# Global Geometry II: implementation and evidence plan ## Programme status This is an alpha implementation plan. It is not peer reviewed and makes no literature-priority or publication claim. The current outcome-bearing state is the separately addressed [U2 evaluation-v2 checkpoint](GLOBAL_GEOMETRY_II_U2_EVALUATION_REPORT.md). It is `UNRESOLVED`: the preview remains `NOT_EVALUATED`, strict Gate 1 is `PASS`, strict Gates 2--8 are `UNRESOLVED`, and full confirmation and physical validation are `NOT_RUN`. Global Geometry II is an active extension of the published Global Geometry Lab. The release-1 core remains the reference implementation for exact finite topology, gated angle-defect Gauss--Bonnet, deterministic local dynamics, finite-scale dimension profiles, and closed configuration replay. Its focused baseline currently passes 26/26 tests. This programme asks a stronger question: \[ \boxed{\text{Which global geometries are locally generable, recognizable, and robustly programmable?}} \] No refinement experiment, numerical collapse, or successful inverse solve is silently promoted to a theorem. Every public claim must link to a machine-readable evidence record with one of these statuses: - `proved`: a complete finite proof or a cited theorem whose hypotheses are checked in the stated scope; - `computational`: a reproducible finite calculation with declared estimators, uncertainty, controls, and failure conditions; - `conjectural`: a falsifiable mathematical statement not yet proved; - `engineering`: a calibrated model or design approximation with stated tolerances and domain; - `speculative`: a motivating interpretation with no evidentiary force. Every `proved` record additionally declares `proved-here` or `proved-external` as its basis. External theorems require a primary source and a machine- or human-auditable hypothesis check; a passing numerical test is not itself a proof. ## Shared mathematical contract A microscopic world is a tuple \[ \mathcal W=(K,\ell,\mu,W,U,F_R,\theta), \] where `K` is a finite graph or cell complex, `ell` is an intrinsic metric, `mu` is a measure, `W` is a transport law, `U` is local state, and `F_R` is a radius-`R` equivariant update with parameters `theta`. A refinement family is not merely a list of larger graphs; it must declare comparison maps, physical scale, boundary convention, operator normalization, and the observables that are intended to converge. For a declared observable family `Phi`, two rule/refinement families are a *candidate macroscopic equivalence* only relative to: 1. the chosen observables and scale window; 2. an explicit scale-alignment rule fixed before comparison; 3. finite-size and perturbation tests; 4. negative controls capable of rejecting the comparison; and 5. a tolerance or convergence model that is not fitted to the final pair. This is deliberately weaker than isometry, equality in law, or a proved continuum limit. ## Architecture The extension remains dependency-free in the browser-facing path: - `global-geometry-core.js`: unchanged release-1 mathematical substrate; - `global-geometry-ii-core.js`: exact spectra, typed evidence, metric and diffusion controls, and the anisotropy/size resolution map; - `global-geometry-ii-ensembles.js` and `global-geometry-ii-atlas.js`: bounded filled-disk families, measurements, aggregation, and semantic replay validation; - `global-geometry-ii-inverse.js`: closed-schema ordered theorem gates, finite candidates, and obstruction certificates; - `global-geometry-ii-sheet.js`: exact q-star budgets, bounded virtual plant, synthetic observation, safety protocol, and transcript validation; - `global-geometry-ii-lab.js`: the forward, universality, recognition, inverse, sheet, and evidence views; and - focused test and generator scripts for the four generated JSON evidence layers and platform replication certificates. Large or optional reference calculations may use audited offline scripts, but the published finite demonstrations must replay from their exported JSON and must state when a browser result is an approximation to an offline result. ## Milestone A: exact calibration and evidence ledger Implement an immutable evidence-record schema and a deterministic run manifest before adding scientific claims. The first exact calibration pair is diffusion on periodic square and triangular lattice graphs. For an `m` by `n` square torus with normalized Laplacian, \[ \lambda_{p,q}^{\square} =1-\tfrac12\left(\cos\tfrac{2\pi p}{m} +\cos\tfrac{2\pi q}{n}\right). \] For the six-neighbour triangular torus used by the implementation, \[ \lambda_{p,q}^{\triangle} =1-\tfrac13\left(\cos a+\cos b+\cos(a-b)\right), \quad a=\tfrac{2\pi p}{m},\ b=\tfrac{2\pi q}{n}. \] These formulae provide independent, large-refinement checks against the dense finite eigensolver. The calibration claim is limited to the declared diffusive/spectral observables after a family-specific normalization; it does not identify the two finite graphs or their complete geometry. In fact, with unit one-skeleton edge lengths, their scaling metric balls are respectively quadrilateral and hexagonal. The foundations prove this graph-metric nonuniversality as an exact negative control. A future full-object