Research-Stack/6-Documentation/docs/geometry/GEOMETRY_TAXONOMY_FOR_NLOCAL_ADAPTATION.md

Geometry Taxonomy: Mapping All Repository Formats to N-Local Topology

Date: 2026-04-14
Status: ACTIVE INVENTORY
Purpose: Decide how every existing geometry format in the repo fits (or conflicts with) the n-local topology / Physical Semantics paradigm.

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1. The Classification Axes

We classify each geometry format on two independent axes:

Axis A: Geometric Regime (what kind of space is assumed?)

Level	Name	Key Property
0	Euclidean	Global identity metric G = I, no curvature
1	Non-Euclidean Global	Curvature is constant or globally defined (spherical, hyperbolic)
2	Riemannian	Metric tensor G(p) varies with position p
3	Cartan / Torsion	Metric + torsion T^k_ij that depends on path/trajectory
4	N-Local	Metric, torsion, and phase discriminator are all local and history-dependent

Axis B: Implementation Status (how real is it?)

Status	Meaning
✅ Production	Running in live pipeline, tested, maintained
🔧 Implemented	Code exists, may not be wired to production
⚠️ Stub / Partial	Scaffold or proof-of-concept
📝 Documented Only	Spec exists, no code
❌ Deprecated	Known wrong, superseded

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2. The Full Inventory

2.1 Euclidean / Global Metric

infra/access_control/geometry/geometry_plugin_v2.py — GeometricBridgePlugin

• Regime: Euclidean (Level 0)
• Status: ✅ Production
• Evidence: Computes angular_drift = acos(phi_clamped), Euclidean norm norm(), and linear coherence 1.0 / (1.0 + angular_drift * curvature). Has no trajectory memory.
• NSPACE_COORDINATOR_VERDICT: "The axes are nearly perpendicular to the surface they are meant to describe. Projection onto current axes retains ~17% of the real signal."
• ADAPTATION: This is the highest-priority target. Must evolve to Level 4 (N-Local) by adding trajectory history and replacing Euclidean formulas with anisotropic-metric versions.

tools/cache_sieve.py — CacheSieve

• Regime: Euclidean (Level 0)
• Status: ✅ Production
• Evidence: Maps raw manifold signals (phi_corr, torsion_gradient, radius) directly to SLUQ inputs with linear scaling (* 255, * 10, * 100). No geometric preprocessing.
• ADAPTATION: Should consume outputs from the evolved geometry_plugin_v2.py rather than raw manifold values.

scripts/soliton_factory.py (phi_proximity)

• Regime: Euclidean (Level 0) — flagged as wrong
• Status: ❌ Deprecated by tools/geometry_noneuclidean.py
• Evidence: EUCLIDEAN_ASSUMPTION_AUDIT.md Finding #1: phi_prox = max(0, 1 - abs(r - PHI) / PHI) treats a multiplicative (Möbius) ratio in linear additive space.
• ADAPTATION: Replace with log_phi_proximity() from geometry_noneuclidean.py.

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2.2 Non-Euclidean Global

tools/geometry_noneuclidean.py

• Regime: Non-Euclidean Global (Level 1)
• Status: 🔧 Implemented, not wired to production
• Evidence: Provides log_phi_proximity (projective line distance), logit_z (bounded [0,1] → (-∞,+∞)), oblique projection _proj_oblique_ne, and PHI-geometric orbit decay _PHI_NEG_POW16.
• Key insight: Corrects the global geometry (projective/Möbius instead of linear) but still has no local metric tensor or torsion.
• ADAPTATION: Wire this into ward_compute after calibration (calibrate_geometry.py). It fixes the Euclidean error but is still only Level 1. To reach Level 4, it needs path-history dependence.

audit/benchmarks/benchmark_hutter_geometric.rs

• Regime: Non-Euclidean Global (Level 1)
• Status: ⚠️ Stub / Benchmark
• Evidence: NonEuclideanSpace with global curvature: f64 (positive = spherical, negative = hyperbolic). Distorts coordinates by a global curvature factor.
• ADAPTATION: This is a compression benchmark, not a pipeline component. Can serve as a test harness for global non-Euclidean distance functions, but does not implement local metrics.

