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

In plain language

The paper set out to see whether turning smooth physical processes into binary computer steps always costs energy. It found that any such translation creates a minimum heat loss called Approximation Entropy, which for a superconducting transmon at 15 mK equals about 8.2 × 10⁻²⁵ joules per logical step. It also predicts a measurable heat change ΔQ that grows with the negative log of the error rate α_r, and that the best‑tuned transmon variant has α*≈3 %. Using three‑state (ternary) encoding cuts the base entropy by roughly 60 %, and a logarithmic time‑stepping method halves the number of sub‑gates needed without changing hardware. Overall, the total heat per gate from all five entropy sources is about 1.4 × 10⁻²⁴ J per gate, and the fractal‑boundary overhead is roughly 50 times larger than a simple binary layout. This matters because it tells engineers exactly how much extra energy they must budget when designing quantum computers, and shows that smarter encoding and timing tricks can recover a large fraction of that cost.

Restated for a non-expert reader; the original wording follows. Still uses some technical language.

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