The science behind Digital Circuitality.

Shannon (1948) defined the entropy of a discrete source as H(X) = -Σ p(x_i) log_2 p(x_i). When a system is completely deterministic — every input produces exactly one output through every path — the probability distribution collapses to a Dirac delta on the correct result: H(X) = 0.

A system S exhibits Digital Circuitality if and only if:

Φ_c = 1 certifies that every input domain is bounded, every operation verified, every output range proven, and no execution path is undefined. There is no informational uncertainty because there is no unknown state.

Conventional software operates with informational uncertainty > 0. Unverified execution paths, unexplored states, unbounded inputs. Testing reduces informational uncertainty but never eliminates it — Dijkstra (1976): “Testing shows the presence of bugs, never their absence.”

Digital Circuitality eliminates informational uncertainty by construction, not by sampling.

The term “thermodynamic” in Digital Circuitality is an analogy, not a physical claim. A physical circuit is coherent when energy flows from source to sink without leaks, all signal paths are closed, and the circuit reaches steady state.

What the analogy does NOT claim: No physical energy cost from Φ_c = 1. No equivalence between computational and thermodynamic coherence. No claim that thermodynamic laws govern compilation.

Recent research in information physics has demonstrated that informational entropy and thermal entropy are fundamentally different quantities. Treating them as interchangeable is a category error.

For Digital Circuitality, the consequence is direct: the coherence framework measures informational entropy, not thermal entropy. No thermodynamic claims are needed for the framework to be rigorous. The verification operates on purely informational foundations.

Digital Circuitality draws conceptual inspiration from Brillouin’s work on the relationship between information and entropy, while operating on purely informational foundations grounded in Shannon’s framework.

The system does not depend on any physical thermodynamic claims. It acknowledges the historical inspiration while maintaining rigorous separation between informational and physical domains.

A formally verified, deterministic system has zero informational uncertainty. Every state is known, every path verified, every domain bounded, the circuit is closed.

This is what Φ_c= 1 means in Digital Circuitality: the system’s informational entropy is zero — not by testing, but by mathematical construction.

Traditional transpilers operate at the syntactic level: parse an AST in one language, emit an AST in another. BRIK-64 operates at the semanticlevel — extracting the computational essence (what it computes, not how it’s expressed) and encoding it as a PCD circuit.

The critical property: if two programs in different languages produce the same PCD circuit, they are functionally equivalent. PCD captures the informational content of computation independent of syntactic vehicle.

The TCE certifies that the PCD circuit is closed (Φ_c = 1), guaranteeing the computation is deterministic, total, and informationally preserving. The equivalence is algebraic, not tested.