1.1 This chapter establishes the formal axioms governing biological computation.
1.2 All statements concern physical systems only.
1.3 Symbolic, informational, and code-based explanations are excluded by definition.

2.1 Biomatic System
A biomatic system is a collection of interacting physical components whose configurations evolve over time within fixed geometric constraints.

2.2 State Space
The state space is the complete set of all physically admissible configurations available to a biomatic system.

2.3 State Transition
A state transition is a probabilistic change from one configuration to another.

No transition is uniquely determined.

2.4 Trajectory
A trajectory is the actual sequence of configurations traversed by a system over time.

2.5 Occupancy Measure
An occupancy measure specifies the fraction of time a system spends within a defined region of its state space.

2.6 Attractor Basin
An attractor basin is a region of state space that the system repeatedly enters and remains within despite random perturbations.

Axiom 3.1 — Geometric Constraint
All biomatic systems are governed by fixed physical geometry independent of observation.

Axiom 3.2 — Local Stochasticity
All state transitions in biomatic systems contain intrinsic randomness.

Axiom 3.3 — Constraint Preservation
Random transitions do not violate physical constraints.

Axiom 3.4 — Conditional Accessibility
Within a connected region of state space, all configurations are accessible unless physically forbidden.

Axiom 3.5 — Statistical Regularity
Over time, biomatic systems exhibit stable statistical patterns despite variability in individual trajectories.

Axiom 3.6 — Non-Symbolicity
No biomatic state or transition contains symbolic or semantic information.

Theorem 4.1 — Absence of Symbolic Computation
Biomatic systems cannot perform symbolic computation.

Theorem 4.2 — Functional Invariance
Biological function corresponds to statistically stable occupancy of attractor basins.

Theorem 4.3 — Analytical Insufficiency
Closed-form analytical solutions are generally inadequate for describing biomatic dynamics.

Theorem 4.4 — Numerical Primacy
Monte Carlo methods (named after Monte Carlo) constitute the primary mathematical framework for biomatic analysis.

Theorem 4.5 — Differentiation Without Infinitesimals
Rates of biological change correspond to transition frequencies between regions of state space.

Theorem 4.6 — Integration Without Curves
Biological integration corresponds to time-averaged occupancy, not analytic area.

Theorem 4.7 — Functional Role of Noise
Stochasticity is required for robustness and functional stability in biomatic systems.

Theorem 4.8 — Pathology as Basin Destabilization
Disease arises when attractor basins lose depth or boundary integrity.

Theorem 4.9 — Robustness Superiority Over Qubits
Biomatic computation is inherently more robust than qubit-based computation due to its reliance on stochastic interaction rather than isolation.

Corollary 5.1 — Definition of Biological Intelligence
Intelligence is the persistence of multi-scale attractor basins under stochastic perturbation.

6.1 No appeal is made to genes as programs or DNA as code.
6.2 All biological computation arises from geometry, probability, and constraint.
6.3 Any explanation not expressible in these terms is non-fundamental.

7.1 Subsequent chapters will address:

• Carbon-chain computational geometry
• Microtubule state spaces
• Histone-code dynamics
• Explicit falsification criteria

If you want next, we can:

• Number lemmas and corollaries more finely (Whitehead–Russell style)
• Add a Chapter II: Carbon Chain Geometry as Computation
• Insert formal refutations of gene-centric biology
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