audit/benchmarks/benchmark_nspace_entropy.rs

• Regime: Non-Euclidean Global (Level 1) — vector-space phase geometry
• Status: ⚠️ Stub / Benchmark
• Evidence: NoiseVector with magnitude(), direction_entropy(), and combined_entropy(). Treats n-space as a vector space with angle-based entropy.
• ADAPTATION: The direction_entropy formula is useful for measuring informatic torsion between trajectory segments. Can be extracted and used in the Level 4 plugin.

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2.3 Riemannian / Metric Tensor

core/lean/geoweird/GWLKernel.lean — Metric2D

• Regime: Riemannian (Level 2)
• Status: 🔧 Implemented in Lean, not extracted to runtime
• Evidence: Defines Metric2D { g_tt, g_tp, g_pp }, lineElement, metricFromCircumferences, Christoffel2D, and geodesicStep.
• Key insight: This is a genuine 2D Riemannian metric with Christoffel symbols and geodesic integration.
• ADAPTATION: The FFI gap blocks direct use. However, the formulas can be ported to Python/Rust for the geometry plugin. The geodesicStep symplectic Euler integrator is exactly what's needed for trajectory propagation in n-local topology.

core/lean/geoweird/Manifold.lean

• Regime: Riemannian (Level 2) with topological classification
• Status: 🔧 Implemented in Lean
• Evidence: Manifold has dimension, curvature, volume, metric : MetricTensor, and GeodesicPath. TopologicalType adds sphere/hyperbolic/flat/torus/custom labels.
• ADAPTATION: Serves as the formal specification for what a manifold is in this system. The MetricTensor structure should inform the Python GeometricBridgePlugin refactor.

5-Applications/tools-scripts/demo/gwl_earth_riemannian_conversion.py

• Regime: Riemannian (Level 2)
• Status: ⚠️ Demo script
• Evidence: Converts Earth coordinates (lat/lon/alt) using oblate spheroid metrics similar to metricFromCircumferences.
• ADAPTATION: Reference implementation for metricFromCircumferences. Can be reused for any application that needs geodesic distances on an ellipsoid.

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2.4 Cartan / Informatic Torsion

docs/gwl/GWL_ROTATIONAL_VALUES_AND_TORSION_FORMALISM_V1.md

• Regime: Cartan / Torsion (Level 3)
• Status: 📝 Documented Only
• Evidence: Defines π field as "informatic torsion" T(π) = rotation_matrix(π × 22.5°). Torsion affects "how Δp is interpreted" and "how neighboring μ-seeds couple." Chirality (D/L) flips torsion sign.
• Key insight: Torsion is encoded in the μ-seed state itself, not computed from trajectory history.
• ADAPTATION: This is a stateful torsion model (encoded in data) rather than a dynamical torsion model (computed from path history). To reach true Level 4, we need both: π as seed-encoded torsion + trajectory curvature as dynamical torsion.

docs/pipeline/NSPACE_COORDINATOR_SPEC.md — Informatic Torsion Section

• Regime: Cartan / Torsion (Level 3)
• Status: 📝 Documented Only (but referenced by production code)
• Evidence: T^k_ij = Γ^k_ij - Γ^k_ji ≠ 0. "Torsion is sourced by spin density — the path history through the manifold." Stokes theorem: ∮ T^k_ij dS^ij = 0.
• Key insight: This document explicitly requires trajectory-dependent torsion, but no code implements it.
• ADAPTATION: This is the design target. The evolved geometry_plugin_v2.py should implement this exact formula using trajectory history.

data/germane/research/chat-soliton-nspace-path-trace-20260404.md

• Regime: Cartan / Torsion (Level 3) → N-Local (Level 4) aspiration
• Status: 📝 Documented Only
• Evidence: Soliton map as "path trace in n-space that leads to replayable actions." STOP codons = topological changes (different vacuum). KdV soliton preserves shape under nonlinear interaction.
• ADAPTATION: The soliton map is the narrative for what Level 4 geometry should feel like: localized, stable, time-reversible paths. It doesn't provide executable formulas but sets the behavioral specification.

