The p-Adic Temperley–Lieb Parameter: Valuation-Theoretic Constraints and an Adelic Accuracy Framework for Quantum Predictions

In the paper's words

The Temperley–Lieb (TL) algebra governs a broad class of exactly solvable quantum systems, from anyonic chains to braid-based teleportation circuits, through its scalar loop parameter δ. We investigate the proposal that a p-adic Temperley–Lieb parameter — a number-theoretic functional of δ — can predict quantum system behavior with accuracy of at least 95%. We construct the only mathematically canonical candidate adapted to the p-adic norm, δ_p = p + p⁻¹ ∈ ℚ_p (the analogue of δ = q + q⁻¹ with q the uniformizer), and analyze it with explicit valuation arithmetic.

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

Numbers it reports

1. We then derive an explicit accuracy budget showing that two decimal digits of Archimedean control guarantee accuracy ≥ 0.9937651 at the worst-case unit value, and — under an explicitly stated joint-control assumption — that three ternary digits of p-adic precision suffice for the 95% target.

   from Abstract

2. A proof of the adelic obstruction and an explicit accuracy budget relating p-adic precision to Archimedean prediction error, with the digit budgets required to reach the 95% target under stated assumptions (Sections 4–5).

   from 1. Introduction

3. Projection (conditional on Assumption J): under $|\varepsilon |_\infty \leq |\varepsilon |_{p} (\alpha = 1, C = 1, p = 3), k = 3$ ternary digits suffice for the 95% target at $r = 5$ and $r = 7$; the budget is 3–6 digits for $\alpha \in [1/2, 1]$.

   from 5. Results

4. We proved the adelic obstruction — p-adic precision cannot control Archimedean error — and derived an explicit accuracy budget: two decimal digits of Archimedean control guarantee accuracy ≥ 0.9937651 over the unit family, and, under the explicitly stated Assumption J, three to six ternary digits of p-adic precision suffice for the 95% target.

   from 7. Conclusion

5. We investigate the proposal that a p-adic Temperley–Lieb parameter — a number-theoretic functional of $\delta$ — can predict quantum system behavior with accuracy of at least 95%.

   from Abstract

6. The research idea under examination asserts that a "p-adic Temperley–Lieb parameter" predicts quantum system behavior with accuracy of at least 95%.

   from 1. Introduction

How it is done

1. We construct the only mathematically canonical candidate adapted to the p-adic norm, $\delta _{p} = p + p^{-1} \in \mathbb{Q}_{p}$ (the analogue of $\delta = q + q^{-1}$ with q the uniformizer), and analyze it with explicit valuation arithmetic.

   from Abstract

2. A proof of the adelic obstruction and an explicit accuracy budget relating p-adic precision to Archimedean prediction error, with the digit budgets required to reach the 95% target under stated assumptions (Sections 4–5).

   from 1. Introduction

3. This uniqueness theorem matters here for two reasons: it licenses evaluating any putative p-adic invariant on a finite generating set (making computation finite), and it identifies the precise scalars — Markov trace values — on which non-trivial p-adic data can live once the loop parameter itself proves p-adically trivial.

   from 2. Background and Related Work

4. 3.1 Definition of the parameter.

   from 3. Methods

What would falsify it

1. We investigate the proposal that a p-adic Temperley–Lieb parameter — a number-theoretic functional of $\delta$ — can predict quantum system behavior with accuracy of at least 95%.

   from Abstract

2. The strong claim of universal $\geq 95\%$ predictive accuracy is not supported; what survives is a precise, falsifiable algebraic program with a conditional accuracy budget.

   from Abstract

3. The research idea under examination asserts that a "p-adic Temperley–Lieb parameter" predicts quantum system behavior with accuracy of at least 95%.

   from 1. Introduction

Where the claim stops

1. Second, the choice $\delta _{p} = p + p^{-1}$ is canonical but not unique; alternatives (e.g., $\delta = q + q^{-1}$ with q a p-adic root of unity, which exists in $\mathbb{Q}_{p}$ only when $p \equiv 1 \bmod$ order) would change every computation.

   from 6. Discussion

2. (iv) Show that Assumption J fails empirically for TL-type systems, voiding the conditional digit budget.

   from 6. Discussion

3. We proved the adelic obstruction — p-adic precision cannot control Archimedean error — and derived an explicit accuracy budget: two decimal digits of Archimedean control guarantee accuracy ≥ 0.9937651 over the unit family, and, under the explicitly stated Assumption J, three to six ternary digits of p-adic precision suffice for the 95% target.

   from 7. Conclusion

Where to read

1. Abstract278 words · 1 min
2. 1. Introduction314 words · 1 min
3. 2. Background and Related Work640 words · 3 min
4. 3. Methods360 words · 2 min
5. 4. Analysis1,324 words · 6 min
6. 5. Results373 words · 2 min
7. 6. Discussion530 words · 2 min
8. 7. Conclusion177 words · 1 min
9. Appendix A. Divergence report231 words · 1 min