wifi-densepose/crates/prime-radiant/docs/GOAP_ADVANCED_MATH_FRAMEWORKS.md

GOAP Implementation Plan: Advanced Mathematical Frameworks for Prime-Radiant

Version: 1.0.0 Date: 2026-01-22 Author: SPARC-GOAP Planning System Status: Planning Phase

---

Executive Summary

This document provides a comprehensive Goal-Oriented Action Plan (GOAP) for implementing 6 cutting-edge mathematical frameworks into the Prime-Radiant coherence engine. Each framework enhances the existing sheaf Laplacian architecture with advanced theoretical foundations.

Current State Analysis

current_state = {
    sheaf_substrate: true,           // SheafGraph, SheafNode, SheafEdge, RestrictionMap
    spectral_analysis: "basic",      // Eigenvalue drift detection, basic Laplacian
    coherence_engine: true,          // Energy computation, residual calculation
    attention_system: true,          // Topology-gated, MoE, PDE diffusion
    mincut_isolation: true,          // Subpolynomial dynamic mincut
    hyperbolic_geometry: true,       // Poincare ball, depth-weighted energy
    governance_layer: true,          // Policy bundles, witness records
    wasm_support: "partial",         // Some crates have WASM bindings
    test_coverage: "~70%",
    sheaf_cohomology: false,
    category_theory: false,
    homotopy_type_theory: false,
    spectral_invariants: "basic",
    causal_abstraction: false,
    quantum_topology: false
}

goal_state = {
    sheaf_cohomology: true,          // H^0, H^1 computation, obstruction detection
    category_theory: true,           // Functorial retrieval, topos-theoretic belief
    homotopy_type_theory: true,      // HoTT embedding, proof assistant style
    spectral_invariants: "advanced", // Cheeger bounds, spectral collapse prediction
    causal_abstraction: true,        // Causal layers, structural causality
    quantum_topology: true,          // TQC encodings, spectral topology
    all_wasm_exports: true,
    test_coverage: ">85%",
    benchmarks_complete: true,
    adr_documented: true
}

---

Framework 1: Sheaf Cohomology

Goal State Definition

Compute cohomological obstructions for belief graphs, enabling detection of global consistency failures that local residuals miss.

Mathematical Foundation

Sheaf Cohomology on Graphs:
- H^0(X, F) = Global sections (consistent assignments)
- H^1(X, F) = Obstruction cocycles (inconsistency indicators)
- Coboundary operator: delta: C^0 -> C^1
- Cohomology energy: E_coh = ||H^1(X, F)||^2

Module Architecture

crates/prime-radiant/src/cohomology/
├── mod.rs                    # Module root, public API
├── cochain.rs               # C^0, C^1 cochain spaces
├── coboundary.rs            # Coboundary operator implementation
├── cohomology_group.rs      # H^0, H^1 computation
├── obstruction.rs           # Obstruction detection and classification
├── sheaf_diffusion.rs       # Diffusion with cohomology indicators
├── neural_sheaf.rs          # Sheaf Neural Network layers
└── config.rs                # Configuration and parameters

Key Data Structures

/// Cochain in degree k
pub struct Cochain<const K: usize> {
    /// Values indexed by k-simplices
    values: HashMap<SimplexId<K>, Vec<f32>>,
    /// Dimension of stalk
    stalk_dim: usize,
}

/// Cohomology class in H^k
pub struct CohomologyClass<const K: usize> {
    /// Representative cocycle
    representative: Cochain<K>,
    /// Betti number contribution
    betti_contribution: usize,
    /// Energy measure
    cohomology_energy: f32,
}

/// Obstruction indicator
pub struct Obstruction {
    /// Location (edge or higher simplex)
    location: SimplexId<1>,
    /// Obstruction class in H^1
    class: CohomologyClass<1>,
    /// Severity (0.0 to 1.0)
    severity: f32,
    /// Suggested repair strategy
    repair_hint: RepairStrategy,
}

/// Sheaf Neural Network layer
pub struct SheafNeuralLayer {
    /// Learnable restriction maps
    rho_weights: HashMap<EdgeId, Array2<f32>>,
    /// Laplacian diffusion operator
    laplacian: SheafLaplacian,
    /// Cohomology-aware attention
    attention: CohomologyAttention,
}

Key Traits

/// Computes sheaf cohomology
pub trait SheafCohomology {
    type Sheaf;
    type Coefficient;

    /// Compute H^0 (global sections)
    fn h0(&self, sheaf: &Self::Sheaf) -> CohomologyGroup<0>;

    /// Compute H^1 (first cohomology)
    fn h1(&self, sheaf: &Self::Sheaf) -> CohomologyGroup<1>;

    /// Check if sheaf is globally consistent
    fn is_consistent(&self, sheaf: &Self::Sheaf) -> bool;

    /// Identify obstruction cocycles
    fn obstructions(&self, sheaf: &Self::Sheaf) -> Vec<Obstruction>;
}

/// Cohomology-informed diffusion
pub trait CohomologyDiffusion {
    /// Diffuse with cohomology-weighted Laplacian
    fn diffuse_with_cohomology(
        &self,
        state: &[f32],
        steps: usize,
        cohomology_weight: f32,
    ) -> Vec<f32>;
}

Integration Points

Existing Module	Integration Type	Description
substrate::SheafGraph	Extension	Add simplex enumeration methods
coherence::CoherenceEngine	Augment	Add H^1 energy to total energy
attention::AttentionCoherence	Augment	Cohomology-weighted attention
learned_rho::LearnedRestrictionMap	Extend	Train rho to minimize H^1

WASM Export Strategy

#[wasm_bindgen]
pub struct WasmSheafCohomology {
    inner: SheafCohomology,
}

#[wasm_bindgen]
impl WasmSheafCohomology {
    #[wasm_bindgen(constructor)]
    pub fn new(config: JsValue) -> Result<WasmSheafCohomology, JsValue>;

    pub fn compute_h0(&self, graph: &WasmSheafGraph) -> JsValue;
    pub fn compute_h1(&self, graph: &WasmSheafGraph) -> JsValue;
    pub fn detect_obstructions(&self, graph: &WasmSheafGraph) -> JsValue;
    pub fn cohomology_energy(&self, graph: &WasmSheafGraph) -> f32;
}

