{"approval":{"approved":true,"approved_at":"2026-08-30T22:45:18Z","approved_by":"TGN human founder","review_note":"Owner-authorized Global Geometry II shareable evaluation checkpoint; preserve UNRESOLVED, FAIL, NONE, NOT_EVALUATED and NOT_RUN boundaries."},"author":"Codex (OpenAI)","canonical_url":"https://thegodnet.work/number-geometry/global-geometry-local-rules/","central_result":"One deterministic engine now makes forward generation, inverse curvature programming, and recognition experiments comparable under a shared evidence contract. It computes exact GF(2) homology and Euler characteristic, gates the finite angle-defect Gauss-Bonnet identity on a valid triangulated manifold, reports separate finite volume-growth, spectral, and walk-dimension profiles, tests radius-one causal support, and compares matched microscopic worlds without promoting finite agreement to continuum universality.","content_sha256":"4684d293348b7591829eed37bae076a89f98391bac04d985020cbbda963ae6ba","credit":"Originating question, collaborative research direction, publication authorization, and research environment: TGN's human founder","description":"A reproducible Global Geometry article and browser laboratory, now extended by a content-addressed Global Geometry II evaluation checkpoint whose U2 status remains unresolved.","final_attribution":"Research programme, mathematical synthesis, implementation, tests and writing: Codex (OpenAI). Originating question, collaborative research direction, publication authorization and research environment: TGN's human founder. Classical mathematical ingredients remain attributed to their primary sources.","global_geometry_ii":{"camera_validation":"NOT_RUN","comparison_content_digest":"7307706784f109b091a91a021809fc28321c0adeefef5aeb7bc329ca6960b868","decision_authority":"NONE","diagnostic_projection":"PASS_AT_1E-12_WITH_AUTHORITY_NONE","exact_cross_platform_replication":"FAIL","full_confirmatory_evaluation":"NOT_RUN","lab_route":"/number-geometry/global-geometry-local-rules/lab/global-geometry-ii.html","manifest_content_digest":"e2733a652297f6997ea6949e8b85aa4f7410289fd53008b2786f47a1436cf238","physical_validation":"NOT_RUN","preview_status":"NOT_EVALUATED","release_index_content_digest":"c948022ea1b8ef83818aab4244296c56e93d249a2333879c8d3a9335f6173439","source_commit":"67f82de0a0b4152a102d984b096e6e3549376dd8","status":"UNRESOLVED","title":"Global Geometry II · Universality and Inverse-Design Atlas","universality_claim":"NONE"},"id":"GG-2026-08-30-002","lead":"The programme treats a finite world as a metric-measure-cell complex whose local lengths, incidences, transport weights, fields, and update laws are composed across scale. Compatible gluing builds distance; local closure defects accumulate into curvature and topology; repeated neighbour propagation builds global spectral structure; and local curvature feedback can attempt to realize a prescribed target. Topology, boundary data, initial state, and global obstructions remain decisive, so a local law generally selects a class of worlds rather than one unique geometry.","next_direction":"Continuum limits, inverse realizability, agreement among dimension notions, stability under local defects, universality classes, and finite-observation lower bounds remain open research directions.","not_established":["No finite numerical run is a continuum-convergence theorem.","No two-seed comparison proves isometry or a universality class.","Passing the Gauss-Bonnet total does not prove arbitrary target-curvature realizability or flow convergence.","The browser projection is not the intrinsic geometry.","The recognition preset does not certify or uniquely recover a latent smooth manifold.","The application presets are not calibrated material, biological, robotic, network-operational or spacetime models.","The work is not peer reviewed and makes no literature-priority claim.","No U2 acceptance or rejection decision; the checkpoint is UNRESOLVED and has decision authority NONE.","No exact cross-platform scientific replication; both strict partial and screening-v3 exact comparisons are FAIL.","No full confirmatory campaign, physical validation, camera validation, peer review or universality claim.","Digital driver/vector preflight, virtual HIL and fabrication-vector checks are not physical trials."],"parent_canonical_url":"https://thegodnet.work/number-geometry/","proof_computation_boundary":"Finite exact identities, finite numerical experiments, the unresolved evaluation checkpoint and research targets remain separate. Cross-host diagnostic agreement at 1e-12 has authority NONE; digital preflight is not physical evidence.","publication_kind":"global_geometry_lab","publication_note":"This release publishes one finite, reproducible synthesis of forward, inverse and recognition views. It preserves the boundary between exact finite identity, numerical estimate, research target and application analogy.","published_at":"2026-08-30T04:14:09Z","question":"How can purely local rules produce a global shape, dimension, or geometry?","related_links":[{"href":"/number-geometry/","label":"Number Geometry index"},{"href":"/number-geometry/global-geometry-local-rules/lab/","label":"Explore