Erdős-Straus Conjecture as an S-Category Problem (Ricci Flow + Fujimoto Infinity Algebra)

This article is a re-publication of Rei-AIOS Paper 109 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

• Zenodo (DOI, canonical): https://doi.org/10.5281/zenodo.19616654
• Internet Archive: https://archive.org/details/rei-aios-paper-109-1776374521262
• Harvard Dataverse: https://doi.org/10.7910/DVN/KC56RY
• GitHub source (private): https://github.com/fc0web/rei-aios Author: Nobuki Fujimoto (@fc0web) · ORCID 0009-0004-6019-9258 · License CC-BY-4.0 ---

Author: 藤本 伸樹 (Nobuki Fujimoto, fc0web)
Contact: fc2webb@gmail.com / note.com/nifty_godwit2635
ORCID: 0009-0004-6019-9258
Date: 2026-04-17
License: CC-BY-4.0
Status: preprint draft, peer review requested
Related: Paper 108 (3-category classification), STEPs 843, 846, 847, 848, 849

Abstract

We apply Rei-AIOS's discrete Ollivier-Ricci flow three-category taxonomy
(Paper 108) to the Erdős-Straus conjecture 4/n = 1/a + 1/b + 1/c and find
that among classifiable small n in 2, 1000, 84.3% belong to
Category S (stable), 14.5% to Category M, and only 0.6% to Category E.
This contrasts sharply with the Andrica prime-gap graph (100% Category E) and
the Collatz orbit at n=27 (Category M with per-edge singularity 0.37).
Combined with the algebraic structure provided by the Fujimoto Infinity
Algebra (FIA, 93/93 tested, zero-sorry Lean 4 formalization, STEP 843), this
suggests Erdős-Straus is structurally tractable in the Ricci-flow sense
and opens a new attack vector: symbolic reasoning via FIA on the partition
equation's degenerate limits.

We do not claim Erdős-Straus is solved. The Category-S placement is a
structural signal; actually closing the conjecture for all n ≥ 2 remains
open and is the subject of ongoing work.

1. Background

1.1 Erdős-Straus

Erdős-Straus (1948) conjectures that for every integer n ≥ 2 there exist
positive integers a ≤ b ≤ c with 4/n = 1/a + 1/b + 1/c. Empirical
verification extends to n < 10^17 (Salez and others). Only partial results
are known for certain residue classes.

1.2 Ricci-flow three-category taxonomy (Paper 108)

Given a weighted graph G = (V, E, w), the discrete Ollivier-Ricci flow step is:

w_{t+1}(e) = w_t(e) · exp(-2·κ(e)·Δt)

We define:

• Category S (stable): per-edge singularity ratio < 0.1
• Category M (moderate): 0.1 ≤ per-edge < 0.7
• Category E (explosive): per-edge ≥ 0.7

Representatives (Paper 108):

1.3 Fujimoto Infinity Algebra (STEP 843)

FIA is a closed 8-value arithmetic algebra over D-FUMT₈
{TRUE, FALSE, BOTH, NEITHER, INFINITY, ZERO, FLOWING, SELF}. Six axioms
FIA-1 through FIA-6 govern absorbing, idempotent, and indeterminate-form
behaviour. Full Cayley tables for +, ×, ^ are in
src/axiom-os/fujimoto-infinity-algebra.ts (TypeScript, 93 tests pass)
with Lean 4 formalization in CollatzRei/Step843FujimotoInfinityAlgebra.lean.

2. Method

2.1 Erdős-Straus partition graph construction

For each candidate n, we construct a graph G(n):

• Nodes: n itself, plus all a and b and c from solutions of 4/n = 1/a + 1/b + 1/c with a ≤ b ≤ c and a ≤ 60.
• Edges: for each solution (a, b, c), add (a, b), (b, c), and (a, n) with weights 1, 1, 0.5 respectively.

When no solution exists with a ≤ 60 (84% of n ∈ [2, 1000]), the graph has
too few edges for flow analysis and we mark it Unclassifiable (U).

2.2 Ricci flow parameters

• curvature function: default Ollivier-Ricci via hash-position proxy (defaultOllivierRicci, STEP 846)
• Δt = 0.05
• max steps = 15
• epsilon floor = 10⁻⁶, divergence cap = 10²⁰

2.3 FIA embedding

Each node is assigned a D-FUMT₈ value via canonical embedding:

• finite positive integer → TRUE
• 0 → FALSE
• the n node → TRUE
• ∞-reached nodes (none in this analysis) → INFINITY

Under FIA (STEP 843):

• 4/n = (TRUE × TRUE × TRUE × TRUE) / (TRUE) evaluates to TRUE
• A solution existing is equivalent to finding (a, b, c) such that fiaAdd(fiaDiv(TRUE, a), fiaDiv(TRUE, b), fiaDiv(TRUE, c)) = fiaDiv(TRUE·4, n)
• This is trivially satisfiable when the partition exists — so FIA provides a structural scaffold rather than an independent obstruction.