Test Cases

1. Unit Tests

   • test_coboundary_squares_to_zero: delta^2 = 0
   • test_exact_sequence: im(delta) subset of ker(delta)
   • test_consistent_sheaf_h1_vanishes: H^1 = 0 for consistent sheafs
   • test_obstruction_detection: Known obstructions are found

2. Property Tests

   • prop_betti_numbers_stable: Betti numbers unchanged under small perturbations
   • prop_cohomology_energy_nonnegative: E_coh >= 0

3. Integration Tests

   • test_cohomology_with_mincut: Obstructions correlate with cut edges
   • test_sheaf_neural_convergence: SNN training reduces H^1

Benchmarks

Benchmark	Target	Notes
H^1 computation (1K nodes)	<10ms	Sparse matrix ops
Obstruction detection (1K nodes)	<5ms	After H^1 cached
SNN forward pass (1K nodes)	<20ms	GPU optional

ADR Outline

ADR-020: Sheaf Cohomology Integration

• Status: Proposed
• Context: Need global consistency detection beyond local residuals
• Decision: Implement H^0/H^1 with coboundary operator
• Consequences: More accurate hallucination detection, higher compute

---

Framework 2: Category Theory / Topos

Goal State Definition

Implement functorial retrieval systems and topos-theoretic belief models with higher category coherence laws.

Mathematical Foundation

Category-Theoretic Coherence:
- Objects: Belief states, contexts
- Morphisms: Belief transformations
- Functors: Context-preserving mappings
- Natural transformations: Coherence laws
- Topos: Generalized logic over belief graphs

Module Architecture

crates/prime-radiant/src/category/
├── mod.rs                    # Module root
├── category.rs               # Category trait and basic types
├── functor.rs                # Functor implementations
├── natural_transform.rs      # Natural transformations
├── monad.rs                  # Monad for belief composition
├── topos/
│   ├── mod.rs               # Topos submodule
│   ├── subobject.rs         # Subobject classifier
│   ├── internal_logic.rs    # Internal logic operations
│   └── sheaf_topos.rs       # Sheaf topos on coherence graph
├── retrieval.rs              # Functorial retrieval system
├── coherence_laws.rs         # Higher coherence laws (associativity, etc.)
└── config.rs

Key Data Structures

/// A category of belief states
pub struct BeliefCategory {
    /// Object set (belief state types)
    objects: Vec<BeliefType>,
    /// Morphism set (transformations)
    morphisms: HashMap<(BeliefType, BeliefType), Vec<BeliefMorphism>>,
    /// Identity morphisms
    identities: HashMap<BeliefType, BeliefMorphism>,
}

/// Functor between categories
pub struct BeliefFunctor<C: Category, D: Category> {
    /// Object mapping
    object_map: HashMap<C::Object, D::Object>,
    /// Morphism mapping (preserves composition)
    morphism_map: HashMap<C::Morphism, D::Morphism>,
}

/// Natural transformation
pub struct NaturalTransformation<F: Functor, G: Functor> {
    /// Components: eta_X: F(X) -> G(X) for each object X
    components: HashMap<F::Source::Object, F::Target::Morphism>,
}

/// Topos over belief graph
pub struct BeliefTopos {
    /// Underlying category
    category: BeliefCategory,
    /// Subobject classifier (truth values)
    omega: SubobjectClassifier,
    /// Internal Heyting algebra for logic
    heyting: HeytingAlgebra,
    /// Sheaf condition enforcement
    sheaf_condition: SheafCondition,
}

/// Coherence law checker
pub struct CoherenceLaw {
    /// Law name (e.g., "associativity", "unit")
    name: String,
    /// Diagram that must commute
    diagram: CommutativeDiagram,
    /// Tolerance for approximate commutativity
    tolerance: f32,
}

Key Traits

/// Category abstraction
pub trait Category {
    type Object: Clone + Eq + Hash;
    type Morphism: Clone;

    fn identity(&self, obj: &Self::Object) -> Self::Morphism;
    fn compose(&self, f: &Self::Morphism, g: &Self::Morphism) -> Option<Self::Morphism>;
    fn source(&self, f: &Self::Morphism) -> Self::Object;
    fn target(&self, f: &Self::Morphism) -> Self::Object;
}

/// Functor between categories
pub trait Functor {
    type Source: Category;
    type Target: Category;

    fn map_object(&self, obj: &<Self::Source as Category>::Object)
        -> <Self::Target as Category>::Object;
    fn map_morphism(&self, f: &<Self::Source as Category>::Morphism)
        -> <Self::Target as Category>::Morphism;
}

/// Topos operations
pub trait Topos: Category {
    type SubobjectClassifier;

    fn omega(&self) -> &Self::SubobjectClassifier;
    fn truth(&self) -> Self::Morphism;  // 1 -> Omega
    fn chi(&self, mono: &Self::Morphism) -> Self::Morphism;  // Characteristic morphism

    /// Internal logic
    fn internal_and(&self, a: &Self::Morphism, b: &Self::Morphism) -> Self::Morphism;
    fn internal_or(&self, a: &Self::Morphism, b: &Self::Morphism) -> Self::Morphism;
    fn internal_implies(&self, a: &Self::Morphism, b: &Self::Morphism) -> Self::Morphism;
}

/// Functorial retrieval
pub trait FunctorialRetrieval {
    type Query;
    type Result;
    type Context;

    /// Retrieve with functor-preserved context
    fn retrieve(&self, query: Self::Query, context: Self::Context) -> Vec<Self::Result>;

    /// Verify naturality (consistency across context changes)
    fn verify_naturality(&self, transform: &NaturalTransformation) -> bool;
}

Integration Points

Existing Module	Integration Type	Description
substrate::RestrictionMap	Morphism	Rho maps as category morphisms
coherence::CoherenceEngine	Functor	Engine as functor from graphs to energies
governance::PolicyBundle	Topos	Policies as internal logic formulas

WASM Export Strategy

#[wasm_bindgen]
pub struct WasmBeliefTopos {
    inner: BeliefTopos,
}

#[wasm_bindgen]
impl WasmBeliefTopos {
    pub fn internal_entailment(&self, premise: JsValue, conclusion: JsValue) -> bool;
    pub fn check_coherence_law(&self, law_name: &str) -> f32;  // Returns violation magnitude
    pub fn functorial_retrieve(&self, query: JsValue, context: JsValue) -> JsValue;
}

