A compositional algebra for coherence-unit configurations: closure, entropy balance, and inertia additivity

Twelve compositional operations that appear, under different names, in physics, chemistry, biology and social systems — union, coupling, fusion, fission, absorption, dissolution, among others — collapse into five primitives (forming the set, coupling, decoupling, marginalizing, copying, relaxing) and a single linear-algebra identity: the Schur block-determinant formula,applied to a configuration object Γ∈ M₄(R). From that single identity, without a per-operation postulate, follows the complete entropy balance, an exact minimum-work bound (a Jarzynski-Crooks analogue), and a domain-agnostic measure of irreducible structural cohesion. Around that core, the paper proves eleven theorems. Removing the restriction to positive-definite matrices, Haynsworth's (1968) inertia additivity extends the algebra to any signature, revealing that the determinant-sign classification used throughout is the coarsest possible partition of a symmetric matrix's inertia, for any dimension (Corollary 8.1). A related result (Theorem 9) showsthat no conjugation-invariant potential can structurally distinguish sub-sectors sharing that phase. Two further theorems (10 and 11) characterize exactly when and how eliminating a component creates or preserves antisymmetric (bivector) structure in the survivor — the second formalized in Lean 4 with no `sorry`. A more incipient line (Appendix A) proposes that the full algebra is acategory enriched over the same entropy balance, in the sense of Lawvere (1973); documented as work in progress, not a closed result. The paper does not claim to derive particle physics: its illustrative correspondences (Coulomb/Lorentz force, ideal gas, chemical bonding) show the algebra's applicability to already-known cases, not a new derivation of those phenomena.