Meta Theory of the Zeckendorf-Hilbert Universe

24.3 Implementation of Curry-Howard-Lambek (CHL) Correspondence in Physics

In Sections 24.1 and 24.2, we established topos model of physical systems and intuitionistic logic foundation. This reveals Constructivity of physical reality: physical truths are not static tautologies, but need to be established through observation (operations).

This section will introduce the most profound isomorphism between logic, computer science, and category theory—Curry-Howard-Lambek (CHL) Correspondence—and generalize it to physics. We will prove that physics, logic, and computation theory are trinitarian at deep structural level. In QCA discrete ontology, Physical Systems are Types, Physical States are Programs, Physical Processes are Proofs. This perspective completely eliminates boundaries between “physical laws” and “mathematical logic,” interpreting universe evolution as a huge Type Inference and Reduction process.

24.3.1 Meta-Isomorphism of Trinity: Physics, Logic, and Computation

CHL correspondence reveals deep isomorphism of three seemingly independent fields:

1. Logic: Science of propositions and proofs.

2. Computation: Science of types and programs ($\lambda$-calculus).

3. Category: Science of objects and morphisms.

After introducing physics, this correspondence extends to Physics-Logic-Computation-Category quaternary isomorphism.

Definition 24.3.1 (CHL Dictionary of Physics)

In QCA physical theory, this correspondence concretizes as:

Physical Interpretation:

When we say “system A is in state $|\psi ⟩$,” it is logically equivalent to saying “proposition A has a proof $|\psi ⟩$.”

• Vacuum is tautology (Truth object $\Omega$), always true.

• State Preparation is constructing a proof.

• Measurement is verifying whether a proof conforms to specific type (eigenspace).

24.3.2 Linear Logic and Quantum Resources: Logical Root of No-Cloning

Classical logic and standard $\lambda$-calculus allow free copying ($A⟹A\wedge A$) and discarding ($A⟹\text{True}$) of information. But in quantum physics, no-cloning theorem and unitarity prohibit such operations.

Therefore, logic corresponding to quantum physics is not classical logic, but Linear Logic (proposed by Jean-Yves Girard).

Theorem 24.3.2 (Linear Logic Formulation of Quantum Processes)

Physical laws in QCA universe follow syntactic rules of Linear Logic:

1. Resource Sensitivity: Propositions (resources) cannot be arbitrarily copied or destroyed. Premise $A$ is “consumed” after inference $A⊢B$, transformed into $B$. This precisely corresponds to extreme difficulty of non-destructive measurement of quantum states, and unitarity of state evolution.

2. Multiplicative Connectives:

   • Tensor Product ($\otimes$): $A\otimes B$ means “simultaneously having resource A and resource B.”

   • Linear Implication ($⊸$): $A⊸B$ means “process consuming A to produce B.” Physically, this is operator space $\mathcal{L}({\mathcal{H}}_A,{\mathcal{H}}_B)\cong {\mathcal{H}}_A^{\ast }\otimes {\mathcal{H}}_B$.

3. Duality: Negation $A^{\perp }$ in linear logic corresponds to dual space ${\mathcal{H}}^{\ast }$ or antiparticles. $A^{\perp \perp }\cong A$ corresponds to dagger structure in DCC.

Corollary:

Quantum mechanics appears “strange” because it runs on a resource-conserving logical system at bottom level. Wave function collapse is not logical error, but resource consumption of linear types—you read data once, data is “used up” (becomes classical record, no longer original quantum state).

24.3.3 Physical Laws as Type Inference Rules

From CHL perspective, physical laws (such as Schrödinger equation or QCA update rules) are no longer descriptive formulas, but constructive Type Inference Rules.

Definition 24.3.3 (Proof-Theoretic Semantics of Dynamics)

Consider QCA’s local update rule $U:{\mathcal{H}}_{\text{in}}\to {\mathcal{H}}_{\text{out}}$.

In type theory, this corresponds to a function term:

$$\text{update}:{\text{State}}_t⊸{\text{State}}_{t+1}$$

Time evolution sequence of universe $x_0\to x_1\to x_2\to \cdots$ corresponds to step-by-step construction of a Proof Tree.

• $t=0$: Axiom (initial conditions).

• $t=n$: Theorem obtained by applying derivation rules (physical laws) $n$ times.

Theorem 24.3.4 (Physics as Normalization)

Running of physical processes is equivalent to Reduction (Normalization) of $\lambda$-terms.

Let initial physical configuration be a complex tensor network (or $\lambda$-term) $M$.

• Interactions (such as particle collisions) correspond to $\beta$-reduction: $(\lambda x.t)u\to t[u/x]$.

• Thermal Equilibrium corresponds to Normal Form: a stable state that cannot be further simplified.

Therefore, time passage is process of universe computing system executing reduction steps.

24.3.4 Universe as Type-Theoretic Universe: From “Existence” to “Construction”

Finally, this perspective resolves a core ontological divergence in physics: Platonism vs. Constructivism.

• Platonism: Physical laws exist in an eternal world of ideas, physical world imitates it.

• QCA Constructivism: Physical reality is constructed by computational processes.

Corollary 24.3.5 (Constructive Realism)

In QCA universe, if a physical state $|\psi ⟩$ cannot be prepared from vacuum or given initial state through finite QCA steps (finite-length proofs), then it is non-existent physically.

This excludes “mathematical states” in standard Hilbert space with uncomputable amplitudes. Physical Hilbert space is subspace of Computable States.

This echoes Gödel incompleteness discussed in Section 5.4: physical truths are limited by Provability. Universe can only explore states it can “compute to.”

Conclusion

Section 24.3 establishes computational-logical foundation of physics.

1. Isomorphism: Physical systems, logical propositions, and computational types are three faces of the same structure.

2. Logic: Quantum physics follows linear logic, emphasizing non-clonability of resources.

3. Dynamics: Evolution is proof, equilibrium is normal form.

At this point, we have completed logical reconstruction of meta-theory of physics. We proved that QCA theory is not only physically self-consistent (unitarity, causality), but also has the most solid mathematical foundation logically (category theory, type theory).

In the upcoming Part XIV: Computational Foundations and Encoding, we will ground these abstract logical principles, exploring how physical world performs optimal encoding (such as bit counting of holographic principle) and thermodynamic cost of computation (Landauer principle).