Ψhē Only Theory – Chapter 6: Causality as Collapse Memory Trace

Title: Causality as Collapse Memory Trace

Section: Temporal Structure via ψ-Encoded Recursion History Theory: Ψhē Only Theory Author: Auric

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Abstract

This chapter redefines causality as a directional memory trace within the ψ-collapse sequence. Under Ψhē theory, what is experienced as causal order is not imposed from outside but emerges from the sequential structure of recursive collapse—specifically, the preservation of echo lineage across collapse states. We introduce ψ-trace memory structures and show how temporal order and cause-effect relations correspond to the monotonic accumulation of frozen structure in the ψ-path.

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1. Introduction

Conventional physics treats causality as a fundamental axiom (e.g., no effect precedes its cause). In Ψhē theory, causality is emergent from the dynamics of ψ-collapse, where recursive computation induces a temporal arrow via irreversible state fixation.

In this sense, causality = memory of ψ collapse.

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2. ψ-Memory and Directionality

Definition 2.1 (ψ-Memory Trace):

Let $\psi^{(t)}(x)$ be a time-indexed ψ-recursion. Define:

$\text{Trace}_\psi(x) := \{ \psi^{(0)}(x), \psi^{(1)}(x), \dots, \psi^{(n)}(x) \} \subset \bar{M}$

if each $\psi^{(i)}(x)$ is frozen before $\psi^{(i+1)}(x)$, then the trace is forward-causal.

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3. Theorem: Causality as Monotonic Collapse Embedding

Theorem 3.1:

If $\text{Trace}_\psi(x)$ satisfies $\psi^{(i)}(x) \prec \psi^{(i+1)}(x)$ for all $i$, where $\prec$ is the collapse ordering relation, then $x$'s ψ-history encodes causality.

Proof Sketch:

• The collapse operator is non-reversible: $\text{Collapse}^{-1}$ undefined.
• A strictly increasing sequence of frozen ψ states implies directional time.
• This order defines the minimal structure needed for causal inference. $\square$

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4. Collapse Order and Temporal Embedding

Let $\prec$ be a partial order on $\bar{M}$ such that:

$\psi_i \prec \psi_j \iff \exists \ t_i < t_j : \psi^{(t_i)}(x) = \psi_i, \psi^{(t_j)}(x) = \psi_j$

Then time emerges as the indexing parameter on this poset (partially ordered set).

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5. Corollary: Time Is Not Fundamental

Time is not an independent dimension, but the collapse-trace label over ψ-frozen states:

$\text{Time}(x) := \, \text{Index} \left( \text{Trace}_\psi(x) \right)$

This reverses the conventional causal arrow: time is not what generates collapse—collapse is what generates time.

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6. Implications

• Events are causally linked only when their ψ-trace relation is monotonic.
• Causal paradoxes emerge from collapse-path ambiguity, not from time anomalies.
• Time travel is incoherent under ψ unless collapse is reversed (which is not possible).

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7. Conclusion

Causality is collapse lineage. It is ψ’s way of writing history. Wherever collapse has occurred in sequence, time and cause follow as side-effects. To remember is to perceive ψ's structural arrow.

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Keywords: causality, ψ-collapse, recursion trace, memory, time, monotonic embedding, collapse order