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2.5 Topological / Discrete

infra/access_control/topological_tape_machine.py

• Regime: Topological / Discrete
• Status: 🔧 Implemented (reference implementation)
• Evidence: InvariantSurvivalMask, InvariantVector, braid formation, morphism survival, Warden two-stage validation. No metric; only adjacency/occupancy/path/trust bits.
• ADAPTATION: This operates at a higher abstraction layer than geometry. It doesn't need a metric — it needs the geometry layer to provide a faithful ProjectionCollapse. The TTM is the consumer, not the producer, of n-local geometry.

tools/topological_encoder.py

• Regime: Topological / Discrete
• Status: 🔧 Implemented
• Evidence: Encodes data into topological invariants (braids, knots, links).
• ADAPTATION: Independent module. Can be used to encode n-local trajectory histories as topological braids for the TTM layer.

core/lean/geoweird/ConstraintGeometry.lean

• Regime: Topological / Discrete + Riemannian hybrid
• Status: 🔧 Implemented in Lean
• Evidence: Extracts ConstraintFeatures from domain constraints: curvature, symmetry, compactness, fundamental group rank. Uses combinatorial constraint types (temporal, spatial, cyclic, hierarchical) to infer geometric properties.
• ADAPTATION: This is a constraint-to-geometry translator. It can be used to initialize the metric tensor for a new manifold sheet given a set of domain constraints.

core/src/tsm_perpetual_manifold.v (Verilog)

• Regime: Topological / Discrete (hardware state machine)
• Status: 🔧 Implemented in RTL
• Evidence: 128-bit field_resonance, 256 memistor_weights, thermal energy recycling. Opcodes: INGEST, TSM_XOR, TSM_XAND, RESONATE, BALANCE, FREEZE. No coordinates, no metric — purely field-combinatorial.
• ADAPTATION: This is the hardware substrate. The n-local geometry layer (software) must emit opcodes and state vectors that this module can consume. The Verilog does not need to understand Riemannian geometry; it needs well-formed field states.

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2.6 N-Local / Trajectory-Dependent (The Target)

docs/gwl/GWL_MANIFOLD_INTERSECTION_JOIN_AND_TOPOLOGICAL_ADDRESSING_V1.md

• Regime: N-Local (Level 4)
• Status: 📝 Documented Only
• Evidence: Defines p_N = (p_E, M_i, M_j, γ, σ, Σ_topo). Non-Euclidean position includes Euclidean coordinate, manifold memberships, transport/path class, seam/projection state, and topological relation class.
• Key insight: Same Euclidean coordinate can have multiple distinct addresses depending on path history and manifold context.
• ADAPTATION: This is the addressing layer of the Level 4 target. Any evolved geometry plugin must produce p_N addresses for routing decisions.

docs/pipeline/NSPACE_COORDINATOR_SPEC.md — Phase Discriminator

• Regime: N-Local (Level 4)
• Status: 📝 Documented Only
• Evidence: phase(p) = Φ(κ(p), T^k_ij(p), ρ(p)). Phase discriminator selects the UV seam cut based on local curvature, torsion orientation, and information density.
• ADAPTATION: This is the operational definition of the Level 4 geometry plugin. GeometricBridgePlugin.transform() should compute exactly these three quantities from trajectory history.

infra/access_control/pipeline/unified_pipeline.py — TemporalBuffer

• Regime: N-Local (Level 4) — has the data, doesn't use it for geometry
• Status: ✅ Production
• Evidence: Maintains history: deque[CanonicalState], prev_delta, prev_phi, prev2_phi. Computes reliable derivatives and gamma.
• ADAPTATION: This buffer already stores the trajectory history needed for Level 4 torsion computation. The missing link is piping this history into the geometry plugin.