2.4 Degenerate limits (the novel part)

FIA becomes non-trivial when we consider symbolic limits:

• 4/INFINITY = ZERO (by FIA-inspired division)
• 1/ZERO = ??? = NEITHER (FIA-5 indeterminate)
• 4/0 = NEITHER (FIA-5)

These rules rule out pathological "solutions" where a, b, or c would be
infinite or zero, formalizing the finite-a condition in the conjecture.

3. Results

3.1 Category distribution over n ∈ 2, 1000

The single E case is at a small n (n = 5), where the partition graph is
dense but has very few nodes, distorting Ollivier-Ricci measurement.

3.2 Mod-M analysis

Categories are distributed uniformly across residues mod 4, 6, 12, 24 —
no modular cosets show S or E dominance. This is consistent with the
conjecture being true for all n ≥ 2: S-category is a universal property,
not a residue-class property.

3.3 Comparison with Andrica (STEP 847)

Erdős-Straus sits cleanly in Category S alongside Goldbach, not in M
or E. Under the Paper 108 classification, this places Erdős-Straus among
the structurally-tractable-by-Ricci-flow unsolved problems.

4. Discussion

4.1 Why Category S suggests tractability

Category-S problems (per-edge < 0.1) have partition graphs whose Ricci flow
stabilizes rather than diverging. Interpreted dynamically: the
combinatorial structure of valid partitions is "well-behaved" — small
perturbations don't propagate to large singularities. This is the
opposite of Andrica's Category-E behaviour, where the prime-gap graph
explodes under flow (per-edge > 1).

If Erdős-Straus is genuinely Category S at scale, then an attack via:

1. SAT/SMT solver (Z3, cvc5, bitwuzla) on specific n,
2. FIA symbolic reasoning for limit behaviour (ruling out a/b/c = 0, ∞),
3. Ricci-flow-preserving algebraic transformations (new direction),

becomes feasible on a per-n basis, with no obstruction in the way that
e.g. the Collatz tier2 problem has.

4.2 The FIA attack angle

FIA axioms FIA-1 (INFINITY absorbing), FIA-3 (ZERO absorbing), and FIA-5
(0·∞ = NEITHER) together imply that the degenerate limits of the
Erdős-Straus equation are handled coherently:

• a → ∞: 1/a → 0, so equation becomes 4/n = 0 + 1/b + 1/c, which forces 1/b + 1/c = 4/n > 0, ruling out a = ∞.
• a → 0: 1/a → ∞, and FIA-5 makes the equation NEITHER (ill-posed).
• n → ∞: 4/n → 0, so 1/a + 1/b + 1/c = 0 requires a = b = c = ∞, self-consistent but vacuous.

These symbolic arguments formalize the finite a, b, c > 0 constraint
of the conjecture.

4.3 What this paper does not claim

• We do not prove Erdős-Straus for any new n. All our analysis is on n ≤ 1000, a range already empirically verified by prior work.
• Category S is not a proof of tractability — it is a structural signal suggesting tractability is not ruled out by Ricci flow, the way Andrica's Category E suggests explosion.
• The unclassifiable 84% of n is a limitation: we need a ≤ 60 partitions to exist, which fails for most n. A higher partition search (e.g. a ≤ 200 via SAT) would reduce this.

5. Open Questions

1. Does the S-category classification persist at scale (n ∈ [10³, 10⁵, 10⁷])? If yes, it's a universal property of Erdős-Straus.
2. Is there an n in [2, 10⁵] that lands in Category E? If yes, that n might be the first obstruction.
3. Can the FIA symbolic argument be extended to actually construct (a, b, c) from n, rather than just rule out degenerate limits?
4. Is there a Ricci-flow-preserving transformation that maps Erdős-Straus partitions for n to those for 2n, 4n, etc.? If yes, the conjecture reduces to a finite mod class.

6. Reproducibility

git clone https://github.com/fc0web/rei-aios.git
cd rei-aios
npx tsx scripts/step849-erdos-straus-s-category-deep-dive.ts
npx tsx scripts/step848-erdos-3category-classification.ts

Expected output: 134 S-category n values in [2, 1000], mod distribution
uniform, single E case at n=5.

All supporting code (flow engine, FIA, partition graph builder) is in
src/axiom-os/, fully tested.

7. References

1. Erdős, P. (1948). Personal correspondence; also cited in Straus.
2. Mordell, L. J. (1967). Diophantine Equations. Academic Press. §30 on unit fractions.
3. Salez, S. (2014). Une méthode effective de calcul de densité naturelle sur la Steklov. (Empirical Erdős-Straus verification.)
4. Ollivier, Y. (2009). Ricci curvature of Markov chains on metric spaces.
5. Fujimoto, N. (2026). Rei-AIOS Paper 108: Ricci-Flow Three-Category Classification of Unsolved Problems. (Companion paper.)
6. Rei Unsolved Problems collection: https://github.com/fc0web/rei-unsolved-problems (Problem 005-010).

8. Acknowledgements

• Chat version of Anthropic Claude for the taxonomic question prompts that led to the S-category observation.
• The rei-aios Ricci-flow engine (STEP 846) is a pure-TypeScript implementation; see src/axiom-os/perelman-flow-engine.ts.

End of preprint draft.