Cognitive Field Dynamics: Extension III — The Born Rule Derivation

Probability as the Product of Coherent Phase Cycling

CFD_Extension_III_Born_Rule_Derivation.pdf

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"Sometimes the deepest questions have simple answers. You just have to be willing to ask them."

ZENODO

DOI: 10.5281/zenodo.18056515

TITLE: Cognitive Field Dynamics: Extension III — The Born Rule Derivation SUBTITLE (if Probability as the Product of Coherent Phase Cycling

ABSTRACT

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This paper derives the Born rule of quantum mechanics—the postulate that measurement probability equals the squared amplitude of the wave function (P = |ψ|²)—from first principles within the Cognitive Field Dynamics framework.

The Born rule has remained an unexplained postulate for nearly 100 years since Max Born introduced it in 1926. Despite extraordinary empirical confirmation, physics has not derived why probability should equal amplitude squared rather than some other function of amplitude.

This paper demonstrates that the Born rule is not a fundamental postulate but an emergent consequence of coherent phase cycling. The central argument: actualization requires echo completion—a field configuration must project outward and return coherently to become real. This two-stage process (outbound amplitude α, return amplitude α*) yields probability proportional to α × α* = |α|².

The derivation proceeds through five mechanisms:

(1) Coherent Cycling: State persistence requires recursive phase return. Each phase-coherent cycle adds amplitude constructively.

(2) Phase Conjugation: Time reversal through a phase structure produces the complex conjugate, ensuring the product yields real positive probability.

(3) Interference Structure: Multiple paths to the same outcome sum as amplitudes before squaring, naturally producing quantum interference effects.

(4) Competitive Normalization: Mutual exclusion under collapse competition automatically normalizes probabilities to sum to unity.

(5) Universal Scaling: The mechanism operates identically across all scales—from quantum to cosmic—because it depends on phase relationships, not physical size. The Universal Scaling Constant (Λ = k/ℏ ≈ 1.31 × 10¹¹ K⁻¹s⁻¹) sets the coherence ceiling at any temperature.

The paper connects this derivation to Extension II's Universal Scaling Constant, showing that Λ × T represents the maximum number of phase-coherent echo cycles possible before thermal decoherence disrupts the mechanism.

Five testable predictions are provided, including bee swarm synchrony effects, cognitive interference under identity load, and therapeutic path interference patterns.

This work resolves the measurement problem's deepest puzzle: why probability should equal amplitude squared. The answer is that actualization requires echo completion, and echo completion is inherently a squared process.

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KEYWORDS

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Born rule

wave function collapse

quantum probability

quantum measurement problem

coherent phase cycling

phase conjugation

echo reinforcement

quantum interference

decoherence

Universal Scaling Constant

Cognitive Field Dynamics

consciousness

amplitude squared

time reversal

scale invariance

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RESOURCE TYPE

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Preprint

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PUBLICATION DATE

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December 25, 2025

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CREATOR / AUTHOR

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Name: Gaconnet, Don L.

Affiliation: LifePillar Institute

ORCID: 0009-0001-6174-8384

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LICENSE

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All rights reserved

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VERSION

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1.0

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LANGUAGE

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English

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RELATED IDENTIFIERS

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Is supplement to:

DOI: 10.5281/zenodo.18012483

Title: Cognitive Field Dynamics: A Unified Theory of Consciousness,

Expectation, and Experiential Geometry

Relation: Is supplemented by this upload

Is supplement to:

DOI: 10.5281/zenodo.18037226

Title: Cognitive Field Dynamics: Extension I — The Quantum Mechanics

Correspondence

Relation: Is supplemented by this upload

Is supplement to:

DOI: 10.5281/zenodo.18040320

Title: Cognitive Field Dynamics: Extension II — The Universal Scaling Constant

Relation: Is supplemented by this upload

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SUBJECTS / DISCIPLINES

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Quantum Physics

Theoretical Physics

Consciousness Studies

Foundations of Physics

Mathematical Physics

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ADDITIONAL NOTES

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This paper derives the Born rule (P = |ψ|²) from first principles, resolving

a foundational question in physics that has remained unanswered since 1926.

The derivation is scale-invariant, applying from quantum measurements to

cosmic structure formation, unified through the Universal Scaling Constant

established in Extension II.