[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"project-95882":3},{"id":4,"name":5,"fullName":6,"owner":7,"repo":5,"description":8,"homepage":9,"htmlUrl":9,"language":10,"languages":9,"totalLinesOfCode":9,"stars":11,"forks":12,"watchers":13,"openIssues":14,"contributorsCount":15,"subscribersCount":15,"size":15,"stars1d":15,"stars7d":15,"stars30d":16,"stars90d":15,"forks30d":15,"starsTrendScore":15,"compositeScore":17,"rankGlobal":9,"rankLanguage":9,"license":18,"archived":19,"fork":19,"defaultBranch":20,"hasWiki":21,"hasPages":19,"topics":22,"createdAt":9,"pushedAt":9,"updatedAt":23,"readmeContent":24,"aiSummary":25,"trendingCount":15,"starSnapshotCount":15,"syncStatus":26,"lastSyncTime":27,"discoverSource":28},95882,"PrimeGaps186","openai\u002FPrimeGaps186","openai","Conditional Lean formalization and numerical certificate for prime gaps at most 186.",null,"Lean",149,11,3,4,0,30,3.24,"Apache License 2.0",false,"main",true,[],"2026-09-21 02:04:28","# Prime Gaps at Most 186\n\nThis repository contains a Lean 4 formalization of a prime-gap bound and a\nPython numerical certificate. The Lean results remain **conditional on three explicit input axioms**;\nthe cited mathematical estimates and numerical computations have not been\nturned into Lean proofs of those inputs.\n\n## The result\n\nFor the sequence of primes $p_n$, the target bound is\n\n```math\n\\liminf_{n\\to\\infty}(p_{n+1}-p_n)\\le 186.\n```\n\nThe development derives $\\mathrm{DHL}[40,2]$ from the inputs below: every\nadmissible set of forty integer shifts has infinitely many translates containing\nat least two primes. Admissibility means omitting a residue class modulo every\nprime. Applying this to the included tuple of diameter 186 gives the gap bound.\n\nThe main declarations in [PrimeGaps186.lean](PrimeGaps186.lean), in namespace\n`PrimeGap186`, are:\n\n| Declaration | Result |\n| --- | --- |\n| `dhl_40_2` | $\\mathrm{DHL}[40,2]$ for every admissible integer tuple. |\n| `infinite_two_prime_translates_admissibleTuple` | Infinitely many two-prime translates of the explicit tuple. |\n| `primeGapLiminf_le_186` | The consecutive-prime gap bound. |\n\n## Assumed Deligne-type estimates\n\nFor a prime $p$, write $e_p(x)=\\exp(2\\pi i\\widetilde{x}\u002Fp)$, where\n$\\widetilde{x}$ is any integer representative of $x\\in\\mathbb{F}_p$. Define\n\n```math\n\\mathrm{Kl}_3(c;p)\n=\\frac1p\\sum_{\\substack{x_1,x_2,x_3\\in\\mathbb{F}_p\\\\x_1x_2x_3=c}}\ne_p(x_1+x_2+x_3),\n```\n\n```math\nK_2(c;p)=\\sum_{u\\in\\mathbb{F}_p^\\times}e_p(u+c\u002Fu).\n```\n\nThe axiom `PrimeGap186.kloosterman3_bound` assumes the following bound for\n**every prime** $p$ and all $c\\in\\mathbb{F}_p^\\times$:\n\n```math\n\\left|\\mathrm{Kl}_3(c;p)\\right|\\le 3.\n```\n\nThis follows from Deligne's theorem as stated in Nicholas M. Katz,\n[*Gauss Sums, Kloosterman Sums, and Monodromy Groups*, Annals of Mathematics\nStudies 116, Princeton University Press (1988), Theorem 4.1.1(1)–(2), p. 49](https:\u002F\u002Fweb.math.princeton.edu\u002F~nmk\u002FKatz-GKM.pdf#page=29).\nWith $n=3$, trivial multiplicative characters, and $b_1=b_2=b_3=1$, rank three\nand weight two give the raw bound $3p$; our normalization divides by $p$.