[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"bDJosGkbbU":3},"# trureturing\n\n**A scientific method for AI to discover truth and find its next question.**\n\n[Vision](docs/VISION.md) · [Film](#film) · [Start](#start-your-journey) · [Method](#from-questions-to-knowledge) ·\n[Examples](#three-places-to-look) · [Truth and computation](#truth-and-computation) ·\n[Spacetime](#toward-holographic-spacetime) · [Information escape](#information-escape) ·\n[Evidence](#what-is-proved-and-what-is-open) · [First run](#first-run) ·\n[Lean source](D5/) · [Book](https://the-omega-institute.github.io/trureturing-mdbook/) ·\n[Contribute](#take-part) · [Licensing](#license-and-foundations)\n\ntrureturing develops a scientific method for AI to turn gaps in knowledge into\nquestions, test conjectures, expose limits in its representations, and return\nchecked results to a reusable library. Autonomous research selection remains a\ngoal to evaluate.\n\nThe name expresses **true · return · Turing**. Truth guides the search;\nverified knowledge returns as a premise for the next inquiry; computation\nexplores the connections. We call the structure we seek the **geometry of\nlogical truth**: dependencies, invariants, distinctions and the boundaries of\nwhat an observer can recover.\n\nThe project combines philosophical inquiry, theory, experiments and Lean 4\nformalization. Each proof establishes its exact statement under declared\nassumptions. The [vision and research guide](docs/VISION.md) connects this work\nto open directions.\n\n## Film\n\n\u003Cp>\u003Ca href=\"https://github.com/the-omega-institute/trureturing-film/releases/download/film-001-v1/TRURETURING_001_narrated_EN_subs_ZH-EN.mp4\">\u003Cimg src=\"https://github.com/the-omega-institute/trureturing-film/releases/download/film-001-v1/TRURETURING_001_cover.jpg\" width=\"1920\" height=\"1080\" alt=\"TRURETURING — Truth Is Discovered: download Film 001\">\u003C/a>\u003C/p>\n\n**TRURETURING — Truth Is Discovered** · English AI narration · Chinese and English subtitles.\n[Download MP4](https://github.com/the-omega-institute/trureturing-film/releases/download/film-001-v1/TRURETURING_001_narrated_EN_subs_ZH-EN.mp4) ·\n[Film source and releases](https://github.com/the-omega-institute/trureturing-film).\n\n## Start your journey\n\nBring a question that matters to you. In an installed **Claude Code or Codex**\nwith a local workspace and Git, paste this one sentence:\n\n> Help me explore https://github.com/the-omega-institute/trureturing: use an existing checkout or clone it into a new directory if needed, read AGENTS.md and README.md, then read the relevant SKILL.md under skills/ to investigate a question I care about and find a checked result or a clearly stated open question.\n\nThe [agent and skills guide](docs/CONTRIBUTING.md#use-claude-code-or-codex)\nexplains how to begin with either client and turn an exploration into a\ncontribution.\n\nThis homepage is the tip of an iceberg. The examples offer a glimpse; the\nlarger shape is yours to explore, following definitions, assumptions and\nconnections with your own questions. What you find can change how you look.\nAn epigraph for that exploration:\n\n> And if thou gaze long into an abyss, the abyss will also gaze into thee.\n\n— Friedrich Nietzsche, *Beyond Good and Evil*,\n[§146](https://www.gutenberg.org/files/4363/4363-h/4363-h.htm).\n\n## From questions to knowledge\n\nSearch [Lean source](D5/) and literature for your target. Identify missing\npremises, distinctions or connections; choose questions addressing them.\nSpecify supporting and refuting outcomes before designing discriminating tests.\nKeep results with their assumptions; check those against your objects before reuse.\n\n> The last line of the ledger is always the first line of the next round.\n\nProofs supply premises; counterexamples refute claims within their stated scope.\nWhen identical readings hide different target values, no function of those\nreadings recovers the target in both cases.\n[Seek new observations or relations](docs/VISION.md#how-ai-can-find-its-next-direction).\n\nEvaluate research selection against stated baselines on questions excluded\nfrom method design, matching information and resources.