Radix to Ultrametrics to Page-Wootters to Wheeler-DeWitt to Bruhat-Tits: A Convergent Synthesis

In the paper's words

Convergent synthesis bridging discrete p-adic ultrametrics with continuous quantum geometry via Page-Wootters, Wheeler-DeWitt, and Bruhat-Tits buildings.

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

Numbers it reports

1. 132,720 triplets, 33.0% violation rate.

   from 6.5.1 Ultrametricity of Conditional State Distances [CODE-EXECUTED]

2. Three mechanisms: (1) p=2 for binary QM (spin-1/2, qubits) [established]; $(2) D = p^{k}$ clock dimension factorization (Aniello & Guglielmi 2025); $(3) B(G,Q_{p})$ determined by G with p as free parameter [speculative].

   from 6.5.2 Physical Radix Selection [CODE-EXECUTED]

3. Simulated Planck data: 3% p-adic amplitude = 0.03-sigma (undetectable).

   from 6.5.5 Experimental Signatures & CMB Detectability [CODE-EXECUTED]

4. Signal must exceed 5% for Planck detection.

   from 6.5.5 Experimental Signatures & CMB Detectability [CODE-EXECUTED]

5. Confidence assessment: $\sim$70% of this synthesis rests on established mathematics and physics. $\sim$30% is conjectural.

   from 8. Verdict

6. D=4 Fourier clock states with N=16 produced 224 violations in 560 triplets (40%).

   from Retraction: D=4/N=8 Pass

How it is done

1. Ostrowski's theorem [established]: Every non-trivial absolute value on $\mathbb{Q}$ is equivalent to either the standard Archimedean absolute value $|\cdot|_\infty$ or a $p$-adic absolute value $|\cdot|_p$ for some prime $p$.

   from 2.1 Radix and the $p$-adic Numbers

2. The clock spectrum must be gapped for a well-defined time observable (Pauli's theorem).

   from 4.2 The Radix Encodes the Clock Spectrum

3. Domain Key Result Representative References $p$-adic numbers Ostrowski's theorem; ultrametricity Mathematical canon Bruhat-Tits buildings Simplicial geometry of $p$-adic groups Bruhat & Tits (1972) Wheeler-DeWitt Timelessness from diffeomorphism invariance DeWitt (1967); Isham (1992) Page-Wootters Emergent time from correlations Page & Wootters (1983); Moreva et al.

   from 5.1 What is Established

4. THEOREM ATTEMPT (FALSIFIED): The claim that D=4 Fourier clock states produce ultrametric correlation distances for all sampling densities is FALSE.

   from 6.5.7 Formal Analysis: D=4 Special Case Theorem [CODE-EXECUTED]

What would falsify it

1. The prime $p$ — formerly a purely mathematical parameter — becomes a physical observable of the clock subsystem [my conjecture] .

   from 4.2 The Radix Encodes the Clock Spectrum

2. The clock spectrum must be gapped for a well-defined time observable (Pauli's theorem).

   from 4.2 The Radix Encodes the Clock Spectrum

3. Claim Status Conditional state distance is ultrametric [my conjecture] — not proven, but structurally motivated Radix $p$ is a physical clock observable [my conjecture] — no experimental evidence Bruhat-Tits buildings encode clock frame changes [my conjecture] — interpretational, not yet formalized CMB correlations exhibit $p$-adic scaling [speculative] — no data analysis performed

   from 5.2 What is Speculative or Conjectural

Where the claim stops

1. The MERA network naturally lives on a tree that approximates a Bruhat-Tits building in the continuum limit — a connection explored by Bhattacharyya, Hung et al.

   from 4.1 Correlation Distance as Ultrametric

2. Claim Status Conditional state distance is ultrametric [my conjecture] — not proven, but structurally motivated Radix $p$ is a physical clock observable [my conjecture] — no experimental evidence Bruhat-Tits buildings encode clock frame changes [my conjecture] — interpretational, not yet formalized CMB correlations exhibit $p$-adic scaling [speculative] — no data analysis performed

   from 5.2 What is Speculative or Conjectural

3. It has fewer violations but still fails.

   from 6.5.1 Ultrametricity of Conditional State Distances [CODE-EXECUTED]

Where to read

1. Abstract (50 words)52 words · 1 min
2. 1. Introduction: The Chain150 words · 1 min
3. 2. Mathematical Primitives394 words · 2 min
4. 3. The Problem of Time in Quantum Gravity295 words · 1 min
5. 4. The Bridge: Why Ultrametrics?642 words · 3 min
6. 5. Evidence Summary: Literature Landscape228 words · 1 min
7. 6. Testable Predictions211 words · 1 min
8. 6.5 Computational Verification (2026-07-01 -- Corrected 2026-07-01)374 words · 2 min
9. 7. Open Problems193 words · 1 min
10. 8. Verdict320 words · 1 min
11. Changelog9 words · 1 min
12. Correction Addendum — v1.1 (2026-07-01)344 words · 1 min