The Idempotent Core: Quantity as Broken Distinction and the Hidden Assumptions of Arithmetic and Algebra

In the paper's words

A critique-first paper on arithmetic's hidden premise: 1+1=2 presupposes two distinguishable marks; idempotence (AA=A) is the primitive law; quantity is created distinction, not repetition. Reconstruction program (Z from ! modality) labeled conjecture with falsifiability conditions.

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. The critique, stated precisely: addition is a theorem about distinguishable marks, not a primitive truth about repetition. [my conjecture] This claim is disconfirmed if a formulation of the natural numbers is produced in which the successor of a numeral is the same mark as its predecessor — that is, if repetition alone generates counting without any new distinction.

   from 3. The Hidden Premise of Addition

2. The evidential weight of the reconstruction is therefore zero until the proof exists; the critique of Section 3, by contrast, does not depend on the reconstruction and stands on its own.

   from 5.3 The integers by group completion

3. The Bruhat–Tits tree provides the geometric model: the tree of balls, in which depth along a branch corresponds to divisibility by higher powers of $p$. [established — Ostrowski 1916; Bruhat–Tits; see 6] Ostrowski's theorem classifies the completions of the rationals: exactly one Archimedean completion and one non-Archimedean completion for each prime.

   from 6. Valuation as Depth

4. The strategic position of the paper is therefore: the critique stands now; the derivation is a program with a clear proof obligation.

   from 7. Discussion and Limits

What would falsify it

1. Claims marked [my conjecture] carry falsifiability conditions and are not presented as established results.

   from Abstract

2. Section 5 outlines the reconstruction program — the route from the idempotent mark to the ring of integers through linear logic — and labels its central conjectures with their falsifiability conditions.

   from 1. Introduction

3. The critique, stated precisely: addition is a theorem about distinguishable marks, not a primitive truth about repetition. [my conjecture] This claim is disconfirmed if a formulation of the natural numbers is produced in which the successor of a numeral is the same mark as its predecessor — that is, if repetition alone generates counting without any new distinction.

   from 3. The Hidden Premise of Addition

Where the claim stops

1. The critique does not depend on the reconstruction program.

   from 7. Discussion and Limits

2. Until they are proven, they carry zero evidential weight as derivations: the framework was built to produce the integers, so producing them is not yet evidence.

   from 7. Discussion and Limits

3. The Peano axioms define the natural numbers by a successor operation, but they do not explain what makes one number distinct from another.

   from 1. Introduction

Where to read

1. Abstract203 words · 1 min
2. 1. Introduction281 words · 1 min
3. 2. The Calculus of Indications as Ground154 words · 1 min
4. 3. The Hidden Premise of Addition368 words · 2 min
5. 4. Multiplication as Branching140 words · 1 min
6. 5. The Reconstruction Program: From the Idempotent Mark to the Integers692 words · 3 min
7. 6. Valuation as Depth157 words · 1 min
8. 7. Discussion and Limits205 words · 1 min
9. 8. Conclusion104 words · 1 min
10. Declarations130 words · 1 min