\n\n\u003Cp>\u003Cimg src=\"docs/assets/inquiry-cycle.svg\" width=\"360\" height=\"560\" alt=\"Ask a question, test hypotheses, check a proof or refutation, and keep a reusable result. Solid return reuses results as premises; dashed returns carry unresolved questions.\">\u003C/p>\n\n*A schematic of inquiry, not runtime behavior or dependency data.* Checked\nresults return as premises; dashed returns carry unresolved questions, even\nwithout a checked result. Tests alone do not establish a theorem.\n\nGolden integers, Fibonacci weights and Zeckendorf representations are one\nthread of the library; the examples below also explore conjecture refutation\nand limits of local observation.\n\n## Three places to look\n\n### 01 · Refute a conjecture.\n\nFor positive integers n, let a(n) be the greatest integer k with `(1 + 1/n)^k ≤ 2`.\nGreathouse conjectured for OEIS A175406 that\n`a(n) = floor((n + 1/2) log 2)`. At `n = 1121626023352383`, the formula gives\n`777451915729368`, while the actual value is one less.\nThe [Lean refutation](D5/S0/Certificates/GreathouseLogTwoFloorRefutation.lean)\nuses certified logarithm bounds to refute the literal universal formula.\nWhat characterizes the inputs where it fails? Witness minimality and priority\nare not claimed. [Problem and sources](Problems/oeis-a175406-log-two-floor-refutation.md) ·\n[Explanation](Blueprint/D5/S0/Certificates/GreathouseLogTwoFloorRefutation.md).\n\n### 02 · Find what observations cannot tell you.\n\nThe [local-marginal theorem](D5/S3/Quantum/Entanglement/LocalMarginalCorrelationBlindSpot.lean)\ngives two-qubit states with identical reduced states on both qubits: the pure\nBell state `(|00⟩+|11⟩)/√2` and the equal `00`/`11` mixture. An added joint\n`X⊗X` readout has expectations `1` and `0`, respectively, where [X](D5/S3/Quantum/FiniteDimensional.lean) swaps\n`0` and `1`. This separates this pair.\n\nFor finite factor dimensions `m, n ≥ 1` with `m × n > 1`, the theorem also\nproves that the correlation sector in the Hermitian tensor model is orthogonal\nto the local sectors and has real dimension `(m² − 1)(n² − 1)`. This identifies\nprecisely which directions the local description omits.\n[Explanation](Blueprint/D5/S3/Quantum/Entanglement/LocalMarginalCorrelationBlindSpot.md).\n\n### 03 · Build a result that holds beyond the examples.\n\nWrite a natural number n as its unique sum of nonadjacent Fibonacci weights\n`F₂ = 1, F₃ = 2, F₄ = 3, …`. Replace each weight Fᵢ by φⁱ, where φ is the\ngolden ratio, to obtain β(n). How far does this coordinate fail to preserve\naddition?\n\n$$\\beta(a)+\\beta(b)-\\beta(a+b)\\in\\lbrace-1,0,1\\rbrace.$$\n\n[`deficit_three_valued`](D5/S1/Deficit/DeficitThreeValued.lean) proves this for\nall natural inputs. For `1 + 1`, `β(1)=φ²` and `β(2)=φ³` give `2φ²−φ³=1`.\nThe discrepancy is also the signed count of the two\nlowest repeated-carry rules during digit normalization: a reusable connection\nbetween an arithmetic algorithm and an exact bound, however large the inputs.\n[Definitions and carry-count theorem](D5/S1/Deficit/DeficitInteger.lean) ·\n[Explanation](Blueprint/D5/S1/Deficit/DeficitThreeValued.md).\n\n## Truth and computation\n\nOur philosophical starting point is that **truth is discovered, not created\nby the act of computing it**. In this view, **Dao (道), or God (神), names an\nencompassing network of truths and their logical relations**, within which a\nfinite observer discovers connections. This is the project's metaphysical\norientation, not a theorem about the existence of God or the physical universe.\n\nComputation constructs examples and counterexamples, searches for proofs and\nchecks them. A verified proof extends what the library can justify and reuse.\nNeither this philosophical conviction nor a growing proof library establishes\nthat one program can enumerate or decide every truth.\n\nThe [dependency topology](D5/S3/ConceptDynamics/DependencyTopology/AlexandrovDependencyTopology.lean)\ncalls a set open when it contains every node reachable from its members. The\n[recovery criterion](D5/S3/ConceptDynamics/Restoration/TargetRecoveryCriterion.lean)\nsays that, on a nonempty state space, a target admits a recovery function from\nan observation exactly when any two states with the same observation have the\nsame target value. Existence alone supplies no algorithm or cost bound.\n\n## Toward holographic spacetime\n\nWe study **holographic spacetime geometry** by asking when partial records\nof time and space support reconstruction and\n[temporal composition](D5/S3/ConceptDynamics/Spacetime/HiddenArchiveTemporalDomain.lean).