Project Rosetta: The Approximation Entropy & The Fractal Limits of Digital Physics — v2.0

In the paper's words

V2.0 / Rosetta v4.0 with Planck-Radix Correction + Qudit Manifesto. Corrects V1.0: transmon spectrum IS discrete (Planck quantization), binary truncation is a radix error not physics error. Transmon is a qudit (d~12), not a qubit.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. This framework makes specific, testable predictions: $\Delta Q \gt 0$ in calorimetry $\Delta Q \propto -\ln(\alpha_r)$ across transmon variants $\alpha^* \approx 3\%$ Fractal Boundary Trotter Wall $\approx 50\times$ overhead Logarithmic time-stepping yields 50% Trotter reduction If $\Delta Q = 0$ -- if digital translation is exact -- the entire framework collapses.

   from Falsifiability

2. For the superconducting transmon qubit, we derive $S_A \approx k_B T \ln(1/\alpha_r)$ and define Rosetta's Constant $R = k_B T \ln(1/\alpha_r) \approx 8.2 \times 10^{-25}$ Joules per logical translation at 15 mK.

   from Abstract

3. Ternary encoding reduces $S_{\text{Base}}$ by 60%.

   from 7. Hidden Axioms Layer (HAL)

4. Logarithmic time-stepping reduces sub-gates by ~50% with no hardware modifications.

   from 7. Hidden Axioms Layer (HAL)

5. Algebraization Complexity $\cos(\phi)$ requires degree-224 polynomial for spectral fidelity $10^{-4}$ 3.

   from 1.2 Statement of Results

6. With all five entropy sources (§7): $Q_{\text{total}} \approx 1.4 \times 10^{-24}$ J/gate.

   from 4.2 Rosetta's Constant $R$

How it is done

1. Through a systematic metamathematical analysis connecting symplectic geometry, transcendental function theory, quantum thermodynamics, and complexity theory, we prove that translating continuous bosonic dynamics into discrete digital logic carries an irreducible thermodynamic cost -- Approximation Entropy $S_A$.

   from Abstract

2. Categorical Foundations The functor $F: \mathcal{P} \to \mathcal{C}$ is non-exact (Theorem 1) 2.

   from 1.2 Statement of Results

3. De Gosson [-] first framed the conflict between geometry and algebra in quantum information using Gromov's non-squeezing theorem, but stopped at the existence argument without quantification.

   from 1.3 Prior Art

4. Theorem 1. $F: \mathcal{P} \to \mathcal{C}$ is not exact -- there exist invariants preserved in $\mathcal{P}$ with no image in $\mathcal{C}$.

   from 2.3 Theorem 1: Non-Exactness

What would falsify it

1. We identify the Hidden Axioms Layer -- Anthropocentricity, Radix Contingency, and the Archimedean Continuum Fallacy -- and propose a falsifiable experimental protocol: the Translation Calorimeter , a cryogenic measurement of the differential heat between analog evolution and digital gate translation on a single transmon.

   from Abstract

2. Experimental Protocol Translation Calorimeter -- falsifiable measurement of $S_A$ HAL.

   from 1.2 Statement of Results

3. Two experiments on the same transmon at the same temperature: Experiment Method Predicted Heat A (Analog) Natural evolution under $H_0$ $Q_A \approx 0$ B (Digital) Trotterized gate sequence $Q_B \approx k_B T \cdot S_A$ Falsification: If $\Delta Q = Q_B - Q_A = 0$ (within noise), the framework is refuted.

   from 6. The Translation Calorimeter

Where the claim stops

1. The Approximation Entropy $S_A$ quantifies the distance between computational control and physical understanding -- in Joules.

   from 10. Conclusion

2. Digital computation has been elevated from an engineering convenience to an ontological assumption -- the belief that physical reality is, at bottom, computable in binary.

   from Abstract

3. Through a systematic metamathematical analysis connecting symplectic geometry, transcendental function theory, quantum thermodynamics, and complexity theory, we prove that translating continuous bosonic dynamics into discrete digital logic carries an irreducible thermodynamic cost -- Approximation Entropy $S_A$.

   from Abstract

Where to read

1. Abstract162 words · 1 min
2. 1. Introduction229 words · 1 min
3. 2. Categorical Foundations149 words · 1 min
4. 3. Algebraization Complexity85 words · 1 min
5. 4. Thermodynamics of Translation140 words · 1 min
6. 5. The Digital Speed Limit88 words · 1 min
7. 6. The Translation Calorimeter85 words · 1 min
8. 7. Hidden Axioms Layer (HAL)125 words · 1 min
9. 8. Discussion121 words · 1 min
10. 9. Note: An Alternative Radix Framing293 words · 1 min
11. 10. Conclusion38 words · 1 min