A No-Go Theorem (NGT) for Reversible Measurement from the Theory of Entropicity (ToE)

Abstract

We present a compact No-Go Theorem (NGT) derived within the Theory of Entropicity (ToE) demonstrating that no physical process can simultaneously yield a stable, distinguishable outcome and remain entropically reversible. The result follows from treating entropy as a universal physical field endowed with geometric curvature and from the Obidi Curvature Invariant (OCI), which fixes ln 2 as the minimal nonzero curvature required for distinguishability. We show that this invariant enforces irreversibility, yields wave-function collapse as a curvature-stabilization process, fixes a collapse timescale through entropy production rates, and implies quantization of the entropy field. The theorem replaces the quantum measurement postulate with a geometric necessity.

Keywords

Theory of Entropicity (ToE); Obidi Curvature Invariant (OCI); No-Go Theorem (NGT); Measurement Irreversibility; Quantum Foundations

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1. Introduction

The coexistence of reversible microscopic laws with irreversible measurement outcomes remains a central inconsistency in fundamental physics. Existing approaches either postulate collapse, relocate irreversibility to environments, or deny single outcomes altogether. In contrast, the Theory of Entropicity (ToE) posits entropy as a physical field whose geometry governs distinguishability and observability. From this single premise we derive a no-go theorem forbidding reversible measurement.

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2. Entropic Geometry and the Obidi Curvature Invariant

Physical states form a convex set, and operational distinguishability is quantified by divergences that are positive, convex, and monotone. These minimal requirements induce a divergence geometry whose second variation defines an entropic curvature. For the minimal binary bifurcation of a convex state into two distinguishable alternatives, the divergence cost is exactly ln 2. This value is invariant under dynamics, scale, and representation.

Obidi Curvature Invariant (OCI): The minimal nonzero entropic curvature compatible with distinguishability is ln 2.

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3. No-Go Theorem for Reversible Measurement

Theorem. No physical process can both (i) generate a stable, distinguishable outcome and (ii) remain entropically reversible.

Proof (sketch). A measurement is any process producing operationally distinguishable outcomes. Distinguishability requires positive divergence from the pre-measurement state. By OCI, the induced entropic curvature satisfies R ≥ ln 2. Reversible processes require vanishing net curvature. The two conditions are incompatible. ∎

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4. Collapse as Entropic Curvature Stabilization

Prior to measurement, superposed states correspond to flat entropic geometry with respect to outcome distinctions. During interaction, entropy production accumulates curvature. Collapse occurs precisely when accumulated curvature reaches the OCI threshold ln 2, stabilizing one branch and suppressing alternatives. Collapse is therefore a geometric transition, not a postulate.

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5. Collapse Timescale

Let Φ_int denote the entropy production rate of a measurement interaction. The entropic curvature evolves as dR/dt = Φ_int. Collapse occurs at R = ln 2, yielding the timescale

This predicts delayed collapse for weak measurements, rapid collapse for strong measurements, and suppression in Zeno-type regimes.

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6. Quantization of the Entropy Field

OCI enforces a minimum excitation ΔS_min = ln 2, implying that the entropy field is fundamentally quantized. Its quanta represent irreducible units of distinguishability and irreversibility. Canonical structure follows naturally, with ln 2 playing the role of the primitive action scale in entropic space.

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7. Implications

Because entropic curvature sources physical geometry, quantized entropy forbids arbitrarily fine-grained spacetime curvature and infinite information density. Spacetime must therefore be finite and emergent, bounded by total entropic content. Singularities are replaced by entropy saturation surfaces.

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8. Conclusion

From a single invariant—ln 2 as the minimal curvature of distinguishability—we have derived a no-go theorem forbidding reversible measurement, obtained wave-function collapse as a geometric necessity, fixed a collapse timescale, and implied entropy quantization and spacetime finiteness. The measurement problem is resolved not interpretively but structurally, by the geometry of entropy itself.

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