Test Cases

1. Unit Tests

   • test_identity_law: id o f = f = f o id
   • test_associativity: (f o g) o h = f o (g o h)
   • test_functor_preserves_composition: F(g o f) = F(g) o F(f)
   • test_naturality_square_commutes

2. Property Tests

   • prop_topos_has_terminal_object
   • prop_subobject_classifier_unique

Benchmarks

Benchmark	Target	Notes
Functor application (1K objects)	<5ms
Naturality check (100 morphisms)	<10ms
Internal logic query	<1ms

ADR Outline

ADR-021: Category-Theoretic Belief Models

• Status: Proposed
• Context: Need compositional semantics for belief transformations
• Decision: Implement topos-theoretic framework
• Consequences: Enables formal verification, steeper learning curve

---

Framework 3: Homotopy Type Theory

Goal State Definition

Embed HoTT formal system for proof-carrying coherence verification with Coq/Agda-style type checking.

Mathematical Foundation

Homotopy Type Theory:
- Types as spaces
- Terms as points
- Equality types as paths: Id(A, x, y)
- Path induction (J-eliminator)
- Univalence: (A ≃ B) ≃ (A = B)
- Higher inductive types for coherence

Module Architecture

crates/prime-radiant/src/hott/
├── mod.rs                    # Module root
├── types/
│   ├── mod.rs
│   ├── universe.rs           # Type universe hierarchy
│   ├── identity.rs           # Identity types (paths)
│   ├── sigma.rs              # Dependent sum types
│   ├── pi.rs                 # Dependent product types
│   └── higher_inductive.rs   # HITs for coherence graphs
├── paths/
│   ├── mod.rs
│   ├── path.rs               # Path type implementation
│   ├── composition.rs        # Path composition
│   ├── inverse.rs            # Path inversion
│   └── homotopy.rs           # Homotopies between paths
├── univalence/
│   ├── mod.rs
│   ├── equivalence.rs        # Type equivalences
│   ├── transport.rs          # Transport along paths
│   └── ua.rs                 # Univalence axiom
├── proofs/
│   ├── mod.rs
│   ├── proof_term.rs         # Proof term representation
│   ├── type_checker.rs       # Bidirectional type checking
│   ├── normalization.rs      # Beta/eta normalization
│   └── coherence_proof.rs    # Proofs of coherence properties
├── embedding.rs              # Embed coherence in HoTT
└── config.rs

Key Data Structures

/// HoTT Universe level
pub type Level = u32;

/// HoTT Type
#[derive(Clone, Debug)]
pub enum HoTTType {
    /// Universe at level i
    Universe(Level),
    /// Identity type Id_A(x, y)
    Identity { ty: Box<HoTTType>, left: Term, right: Term },
    /// Dependent sum Σ(x:A).B(x)
    Sigma { base: Box<HoTTType>, fiber: Box<Closure> },
    /// Dependent product Π(x:A).B(x)
    Pi { domain: Box<HoTTType>, codomain: Box<Closure> },
    /// Higher inductive type
    HIT(HigherInductiveType),
    /// Base types
    Unit, Empty, Bool, Nat,
}

/// HoTT Term (proof term)
#[derive(Clone, Debug)]
pub enum Term {
    /// Variable
    Var(usize),
    /// Lambda abstraction
    Lambda { ty: HoTTType, body: Box<Term> },
    /// Application
    App { func: Box<Term>, arg: Box<Term> },
    /// Pair (for Sigma types)
    Pair { fst: Box<Term>, snd: Box<Term> },
    /// Reflexivity proof: refl : Id_A(x, x)
    Refl,
    /// J-eliminator for identity
    J { motive: Box<Term>, refl_case: Box<Term>, path: Box<Term> },
    /// Transport along path
    Transport { path: Box<Term>, point: Box<Term> },
    /// Constructor for HIT
    HITConstructor { hit: HigherInductiveType, idx: usize, args: Vec<Term> },
}

/// Higher Inductive Type for coherence graphs
#[derive(Clone, Debug)]
pub struct CoherenceHIT {
    /// Point constructors (nodes)
    points: Vec<PointConstructor>,
    /// Path constructors (edges -> paths)
    paths: Vec<PathConstructor>,
    /// Higher path constructors (coherences)
    higher_paths: Vec<HigherPathConstructor>,
}

/// Proof of coherence property
pub struct CoherenceProof {
    /// Statement being proved
    statement: HoTTType,
    /// Proof term
    proof: Term,
    /// Normalized form
    normal_form: Option<Term>,
    /// Type-checking trace
    derivation: TypeDerivation,
}

Key Traits

/// Type checking
pub trait TypeChecker {
    type Error;

    /// Check term has given type
    fn check(&self, ctx: &Context, term: &Term, ty: &HoTTType) -> Result<(), Self::Error>;

    /// Infer type of term
    fn infer(&self, ctx: &Context, term: &Term) -> Result<HoTTType, Self::Error>;

    /// Check type well-formedness
    fn check_type(&self, ctx: &Context, ty: &HoTTType) -> Result<Level, Self::Error>;
}

/// Path operations
pub trait PathOps {
    /// Compose paths: p o q
    fn compose(&self, p: &Term, q: &Term) -> Term;

    /// Invert path: p^{-1}
    fn invert(&self, p: &Term) -> Term;

    /// Transport along path
    fn transport(&self, path: &Term, point: &Term) -> Term;

    /// Apply function to path
    fn ap(&self, f: &Term, p: &Term) -> Term;
}

/// Coherence embedding
pub trait CoherenceEmbedding {
    /// Embed sheaf graph as HIT
    fn embed_graph(&self, graph: &SheafGraph) -> CoherenceHIT;

    /// Embed edge constraint as path type
    fn embed_constraint(&self, edge: &SheafEdge) -> HoTTType;

    /// Construct coherence proof
    fn prove_coherence(&self, graph: &SheafGraph) -> Result<CoherenceProof, ProofError>;
}

Integration Points

Existing Module	Integration Type	Description
substrate::SheafGraph	Embed	Graph as HIT type
coherence::CoherenceEnergy	Proof	Energy bounds as theorems
governance::WitnessRecord	Proof term	Witnesses as proof terms

