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Comment on Marek Czachor article entitled "On Relativity of Quantumness as Implied by Relativity of Arithmetic and Probability"

Krzysztof Sienicki, Mikołaj Sienicki

Abstract

Czachor's model of hierarchical arithmetics begins with a valid formal premise but fixes the key probability mapping g by importing the Born rule and Fubini-Study metric from standard quantum mechanics, where Born probabilities are Kolmogorov within a fixed measurement context. This g is then applied in a non-Newtonian hidden-variable setting, producing a hybrid framework whose agreement with quantum correlations is built in by design, not derived from new physics, and thus does not constitute a genuine counterexample to Bell's theorem

Comment on Marek Czachor article entitled "On Relativity of Quantumness as Implied by Relativity of Arithmetic and Probability"

Abstract

Czachor's model of hierarchical arithmetics begins with a valid formal premise but fixes the key probability mapping g by importing the Born rule and Fubini-Study metric from standard quantum mechanics, where Born probabilities are Kolmogorov within a fixed measurement context. This g is then applied in a non-Newtonian hidden-variable setting, producing a hybrid framework whose agreement with quantum correlations is built in by design, not derived from new physics, and thus does not constitute a genuine counterexample to Bell's theorem

Paper Structure

This paper contains 8 sections, 14 equations, 1 figure.

Table of Contents

  1. Step 1 — The purely formal setup
  2. Step 2 — Connecting hidden and observable levels
  3. Step 3 — Calibrating the observable layer with standard QM
  4. Step 4 — Choosing a specific link $g$
  5. Step 5 — About Bell’s theorem
  6. Step 6 — Where the frameworks get mixed
  7. Step 7 — What follows from this
  8. Conclusion

Figures (1)

  • Figure 1: Example mappings $g(p)$ satisfying Assumption 1 (complement preservation), continuity, and strict monotonicity. Czachor’s choice $g(p)=\sin^2\!(\frac{\pi p}{2})$ (solid) is one among many admissible forms, including the polynomial $3p^2-2p^3$ (dashed) and a distinct, symmetric sine-embedded variant $g_{\text{alt}}(p)=\sin^{2}\!(\frac{\pi}{4}[1+\sin(\pi(p-\frac{1}{2}))])$ (dotted). All three interpolate $(0,0)$ to $(1,1)$.