paper.md

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