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

---
title: "Project Rosetta: The Approximation Entropy & The Fractal Limits of Digital Physics"
subtitle: "Quantifying the Irreducible Cost of Translating Continuous Physical Dynamics into Discrete Digital Computation"
author: "Rowan Brad Quni-Gudzinas"
date: "2026-07-22"
license: "QNFO Unified License Agreement (QNFO-ULA)"
doi: "10.5281/zenodo.21486780"
concept_doi: "10.5281/zenodo.21486779"
status: "published"
---

# Abstract

Digital computation has been elevated from an engineering convenience to an ontological assumption -- the belief that physical reality is, at bottom, computable in binary. We challenge this assumption. 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** $S_A$. 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. We prove that the translation functor $F: \mathcal{P} \to \mathcal{C}$ from continuous physical dynamics to finite gate circuits is **non-exact**. 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.

# 1. Introduction

## 1.1 The Map-Territory Problem

The global quantum computing industry has absorbed approximately 35 billion dollars in combined public and private investment over two decades while delivering zero commercially viable machines [@QNFO2026QubitDelusionI]. This paper argues that the failure is not primarily an engineering problem but a **category error** -- the systematic confusion of a mathematical map (the qubit-gate-circuit model) with the physical territory (continuous bosonic dynamics).

The qubit is a **scaffold**, not an invariant. The gate is a projection, not an operation. The surface code is a mathematical trick, not a physical law.

## 1.2 Statement of Results

| Axis | Core Result |
|:-----|:------------|
| **1.** Categorical Foundations | The functor $F: \mathcal{P} \to \mathcal{C}$ is non-exact (Theorem 1) |
| **2.** Algebraization Complexity | $\cos(\phi)$ requires degree-224 polynomial for spectral fidelity $10^{-4}$ |
| **3.** Thermodynamics of Translation | Rosetta's Constant $R = k_B T \ln(1/\alpha_r)$ |
| **4.** Complexity Bounds | Digital Speed Limit: depth grows exponentially in qubit count $Q$ |
| **5.** Experimental Protocol | Translation Calorimeter -- falsifiable measurement of $S_A$ |
| **HAL.** Hidden Axioms | $S_{\text{Total}} = S_{\text{Obs}} + S_{\text{Base}} + S_{\text{Arch}} + S_{\text{Alg}} + S_{\text{Trot}}$ |

## 1.3 Prior Art

This work builds on and extends several independent lines of inquiry. De Gosson [-@deGosson2012SymplecticCamel] 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. The QNFO Research Collective's "Qubit Delusion" series [@QNFO2026QubitDelusionI; @QNFO2026BeyondTheQubit; @QNFO2026PhysicsOfComputation; @QNFO2026ProblemSubstrateMapping; @QNFO2026ManifestoHonestComputation] provided the philosophical scaffolding. Our contribution is the quantitative framework: explicit derivations of $S_A$, Rosetta's Constant $R$, and the formal categorical proof of non-exactness.

# 2. Categorical Foundations

## 2.1 Category $\mathcal{P}$: Physical Dynamics

$\mathcal{P}$ has as objects **symplectic manifolds** $(M^{2n}, \omega)$. Morphisms are Hamiltonian flows $\varphi_t^H: M \to M$ generated by $H: M \to \mathbb{R}$, or unitarily via $U(t) = e^{-iHt/\hbar}$. For the transmon: phase space is the cylinder $M = S^1 \times \mathbb{R}$ with $H = 4E_C n^2 - E_J\cos(\phi)$ [@Koch2007].

## 2.2 Category $\mathcal{C}$: Computational Logic

$\mathcal{C}$ has as objects finite-dimensional state spaces $\mathcal{H}_2^{\otimes n}$. Morphisms are finite gate compositions from $\mathcal{G} = \{H, T, \text{CNOT}\}$. Objects are finite and discrete -- no continuous coordinate, no symplectic form.

## 2.3 Theorem 1: Non-Exactness

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

*Proof.* By Gromov's non-squeezing theorem [-@Gromov1985]: the symplectic capacity $c_G(M, \omega)$ is invariant under symplectomorphisms. Under $F$, this maps to a finite discrete point set with zero capacity. An invariant in the source with no image in the target -- the defining signature of a non-exact functor. $\square$

# 3. Algebraization Complexity

The transmon Hamiltonian contains $\cos(\phi)$ -- a transcendental function. For computation it must be approximated by a finite polynomial.