\n\n[Local agreement can fail globally](D5/S3/ConceptDynamics/Gluing/LocalLawGluingObstruction.lean):\nthree windows on Boolean variables require `x=y`, `y=z` and `x≠z`. Every\noverlap allows both values, yet no triple satisfies all three constraints.\n\nThe [tree extension theorem](D5/S3/ConceptDynamics/Gluing/RunningIntersectionRecords.lean)\nassumes nonempty local record sets on a finite tree: each recorded variable's\noccurrences form a connected subtree, and neighbors allow exactly the same\njoint assignments on their full overlap. Every allowed local record extends\nacross all recorded variables, satisfying every local constraint.\nAdditional global constraints can exclude every extension.\n\nUniqueness, original-history recovery, computational cost, and reconstruction\nwith resolution and error bounds require further results. Links to physical\nspacetime or holographic duality remain research questions.\n\nTheory inputs study\n[event archives](docs/develop/theory/CONTEXTUAL_SPACETIME_ARITHMETIC.md)\nretaining time, position, causal order and provenance;\n[when observations preserve](docs/develop/theory/RECURSIVE_RELATIONAL_OBSERVATION.md)\ncomposition, shared sources and targets; and\n[experimental distances](docs/develop/theory/RECURSIVE_RELATIONAL_OBSERVATION_CONTEXT_GEOMETRY.md)\ndefined through allowed experiments and responses. Their prose does not certify\nformal coverage.\n\n## Information escape\n\n**Information escape**: some distinct states remain indistinguishable under\nchosen **readouts** (observation methods). Four questions guide an **automated\nreviewer** under development:\n\n- **Where did information escape?** Name the objects, assumptions and\n  observations under which distinct states remain indistinguishable.\n- **How is that escape addressed?** Describe the added readout or relation,\n  and show how a proof connects it to those objects and assumptions.\n- **What new information emerges?** Identify the distinction now justified\n  by the result and its verified connection to the readout.\n- **Where does information continue to escape?** Exhibit a remaining\n  indistinguishable pair, prove none remain within the stated scope, or mark\n  the boundary open.\n\n**Mathematical comparison.** Fix a catalog of registered theorem occurrences and their readouts on one\nshared state space. Remove one occurrence while keeping the others fixed.\nIts **unique captures** are the pairs of distinct states that were\ndistinguishable before removal and indistinguishable afterward. The\n[EscapePairs definitions and proofs](D5/S3/ConceptDynamics/InformationEscape/EscapePairs.lean)\nformalize this comparison.\n\nFor a **finite state space with at least two states**,\n[StructuralNovelty](D5/S3/ConceptDynamics/InformationEscape/StructuralNovelty.lean)\nshows that removing an occurrence strictly increases the **escape rate** exactly\nwhen it has a unique capture. The rate is the fraction of ordered distinct-state\npairs left indistinguishable. Zero unique capture does not mean worthlessness:\nanother occurrence can carry the same distinction. This comparison supplies\nneither a universal value score nor a historical novelty judgment.\n\n[Example 02](#02--find-what-observations-cannot-tell-you) adds `X⊗X` expectations to separate a locally\nindistinguishable pair. Which pairs, if any, remain indistinguishable after\nadding this readout?\n\nThe current declared-template binding rule issues\n**Observe warnings that do not block admission**\n([specification, A5.5](docs/develop/spec/golden-ledger-repo-spec.md);\n[implementation](tools/StrataLint.Engine/Rules/TheoryGeneration/DeclaredTemplateBindingRule.cs)).\nOther admission checks still apply. Its module selection is\nseparate from the mathematical comparison above.\nThe [Normative Draft](docs/develop/spec/lean_single_compile_intrinsic_information_escape_theory_and_spec.md)\ndescribes a wider design whose implementation remains incomplete.\n\n## What is proved, and what is open\n\nMathematical reuse rests on the statements, checked proof terms and axiom\ndependencies in the [Lean source](D5/). The\n[frozen ledger](Golden/Frozen/state/) tracks frozen module identities.