Measurement as Place-Crossing: Completion-Selective Adelic Projection

In the paper's words

We propose a structural reframing of the quantum measurement problem in number-theoretic terms. By Ostrowski's theorem, the completions of Q consist of a single Archimedean place (R) and countably many p-adic places (Q_p); standard quantum mechanics is formulated exclusively over the Archimedean completion. We conjecture that a quantum state admits a coherent adelic extension across all places, and that measurement is a place-crossing event: coupling to a place-specific apparatus acts as a completion-selective projection that suppresses amplitudes on non-selected completions. [...]

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

Numbers it reports

1. Superconducting quantum processors with high-fidelity readout, such as the 105-qubit Tianyan-287 platform with readout fidelity $98.7\%$ [7], illustrate the precision with which measurement statistics are now verified; any adelic correction to Born statistics would be constrained far below this level, which we quantify as a bound in Section 5.

   from 2. Background and Related Work

2. With $\Gamma_v = \lambda_v^2 N_b \sigma_A^2$: $$ \Gamma_{\infty} = \lambda_{\infty}^2 N_b \sigma_A^2 = \lambda_{\infty}^2 \cdot 10^{23} \cdot 1 = 10^{23}\lambda_{\infty}^2, $$ $$ \Gamma_3 = \lambda_3^2 N_b \sigma_A^2 = \left(\frac{\lambda_{\infty}}{2}\right)^2 \cdot 10^{23} = \frac{10^{23}\lambda_{\infty}^2}{4} = 0.25 \times 10^{23}\lambda_{\infty}^2. $$ Hence the rate ratio: $$ \frac{\Gamma_{\infty}}{\Gamma_3} = \frac{10^{23}\lambda_{\infty}^2}{0.25 \times 10^{23}\lambda_{\infty}^2} = \frac{1}{0.25} = 4. $$ Derivation 2 (inter-place coherence decay).

   from 4. Analysis

3. The Tianyan-287 platform reports readout fidelity $98.7\%$ [7], i.e., outcome-error probability $\epsilon = 1 - 0.987 = 0.013 = 1.3 \times 10^{-2}$.

   from 4. Analysis

4. Requiring adelic deviations below the readout-error floor $\epsilon = 1.3 \times 10^{-2}$ of the 105-qubit Tianyan-287 platform [7] requires suppression time $t \ge 58.41$ model units ($\approx 3.65$ coherence times) in the worst-case equal superposition (Derivation 5).

   from 5. Results

5. If the model-time unit is identified with a microscopic collision timescale $\tau \sim 10^{-13}\,\mathrm{s}$ (a standard order-of-magnitude for molecular timescales; this identification is an assumption, not a derivation), then $T_{\times} \sim 16 \times 10^{-13}\,\mathrm{s} = 1.6 \times 10^{-12}\,\mathrm{s}$ and full suppression to the [7] bound occurs within $t \sim 58.41 \times 10^{-13}\,\mathrm{s} \approx 5.84 \times 10^{-12}\,\mathrm{s}$.

   from 5. Results

6. (This identification of model time with physical time is an assumption; the factor-$100$ uncertainty from $\lambda_{\infty}$ applies.) The uncertainty in this projection is dominated by the assumed $\tau$ and by the coupling $\lambda_{\infty} = 10^{-12}$, which enters $\Gamma$ quadratically: a factor-$10$ uncertainty in $\lambda_{\infty}$ gives a factor-$100$ uncertainty in $T_{\times}$.

   from 5. Results

How it is done

1. By Ostrowski's theorem, the completions of $\mathbb{Q}$ consist of a single Archimedean place ($\mathbb{R}$) and countably many $p$-adic places ($\mathbb{Q}_p$); standard quantum mechanics is formulated exclusively over the Archimedean completion.

   from Abstract

2. Ostrowski's theorem states that the only non-trivial absolute values on $\mathbb{Q}$, up to equivalence, are the usual real absolute value $|\cdot|_{\infty}$ and the $p$-adic absolute values $|\cdot|_p$ for primes $p$ [9], [10].

   from 1. Introduction

3. Ostrowski's theorem and the mutual singularity of completions are catalogued in the QNFO taxonomy [9], [10], which documents "completion failures" arising when Archimedean measure-theoretic intuitions are exported without justification, and frames the "completion problem" as a foundational question for physics.

   from 2. Background and Related Work

4. We restrict to two places, $v = \infty$ (real) and $v = 3$ (ternary), with a two-level system at each place.

   from 3.3 Solvable model: two-place system

What would falsify it

1. We discuss falsifiability, the relation to adelic path integrals and equidistribution theory, and the framework's central open assumption: the place-prior that selects the Archimedean completion.

   from Abstract

2. The model's physical status is conjectural; Section 6 states what would falsify it.

   from 1. Introduction

3. The observable statistics in the selected completion are computed from $\rho_{\infty}(t)$, and we verify they converge to the Born distribution $|\alpha|^2, |\beta|^2$ as inter-place coherence vanishes — the mechanism being the same local-incoherence resolution of [6], transplanted to the place index.

   from 3.3 Solvable model: two-place system

Where the claim stops

1. If the prior is not uniform, the Born-rule analogy fails or requires a weighted prior whose origin is unexplained — arguably relocating rather than solving the measurement problem.

   from 6. Discussion

2. The identification of model time with physical time is an unproven assumption (Section 5, projection).

   from 6. Discussion

3. (ii) If the product-formula coherence condition fails for interacting (non-free) systems, the adelic state space may not exist beyond free theories, collapsing the program to the free-case results of the Dragovich line [11].

   from 6. Discussion

Where to read

1. Abstract235 words · 1 min
2. 1. Introduction431 words · 2 min
3. 2. Background and Related Work546 words · 2 min
4. 3. Methods542 words · 2 min
5. 4. Analysis732 words · 3 min
6. 5. Results307 words · 1 min
7. 6. Discussion638 words · 3 min
8. 7. Conclusion128 words · 1 min
9. Appendix A. Divergence report309 words · 1 min
10. Appendix B. Claim attribution366 words · 2 min