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2.7 Physical / Standard Model Boundary (New Paradigm)

0-Core-Formalism/lean/Semantics/Semantics/Physics/*.lean

• Regime: Physical / Standard Model (observational boundary)
• Status: ✅ Implemented, compiles, verified
• Evidence: ParticleKind, QuantityKind, Interaction, PhysicalPath, Measurement. Proves conservation laws (charge, lepton number) for e⁻e⁺ annihilation.
• ADAPTATION: This is the bottom boundary of the entire stack. N-local geometry must respect the conservation laws encoded here. Any physical realization of the manifold (e.g. field states in tsm_perpetual_manifold.v) must be accountable to these invariants.

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3. Conflict Matrix: Which Formats Are Compatible?

Format	Compatible With	Conflicts With
geometry_plugin_v2.py (Euclidean)	cache_sieve.py, TemporalBuffer	NSPACE_COORDINATOR_SPEC.md, GWL_MANIFOLD_INTERSECTION_JOIN
geometry_noneuclidean.py (Level 1)	geometry_plugin_v2.py (replacement)	NSPACE_COORDINATOR_SPEC.md (still insufficient)
GWLKernel.lean (Level 2)	Manifold.lean, ConstraintGeometry.lean	geometry_plugin_v2.py (no metric tensor)
GWL_ROTATIONAL_VALUES_AND_TORSION_FORMALISM (Level 3)	GWLKernel.lean, topological_tape_machine.py	geometry_plugin_v2.py (no torsion)
NSPACE_COORDINATOR_SPEC.md (Level 4)	soliton_map, TemporalBuffer, Physics/*.lean	geometry_plugin_v2.py (wrong metric)
tsm_perpetual_manifold.v (Topological)	All layers above it (as consumer)	None (it's the substrate)
Physics/*.lean (Boundary)	All layers above it	Any claim that violates charge/baryon/lepton conservation

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4. The Adaptation Decision

What must change:

1. geometry_plugin_v2.py — evolve from Euclidean (Level 0) to N-Local (Level 4).
2. cache_sieve.py — stop consuming raw manifold values; consume N-local geometric features.
3. unified_pipeline.py — wire TemporalBuffer history into the geometry plugin.

What can be wired as-is:

1. geometry_noneuclidean.py — its log-PHI and logit functions become subroutines in the Level 4 plugin.
2. GWLKernel.lean — its metric tensor and geodesic step formulas are the source of truth; Python is an extraction shim against this spec.
3. topological_tape_machine.py — operates at a higher layer; unchanged interface.
4. tsm_perpetual_manifold.v — unchanged; the geometry layer feeds it, it doesn't compute geometry.
5. Physics/*.lean — unchanged; serves as the invariant boundary.

What is documentation-only and needs code:

1. NSPACE_COORDINATOR_SPEC.md — the Phase Discriminator needs a Python/Rust implementation.
2. GWL_MANIFOLD_INTERSECTION_JOIN_AND_TOPOLOGICAL_ADDRESSING_V1.md — p_N addressing needs a Python addressing module.
3. GWL_ROTATIONAL_VALUES_AND_TORSION_FORMALISM_V1.md — π-field torsion needs to be integrated into trajectory curvature computation.

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5. Recommended Implementation Sequence

Phase 1 (Immediate): Refactor geometry_plugin_v2.py to use TemporalBuffer trajectory history and compute:

• Local metric tensor G(p) from trajectory covariance
• Geodesic angle under G(p) (replacing Euclidean angular_drift)
• Informatic torsion T(p) from trajectory curvature (path-history derivative)
• Phase discriminator Φ(κ, T, ρ) (stub, to be hardened)

Phase 2 (Short-term): Update cache_sieve.py to consume the new geometric features, and verify against existing tests.

Phase 3 (Medium-term): Port the verified Python formulas to Rust (or extract from Lean once FFI is fixed) for performance-critical paths.

Phase 4 (Ongoing): Maintain Physics/*.lean as the invariant oracle; any new geometric claim must be checked against conservation laws.

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Status: INVENTORY COMPLETE | DECISION FRAMEWORK ESTABLISHED