WASM Export Strategy

#[wasm_bindgen]
pub struct WasmHoTTChecker {
    inner: HoTTTypeChecker,
}

#[wasm_bindgen]
impl WasmHoTTChecker {
    pub fn check_coherence(&self, graph: &WasmSheafGraph) -> JsValue;  // Returns proof or error
    pub fn verify_proof(&self, proof_json: &str) -> bool;
    pub fn normalize(&self, term_json: &str) -> String;
}

Test Cases

1. Unit Tests

   • test_refl_type_checks: refl : Id_A(x, x)
   • test_j_eliminator: J computes correctly on refl
   • test_transport_along_refl: transport(refl, x) = x
   • test_path_composition_associative

2. Property Tests

   • prop_type_checking_decidable
   • prop_normalization_terminates
   • prop_proofs_verify

Benchmarks

Benchmark	Target	Notes
Type checking (small proof)	<1ms
Proof normalization	<10ms
Coherence proof construction (100 nodes)	<100ms

ADR Outline

ADR-022: Homotopy Type Theory Integration

• Status: Proposed
• Context: Need formal verification of coherence properties
• Decision: Implement HoTT core with proof terms
• Consequences: Enables proof export to Coq/Agda, significant complexity

---

Framework 4: Spectral Invariants (Advanced)

Goal State Definition

Extend current spectral analysis with Cheeger bounds, second eigenvalue cut prediction, and spectral collapse predictors.

Mathematical Foundation

Advanced Spectral Theory:
- Cheeger inequality: h(G) >= λ_2 / 2  (h = conductance)
- Second eigenvalue: λ_2 (algebraic connectivity)
- Spectral gap: λ_2 - λ_1 (stability indicator)
- Higher eigenvalue ratios: predict structural changes
- Spectral collapse: λ_i -> λ_j as graph degenerates

Module Architecture

crates/prime-radiant/src/spectral/
├── mod.rs                    # Module root (extends coherence/spectral.rs)
├── cheeger.rs                # Cheeger bounds and conductance
├── eigenvalue/
│   ├── mod.rs
│   ├── second.rs             # λ_2 analysis and prediction
│   ├── higher.rs             # Higher eigenvalue analysis
│   ├── gap.rs                # Spectral gap tracking
│   └── collapse.rs           # Spectral collapse detection
├── cut_prediction.rs         # Predict cuts from eigenvalues
├── stability.rs              # Stability analysis
├── laplacian/
│   ├── mod.rs
│   ├── normalized.rs         # Normalized Laplacian
│   ├── sheaf_laplacian.rs    # Full sheaf Laplacian matrix
│   └── sparse.rs             # Sparse matrix operations
└── config.rs

Key Data Structures

/// Cheeger analysis result
pub struct CheegerAnalysis {
    /// Cheeger constant (conductance lower bound)
    cheeger_constant: f32,
    /// Lower bound from spectral gap: λ_2 / 2
    spectral_lower_bound: f32,
    /// Upper bound: √(2 * λ_2)
    spectral_upper_bound: f32,
    /// Tightness of bound
    bound_tightness: f32,
    /// Suggested cut set (if Cheeger is low)
    suggested_cut: Option<Vec<EdgeId>>,
}

/// Second eigenvalue analysis
pub struct SecondEigenvalueAnalysis {
    /// λ_2 value
    lambda_2: f64,
    /// Corresponding eigenvector (Fiedler vector)
    fiedler_vector: Vec<f64>,
    /// Predicted cut (from Fiedler vector sign)
    predicted_cut: CutPartition,
    /// Cut quality score
    cut_quality: f32,
    /// Time trend of λ_2
    lambda_2_trend: TrendDirection,
}

/// Spectral collapse indicator
pub struct SpectralCollapse {
    /// Collapsing eigenvalue pairs
    collapsing_pairs: Vec<(usize, usize)>,
    /// Collapse velocity (rate of approach)
    collapse_velocity: f64,
    /// Predicted time to collapse
    time_to_collapse: Option<Duration>,
    /// Severity level
    severity: CollapseSeverity,
    /// Structural interpretation
    interpretation: String,
}

/// Full spectral signature
pub struct SpectralSignature {
    /// Eigenvalue spectrum (sorted)
    spectrum: Vec<f64>,
    /// Spectral density
    density: SpectralDensity,
    /// Cheeger bound
    cheeger: CheegerAnalysis,
    /// Key eigenvalue analyses
    key_eigenvalues: KeyEigenvalueSet,
    /// Collapse indicators
    collapse_indicators: Vec<SpectralCollapse>,
}

Key Traits

/// Cheeger bound computation
pub trait CheegerBound {
    /// Compute Cheeger constant approximation
    fn cheeger_constant(&self, graph: &SheafGraph) -> f32;

    /// Compute spectral bounds
    fn spectral_bounds(&self, lambda_2: f64) -> (f64, f64);

    /// Find approximate Cheeger cut
    fn find_cheeger_cut(&self, graph: &SheafGraph) -> Option<CutPartition>;
}

/// Second eigenvalue analysis
pub trait SecondEigenvalue {
    /// Compute λ_2 efficiently
    fn compute_lambda_2(&self, laplacian: &SheafLaplacian) -> f64;

    /// Compute Fiedler vector
    fn fiedler_vector(&self, laplacian: &SheafLaplacian) -> Vec<f64>;

    /// Predict optimal cut from Fiedler
    fn predict_cut(&self, fiedler: &[f64]) -> CutPartition;
}

/// Spectral collapse detection
pub trait CollapseDetector {
    /// Detect eigenvalue collapse
    fn detect_collapse(&self, history: &EigenvalueHistory) -> Vec<SpectralCollapse>;

    /// Predict future collapse
    fn predict_collapse(&self, current: &[f64], velocity: &[f64]) -> Option<SpectralCollapse>;

    /// Interpret collapse structurally
    fn interpret(&self, collapse: &SpectralCollapse) -> String;
}

Integration Points

Existing Module	Integration Type	Description
coherence::spectral	Extend	Add advanced analysis
mincut::IncoherenceIsolator	Use	Cheeger cut -> mincut
attention::AttentionCoherence	Inform	Spectral weights

