{ "cohort": { "id": "launch_cohort.collatz-predecessor-090.v001", "title": "Collatz Predecessor Exponent-0.90 Publication Cohort" }, "nonClaims": [ "This packet is not proof evidence by itself.", "The formal results do not establish that every Collatz orbit reaches 1.", "No explicit eventual cutoff is provided.", "The target-dependent positive constant in the exponent 901/1000 theorem is not claimed to equal 1.", "No optimality statement is made for exponent 901/1000.", "The native-assisted certificate checks are not described as kernel-clean.", "Preliminary novelty evidence is not final priority clearance." ], "publication": { "acceptedResultStatus": "accepted_results_recorded", "launchStatus": "authorized", "surface": "public_production" }, "computationFootprint": { "forcedPotentialInequalities": 215233605, "lpRows": 129140163, "sourcePaths": [ "artifacts/kl_predecessor/k18_g901_1000/python-verification.json", "artifacts/kl_predecessor/k18_adaptive_potential/python-verification.json" ] }, "proofs": [ { "acceptedResultId": "accepted.collatz.predecessor-lower-bound-090.nat.v001", "acceptedResultPresentation": { "acceptedResultId": "accepted.collatz.predecessor-lower-bound-090.nat.v001", "artifactId": "artifact.collatz.predecessor-lower-bound-090.nat.v001", "resultBoundary": "For each fixed positive target not divisible by 3, the retained Lean declarations give eventual unit-coefficient exponent 9/10 lower bounds for real and natural cutoffs and a target-dependent positive-constant exponent 901/1000 real-cutoff bound. No explicit eventual cutoff is asserted. Non-claim: This target record is not proof evidence by itself. Non-claim: The formal results do not establish that every Collatz orbit reaches 1. Non-claim: The target-dependent positive constant is not claimed to equal 1. Non-claim: No optimality or final novelty claim is made. Non-claim: Native-assisted is not kernel-clean. Non-claim: This accepted-result record does not by itself authorize public export or deployment.", "statusLabel": "Accepted formal theorem", "title": "Accepted Result: Collatz Predecessor Natural-Cutoff x^0.90 Lower Bound" }, "artifactId": "artifact.collatz.predecessor-lower-bound-090.nat.v001", "axiomProfile": "custom_axioms_present", "axioms": [ "Classical.choice", "Erdos1135.KrasikovLagarias.k18AdaptiveForcedPotentialCertificate_check._native.native_decide.ax_1_1", "Erdos1135.KrasikovLagarias.k18Gamma901EncodedCertificate_check._native.native_decide.ax_1_1", "Quot.sound", "propext" ], "buildElapsedMs": 7051, "buildStatus": "passed", "declarationName": "CollatzPredecessor090.predecessor_count_lower_bound_090_nat", "noSorryStatus": "passed", "packageCommit": "5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f", "firstPartySourcePath": "sources/collatz-predecessor-090/commits/5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f/CollatzPredecessor090.lean", "publicSourceUrl": "https://proofatlas.ai/sources/collatz-predecessor-090/commits/5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f/CollatzPredecessor090.lean", "sourceFile": "CollatzPredecessor090.lean", "statementId": "statement.collatz.predecessor-lower-bound-090.nat.v001", "formalText": "theorem predecessor_count_lower_bound_090_nat {target : Nat} (htarget : 0 < target) (hmod : target % 3 ≠ 0) : ∀ᶠ cutoff : Nat in Filter.atTop, (cutoff : Real) ^ ((9 : Real) / 10) ≤ (predecessorCount target cutoff : Real)", "title": "Collatz Predecessor Natural-Cutoff x^0.90 Lower Bound" }, { "acceptedResultId": "accepted.collatz.predecessor-lower-bound-090.real.v001", "acceptedResultPresentation": { "acceptedResultId": "accepted.collatz.predecessor-lower-bound-090.real.v001", "artifactId": "artifact.collatz.predecessor-lower-bound-090.real.v001", "resultBoundary": "For each fixed positive target not divisible by 3, the retained Lean declarations give eventual unit-coefficient exponent 9/10 lower bounds for real and natural cutoffs and a target-dependent positive-constant exponent 901/1000 real-cutoff bound. No explicit eventual cutoff is asserted. Non-claim: This target record is not proof evidence by itself. Non-claim: The formal results do not establish that every Collatz orbit reaches 1. Non-claim: The target-dependent positive constant is not claimed to equal 1. Non-claim: No optimality or final novelty claim is made. Non-claim: Native-assisted is not kernel-clean. Non-claim: This accepted-result record does not by itself authorize public export or