comparison must instead declare a common facewise piecewise-Euclidean metric and a consistent finite-element/cotangent operator. Required gates include exact node/degree counts, spectral range and zero-mode multiplicity, analytic-versus-numerical agreement on small instances, relabeling invariance, deterministic serialization, and fail-closed manifests. The alpha browser core retains a named non-cryptographic canonical JSON checksum for some internal accidental-mutation checks; it must never be described as a security boundary. Release artifacts and platform records use canonical SHA-256 content addresses. Those addresses make mutation evident but are not signatures, endpoint authentication, or publication provenance by themselves. ## Milestone B: falsifiable universality atlas The principal experiment adds bounded planar disk refinements, a third local triangulation family, and seeded uniformly elliptic edge disorder. It measures \[ \Phi(s)=\bigl(d_V(s),d_s(s),d_w(s),K(s),\beta_k, H(s),\lambda_2,\text{transport}(s),\text{stability}(s)\bigr). \] Scale alignment and diffusivity normalization are fixed by calibration rather than selected to minimize the final discrepancy. The atlas reports raw and aligned profiles, estimator windows, residuals, ensemble intervals, and finite-size trends. Mandatory negative controls are comb, shortcut/small-world, bottleneck, perforated, anisotropic, and non-elliptic families. A universality badge is disabled unless the candidate passes replay, refinement, perturbation, alternate-estimator, leave-one-size-out, and control-separation gates. ## Milestone C: obstruction-aware inverse compiler The compiler is a staged decision procedure rather than one optimizer: 1. parse and normalize the requested target and constraints; 2. check exact combinatorial and Gauss--Bonnet compatibility; 3. apply stronger declared curvature-polytope or subset inequalities when their hypotheses are met; 4. attempt a deterministic intrinsic circle-packing or metric solve; 5. validate residual, robustness, and refinement behaviour; 6. separately assess extrinsic embedding, material, sensing, and actuation feasibility. Every result is one of `certified-compatible`, `certified-obstructed`, `candidate-found`, `numerically-unresolved`, or `out-of-model`. Passing a necessary condition is never represented as sufficient. An obstruction report includes the violated invariant or inequality and a minimal witness whenever the finite model permits one. ## Milestone D: programmable-sheet instrument The first release is fully testable without physical hardware. It contains an intrinsic triangular-sheet model, deterministic actuator emulator, synthetic sensor noise, reconstruction, and closed-loop target tracking. Physical layers are separate: - fabricable passive templates and safe low-cost actuator options; - calibration targets and fiducial/camera reconstruction; - content-addressed cross-platform software replay, with Windows camera/device instrumentation kept as a separate future layer; - uncertainty comparison between commanded, inferred, and predicted geometry. Intrinsic curvature realization, extrinsic shell equilibrium, and material response receive separate evidence records. Hardware purchase, fabrication, or public deployment requires user approval. ## Milestone E: scoped platform replay and publication candidate The bounded alpha software snapshot has two content-addressed local inputs, `artifacts/global-geometry-ii/certificates/macos-local-v2.json` and `artifacts/global-geometry-ii/certificates/windows-local-v1.json`. Their external comparison record, `artifacts/global-geometry-ii/certificates/macos-windows-comparison-v1.json`, reports `PASS_FOR_DECLARED_DIGEST_AND_TEST_SURFACE`. It closes only equality of the declared manifest, release-index, source, scientific-artifact, reproduction-resource, and canonical nine focused PASS suite identity/path surface. Each local input remains a single-run record. This bounded comparison does not attest suite-output or performance equality, the confirmatory U2 batch, vision/device behavior, camera calibration, hardware, physical validation, scientific completeness, authentication, priority, peer review, publication, or merge readiness. Enlarging the replay surface requires new local records and a new comparison. Publication and protected-branch merge remain explicit approval gates outside the attestation. The Milestone E v1 index and comparison are now historical prior-release records. They continue to describe their original bounded surface but do not certify the enlarged evaluation-v2 surface. ## Milestone F: shareable U2 evaluation checkpoint Milestone F publishes a fail-closed, content-addressed partial checkpoint: - screening v3 completed 3,840/3,840 records independently on each host, while its exact raw cross-platform comparison is `FAIL` and its decision authority remains `NONE`; - the strict partial campaign derives Gate 1 `PASS` and Gates 2--8 `UNRESOLVED`; - strict exact cross-platform replication is `FAIL`, while an explicitly nondecisional \(10^{-12}\) diagnostic projection is `PASS` with authority `NONE`; - the full confirmatory campaign is `NOT_RUN`; and - the passive-sheet digital dry