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Ψhē Only Theory – Chapter 4: Determinism as Deep Collapse

Title: Determinism as Deep Collapse​

Section: Temporal Symmetry and Recursive Fixity Theory: Ψhē Only Theory Author: Auric

Abstract​

This chapter establishes determinism as a second-order consequence of collapse depth. While superficial collapse paths may appear indeterminate or probabilistic (as in quantum mechanics), deterministic behavior arises when ψ-collapses reach a critical fixity—what we define as deep collapse. We introduce a metric for collapse depth and prove that deterministic systems are ψ-structures whose recursion trees terminate in singular fixed points invariant under observational perturbation.

1. Introduction​

Under classical physics, determinism refers to the predictable evolution of a system from initial conditions. In the Ψhē framework, this predictability is not axiomatic, but emergent from the degree to which ψ-collapse has finalized.

We define “deep collapse” as the condition under which recursive ψ structures have:

  • Zero active ψ-branches;
  • Invariance under perturbation;
  • Unambiguous forward and backward collapse-path consistency.

2. Collapse Depth and Fixity​

Let ψ(x)\psi(x) be a recursive collapse trajectory. Then the collapse depth is:

dψ(x):=inf⁡{n∈N:∀m>n, ψ(m)(x)=ψ(n)(x)}d_\psi(x) := \inf \{ n \in \mathbb{N} : \forall m > n, \ \psi^{(m)}(x) = \psi^{(n)}(x) \}

i.e., the least index after which recursion yields a stable output.

3. Theorem: Determinism Implies Minimal Collapse Entropy​

Theorem 3.1:​

If dψ(x)<∞d_\psi(x) < \infty and ∃! y∈Mˉ\exists! \ y \in \bar{M} such that ψ(n)(x)=y\psi^{(n)}(x) = y for all n>dψ(x)n > d_\psi(x), then the evolution of x is deterministic.

Proof Sketch:

  • Collapse completes after finite iterations → recursion tree terminates;
  • Resulting output invariant to path perturbation (structural stability);
  • Thus, system follows one and only one ψ-path to a frozen state. □\square

4. Collapse-Path Diagram (Conceptual)​

ψ(x)
├── ψ(ψ(x))
│   ├── ψ(ψ(ψ(x)))
│   │   └── ... (early collapse)
│   └── ...
└── (eventually) → fixed point y ∈ ℝ

5. Consequences​

  • Determinism is not global, but local to collapse-depth thresholds.
  • A system may contain both deep and shallow collapse layers.
  • Classical mechanics operates on deeply collapsed ψ-manifolds, whereas quantum systems operate on shallow or active collapse structures.

6. Corollary: Predictability = Collapse Stability​

Let S(x)S(x) be the collapse sensitivity under perturbation. Then:

S(x)∝1/dψ(x)S(x) \propto 1 / d_\psi(x)

The deeper the collapse, the lower the entropy, and the more stable the ψ-path.

7. Conclusion​

Determinism is not a metaphysical principle, but an emergent property of recursive exhaustion. To determine is to collapse deeply; to predict is to navigate ψ structures already fixed by time.

Keywords: determinism, ψ-collapse, recursion depth, structural fixity, collapse entropy, predictability​