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The Pair-Lock Principle: On the Geometry of Complementary Pairs

Kafkas M. Caprazli

Published Dec 23, 2025• 0 citations
Publisher

Abstract

Two is the only channel count where parity finds balance.For n observable channels sharing a conserved total, balance (equal partition) occurs at angle θ_bal(n) = arccos(1/√n), while parity (observed equals hidden) fixes at θ_par = 45°. These coincide if and only if n = 2. Pairs are geometric fixed points. Hidden structure forms perpendicular pairs, restoring the lock at each level. Orthogonality is dynamically stable: non-orthogonal configurations are Pareto inefficient and collapse. This principle identifies a common constraint underlying symplectic structure, Lindblad dynamics, and flavour symmetry: stability at n = 2, stacking at 90°.

DOI
10.5281/zenodo.18027768

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Two is the only channel count where parity finds balance.For n observable channels sharing a conserved total, balance (equal partition) occurs at angle θ_bal(n) = arccos(1/√n), while parity (observed equals hidden) fixes at θ_par = 45°. These coincide if and only if n = 2. Pairs are geometric fixed points. Hidden structure forms perpendicular pairs, restoring the lock at each level. Orthogonality is dynamically stable: non-orthogonal configurations are Pareto inefficient and collapse. This principle identifies a common constraint underlying symplectic structure, Lindblad dynamics, and flavour symmetry: stability at n = 2, stacking at 90°.
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For position-coupled decoherence (Lindblad operator L = x), coherence decay and momentum diffusion partition as cos²θ and sin²θ with measurement quadrature angle θ. The trigonometric identity yields a first complementarity relation: γ + d = 1, where γ = Γ_coh/Γ₀ and d = D_var/D₀ are dimensionless. Deviation from unity signals a hidden channel but does not locate it. We derive a second relation: when S = γ + d < 1, the hidden contribution μ_C = 1 - S has angular position θ_C = arctan(μ_C/S) on the completeness axis. When μ_C = S = 0.5, the hidden channel appears at 45°—the same angle at which roll balance equalizes known channels. This double-45° structure reflects equal partition on orthogonal axes. The first relation predicts τ(45°)/τ(0°) = 2; the second parameterizes incompleteness. Both are testable with existing quadrature-resolved optomechanics technology.
A pedagogical parable illustrating the second complementarity relation for gravitational decoherence. Two children on a seesaw discover their pivot height has shifted evidence their seesaw rests on a master seesaw, with another seesaw at its far end. Their seesaw’s tilt (left-right) is perpendicular to the master’s tilt (up-down); balance reveals the master structure. The master angle θC = arctan(μC/S) encodes the ratio of hidden to known contributions. When hidden equals known, both the seesaw balance angle and the master tilt equal 45—a double-45 structure reflecting equal partition on orthogonal axes.