deployment.", "statusLabel": "Accepted formal theorem", "title": "Accepted Result: Collatz Predecessor Real-Cutoff x^0.90 Lower Bound" }, "artifactId": "artifact.collatz.predecessor-lower-bound-090.real.v001", "axiomProfile": "custom_axioms_present", "axioms": [ "Classical.choice", "Erdos1135.KrasikovLagarias.k18AdaptiveForcedPotentialCertificate_check._native.native_decide.ax_1_1", "Erdos1135.KrasikovLagarias.k18Gamma901EncodedCertificate_check._native.native_decide.ax_1_1", "Quot.sound", "propext" ], "buildElapsedMs": 7215, "buildStatus": "passed", "declarationName": "CollatzPredecessor090.predecessor_count_lower_bound_090", "noSorryStatus": "passed", "packageCommit": "5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f", "firstPartySourcePath": "sources/collatz-predecessor-090/commits/5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f/CollatzPredecessor090.lean", "publicSourceUrl": "https://proofatlas.ai/sources/collatz-predecessor-090/commits/5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f/CollatzPredecessor090.lean", "sourceFile": "CollatzPredecessor090.lean", "statementId": "statement.collatz.predecessor-lower-bound-090.real.v001", "formalText": "theorem predecessor_count_lower_bound_090 {target : Nat} (htarget : 0 < target) (hmod : target % 3 ≠ 0) : ∀ᶠ x in Filter.atTop, x ^ ((9 : Real) / 10) ≤ (predecessorCountReal target x : Real)", "title": "Collatz Predecessor Real-Cutoff x^0.90 Lower Bound" }, { "acceptedResultId": "accepted.collatz.predecessor-lower-bound-0901.constant.v001", "acceptedResultPresentation": { "acceptedResultId": "accepted.collatz.predecessor-lower-bound-0901.constant.v001", "artifactId": "artifact.collatz.predecessor-lower-bound-0901.constant.v001", "resultBoundary": "For each fixed positive target not divisible by 3, the retained Lean declarations give eventual unit-coefficient exponent 9/10 lower bounds for real and natural cutoffs and a target-dependent positive-constant exponent 901/1000 real-cutoff bound. No explicit eventual cutoff is asserted. Non-claim: This target record is not proof evidence by itself. Non-claim: The formal results do not establish that every Collatz orbit reaches 1. Non-claim: The target-dependent positive constant is not claimed to equal 1. Non-claim: No optimality or final novelty claim is made. Non-claim: Native-assisted is not kernel-clean. Non-claim: This accepted-result record does not by itself authorize public export or deployment.", "statusLabel": "Accepted formal theorem", "title": "Accepted Result: Collatz Predecessor Positive-Constant x^0.901 Lower Bound" }, "artifactId": "artifact.collatz.predecessor-lower-bound-0901.constant.v001", "axiomProfile": "custom_axioms_present", "axioms": [ "Classical.choice", "Erdos1135.KrasikovLagarias.k18AdaptiveForcedPotentialCertificate_check._native.native_decide.ax_1_1", "Erdos1135.KrasikovLagarias.k18Gamma901EncodedCertificate_check._native.native_decide.ax_1_1", "Quot.sound", "propext" ], "buildElapsedMs": 8398, "buildStatus": "passed", "declarationName": "CollatzPredecessor090.predecessor_count_lower_bound_0901", "noSorryStatus": "passed", "packageCommit": "5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f", "firstPartySourcePath": "sources/collatz-predecessor-090/commits/5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f/CollatzPredecessor090.lean", "publicSourceUrl": "https://proofatlas.ai/sources/collatz-predecessor-090/commits/5f76a170e74ea5b0c37c56683bd4c1e9d72e5e3f/CollatzPredecessor090.lean", "sourceFile": "CollatzPredecessor090.lean", "statementId": "statement.collatz.predecessor-lower-bound-0901.constant.v001", "formalText": "theorem predecessor_count_lower_bound_0901 {target : Nat} (htarget : 0 < target) (hmod : target % 3 ≠ 0) : ∃ constant : Real, 0 < constant ∧ ∀ᶠ x in Filter.atTop, constant * x ^ ((901 : Real) / 1000) ≤ (predecessorCountReal target x : Real)", "title": "Collatz Predecessor Positive-Constant x^0.901 Lower Bound" } ], "publicSafeSummary": "A standalone Lean 4 development proves that, for every fixed positive target not divisible by 3, the accelerated-Collatz predecessor count is eventually at least x^0.90, in both real- and natural-cutoff forms. A stronger target-dependent positive-constant x^0.901 bound supplies the exponent reserve. Lean checks the exact theorem chain; two disclosed native finite checks are independently replayed over 344,373,768 exact inequalities. 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