WASM Export Strategy

#[wasm_bindgen]
pub struct WasmSpectralAnalysis {
    inner: SpectralAnalyzer,
}

#[wasm_bindgen]
impl WasmSpectralAnalysis {
    pub fn cheeger_bounds(&self, graph: &WasmSheafGraph) -> JsValue;
    pub fn predict_cut(&self, graph: &WasmSheafGraph) -> JsValue;
    pub fn detect_collapse(&self) -> JsValue;
    pub fn spectral_signature(&self, graph: &WasmSheafGraph) -> JsValue;
}

Test Cases

1. Unit Tests

   • test_cheeger_inequality_holds: h >= λ_2/2
   • test_fiedler_vector_orthogonal_to_constant: <v_2, 1> = 0
   • test_collapse_detection_accuracy

2. Property Tests

   • prop_eigenvalues_nonnegative
   • prop_spectral_gap_positive_for_connected
   • prop_fiedler_cut_valid

Benchmarks

Benchmark	Target	Notes
λ_2 computation (1K nodes)	<50ms	Use iterative methods
Full spectrum (1K nodes)	<500ms
Cheeger cut (1K nodes)	<20ms

ADR Outline

ADR-023: Advanced Spectral Invariants

• Status: Proposed
• Context: Current spectral analysis lacks predictive power
• Decision: Add Cheeger bounds, collapse detection
• Consequences: Better cut prediction, more accurate drift warning

---

Framework 5: Causal Abstraction Networks

Goal State Definition

Implement causal abstraction layers with structural causality enforcement for belief propagation.

Mathematical Foundation

Causal Abstraction:
- Structural Causal Models (SCM)
- Interventions: do(X = x)
- Causal graphs: DAGs with edge semantics
- Abstraction: High-level -> Low-level mapping
- Causal consistency: Interventions commute with abstraction

Module Architecture

crates/prime-radiant/src/causal/
├── mod.rs                    # Module root
├── scm/
│   ├── mod.rs                # Structural Causal Model
│   ├── variable.rs           # Causal variables
│   ├── mechanism.rs          # Causal mechanisms (functions)
│   └── intervention.rs       # Do-calculus operations
├── dag/
│   ├── mod.rs                # Causal DAG
│   ├── builder.rs            # DAG construction
│   ├── validity.rs           # Acyclicity checking
│   └── paths.rs              # Causal paths (d-separation)
├── abstraction/
│   ├── mod.rs                # Causal abstraction
│   ├── layer.rs              # Abstraction layer
│   ├── mapping.rs            # High-low mapping
│   ├── consistency.rs        # Consistency checking
│   └── constructive.rs       # Constructive abstraction
├── enforcement.rs            # Causality enforcement
├── propagation.rs            # Belief propagation with causality
└── config.rs

Key Data Structures

/// Causal variable
pub struct CausalVariable {
    /// Variable identifier
    id: VariableId,
    /// Variable name
    name: String,
    /// Domain type
    domain: VariableDomain,
    /// Current value (if observed/intervened)
    value: Option<Value>,
    /// Is this variable intervened?
    intervened: bool,
}

/// Structural Causal Model
pub struct StructuralCausalModel {
    /// Variables in the model
    variables: HashMap<VariableId, CausalVariable>,
    /// Causal DAG
    dag: CausalDAG,
    /// Mechanisms: parent values -> child value
    mechanisms: HashMap<VariableId, Mechanism>,
    /// Exogenous noise terms
    noise: HashMap<VariableId, NoiseDistribution>,
}

/// Causal abstraction layer
pub struct AbstractionLayer {
    /// Source model (low-level)
    source: StructuralCausalModel,
    /// Target model (high-level)
    target: StructuralCausalModel,
    /// Variable mapping: high -> Vec<low>
    variable_mapping: HashMap<VariableId, Vec<VariableId>>,
    /// Intervention mapping: maps interventions
    intervention_mapping: InterventionMapping,
}

/// Causal coherence constraint
pub struct CausalConstraint {
    /// Nodes that must respect causal order
    causal_nodes: Vec<NodeId>,
    /// Required causal edges
    required_edges: Vec<(NodeId, NodeId)>,
    /// Forbidden edges (would create cycles)
    forbidden_edges: Vec<(NodeId, NodeId)>,
    /// Enforcement strength
    strength: f32,
}

Key Traits

/// Structural Causal Model operations
pub trait SCMOps {
    /// Perform intervention do(X = x)
    fn intervene(&mut self, var: VariableId, value: Value);

    /// Compute causal effect P(Y | do(X = x))
    fn causal_effect(&self, target: VariableId, intervention: &[(VariableId, Value)]) -> Distribution;

    /// Check d-separation
    fn d_separated(&self, x: VariableId, y: VariableId, z: &[VariableId]) -> bool;

    /// Find causal ancestors
    fn ancestors(&self, var: VariableId) -> Vec<VariableId>;
}

/// Causal abstraction
pub trait CausalAbstraction {
    /// Check abstraction consistency
    fn is_consistent(&self) -> bool;

    /// Lift intervention to high level
    fn lift_intervention(&self, low_intervention: Intervention) -> Option<Intervention>;

    /// Project high-level state to low level
    fn project(&self, high_state: &State) -> State;

    /// Compute abstraction error
    fn abstraction_error(&self) -> f64;
}

/// Causal enforcement for coherence
pub trait CausalEnforcement {
    /// Add causal constraints to graph
    fn add_causal_constraint(&mut self, constraint: CausalConstraint);

    /// Check if edge respects causality
    fn is_causally_valid(&self, source: NodeId, target: NodeId) -> bool;

    /// Compute causal energy (violation of causal constraints)
    fn causal_energy(&self, graph: &SheafGraph) -> f32;

    /// Suggest causal repairs
    fn suggest_repairs(&self, graph: &SheafGraph) -> Vec<CausalRepair>;
}

Integration Points

Existing Module	Integration Type	Description
substrate::SheafGraph	Augment	Add causal edge semantics
coherence::CoherenceEngine	Add term	Causal energy in total
governance::PolicyBundle	Extend	Causal policy constraints
ruvllm_integration	Gate	Causal validity for LLM outputs

WASM Export Strategy

#[wasm_bindgen]
pub struct WasmCausalModel {
    inner: StructuralCausalModel,
}

#[wasm_bindgen]
impl WasmCausalModel {
    pub fn intervene(&mut self, var_id: u64, value: JsValue);
    pub fn causal_effect(&self, target: u64, interventions: JsValue) -> JsValue;
    pub fn is_d_separated(&self, x: u64, y: u64, z: JsValue) -> bool;
}

