• Paper 109 — Erdős-Straus Conjecture as an S-Category Problem: A Ricci-Flow + Fujimoto Infinity Algebra Attack Angle
  • Abstract
  • 1. Background
    • 1.1 Erdős-Straus
    • 1.2 Ricci-flow three-category taxonomy (Paper 108)
    • 1.3 Fujimoto Infinity Algebra (STEP 843)
  • 2. Method
    • 2.1 Erdős-Straus partition graph construction
    • 2.2 Ricci flow parameters
    • 2.3 FIA embedding
    • 2.4 Degenerate limits (the novel part)
  • 3. Results
    • 3.1 Category distribution over n ∈ [2, 1000] (STEP 849)
    • 3.2 Mod-M analysis
    • 3.3 Comparison with Andrica (STEP 847)
  • 4. Discussion
    • 4.1 Why Category S suggests tractability
    • 4.2 The FIA attack angle
    • 4.3 What this paper does not claim
  • 5. Open Questions
  • 6. Reproducibility
  • 7. References
  • 8. Acknowledgements

Canonical DOI: https://doi.org/10.5281/zenodo.19616654
Author: Nobuki Fujimoto (ORCID 0009-0004-6019-9258)
License: CC-BY-4.0

§Paper 109 — Erdős-Straus Conjecture as an S-Category Problem: A Ricci-Flow + Fujimoto Infinity Algebra Attack Angle

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] (i.e. those for which the partition graph has at least 3 edges with a ≤ 60), 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] (STEP 849)

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.