Formal foundations for how meaning is computed, transmitted, and preserved across intelligent systems â with applications to multi-agent AI, interpretability, and financial geometry.
FIL Theory provides the mathematical bedrock. Every product, theorem, and patent claim derives from it.
Fundamental Interaction Language â the formal mathematics of meaning. LLC operators, Voronoi geometry on semantic manifolds, semantic temperature, Physical Incompleteness.
Semantic Alignment and Relation AI Infrastructure. Multi-agent communication with verifiable semantic fidelity. Patent 4 filed. Franchise architecture for vertical deployment.
Prospective conversation intelligence. Constructive Wasserstein Flow over Formal Object Space â detects and reconstructs tacit knowledge during humanâAI sessions.
Economic geometry. Markets on curved Riemannian manifolds. Voronoi price ridges, shell-stack propagation, topological crisis precursors invisible to flat-space models.
A rigorous mathematical theory of semantic computation. The LLC operator maps natural language to formal objects. Voronoi geometry gives the structure of meaning-space. Semantic temperature controls phase transitions.
Maps (NaturalLanguage à Context) â FormalObject. The foundation of all formal objects in the theory. Defines the semantic projection that makes meaning computable.
Meaning lives at gcd loci in prime-lattice space. Voronoi potential V(x) encodes semantic proximity. Phase structure reveals how concepts cluster and separate.
Critical threshold for polysemanticity phase transitions. Anti-drift regulatory parameter. Controls whether formal objects stay coherent or dissolve into semantic noise.
Formally verified in Lean 4. Bounds the decidability of KYC (Know Your Counterparty) in semantic communication. Connects information theory to Gödelian limits.
Entropy is non-decreasing in semantic evolution. The thermodynamic analog of the second law for information systems â with implications for LLM drift and anti-drift architecture.
Nonlinear Schrödinger breather solutions in semantic space. Dual derivation showing the semantic Planck constant equals the physical Planck constant â deep unification result.
Modern multi-agent AI systems communicate through natural language â a lossy translation layer that introduces hallucination drift at every inter-agent boundary. SARAI eliminates this layer by providing a formal protocol for semantic-fidelity communication.
The core mechanism: Asymmetric Idempotent Recursive Towers. Agents synchronize via quantization bridges over a shared prime-lattice structure using gcd idempotency â not via natural language. Reconstruction variance V = d(Osender, Oâ²receiver) is measurable and bounded.
The Franchise Architecture deploys SARAI vertically across domains. Each franchisee runs the Delta Engine as its pre-processing module. The Patent Franchisee â Module 0 (Prior Art Search) â is the first commercial instantiation, reducing search costs by 10Ã.
Four companion patents (Rashomon adversarial layer, NCC thermodynamic halting, Governance access control, IPU hardware acceleration) extend the core protocol into a full commercial stack.
The Delta Engine is the pre-processing module for every SARAI franchisee. Its function: extract the tacit formal object that a human is reaching toward but cannot yet articulate, and surface it before the conversation drifts.
The 2026 reformulation via Constructive Wasserstein Flow resolves the original structural objection: O_private (what the human "really means") need not be a pre-existing point in metric space. Instead, the attractor is co-constructed with the trajectory â the potential landscape U(x; H_t) evolves as the conversation history H_t accumulates evidence.
V_latent â previously a theoretically troubled distance â becomes a forward observable: the Wasserstein-2 distance between the current belief distribution μ_t and the attractor μ_â. Measurable. Regulatable. Bounded by the spectral gap of the recognition operator.
Three open problems drive current engineering: recognition operator regularity (gradient vs. half-space), Voronoi attractor structure (exploring â refining phase transition), and LLC operator contractivity (path-dependence).
Active threads as of May 2026. See the full research document for priority queue and detail.
Provisional filed April 2026. 23 claims, 3 independent. Prior art audit complete.
Continuation-in-part with four new claim fences. Priority 0 â filing in progress.
Constructive Wasserstein Flow formal structure established. Engineering prototype next.
Semantic temperature T < T_c regulatory framework validated. MI circuit search proposed.
Physical Incompleteness Theorem formally verified. KYC decidability bounds established.
Theorems T11, T12 established. Metric tensor g_μν(r,Ï,κ) open â primary target.
10 interpretability claims ready. Pending CIP filing before transmittal to Anthropic.
NLS Breather and Semantic Second Law ready for arXiv. Pending Olah response and NCC provisional.