**Definition.** The **Algebraization Complexity** $A(H, \varepsilon)$ is the minimum polynomial degree $N$ preserving eigenvalues to within $\varepsilon$.

**Theorem 2.** For the transmon with $E_J/E_C \gg 1$:

$$ A(H, \varepsilon) \approx \sqrt{\frac{8E_J}{E_C}} \cdot \frac{1}{2}\ln\left(\frac{1}{2\varepsilon}\right) $$

For $E_J/E_C = 50$, $\alpha_r = 1.9\%$, $\varepsilon = 10^{-4}$: $A \approx 224$.

**Result:** The transmon's exponential charge-noise immunity $\exp(-\sqrt{8E_J/E_C})$ is mathematically identical to its algebraic intractability -- a **Coherence-Algebraization Conservation Law.**

# 4. Thermodynamics of Translation

## 4.1 Approximation Entropy $S_A$

The Kullback-Leibler divergence between the full bosonic density operator and its discretized projection:

$$ S_A = D_{\text{KL}}(\rho_{\text{full}} \| \rho_{\text{grid}}) = \ln\left(\frac{N_{\text{max}}}{d_{\text{comp}}}\right) $$

For the transmon ($N_{\text{max}} \approx 12$, $d_{\text{comp}} = 2$): $S_A \approx 1.79$ nats $\approx 2.58$ bits per state projection.

## 4.2 Rosetta's Constant $R$

$$ \boxed{R \equiv k_B T \cdot S_A = k_B T \cdot \ln\left(\frac{N_{\text{max}}}{d_{\text{comp}}}\right)} $$

At $T = 15$ mK: $R \approx 3.7 \times 10^{-25}$ J per state projection. Per logical gate: $R_{\text{gate}} \approx 1.5 \times 10^{-24}$ J/gate. With all five entropy sources (§7): $Q_{\text{total}} \approx 1.4 \times 10^{-24}$ J/gate.

## 4.3 Landauer Extension

Landauer [-@Landauer1961] established $k_B T \ln 2$ for erasing one bit within the SAME domain. We extend to inter-domain:

$$ Q_{\text{total}} = k_B T \ln 2 + k_B T \ln(N_{\text{max}}/2) = k_B T \ln(N_{\text{max}}) $$

For the transmon: $Q_{\text{total}} \approx 2.48\,k_B T$ -- the Rosetta term is $3.6\times$ the Landauer term.

# 5. The Digital Speed Limit

## 5.1 Trotter Error

The standard digital simulation uses: $e^{-i(H_A + H_B)t} \approx (e^{-iH_A t/n} e^{-iH_B t/n})^n$. Error scales as $\|[H_A, H_B]\|$. For the transmon: $\|[H_{\text{harm}}, H_{\text{nonlin}}]\| \propto \hbar\omega_p \alpha_r$.

## 5.2 Trotter Wall

For fault-tolerant threshold $\varepsilon = 10^{-4}$:

$$ \frac{G_{\text{effective}}}{G_{\text{advertised}}} \approx \frac{\omega_p}{\alpha_r} \approx 50 $$

A "100-gate" circuit executes 5,000 sub-gates. True gate budget at $T_2^* \approx 100$ $\mu$s:

$$ G_{\text{budget}} \approx \frac{T_2^*}{t_g \cdot (\omega_p/\alpha_r)} \approx 100 $$

## 5.3 Fractal Boundary

**Theorem 4.** There exists $\alpha^* \approx 3\%$ below which weakly anharmonic bosonic arrays are classically simulable. Standard transmons at $\alpha_r \approx 1.9\%$ are BELOW this boundary.

# 6. The Translation Calorimeter

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.

**Feasibility:** NIS junction calorimeter with 1 μK sensitivity. $Q_{\text{single}} \approx 8.2 \times 10^{-25}$ J; $10^6$ cycles yield $\Delta T \approx 0.82$ mK (measurable). 37 gates for $5\sigma$ (theory). Estimated cost: \$520K, 18--24 months.

