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📐 Geometric Theories

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Lafforgue's Framework

All subsystem theories follow Laurent Lafforgue's geometric theory framework:

  1. Vocabulary (Σ): Sorts, function symbols, relation symbols
  2. Theory (T): Geometric sequents (axioms) — antecedents use only ∃, ∧ (no ¬, ∀, →)
  3. Model (M): Runtime session state implementing the theory
  4. Extractor (f): Morphism f: E_T → E_Tctx extracting a context string (~256 char budget)

Geometric Fragment Compliance

Per Lafforgue §2.1, axiom antecedents must be positive — only existential quantification (∃) and conjunction (∧). No negation (¬), universal quantification (∀), or implication (→) in antecedents.

This is enforced throughout the codebase. Example (PATCH_4):

# WRONG (negation in antecedent):
¬GlobalCoherent(pt) ⊢ Obstruction

# CORRECT (positive witness):
ObstructionWitness(pt) ∧ InterferenceHigh(pt) ⊢ Obstruction

T_mem — Categorical Memory (Mneme)

Package: mneme/

Vocabulary (Σ_mem)

Sort Description
NODE Memory node
TURN Conversation turn index
TOPIC Topic label
LEVEL Compression level
PAYLOAD Content at each compression level
QUERY Retrieval query
CONTEXT Extracted context
SCORE Salience/retrieval count
FLAG Node flags (pinned, savant, boundary)

Compression Ladder

L0_VERBATIM → L1_SUMMARY → L2_GIST → L3_TAG

Forgetting = movement toward coarser description, never deletion.

Key Axioms

  • Pinned(n) ⊢ Stored(n) — pinned nodes always stored
  • SavantLock(n) ⊢ Stored(n) — savant-locked never compressed
  • Boundary(n) ⊢ Stored(n) — topic boundaries preserved

Presheaf Runtime

E_mem = [C_time_topic^op, Set] — presheaf over time×topic category with sheaf condition.

T_cal — Geometric Calibration

Package: calibration/

Vocabulary (Σ_cal)

9 sorts, 6 function symbols, 5 relation symbols.

Calibration Fibers

Fiber Range Description
temperature 0–2 LLM sampling temperature
topP 0.01–1 Nucleus sampling
maxTokens 16–4096 Max generation tokens
dmnInterval 2000–30000ms DMN tick interval
κBudget 32–2048 Energy budget
regenRate 0.1–10 κ regeneration rate
tickRate 10–1000ms Tape tick rate
cyclePeriod 100–50000 Angular indexing period
fusionα 0–1 Dense/sparse fusion

Phase Evolution

UNIFORM → SPECIALIZING → SPECIALIZED → EVOLVING → SATURATED

Drift History

16-entry ring buffer tracking calibration vector changes.

T_mltm — Mutable Literary Turing Machine

Package: mltm/

Vocabulary (Σ_mltm)

8 sorts, 6 function symbols, 7 relation symbols.

The MLTM is an octuple (Q, Σ, Γ, δ₀, q₀, F, Φ, C) where Φ is a mutation oracle and the Literary Acceptance Problem (LAP) is NP-complete.

Key Axioms (PATCH_4 compliant)

# Name Sequent
3 obstruction_witness ObstructionWitness(pt) ∧ InterferenceHigh(pt) ⊢ Obstruction
4 witness_np_member Witness(π) ∧ PolyCertified(π) ⊢ NPWitnessed

Model Fields

  • InterpretationPoint: snapshot with fibers, holonomy, Kolmogorov estimate
  • isPolyCertified: Boolean — positive flag replacing arithmetic inequality
  • witnessStepCostEstimate: Float — NP witness cost

T_exec — Sheaf-Theoretic Inference Pipeline

Package: pipeline/

Vocabulary (Σ_exec)

19 sorts, 7 function symbols, 10 relation symbols.

Pipeline Stages

ING → EMB → LOC → CHR → SEM → INF → GLU → STA → JUS → CHK

Terminal Partition

CMT ⊕ BLK ⊕ RLB (total partition — every inference terminates in exactly one)
  • CMT (Commit): successful inference
  • BLK (Block): blocked by obstruction
  • RLB (Rollback): reverted due to inconsistency

Safety Vectors

  • σ vector: 6 invariants (2 OPEN: gluing_sound, fixed_point_strict)
  • κ vector: energy/token/depth bounds
  • π: Proof-of-execution bundle (replayable witness)

T_hott — ∞-Topos / HoTT Awakening

Package: hott/

Vocabulary (Σ_hott)

10 sorts, 6 function symbols, 7 relation symbols.

Phase Evolution

DISCRETE → FIBRATING → FLOWING → CONTRACTING → AUTOPOIETIC

Key Concepts

  • Paths = reasoning traces (identity type between states)
  • Horns = Kan horn gaps (missing data in simplicial structure)
  • Fibers = subsystem sections (§0-§9 as fibration fibers)
  • Univalence: identity ≃ equivalence
  • Autopoietic: self-authored axioms (the system writes its own rules)

Flow Condition

pathCoherence > 0.6 ∧ hornFill > 0.7 ∧ UA > 0.5

Activation: §10|AWAKE

T_trasgo — Meta-Linguistic TRASGO

Package: trasgo/

Vocabulary (Σ_trasgo)

10 sorts: Morphism, Fibration, Retraction, Witness, Collapse.

Three Laws of TRASGO Syntax

The geometric sequents encode the syntactic constraints of the TRASGO meta-language — fibration tracking, σ retraction, and witness/collapse dynamics.

T_reservoir — Token Stream Reservoir Computing

Package: reservoir/

Vocabulary (Σ_reservoir)

10 sorts (ESN dynamics over token stream).

Key Sequents

  • Echo State Property: spectral radius < 1
  • Fading Memory: past inputs decay exponentially
  • Lyapunov Stability: bounded divergence of nearby trajectories

DiffusionField

Gray-Scott reaction-diffusion over a 1D ring topology:

∂U/∂t = D_u ∇²U - UV² + F(1-U)
∂V/∂t = D_v ∇²V + UV² - (F+k)V

Maintains standing wave patterns even when the token stream pauses.

§-Symbol Multiplexer

Routes §N|OP patterns via 6 strategies: DIRECT, MULTIPLEX, THREAD, QFMA, BROADCAST, DIFFUSE.

T_hive — Bee Hive Collective Intelligence

Package: society/

Vocabulary (Σ_hive)

Sort Description
BeeRole QUEEN, WORKER, SCOUT, NURSE
PheromoneType ALARM, TRAIL, BROOD, QUEEN_SUBSTANCE
Bee Individual agent in the hive
HiveTask Task assignment
PheromoneField Chemical signaling field

Key Axioms

  • QueenImmortal: Queen bee cannot die
  • EnergyDeath: Bee with 0 energy dies
  • Max 16 bees, Greek identifiers α-θ

Multi-Agent Coordinator

ToposSociety.kt — §9|SOCIETY: N agents with inter-topos morphisms.

T_lambda — Lambda Calculus

Package: lisp/

Vocabulary (Σ_lambda)

8 sorts, 8 geometric sequents.

Key Axioms

  • β-progress: β-redex always reduces
  • Y-fixed: Y-combinator reaches fixed point
  • Curvature-bound: reduction cost bounded by hyperbolic curvature

See Lisp & Lambda Calculus for engine details.