Non-Computable Reals as Projection Artifacts: A Reconciled Framework for Tree-to-Manifold Construction of the Continuum

In the paper's words

We develop a formal framework for the conjecture that the continuum is constructed rather than primitive: natural numbers, rationals, and computable reals arise from a single distinction operator via enclosure operations, while the full real line emerges only through a lossy projection — the Monna-map — of a discrete, non-Archimedean Bruhat–Tits tree onto a smooth Archimedean manifold. The central claim under analysis is that non-computable reals are not discovered but created by this projection: they are artifacts with no discrete counterpart, analogous to upsampling artifacts in generative signal and image synthesis. We derive with explicit arithmetic that the computable reals form a set of Lebesgue measure zero, so the artifact set has full measure in [0,1], reinterpreting the measure-theoretic dominance of non-computables as a projection effect. [...]

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

Numbers it reports

1. Under the $(p+1)$-regular presentation (Definition 3.2, second convention; Appendix A, D1), the number of rays of length $n$ from a fixed root-adjacent edge is $p\,(p+1)^{n-1}$ for $n\ge 1$.

   from 4.5 Regular-tree convention counts

2. Check at $n=20$: $(1+400)/2^{20}=401/1{,}048{,}576=0.00038242340087890625\approx 3.82423\times 10^{-4}$, an upper bound consistent with the exact $1.23024\times 10^{-4}$ of Section 4.2.

   from 4.4 Exponential decay of the definable fraction

3. The definable fraction $f_{n}\le (1+n^{2})/2^{n}$ decays exponentially in $n$; at $n=20$ the bound is $\approx 3.82423\times 10^{-4}$ (Section 4.4).

   from 5. Results

How it is done

1. We derive with explicit arithmetic that the computable reals form a set of Lebesgue measure zero, so the artifact set has full measure in $[0,1]$, reinterpreting the measure-theoretic dominance of non-computables as a projection effect.

   from Abstract

2. The supplied summary of [8] does not state specific range theorems we could import, so we use it as a structural analogy only.

   from 2. Background and Related Work

3. Formally, fix a two-value form $V_{2}=\{0,1\}$ and let the distinction operator $\delta$ be a map that, applied to an unmarked space, produces a marked and an unmarked region.

   from 3.1 The distinction operator and enclosure hierarchy

4. Finite-depth trees of depths $d\in\{4,8,16,32\}$ with $p\in\{2,3,5\}$ are to be enumerated; each depth-$d$ ray is mapped to the rational midpoint of its shadow interval; the resulting finite decimal strings are compared to decimal expansions of benchmark computable reals.

   from 3.5 Simulation protocol (proposed, not executed)

What would falsify it

1. We state conjectures, falsification conditions, a proposed simulation protocol, and reconcile divergent conventions among independent formulations of the framework.

   from Abstract

2. Our contributions are: (i) precise definitions of the distinction operator, the enclosure hierarchy, and the Monna-map projection (Section 3); (ii) explicit derivations, with all arithmetic shown, that the computable reals have Lebesgue measure zero, that the artifact set has full measure, that the density of finitely definable rational shadows among tree paths decays exponentially in depth, and exact finite-depth tree counts under two stated conventions (Section 4); (iii) results and labeled projections (Section 5); and (iv) limitations, failure modes, falsification conditions, and a divergence report reconciling three independent formulations (Sections 6, Appendices A–B).

   from 1. Introduction

3. A three-axis framework extends this with Depth (computable reals), Breadth (non-computable reals, described as physically vacuous), and Valuation ($p$-adic completions as information carriers), together with falsifiable predications for quantum error correction and gauge structure [11]; in our construction the Bruhat–Tits tree is the Valuation axis made geometric, and the Monna-map is the passage from Valuation to Depth/Breadth.

   from 2. Background and Related Work

Where the claim stops

1. If non-computable reals admitted limit-definable tree paths (e.g., via oracles or non-constructive choice), C2 would fail and artifacts would be a misnomer.

   from 6. Discussion

2. We quantify the gap between finite-depth tree approximations and their Archimedean shadows: at depth $n=20$, exactly definable rational shadows number $129$ against $2^{20}=1{,}048{,}576$ paths, a definable fraction of $\approx 1.23024\times 10^{-4}$; programs of length at most $100$ number $2^{101}-1=2{,}535{,}301{,}200{,}456{,}458{,}802{,}993{,}406{,}410{,}751$.

   from Abstract

3. The supplied summary of [8] does not state specific range theorems we could import, so we use it as a structural analogy only.

   from 2. Background and Related Work

Where to read

1. Abstract184 words · 1 min
2. 1. Introduction405 words · 2 min
3. 2. Background and Related Work924 words · 4 min
4. 3. Methods682 words · 3 min
5. 4. Analysis569 words · 2 min
6. 5. Results177 words · 1 min
7. 6. Discussion285 words · 1 min
8. 7. Conclusion77 words · 1 min
9. Appendix A. Divergence report145 words · 1 min
10. Appendix B. Claim attribution92 words · 1 min