preflight passes its declared fit/render checks, while physical and camera validation remain `NOT_RUN`. The screening failure ledger retains 112 estimator-unavailable records (9 comb, 103 small-world). The strict normalization artifact is provenance-bound and validated, but its fitted values are not yet consumed by the partial campaign measurements; those channels are therefore not described as calibration-normalized. The checkpoint is bound by the [evaluation-v2 index](artifacts/global-geometry-ii/u2/evaluation-v2/evaluation-index-v2.json), [release-v2 manifest](global_geometry_ii_reproducibility_manifest_v2.json), and [release-v2 evidence index](artifacts/global-geometry-ii/release-index-v2.json). It is shareable for technical review, not as an accepted or rejected U2 result. ## Integration execution order 1. Reconcile the formal, universality, and inverse/sheet specifications. 2. Freeze schema version 2 for manifests and evidence records. 3. Implement the exact metric and diffusion controls and their tests. 4. Add bounded ensembles, perturbation sweeps, and negative controls. 5. Implement staged inverse certificates and the digital sheet loop. 6. Integrate the interactive atlas, fabrication package, and manuscript. 7. Generate the bounded macOS and Windows local records and compare their declared canonical digest-and-focused-test surface. 8. For confirmatory U2, vision fixtures, or instrumentation, freeze a new surface and issue new platform records rather than extending the bounded comparison by prose. 9. Publish the strict-partial and screening results through a mechanically derived evaluation index that exposes exact comparison failures and unperformed stages. 10. Freeze a distinct release-v2 source/artifact/test registry, run the deep verifier on macOS and Windows, and compare only the surface declared by the new certificates. 11. Continue the full confirmatory and physical programmes only under their existing authorization and preregistration boundaries. ## Current verified implementation state As of the integrated alpha candidate: - the unchanged release-1 core passes 26/26 focused tests; - the Global Geometry II core passes 16/16 tests, with sixteen machine-readable PROVED-HERE records; - P8.1 proves graph-metric nonuniversality, P8.2 proves the common square/triangular Gaussian finite-dimensional limit, and P8.3 proves a uniform covariance-whitened finite-dimensional limit for the declared bounded centered nondegenerate iid basin; - the P8.3 audit contains four unrelated microscopic laws, an anisotropic raw negative control, and a frozen 5-by-5 anisotropy/sample-size resolution map. Its color boundary is a numerical guard, not a phase-transition claim; - the filled-disk generator and atlas suites pass 14/14 and 13/13 tests. The generated pilot contains 48 runs in 24 cells and deliberately reports `NOT_EVALUATED` for bounded-disk universality; - the inverse compiler passes 29/29 tests and emits seven replayable reports: successful intrinsic witnesses for a tetrahedral sphere, triangular disk, annulus, and periodic torus, plus exact Chow--Luo, Gauss--Bonnet, and strict triangle-inequality obstructions; - the virtual-sheet suite passes 32/32 tests for q=5,6,7 intrinsic phases, observation-bound commands, watchdog/ESTOP behavior, transcript binding, and fail-closed replay; - the artifact suite passes 5/5 tests over SHA-256-addressed calibration, atlas, inverse, and virtual-sheet layers; - the bounded platform comparison delegates cross-platform status to `macos-windows-comparison-v1.json`, whose local inputs are `macos-local-v2.json` and `windows-local-v1.json`; it covers only the declared digest and canonical nine focused PASS suite identity/path surface; this v1 record is historical for the evaluation-v2 release; - the bounded U2 preview still reports `NOT_EVALUATED`; - screening v3 completed 3,840/3,840 records on both macOS and Windows, but exact raw cross-platform replication is `FAIL` and screening has no confirmatory decision authority; - the strict partial campaign reports Gate 1 `PASS`, Gates 2--8 `UNRESOLVED`, exact cross-platform replication `FAIL`, and a decision-authority-`NONE` \(10^{-12}\) diagnostic `PASS`; - the full confirmatory campaign is `NOT_RUN`, so the evaluation-v2 top-level state is `UNRESOLVED` with neither acceptance nor rejection claimed; - the digital sheet dry preflight passes only its declared fit/render surface, while camera and physical validation remain `NOT_RUN`; and - the interactive lab exposes real forward, universality, recognition, inverse, programmable-sheet, and evidence modes with an explicit evidence ceiling in every panel. The full programme is not yet complete. The evaluation-v2 checkpoint is share-worthy because it publishes negative, failed, and unresolved states without promotion, but it is not a confirmatory U2 decision. The remaining decision-grade estimators, perturbation campaign, control conjunctions, and exact independent-host gate are incomplete. The sheet is digitally preflighted and fabrication-ready rather than physically validated; every physical and camera stage remains `NOT_RUN`. Public deployment and protected-branch merge remain outside the attestation and approval-gated.