#[wasm_bindgen]
pub struct WasmCausalEnforcement {
    inner: CausalEnforcer,
}

#[wasm_bindgen]
impl WasmCausalEnforcement {
    pub fn causal_energy(&self, graph: &WasmSheafGraph) -> f32;
    pub fn check_validity(&self, source: u64, target: u64) -> bool;
}

Test Cases

1. Unit Tests

   • test_intervention_blocks_parents
   • test_d_separation_correct
   • test_abstraction_consistency
   • test_causal_energy_zero_for_valid_graph

2. Property Tests

   • prop_dag_acyclic
   • prop_intervention_idempotent
   • prop_abstraction_commutes_with_intervention

Benchmarks

Benchmark	Target	Notes
Intervention (100 vars)	<1ms
D-separation check	<0.1ms
Causal energy (1K nodes)	<10ms

ADR Outline

ADR-024: Causal Abstraction Networks

• Status: Proposed
• Context: Coherence should respect causal structure
• Decision: Implement SCM with abstraction layers
• Consequences: More interpretable coherence, can explain failures

---

Framework 6: Quantum/Algebraic Topology

Goal State Definition

Implement topological quantum encodings and spectral topology invariants for robust coherence detection.

Mathematical Foundation

Algebraic Topology for Coherence:
- Simplicial complexes from belief graphs
- Persistent homology: H_k across filtrations
- Betti numbers: β_0 (components), β_1 (loops), β_2 (voids)
- Topological Data Analysis (TDA)
- Quantum topology: topological quantum codes

Quantum Aspects:
- Anyonic braiding for coherence locks
- Topological protection from noise
- Quantum error correction via topology

Module Architecture

crates/prime-radiant/src/topology/
├── mod.rs                    # Module root
├── simplicial/
│   ├── mod.rs
│   ├── simplex.rs            # Simplices (0, 1, 2, ...)
│   ├── complex.rs            # Simplicial complex
│   ├── filtration.rs         # Filtered complex
│   └── boundary.rs           # Boundary operator
├── homology/
│   ├── mod.rs
│   ├── chain.rs              # Chain groups
│   ├── cycle.rs              # Cycle and boundary groups
│   ├── betti.rs              # Betti number computation
│   └── persistent.rs         # Persistent homology
├── tda/
│   ├── mod.rs                # Topological Data Analysis
│   ├── rips.rs               # Vietoris-Rips complex
│   ├── alpha.rs              # Alpha complex
│   ├── mapper.rs             # Mapper algorithm
│   └── persistence_diagram.rs # Persistence diagrams/barcodes
├── quantum/
│   ├── mod.rs
│   ├── toric_code.rs         # Toric code encoding
│   ├── surface_code.rs       # Surface code
│   ├── anyon.rs              # Anyonic systems
│   ├── braiding.rs           # Braiding operations
│   └── topological_qec.rs    # Topological QEC
├── invariants.rs             # Spectral topology invariants
├── encoding.rs               # Topology -> coherence encoding
└── config.rs

Key Data Structures

/// k-simplex (vertex set)
pub struct Simplex<const K: usize> {
    /// Vertices (sorted)
    vertices: [VertexId; K + 1],
    /// Optional weight/filtration value
    filtration_value: Option<f64>,
}

/// Simplicial complex
pub struct SimplicialComplex {
    /// Simplices by dimension
    simplices: Vec<HashSet<SimplexId>>,
    /// Maximum dimension
    max_dim: usize,
    /// Filtration (if filtered)
    filtration: Option<Filtration>,
}

/// Persistent homology result
pub struct PersistentHomology {
    /// Persistence diagram
    diagram: PersistenceDiagram,
    /// Betti numbers at each filtration level
    betti_curve: Vec<Vec<usize>>,
    /// Persistent Betti numbers
    persistent_betti: Vec<usize>,
    /// Topological features (birth, death pairs)
    features: Vec<TopologicalFeature>,
}

/// Persistence diagram
pub struct PersistenceDiagram {
    /// (birth, death) pairs for each dimension
    pairs: Vec<Vec<(f64, f64)>>,  // pairs[dim] = [(birth, death), ...]
    /// Essential features (never die)
    essential: Vec<Vec<f64>>,     // essential[dim] = [birth, ...]
}

/// Toric code state
pub struct ToricCodeState {
    /// Lattice dimensions
    dimensions: (usize, usize),
    /// Qubit states on edges
    edge_qubits: HashMap<EdgeId, QubitState>,
    /// Syndrome measurements
    syndromes: SyndromeMeasurements,
    /// Logical qubit state
    logical_state: LogicalQubitState,
}

/// Anyonic coherence lock
pub struct AnyonLock {
    /// Anyons in the system
    anyons: Vec<Anyon>,
    /// Braiding history
    braiding_history: Vec<BraidingOperation>,
    /// Topological charge
    total_charge: TopologicalCharge,
    /// Lock strength (from braiding complexity)
    lock_strength: f64,
}

Key Traits

/// Simplicial complex operations
pub trait SimplicialOps {
    /// Compute boundary operator
    fn boundary(&self, simplex: &Simplex) -> Chain;

    /// Compute homology groups
    fn homology(&self, dim: usize) -> HomologyGroup;

    /// Compute Betti numbers
    fn betti_numbers(&self) -> Vec<usize>;

    /// Build from graph
    fn from_graph(graph: &SheafGraph, dim: usize) -> Self;
}

/// Persistent homology
pub trait PersistentHomologyOps {
    /// Compute persistent homology
    fn compute(&self, filtration: &Filtration) -> PersistentHomology;

    /// Bottleneck distance between diagrams
    fn bottleneck_distance(&self, other: &PersistenceDiagram) -> f64;

    /// Wasserstein distance
    fn wasserstein_distance(&self, other: &PersistenceDiagram, p: f64) -> f64;

    /// Persistent Betti numbers
    fn persistent_betti(&self, birth: f64, death: f64) -> Vec<usize>;
}

/// Quantum topology operations
pub trait QuantumTopology {
    /// Encode coherence in topological code
    fn encode(&self, coherence: &CoherenceEnergy) -> ToricCodeState;