\n\nThe axiom `PrimeGap186.kloosterman2_correlation_bound` assumes the following\nbound for **every prime** $p$ and all $A,B\\in\\mathbb{F}_p^\\times$:\n\n```math\n\\left|\\sum_{t\\in\\mathbb{F}_p\\setminus\\{0,-1\\}}\nK_2(A\u002Ft;p)\\,K_2(B\u002F(t+1);p)\\right|\\le 8p\\sqrt p.\n```\n\nThis is Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel,\n[*The Friedlander–Iwaniec character sum*, 14 June 2013, Proposition 2, p. 1](https:\u002F\u002Fpeople.math.ethz.ch\u002F~kowalski\u002Ffriedlander-iwaniec-sum.pdf#page=1).\nTheir normalized $\\mathrm{Kl}_2(c)$ equals $K_2(c;p)\u002F\\sqrt p$ after\ninverting the summation variable, so their $8\\sqrt p$ bound becomes\n$8p\\sqrt p$ here. No condition $A\\ne B$ is imposed; the two poles are excluded\neven when $A=B$.\n\nThese estimates are established in the cited literature, but remain unproved\ninputs in this Lean development.\n\n## Numerical input and certificate\n\n`PrimeGap186.physical_integral_bounds` assumes 104 outer and 45 inner\nphysical-integral upper bounds, plus three cap bounds.\n\nThe [Python certificate](prime_gap_186_certificate.py) recomputes the trial\nfrom scratch. The tested environment used Python 3.12.13, NumPy 2.2.6,\npython-flint 0.9.0, and a custom FLINT 3.6.0 build with corrected signed\npolynomial convolution (not bundled).\n\n```sh\npython3 -B prime_gap_186_certificate.py --workers 4 --output prime_gap_186_fresh.json\n```\n\nUse a new output path. Keep `PYTHONOPTIMIZE` unset and do not use `-O` or `-OO`.\nMandatory floating-point and signed-convolution checks must pass. A successful\nrun produces a receipt with `passed: true`; it does not discharge any Lean\naxiom.\n\n## Building and verification\n\nThe project pins Lean 4.34.0-rc2 and its Mathlib dependencies. With\n[elan](https:\u002F\u002Fgithub.com\u002Fleanprover\u002Felan) installed, run:\n\n```sh\nlake exe cache get\nlake build PrimeGaps186\n```\n\nThe registered Lean build passed without errors or warnings. Comparator matched\nall three results to `Challenge.lean`, and Nanoda and Lean’s kernel accepted their\nproofs in a local Colima Linux VM. The [configuration](comparator\u002Fmain.json) permits\nthe three documented project axioms plus `propext`, `Quot.sound`, and\n`Classical.choice` (six total); this verifies conditional proofs, not the inputs\nthemselves. The numerical certificate is unchanged from its earlier passing run.\n\n[Challenge.lean](Challenge.lean) specifies the statements and input assumptions,\nwith three intentional theorem placeholders. See the\n[Comparator instructions](comparator\u002FREADME.md) and\n[formalization metadata](formalization.yaml) for the checking setup and status.\n\nProject contributions use [Apache 2.0](LICENSE); existing third-party notices\nremain applicable.\n","该项目是对素数间隙上界≤186这一数学结论的条件化形式化验证，基于Lean 4定理证明语言实现。核心功能包括：形式化推导DHL[40,2]命题、构造直径为186的可容许素数组、并由此导出liminf(pₙ₊₁−pₙ) ≤ 186；技术特点是采用条件化建模——依赖三个明确列出的Kloosterman和估计公理（未在Lean中证明，但引自Deligne定理与Fouvry–Kowalski–Michel文献），辅以Python数值验证脚本提供计算支撑。适用于数论形式化验证、解析数论教学演示、以及对素数分布边界结果进行可信性复现与扩展研究。",2,"2026-09-05 02:30:09","CREATED_QUERY"]