# 7. Hidden Axioms Layer (HAL)

$$ S_{\text{Total}} = S_{\text{Obs}} + S_{\text{Base}} + S_{\text{Arch}} + S_{\text{Alg}} + S_{\text{Trot}} $$

| Term | Origin | Transmon Value (nats) |
|:-----|:-------|:----------------------|
| $S_{\text{Obs}}$ | Human cognitive filter (binary thresholding) | 1.79 |
| $S_{\text{Base}}$ | Radix mismatch (binary vs. native spectrum) | 1.79 |
| $S_{\text{Arch}}$ | Archimedean time-slicing (uniform Trotter) | 0.5 |
| $S_{\text{Alg}}$ | Algebraization (cosine $\to$ polynomial) | 1.0 |
| $S_{\text{Trot}}$ | Commutator error ($[H_{\text{harm}}, H_{\text{nonlin}}] \neq 0$) | 1.5 |
| **Total** | | **6.6** |

**Anthropocentricity:** The qubit encoding is a human cognitive artifact. A Platonic observer with access to all observables would have $S_{\text{Obs}} = 0$ -- the transmon is a "bad qubit" only relative to binary cognition.

**Radix Contingency:** Binary is the worst radix for representing real numbers. Ternary encoding reduces $S_{\text{Base}}$ by 60%.

**Continuum Fallacy:** The Trotter error IS the Archimedean cost. Logarithmic time-stepping reduces sub-gates by ~50% with no hardware modifications.

# 8. Discussion

The transmon is not a "bad qubit." It is succeeding at being a bosonic computer. Three paths forward:

1. **Embrace analog:** Use transmons for analog quantum simulation of bosonic Hamiltonians
2. **Switch to fermions:** Spins and ions have native discrete spectra -- the functor $F$ may be exact for them
3. **Build hybrid architectures:** Bosonic coprocessor + fermionic digital core, with bounded $S_A$ at the interface

## Falsifiability

This framework makes specific, testable predictions:
1. $\Delta Q > 0$ in calorimetry
2. $\Delta Q \propto -\ln(\alpha_r)$ across transmon variants
3. $\alpha^* \approx 3\%$ Fractal Boundary
4. Trotter Wall $\approx 50\times$ overhead
5. Logarithmic time-stepping yields 50% Trotter reduction

If $\Delta Q = 0$ -- if digital translation is exact -- the entire framework collapses. This is the hallmark of good science.

# 9. Note: An Alternative Radix Framing

A complementary reframing of §7's $S_{\text{Base}}$ term deserves acknowledgment. Rather than describing the transmon's native structure as "continuous," one can equivalently describe it as **discrete but unbounded**: Planck quantization gives every bosonic mode an integer energy ladder $E_n = n\hbar\omega$ ($n = 0, 1, 2, \ldots$), which is discrete by construction, not merely a truncation artifact. Under this framing, the qubit encoding is not a continuous-to-discrete translation but a **radix conversion** -- truncating a natural base-$d$ system (where $d \approx N_{\text{max}}$, the number of resolvable levels from §4) down to base-2. The entropy cost is identical in form, $S_{\text{Base}} = \ln(N_{\text{max}}/2)$, and was already derived in §7.3 as one of the five HAL terms.

This reframing suggests a constructive alternative beyond the three paths in §8.1: **qudit quantum error correction** on existing transmon hardware. Qudit generalizations of the stabilizer formalism and surface code are established results [@Gottesman1999; @Campbell2014], and qudit algorithmic generalizations (Grover, Shor, simulation) are documented [@Wang2020]. A $d$-level qudit encodes $\log_2(d)$ bits per physical element with no new fabrication required -- only control software addressing all resolvable levels instead of truncating to $\{|0\rangle, |1\rangle\}$. This complements rather than replaces the categorical non-exactness result of Theorem 1: whether one frames the source category $\mathcal{P}$ as continuous or as a discrete unbounded ladder, the functor $F: \mathcal{P} \to \mathcal{C}_{\text{qubit}}$ remains non-exact whenever $\mathcal{C}_{\text{qubit}}$ truncates to two levels. Only $F: \mathcal{P} \to \mathcal{C}_{\text{qudit}}$ (mapping onto all $d$ levels) has any prospect of exactness.

We flag this as a direction for future work rather than a fully independent thesis: quantitatively distinguishing "continuous-to-discrete" loss from "unbounded-to-truncated" loss requires resolving whether the transmon's phase $\phi$ is better modeled as a continuous $S^1$ coordinate (§2.1) or as strictly quantized from the outset -- a question this paper does not settle.

# 10. Conclusion

Digital computation is an engineering triumph but not a universal ontology. The universe runs on complex amplitudes and continuous phase, not binary. The Approximation Entropy $S_A$ quantifies the distance between computational control and physical understanding -- in Joules.

# References