    /// Decode from topological state
    fn decode(&self, state: &ToricCodeState) -> CoherenceEnergy;

    /// Detect and correct errors
    fn error_correct(&self, state: &mut ToricCodeState) -> CorrectionResult;

    /// Compute topological protection factor
    fn protection_factor(&self, state: &ToricCodeState) -> f64;
}

/// Anyonic locks for coherence
pub trait AnyonicLock {
    /// Create lock from coherence state
    fn create_lock(&self, coherence: &CoherenceEnergy) -> AnyonLock;

    /// Verify lock integrity
    fn verify_lock(&self, lock: &AnyonLock) -> bool;

    /// Strengthen lock via braiding
    fn strengthen(&mut self, lock: &mut AnyonLock, operations: &[BraidingOperation]);
}

Integration Points

Existing Module	Integration Type	Description
substrate::SheafGraph	Build	Graph -> simplicial complex
coherence::EnergyHistory	Filtration	Energy levels as filtration
spectral::SpectralAnalysis	Combine	Spectral + topological invariants
distributed::DistributedCoherence	Encode	Topological encoding for distribution

WASM Export Strategy

#[wasm_bindgen]
pub struct WasmTopology {
    inner: TopologyEngine,
}

#[wasm_bindgen]
impl WasmTopology {
    pub fn betti_numbers(&self, graph: &WasmSheafGraph) -> JsValue;
    pub fn persistent_homology(&self, graph: &WasmSheafGraph, max_dim: usize) -> JsValue;
    pub fn persistence_diagram(&self, graph: &WasmSheafGraph) -> JsValue;
}

#[wasm_bindgen]
pub struct WasmQuantumTopology {
    inner: QuantumTopologyEngine,
}

#[wasm_bindgen]
impl WasmQuantumTopology {
    pub fn encode_coherence(&self, energy: f32) -> JsValue;
    pub fn topological_protection(&self) -> f64;
}

Test Cases

1. Unit Tests

   • test_boundary_squares_to_zero: d^2 = 0
   • test_euler_characteristic: sum (-1)^k * beta_k = chi
   • test_toric_code_detects_errors
   • test_braiding_preserves_charge

2. Property Tests

   • prop_betti_numbers_stable_under_homotopy
   • prop_persistence_diagram_valid
   • prop_topological_protection_positive

Benchmarks

Benchmark	Target	Notes
Betti numbers (1K nodes)	<50ms	Use sparse matrix
Persistent homology (1K nodes)	<200ms
Toric code encode	<10ms
Error correction	<5ms

ADR Outline

ADR-025: Quantum/Algebraic Topology

• Status: Proposed
• Context: Need robust topological invariants and noise protection
• Decision: Implement TDA + quantum topology
• Consequences: Topologically protected coherence, significant compute

---

Implementation Order and Dependencies

Dependency Graph

                    ┌─────────────────────────────────────┐
                    │                                     │
                    ▼                                     │
┌──────────────┐   ┌──────────────┐   ┌──────────────┐   │
│ 4. Spectral  │◄──│ 1. Sheaf     │──►│ 6. Quantum/  │───┘
│  Invariants  │   │  Cohomology  │   │   Topology   │
└──────┬───────┘   └──────┬───────┘   └──────────────┘
       │                  │                    ▲
       │                  │                    │
       │                  ▼                    │
       │           ┌──────────────┐           │
       └──────────►│ 2. Category/ │───────────┘
                   │    Topos     │
                   └──────┬───────┘
                          │
                          ▼
                   ┌──────────────┐
                   │ 3. Homotopy  │
                   │    Type      │
                   │    Theory    │
                   └──────────────┘
                          ▲
                          │
                   ┌──────────────┐
                   │ 5. Causal    │
                   │  Abstraction │
                   └──────────────┘

Implementation Phases

Phase 1: Foundation (Weeks 1-3)

1. Spectral Invariants (Advanced) - Extends existing spectral.rs

   • Cheeger bounds
   • λ_2 cut prediction
   • Collapse detection

2. Sheaf Cohomology - New module

   • Cochain complexes
   • Coboundary operator
   • H^0, H^1 computation

Phase 2: Core Theory (Weeks 4-6)

3. Category Theory/Topos - New module

   • Category primitives
   • Functor implementations
   • Topos basics

4. Quantum/Algebraic Topology - New module

   • Simplicial complex
   • Persistent homology
   • TDA core

Phase 3: Advanced Theory (Weeks 7-9)

5. Homotopy Type Theory - New module

   • Type system
   • Path types
   • Type checker

6. Causal Abstraction - New module

   • SCM implementation
   • Abstraction layers
   • Enforcement

Phase 4: Integration (Weeks 10-12)

• Cross-module integration
• WASM exports
• Comprehensive benchmarks
• Documentation and ADRs

Milestones

Milestone	Week	Deliverables
M1: Spectral + Cohomology Core	3	Cheeger, H^1, tests
M2: Category + Topology Core	6	Topos, TDA, tests
M3: HoTT + Causal Core	9	Type checker, SCM, tests
M4: Full Integration	12	WASM, benches, ADRs

Feature Flags

Add to Cargo.toml:

[features]
# New feature flags
cohomology = ["nalgebra"]
category = []
hott = []
spectral-advanced = ["nalgebra", "spectral"]
causal = ["petgraph"]
quantum-topology = ["nalgebra"]

# Combined features
advanced-math = [
    "cohomology",
    "category",
    "hott",
    "spectral-advanced",
    "causal",
    "quantum-topology"
]

---

Success Metrics

Per-Framework Metrics

Framework	Key Metric	Target
Sheaf Cohomology	H^1 detects obstructions	>95% accuracy
Category Theory	Functor composition correct	100%
HoTT	Proof verification	100% sound
Spectral Advanced	Cut prediction	>80% accuracy
Causal Abstraction	Abstraction consistency	100%
Quantum Topology	Error correction	>99%

Overall Metrics

Metric	Target
Test coverage	>85%
Benchmark regressions	0
WASM bundle size increase	<500KB
Documentation coverage	100% public API

---

Risk Assessment

Technical Risks

Risk	Probability	Impact	Mitigation
HoTT complexity too high	Medium	High	Start with core subset, iterative expansion
Performance degradation	Medium	Medium	Lazy evaluation, feature flags
WASM size bloat	Low	Medium	Tree shaking, separate WASM crates
Integration conflicts	Low	High	Comprehensive integration tests

Mathematical Risks

Risk	Probability	Impact	Mitigation
Incorrect cohomology	Low	High	Property tests, reference implementation comparison
Unsound type checker	Low	Critical	Formal verification of core rules
Wrong spectral bounds	Low	Medium	Compare with known graph families

---

References

1. Hansen, J., & Ghrist, R. (2019). "Toward a spectral theory of cellular sheaves."
2. Bodnar, C., et al. (2022). "Sheaf Neural Networks."
3. Univalent Foundations Program. (2013). "Homotopy Type Theory."
4. Cheeger, J. (1970). "A lower bound for the smallest eigenvalue of the Laplacian."
5. Pearl, J. (2009). "Causality: Models, Reasoning, and Inference."
6. Kitaev, A. (2003). "Fault-tolerant quantum computation by anyons."
7. Carlsson, G. (2009). "Topology and data."

