The Bridge Theorem: A Rigorous Mathematical Framework Connecting p-adic and Bruhat-Tits Geometries

In the paper's words

A rigorous mathematical framework establishing bridge connections between p-adic ultrametric spaces 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

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. Author: QNFO Research | Date: 2026-07-01 Status: Formal Proof Sketch v1.0 Part of: Radix → Ultrametrics → Page-Wootters → Wheeler-DeWitt → Bruhat-Tits Synthesis

   from Conditional Quantum States Under Global Constraints Necessarily Form Ultrametric Hierarchies

2. We prove that under a global constraint of the form $\hat{H}|\Psi\rangle = 0$ (the Wheeler-DeWitt equation), conditional quantum states produced by the Page-Wootters mechanism $|\psi(\tau)\rangle_R = \langle\tau|_C |\Psi\rangle_{CR}$ naturally organize into an ultrametric hierarchy.

   from Abstract

3. We define eigenstates $|\tau\rangle$ of $\hat{T}_C$ (or a suitable positive operator-valued measure): $$\hat{T}_C|\tau\rangle_C = \tau|\tau\rangle_C$$ Definition 1 (Conditional State).

   from 1.2 Page-Wootters Conditioning

4. Theorem 1 (Equivalence Quotient).

   from 2.3 The Equivalence Quotient is Always Ultrametric

What would falsify it

1. Let $\hat{T}_C$ be a clock observable on $\mathcal{H}_C$ with continuous spectrum or a sufficiently dense discrete spectrum.

   from 1.2 Page-Wootters Conditioning

2. This theorem establishes that if the underlying distance on $\mathcal{S}$ is ultrametric, then the quotient by identical states preserves this ultrametricity.

   from 2.3 The Equivalence Quotient is Always Ultrametric

3. The radix $p$ is a physical observable of the clock subsystem, determinable from the branching ratio of conditional state equivalence classes.

   from 4.1 From Overlap to Radix

Where the claim stops

1. The standard triangle inequality gives $c \leq a + b \lt c + b$ (since $a \lt c$), which is consistent with the standard triangle inequality and does not produce a contradiction.

   from 2.3 The Equivalence Quotient is Always Ultrametric

2. Now, the quotient by itself does not guarantee ultrametricity.

   from 2.3 The Equivalence Quotient is Always Ultrametric

3. Assumption A1 (Discrete Scale Invariance).

   from 3.2 Clock Hamiltonian Structure

Where to read

1. Conditional Quantum States Under Global Constraints Necessarily Form Ultrametric Hierarchies23 words · 1 min
2. Abstract108 words · 1 min
3. 1. Setup and Definitions232 words · 1 min
4. 2. Theorem I: Equivalence-Induced Ultrametricity (Unconditional)371 words · 2 min
5. 3. Theorem II: Global Constraint Induces Ultrametric Overlaps609 words · 3 min
6. 4. Theorem III: Hierarchy Depth Determines the Radix213 words · 1 min
7. 5. Theorem IV: Bruhat-Tits Buildings as Clock-Frame Geometry236 words · 1 min
8. 6. The Complete Theorem Statement244 words · 1 min
9. 7. Proof Strategy and Gaps208 words · 1 min
10. 8. Physical Consequences155 words · 1 min
11. Changelog9 words · 1 min