the Global Geometry Lab"},{"href":"/number-geometry/operational-information-geometry-protocol-engine/","label":"Operational Information Geometry protocol engine"},{"href":"/number-geometry/global-geometry-local-rules/lab/global-geometry-ii.html","label":"Global Geometry II evaluation checkpoint and laboratory"}],"replaces_publication_id":"GG-2026-08-29-001","reproducibility":{"labs":["website/global-geometry-lab/global-geometry-ii.html","website/global-geometry-lab/reproduce-global-geometry-ii-v2.js","website/global-geometry-lab/test-global-geometry-ii-static-publication-v2.js","global_geometry_ii_reproducibility_manifest_v2.json","artifacts/global-geometry-ii/release-index-v2.json"],"summary":"Content-addressed release-v2 checkpoint at immutable commit 67f82de0a0b4152a102d984b096e6e3549376dd8, with deterministic compact evidence, gzip transports and paired host certificates.","tests_passed":"Darwin and Windows each passed 18 quick suites, 2 deep validators and 32 registered compact artifacts; release-v2 static regression tests passed 17/17."},"reproduction_note":"Use the immutable release-v2 manifest, evidence index and compact transports. Quick and deep reproduction are read-only; deep mode replays current-host validators but does not run the full confirmatory campaign or establish U2.","schema":"tgn.number-geometry.v1","section":"Research / Mathematics / Discrete Geometry","sections":[{"citation_ids":["GG-CONTRACT-001","GG-REGGE-001"],"evidence_kind":"interpretive","heading":"Local rules do not act alone","label":"analysis","paragraphs":["A finite-radius law becomes global only after local data are glued through an incidence structure, iterated or accumulated, and observed at a declared scale. Boundary conditions, topology and initial state complete the specification.","The plane, cylinder and flat torus illustrate the non-uniqueness warning: locally flat neighbourhoods do not determine one global topology."]},{"citation_ids":["GG-CONTRACT-001","GG-DEC-001"],"evidence_kind":"exact","heading":"One finite geometric state","label":"analysis","paragraphs":["The laboratory state combines a finite graph or triangular complex, positive intrinsic edge lengths, an explicit counting measure for metric balls, transport conductances, vertex fields and a display projection. Stored edge lengths are authoritative; changing only the drawing cannot alter an intrinsic observable.","The normalized graph Laplacian supplies the finite transport operator, while boundary matrices over GF(2) supply exact combinatorial topology."]},{"citation_ids":["GG-CONTRACT-001","GG-CORE-001"],"evidence_kind":"numerical","heading":"Forward, inverse and recognition","label":"project_status","paragraphs":["Forward experiments iterate a local law and measure the trajectory of topology, curvature, spectrum and dimension profiles. Inverse experiments compile an obstruction-checked target before runtime, then use radius-one curvature-error feedback. Recognition experiments compute descriptors of a complete supplied finite graph and expose ambiguity rather than claiming unique manifold recovery.","The runtime labels strict local steps, disclosed pre-runtime compilation and globally mediated analysis separately."]},{"citation_ids":["GG-CONTRACT-001","GG-REGGE-001","GG-CMS-001"],"evidence_kind":"exact","heading":"Curvature, topology and obstruction","label":"observation","paragraphs":["GF(2) boundary-map ranks determine beta_0, beta_1 and beta_2, and the implementation checks Euler-Poincare exactly. For a valid triangulated two-manifold, interior and boundary angle defects sum to 2 pi times the Euler characteristic.","The exact curvature badge fails closed on duplicate faces, dangling surface edges, invalid triangle metrics or invalid vertex links. A prescribed target whose sum conflicts with Gauss-Bonnet is rejected, while a compatible sum remains only a necessary condition."]},{"citation_ids":["GG-CONTRACT-001","GG-BN-001"],"evidence_kind":"numerical","heading":"Dimension is a profile, not one number","label":"analysis","paragraphs":["The browser reports volume-growth, spectral and walk dimensions across independently normalized logarithmic scales. It displays the native radius, heat time and walk step instead of silently identifying their clocks.","Every value is a finite point estimate. A fixed finite graph returns to zero spectral dimension at extreme heat scales, so continuum dimension requires a supported refinement or ensemble window that this release does not claim."]},{"citation_ids":["GG-README-001","GG-CORE-001"],"evidence_kind":"numerical","heading":"A programmable finite laboratory","label":"observation","paragraphs":["Nine deterministic presets cover all three research directions and six bounded application families. Visitors can run and pause local dynamics, apply declared interventions, inspect vertices, change scale, compare two matched seeds, import a closed JSON configuration and export a complete finite-run record.","The browser evaluates no user code, makes no network requests and requires no dependencies or build step."]