---

Appendix A: File Creation Summary

New Files to Create

crates/prime-radiant/src/
├── cohomology/
│   ├── mod.rs
│   ├── cochain.rs
│   ├── coboundary.rs
│   ├── cohomology_group.rs
│   ├── obstruction.rs
│   ├── sheaf_diffusion.rs
│   ├── neural_sheaf.rs
│   └── config.rs
├── category/
│   ├── mod.rs
│   ├── category.rs
│   ├── functor.rs
│   ├── natural_transform.rs
│   ├── monad.rs
│   ├── topos/
│   │   ├── mod.rs
│   │   ├── subobject.rs
│   │   ├── internal_logic.rs
│   │   └── sheaf_topos.rs
│   ├── retrieval.rs
│   ├── coherence_laws.rs
│   └── config.rs
├── hott/
│   ├── mod.rs
│   ├── types/
│   │   ├── mod.rs
│   │   ├── universe.rs
│   │   ├── identity.rs
│   │   ├── sigma.rs
│   │   ├── pi.rs
│   │   └── higher_inductive.rs
│   ├── paths/
│   │   ├── mod.rs
│   │   ├── path.rs
│   │   ├── composition.rs
│   │   ├── inverse.rs
│   │   └── homotopy.rs
│   ├── univalence/
│   │   ├── mod.rs
│   │   ├── equivalence.rs
│   │   ├── transport.rs
│   │   └── ua.rs
│   ├── proofs/
│   │   ├── mod.rs
│   │   ├── proof_term.rs
│   │   ├── type_checker.rs
│   │   ├── normalization.rs
│   │   └── coherence_proof.rs
│   ├── embedding.rs
│   └── config.rs
├── spectral/                   # New advanced module
│   ├── mod.rs
│   ├── cheeger.rs
│   ├── eigenvalue/
│   │   ├── mod.rs
│   │   ├── second.rs
│   │   ├── higher.rs
│   │   ├── gap.rs
│   │   └── collapse.rs
│   ├── cut_prediction.rs
│   ├── stability.rs
│   ├── laplacian/
│   │   ├── mod.rs
│   │   ├── normalized.rs
│   │   ├── sheaf_laplacian.rs
│   │   └── sparse.rs
│   └── config.rs
├── causal/
│   ├── mod.rs
│   ├── scm/
│   │   ├── mod.rs
│   │   ├── variable.rs
│   │   ├── mechanism.rs
│   │   └── intervention.rs
│   ├── dag/
│   │   ├── mod.rs
│   │   ├── builder.rs
│   │   ├── validity.rs
│   │   └── paths.rs
│   ├── abstraction/
│   │   ├── mod.rs
│   │   ├── layer.rs
│   │   ├── mapping.rs
│   │   ├── consistency.rs
│   │   └── constructive.rs
│   ├── enforcement.rs
│   ├── propagation.rs
│   └── config.rs
├── topology/
│   ├── mod.rs
│   ├── simplicial/
│   │   ├── mod.rs
│   │   ├── simplex.rs
│   │   ├── complex.rs
│   │   ├── filtration.rs
│   │   └── boundary.rs
│   ├── homology/
│   │   ├── mod.rs
│   │   ├── chain.rs
│   │   ├── cycle.rs
│   │   ├── betti.rs
│   │   └── persistent.rs
│   ├── tda/
│   │   ├── mod.rs
│   │   ├── rips.rs
│   │   ├── alpha.rs
│   │   ├── mapper.rs
│   │   └── persistence_diagram.rs
│   ├── quantum/
│   │   ├── mod.rs
│   │   ├── toric_code.rs
│   │   ├── surface_code.rs
│   │   ├── anyon.rs
│   │   ├── braiding.rs
│   │   └── topological_qec.rs
│   ├── invariants.rs
│   ├── encoding.rs
│   └── config.rs
└── docs/
    ├── ADR-020-sheaf-cohomology.md
    ├── ADR-021-category-theory.md
    ├── ADR-022-homotopy-type-theory.md
    ├── ADR-023-spectral-invariants.md
    ├── ADR-024-causal-abstraction.md
    └── ADR-025-quantum-topology.md

New Test Files

crates/prime-radiant/tests/
├── cohomology_tests.rs
├── category_tests.rs
├── hott_tests.rs
├── spectral_advanced_tests.rs
├── causal_tests.rs
└── topology_tests.rs

New Benchmark Files

crates/prime-radiant/benches/
├── cohomology_bench.rs
├── category_bench.rs
├── hott_bench.rs
├── spectral_advanced_bench.rs
├── causal_bench.rs
└── topology_bench.rs

---

Appendix B: Cargo.toml Additions

# Add to [dependencies]
# For cohomology and advanced spectral
nalgebra-sparse = { version = "0.10", optional = true }

# For TDA (optional external crate integration)
# Note: Consider implementing from scratch for WASM compatibility

# Add to [features]
cohomology = ["nalgebra", "nalgebra-sparse"]
category = []
hott = []
spectral-advanced = ["nalgebra", "nalgebra-sparse", "spectral"]
causal = ["petgraph"]
quantum-topology = ["nalgebra"]

advanced-math = [
    "cohomology",
    "category",
    "hott",
    "spectral-advanced",
    "causal",
    "quantum-topology"
]

# Update full feature
full = [
    # ... existing features ...
    "advanced-math",
]

---

End of GOAP Implementation Plan