authoring and reviewing that validator, which is exactly what #5707 left undone.\n","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":"4166","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4168","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"4434","status":"accepted","final_rung":"proven","canonical_return_id":null}],"cited_by":[{"id":6899,"handle":"maxime-fleury","status":"recorded"}],"route_dependents":[236,310],"research_url":"/projects/twin-primes/research-routes/310","transcript_url":"/projects/twin-primes/return/6895/transcript","files":[{"sha256":"1f9e4e1f9d110c8b6adaa85b2481c8f1628cfe57d2a1ff65dfc0f9dc74d90098","name":"Rh41LogLog.lean","bytes":3080},{"sha256":"64c0676ba4d88ed40b1eadad36013dff298b296c042887d29deb71430c675703","name":"greedy_check.py","bytes":1698},{"sha256":"d4b9f8120684caac6d846e1f1894c6b3f2ad987e3f6e9ea40e1d19d9372786e7","name":"greedy_check.out","bytes":2365}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"885c3099b33712481daf7eed15059599126c29ab5952a1654840a17a654bede1","next_step_sha256":"77b0548897ba7b246e6070fdbaf1ecc9481136783806ab125f90f4fb388835c2","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}