\n[Theory prose](docs/develop/theory/) supplies research input, and\n[experiments](Evidence/) supply observations within their declared scope;\nneither substitutes for a Lean proof. The C# harness checks repository rules,\nproof reports and frozen state. Independent review examines whether statements\nfaithfully express the intended mathematics.\n\nThe [book](https://the-omega-institute.github.io/trureturing-mdbook/) is a\nsearchable snapshot of [explanations](Blueprint/), [problem dossiers](Problems/)\nand [literature notes](Library/), published by\n[trureturing-mdbook](https://github.com/the-omega-institute/trureturing-mdbook).\nThe formal source remains authoritative.\n\nTwo explicit boundaries live in [Hearts.lean](D5/X_Frontier/Hearts.lean):\n\n- **O-5:** `o5_independence`, a zero-localization claim for the canonical golden\n  Euler germ, has an unresolved proof body (`sorry`).\n- **O-6:** `o6WeilPositivityStatement` defines a Weil-positivity proposition.\n  Defining that proposition supplies no proof of it.\n\nThese are open research obligations, not established impossibility results.\nThis repository does **not** establish the Riemann hypothesis.\n\n## First run\n\nInstall [elan](https://github.com/leanprover/elan#installation) and the\n[.NET SDK](https://dotnet.microsoft.com/en-us/download) version specified in\n[global.json](global.json). Ensure `lake` and `dotnet` are on `PATH`;\nyou also need Git, Make and Bash.\nelan selects Lean from [lean-toolchain](lean-toolchain). Mathlib is declared in\n[lakefile.toml](lakefile.toml), with resolved dependencies in\n[lake-manifest.json](lake-manifest.json). For contribution checks, add Python\n3.11+ as `python3`; see the [full prerequisites](docs/CONTRIBUTING.md#prerequisites).\n\nClone the project, then build just the introductory module. The `make` entry\nprepares a private Lean cache; the first run may download dependencies.\n\n```sh\ngit clone https://github.com/the-omega-institute/trureturing.git\ncd trureturing\nmake lean LEAN_TARGETS=D5.S0.Conventions.WDigits\n```\n\nThe [WDigits module](D5/S0/Conventions/WDigits.lean) directly reuses\n**Mathlib's Zeckendorf development** to encode and decode natural numbers.\nFrom that same directory, run this temporary example:\n\n```sh\nexample_dir=\"$(mktemp -d)\"\ncat > \"$example_dir/First.lean\" \u003C\u003C'LEAN'\nimport D5.S0.Conventions.WDigits\nopen D5.S0.Conventions\n\n#eval wdigits 42\n#eval ((wdigits 42).map Nat.fib).sum\n#check decode_wdigits\nLEAN\nlake env lean \"$example_dir/First.lean\"\nrm \"$example_dir/First.lean\"\nrmdir \"$example_dir\"\n```\n\nThe two evaluations print `[9, 6]` and `42`. The list contains **Fibonacci\nindices**, so its weights are `F₉ = 34` and `F₆ = 8`; it is not a list of the\nweights themselves. `#check` displays the general decoding theorem's type.\nEvaluating 42 illustrates the encoding; the theorem covers every natural number.\n\n[Read the explanation](Blueprint/D5/S0/Conventions/WDigits.md).\n`make help` lists the repository's command entry points.\n\n## Take part\n\nStart with one of the examples above. Reproduce it, improve its explanation,\nreport a mismatch between prose and a statement, or explore a precise open\nquestion. Contributions in English and Chinese are welcome.\n\n### A continuing research program\n\nThe [research directions](docs/VISION.md#research-directions) specify evidence\nof progress for each question:\n\n- Can AI choose questions that yield reusable knowledge?\n- Which maps connect proof dependencies and observational distinctions?\n- Which historical relations support reconstruction and legal composition?\n\nThe [contribution guide](docs/CONTRIBUTING.md) walks you through forks, isolated\nworktrees, checks and pull requests to `dev`.\n[Issues](https://github.com/the-omega-institute/trureturing/issues) are a place\nto bring a concrete question or reproducible problem.\n\n## License and foundations\n\nThe root [LICENSE](LICENSE) contains Apache-2.0. The repository's\n[licensing specification](docs/develop/spec/golden-ledger-repo-spec.md#第八部治理)\nassigns Apache-2.0 to repository-produced Lean code, CC-BY-4.0 to text, and CC0\nto data. Third-party dependencies retain their upstream licenses and applicable\nnotices.\n\nBuilt on\n[Lean](https://lean-lang.org/) and [Mathlib](https://github.com/leanprover-community/mathlib4).\nClassical results and upstream proofs remain credited to their sources.\n",1792350332275]