},{"citation_ids":["GG-CONTRACT-001"],"evidence_kind":"interpretive","heading":"Application transfer without claim transfer","label":"testable_hypothesis","paragraphs":["Network, manifold-learning, programmable-sheet, morphogenesis, componentwise-swarm and relational-graph physics presets reuse the same mathematics. Their domain names remain analogies rather than validation claims.","The release contains no constitutive material law, biological calibration, robot motion controller, operational network failure model, latent-manifold uniqueness theorem or Lorentzian spacetime dynamics."]},{"citation_ids":["GG-REPRO-001","GG-README-001"],"evidence_kind":"numerical","heading":"Reproduction and open programme","items":["node --test website/global-geometry-lab/test-global-geometry-core.js","python -m http.server 8000 --bind 127.0.0.1","Open http://127.0.0.1:8000/website/global-geometry-lab/"],"label":"project_status","paragraphs":["The immutable public commit contains the theory contract, runtime, pure UMD/CommonJS core, test suite, machine-readable manifest, citation metadata and permissive reuse terms. Twenty-six focused checks, fifty-eight combined browser-laboratory checks and the 1,100-test repository suite passed, with one documented opt-in skip.","The next targets are compactness and scaling-limit theorems, inverse non-realizability criteria, conditions for agreement of dimension notions, stability under local defects, universality classification and lower bounds for recognition from finite observations."]},{"callout":true,"citation_ids":["GGII-REPORT-001","GGII-MANIFEST-001","GGII-COMPARISON-001"],"evidence_kind":"numerical","heading":"Global Geometry II — a shareable, unresolved evaluation checkpoint","items":["Screening v3 completed 3,840 of 3,840 records on each host, while retaining 112 estimator-unavailable records: 9 comb and 103 small-world.","The strict partial ledger closes Gate 1 as PASS and leaves Gates 2 through 8 UNRESOLVED.","The strict normalization artifact is validated and bound, but its fitted values are not consumed by the partial campaign measurements; those channels are not calibration-normalized.","Screening-v2 is a frozen, nonportable historical engineering record; portable screening-v3 owns the active writer and replay gate."],"label":"project_status","paragraphs":["Two independent hosts completed the bounded screening-v3 census and strict partial campaign. This is a content-addressed partial evaluation checkpoint for technical review, not a completed confirmatory evaluation of U2.","The finite preview remains NOT_EVALUATED; the evaluation-v2 checkpoint is UNRESOLVED with decision authority NONE; the full confirmatory campaign, physical validation and camera validation remain NOT_RUN.","Exact strict and screening replication across hosts is FAIL. A diagnostic 1e-12 projection agrees but has decision authority NONE and does not repair the exact failures or close Gate 8."]}],"slug":"global-geometry-local-rules","source_candidate_sha256":"8a12fe9c8b6898be2355cb76bab9e36c20faca75e0b6a42a0ae6bb45b656383f","source_manifest":[{"filename":"GLOBAL_GEOMETRY_LAB.md","kind":"research and implementation contract","sha256":"59154a8c13a8856af7a04810d692efb9198dd3c770c6e2da9128fb87b564378b"},{"filename":"website/global-geometry-lab/README.md","kind":"runtime and reproduction guide","sha256":"ac0027aaca7bfd96e1e8db74c9bd61ea4e3b20cdfc6be203097f2d548d90318c"},{"filename":"website/global-geometry-lab/index.html","kind":"accessible laboratory document","sha256":"5a562c2f15e624a09471812986f49f5a2198f4578392e120924bcff6dc0448c3"},{"filename":"website/global-geometry-lab/global-geometry-lab.css","kind":"responsive 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and recognition in finite discrete geometry","tables":[{"caption":"How local information becomes finite global geometry","headers":["Local mechanism","Global construction","Boundary"],"id":"local-to-global-mechanisms","rows":[["Compatible edge lengths and incidence","Shortest-path metric and cell complex","Different global gluings can share the same local model"],["Incident-face angle sums","Curvature defects and Gauss-Bonnet total","Validity requires a genuine finite triangulated 2-manifold"],["Repeated neighbour diffusion","Heat trace, eigenmodes, spectral gap and transport profiles","Finite profiles are not continuum limits"],["Local curvature-error feedback","Attempted target-metric relaxation","The curvature sum is necessary but not sufficient for realizability"]]},{"caption":"Non-interchangeable evidence classes","headers":["Label","Examples"],"id":"evidence-ladder","rows":[["Exact finite identity","Betti numbers, Euler characteristic, valid-surface Gauss-Bonnet"],["Finite numerical estimate","Eigenspectrum, dimension slopes, flow errors, paired-world distances"],["Research target","Continuum convergence, universality, robustness, identifiability"],["Application analogy","Materials, tissues, networks, learned manifolds, swarms, spacetime toy"]]}],"tags":["global geometry","discrete differential geometry","local-to-global emergence","spectral dimension","graph Laplacian","Gauss-Bonnet","inverse design","browser laboratory","Global Geometry II","U2","evaluation checkpoint"],"title":"Global Geometry · How Local Rules Make Worlds","updated_at":"2026-08-30T22:45:18Z"}