A Formal Theory of the Ideal Regulator: Port-Thermodynamics, Closed-Loop Homeostasis, and Asymptotic Stability in Minimal Dissipative Agents

SUMMARY & OVERVIEW In classical thermodynamics, the Ideal Gas Law (PV = nRT) and the Carnot cycle (η = 1 - Tc/Th) establish exact, frictionless reference benchmarks against which all real physical engines and energetic losses are measured. Theoretical biology has historically lacked an equivalent analytical baseline for autonomous dissipative agency. This preprint establishes the mathematical derivation, discrete algorithmic realization, and continuous control-theoretic verification of the Ideal Regulator—the biological analogue of the Ideal Gas and Carnot engine. The model deliberately abstracts away non-universal evolutionary adaptations (such as cellular differentiation and morphological branching) to define the absolute stoichiometric and kinetic ceiling of living agency. KEY THEORETICAL & MATHEMATICAL RESULTS 1. The Tri-Partite Mass-Energy State Space:Matter within the open boundary is strictly partitioned into three mutually exclusive structural classes: High-Energy Low-Polymer nutrients (M_HL), High-Energy High-Polymer catalytic structures and replication tapes (M_HH), and fully degraded Low-Energy Low-Polymer exhaust (M_LL). Mass conservation is proven to be functorially exact across discrete cycles and continuous flows. 2. Resolution of Dual Infinite Regresses:- Structural Regress (Rosen's Constructor Paradox): Broken by formalizing the divide between active, catalyzed fabrication (covalent polymerization, ΔG > 0) and spontaneous thermodynamic self-assembly (non-covalent folding and aggregation, ΔG < 0), eliminating the need for an infinite cascade of constructor machines.- Cognitive Regress (Ryle's Interpreter Paradox): Broken by treating the heritable tape (T) as a materialized non-holonomic constraint operating through the physical cascade: G (latent state) -> G' (sensory perturbation) -[T]-> G'' (transduced effector state) -> action suite (A) and allocation split (α), eliminating the homunculus meta-interpreter. 3. The Master Equation of State:Under continuous feedforward metabolic impedance matching (dc/dt = 0), we derive the differential power balance and the biological equation of state: Π = (q + θ)δ + k_osmequating net available metabolic pressure (Π) directly to structural turnover ((q + θ)δ) and active osmotic counteraction (k_osm). 4. Closed-Loop Point-Attractor Stability:We prove that purely feedforward metabolic matching produces an invariant neutral center manifold with a zero eigenvalue (λ = 0), leaving open-loop systems vulnerable to drift. Augmenting the policy with closed-loop proportional error feedback collapses the center manifold into a globally asymptotically stable 2D point attractor with pronounced fast-slow timescale separation (τ_c ≈ 1.4 s vs. τ_M ≈ 19.7 s). 5. The Discrete 7-Step Arithmetic Ledger:We translate the continuous port-thermodynamics into an exact, cycle-by-cycle arithmetic algorithm. Previous mitotic deadlocks are eliminated through a three-tier hierarchical anabolism priority scheme (Priority 1: Somatic Repair -> Priority 2: Tape Replication -> Priority 3: Pre-Mitotic Container Growth). EMPIRICAL BENCHMARKING ON ESCHERICHIA COLI Calibrated against publicly annotated physiological data (BioNumbers, Stouthamer 1973, Pirt 1965, Neidhardt et al. 1990):- Carnot-Optimal Allocation: Predicts an ideal catabolic split of α* ≈ 15.3% (matching Stouthamer's theoretical limit of 13-15%), compared to real E. coli's empirical split of ~52% (biomass yield ~0.45-0.50 gDW/g glucose).- Prokaryotic Speed Limit: Predicts an absolute mitotic doubling limit of T_div^min ≈ 19.0 minutes under carrier saturation, directly matching E. coli's fastest observed division time (~20 minutes at 37°C in rich broth).- Critical Maintenance Influx: Derives a basal survival threshold of J_in* ≈ 0.0365 g glucose/(gDW·h), or ~9,500 glucose molecules per cell per second, below which cells undergo autophagic starvation lysis. VALIDATION BATTERY The continuous framework is benchmarked across four control-theoretic regimes:- Test 1: Non-Equilibrium Steady State (NESS) asymptotic convergence with constant linear entropy/exhaust production.- Test 2: Sub-critical famine step disturbance buffering via active free-energy capacitor discharge without osmotic lysis.- Test 3: Harmonic environmental disturbance rejection and structural low-pass mechanical filtering.- Test 4: Phase-space topological vector field verification of point attractor stability across all four quadrants.