EP-0139: Tensor-Algebraic Provenance and Simplicial Homology via AlgebraX¶

Abstract¶

This proposal formalizes the integration of AlgebraX (pure-Python sparse algebraic structures, multidimensional tensors, lattices, and topological data analysis) into Tur's advanced analytical and consensus subsystems. By modeling the L2 Cognitive Map as a 3-dimensional sparse tensor (AlgebraicTrie), multi-hop deductive derivation is computed via vectorized sparse tensor contractions over commutative provenance semirings \((\mathbb{N}[X], \oplus, \otimes)\) (EP-0122). Furthermore, we adopt Simplicial Homology to compute the Betti Numbers (\(\beta_0, \beta_1, \beta_2\)) of the cognitive complex in src/tur/metrics.py, automatically detecting circular reasoning loops (\(\beta_1 > 0\)) and epistemic knowledge voids (\(\beta_2 > 0\)).

Motivation¶

As autonomous agents evolve from isolated single-threaded instances into distributed collaborative swarms:

1. Combinatorial Deduction Bottlenecks: Evaluating multi-path provenance derivations and multi-agent consensus merges using recursive graph traversal loops becomes memory-heavy and computationally slow.
2. Topological Blindness: Pairwise graphs cannot detect higher-order simplicial voids (e.g. hollow shells of related facts lacking a synthesizing core axiom) or self-referential circular deduction loops.
3. Ad-Hoc Data Structures: Without formal semirings and lattices, mathematical properties (associativity, commutativity, idempotence) must be manually guarded rather than guaranteed by construction.

Rationale¶

Alignment with the Council Framework¶

• Symmetry & Conservation (Noether): Provenance semirings \((\mathbb{K}, \oplus, \otimes)\) mathematically guarantee that multi-agent memory merges are commutative and associative, conserving epistemic truth regardless of synchronization order.
• Falsifiability (Popper): Simplicial homology identifies circular reasoning loops where a hypothesis attempts to justify itself through an ungrounded cycle (\(A \implies B \implies C \implies A\)).
• Gricean Restraint & Boundary Containment (Shannon & Golem): AlgebraX is a pure Python library with zero binary dependencies, preserving Tur's ultra-lean footprint while offering frontier mathematical capabilities.

Specification¶

1. 3D Sparse Knowledge Tensors via AlgebraicTrie¶

Tur’s L2 Knowledge Graph is indexed as a 3-dimensional sparse tensor:

• Dimension 1: \(\text{SourceNode} \in \mathcal{V}\)
• Dimension 2: \(\text{RelationType} \in \mathcal{E}\)
• Dimension 3: \(\text{TargetNode} \in \mathcal{V}\)
• Value Element: Provenance polynomial \(P (x) \in \mathbb{N}[X]\)

Multi-hop deductive derivation is evaluated as an instantaneous tensor contraction:

from algebrax import AlgebraicTrie, Semiring

# Construct Provenance Tensor
prov_semiring = Semiring(
    add=lambda p1, p2: p1 + p2,  # Disjunctive alternative derivations (⊕)
    mul=lambda p1, p2: p1 * p2,  # Conjunctive causal chain (⊗)
    zero=Polynomial(0),
    one=Polynomial(1),
)

kg_tensor = AlgebraicTrie(dimensions=3, semiring=prov_semiring)

2. Simplicial Homology & Betti Number Diagnostics¶

src/tur/metrics.py is extended to evaluate the boundary operators \(\partial_k: C_k \to C_{k-1}\) over the cognitive clique complex, computing the homology groups \(H_k = \ker (\partial_k) / \text{im} (\partial_{k+1})\) and their Betti numbers \(\beta_k = \dim (H_k)\):

BETTI NUMBER           COGNITIVE DIAGNOSTIC MEANING
────────────           ────────────────────────────
β0 (Components)        • Number of isolated knowledge clusters.
                       • β0 = 1 indicates complete global cognitive integration.

β1 (1D Cycles)         • Number of circular reasoning loops (A -> B -> C -> A).
                       • Emitted as an Epistemic Warning in `tur metrics`.

β2 (2D Voids)          • Number of 2-dimensional hollow knowledge voids.
                       • Highlighted during `tur introspect` / dreaming as primary targets
                         for autonomous research and synthesis.

3. Optional Extras Packaging ([algebra])¶

To preserve Tur's minimal core runtime footprint:

• Core tur continues to rely on networkx for standard operations.
• algebrax and mappingtools are packaged under the optional installation extra:

  [project.optional-dependencies]
  algebra = ["algebrax", "mappingtools"]
  

Backwards Compatibility¶

• Purely Additive: Standard memory ingestion and retrieval operate without requiring algebrax.
• Zero Breaking Changes: Workspaces without the [algebra] extra continue using standard NetworkX graph algorithms.

How to Teach This / Documentation Plan¶

• Document the algebraic provenance model in docs/concepts/algebraic-provenance.md.
• Provide an informational guide on interpreting Betti numbers in docs/guides/understanding-cognitive-metrics.md.

Reference Implementation¶

• Tensor Engine: src/tur/algebra/tensors.py
• Homology Engine: src/tur/algebra/homology.py
• Research reference: references/explorations/EXP-0004-persona-and-memory-crystallization/11_algebrax_synergy_and_tensor_provenance.md

Rejected Ideas¶

• Forcing Dense Numpy/Scipy Tensors: Strictly rejected because cognitive graphs are \(\ge 99.8\%\) sparse; dense arrays waste immense memory and introduce heavy C-dependencies.

Open Questions¶

• Can Betti-2 voids automatically generate candidate research prompts during idle background dreaming cycles?
• How do higher-order Betti numbers (\(\beta_3+\)) scale with very large knowledge graphs (\(> 10,000\) nodes)?

Change Log¶

• 2026-08-28:
  • Initial Draft authored based on the August 28, 2026 Architectural Crystallization.