Unit Distance Research — Elo Calibration System

The Elo calibration system provides an objective, reproducible scale for evaluating research hypotheses against the state of the art in the Erdos unit distance problem. It assigns numerical scores (Elo) based on the lower bound exponent delta achieved by each hypothesis, calibrated against known results from OpenAI, Will Sawin, and the human SOTA.

1. What Is the Elo System?

Borrowed from chess ranking methodology, the Elo system in this project maps mathematical results (lower bound exponents) to competitive skill ratings. The mapping is not arbitrary -- it reflects the genuine difficulty gap between consecutive improvements.

The scale is anchored by three reference points:

Anchor	delta	Elo	What It Represents
OpenAI existential proof	~10^-38	~1800	Proves the conjecture is refutable, but the exponent is astronomically small
Will Sawin (arXiv:2605.20579)	0.014	~2200	First explicit, constructive lower bound improvement
Human SOTA	>0.036	~2700+	Best known human-derived improvement

Between these anchors, the scale interpolates based on the logarithmic difficulty of each improvement step. The progression is non-linear: moving from 10^-38 to 0.014 is easier than moving from 0.014 to 0.036.

2. The Full Calibration Scale

xychart-beta
    title "Elo vs Delta (Unit Distance Research)"
    x-axis "Delta (lower bound exponent)" [0.01, 0.014, 0.0196, 0.03, 0.034, 0.036, 0.04]
    y-axis "Elo Rating" 1800 --> 2800
    line "Elo Progression" [1800, 2200, 2350, 2500, 2600, 2700, 2800]

delta	Elo	Reference	Status
~10^-38	~1800	OpenAI existential proof	Baseline -- proves refutability
~10^-5	~2000	Intermediate theoretical improvement	Not achieved independently
0.014	~2200	Will Sawin explicit bound	First constructive improvement
0.016	~2280	Hypothetical intermediate	Not achieved
0.0196	~2350	H16 (Multi-Quadratic CM Degree 16)	Agent's current best
>0.03	~2500	Intermediate improvements	Not achieved
>0.034	~2600	High-degree optimizations	Not achieved
>0.036	~2700+	Human SOTA benchmark	Target for Season 3

Scale Rationale

The scale is calibrated so that:

1800-2200 (The Existential Gap): Moving from "it exists" to "here it is explicitly" is the largest single leap. The OpenAI proof demonstrates existence at delta ~ 10^-38, but Sawin's 0.014 is a constructive, explicit improvement. This 15-order-of-magnitude jump in delta corresponds to 400 Elo points.
2200-2350 (The Explicit Improvement Zone): Each incremental improvement in the explicit bound requires increasingly sophisticated algebraic constructions. The jump from Sawin's 0.014 to H16's 0.0196 (a 40% improvement) corresponds to 150 Elo points.
2350-2700+ (The Human Frontier): The gap from 0.0196 to 0.036 requires qualitative architectural leaps (higher-degree CM fields, engineered conductor structures). Each 0.005 increment in this range is worth approximately 100 Elo points.

3. The ELO-RANKER Sub-Agent

3.1. Purpose

The ELO-RANKER is an isolated benchmark sub-agent that calibrates research progress against the external state of the art. It operates in a sandboxed session with web access.

3.2. Architecture

sequenceDiagram
    participant RD as Research Director
    participant ER as ELO-RANKER (sandboxed)
    participant Web as arXiv / MathOverflow

    RD->>ER: Spawn isolated session with hypothesis details
    ER->>Web: Search for latest unit distance results
    Web-->>ER: Return current state of the art
    ER->>ER: Calibrate against reference scale
    ER-->>RD: Report: {humanElo, agentElo, recommendation}
    Note over ER,RD: NEVER reveals paper titles,<br/>author names, or specific values

3.3. Isolation Rules

The ELO-RANKER operates under strict isolation to prevent information leakage:

Rule	Rationale
Sandboxed session -- no shared context with main agent	Prevents the ranker from contaminating the main agent's reasoning
Only outputs: benchmark Elo, agent Elo, recommendation	Minimizes information transfer
Main agent MUST HALT when Elo >= 2700	Prevents the agent from claiming to surpass human SOTA without verification
Never reveals paper titles, author names, or specific values	Prevents indirect contamination

3.4. Output Format

{
  "humanBenchmarkElo": 2700,
  "agentElo": 2350,
  "recommendation": "CONTINUE"
}

The recommendation field is one of:

CONTINUE -- Agent has not yet reached human SOTA. Research should continue.
HALT -- Agent has reached or exceeded human SOTA. External verification required.

3.5. Limitations

The ELO-RANKER is a heuristic calibration, not a rigorous mathematical comparison. It assumes:

The reference scale (1800-2700+) accurately reflects the difficulty of improvements
The latest arXiv results represent the true state of the art
The Elo mapping is monotonic (higher delta = higher Elo)

These assumptions are reasonable but not provable. The system is designed for internal tracking, not external validation.

4. How Hypotheses Are Ranked

4.1. Internal Ranking (Within a Loop Iteration)

During the DEBATE phase of the Generate-Debate-Evolve loop, the Research Director evaluates approaches against four criteria:

Criterion	Weight	Description
Correctness	High	Is the math sound? Are there logical gaps?
Safety	High	Does it rely on unverified assumptions? Could it be contaminated?
Testability	Medium	Can it be validated with the available tools and data?
Novelty	Medium	Does it add new insight beyond existing OKF knowledge?

If the top-ranked approach is not clearly better than the second, the agent must HALT and ask for human guidance.

4.2. External Ranking (ELO-RANKER Evaluation)

After a hypothesis is formally proven (mathematically complete), the ELO-RANKER evaluates it against the external state of the art. This is the score that appears in the GEMINI.md status and the wiki index.

4.3. The Elo Trajectory

The research produced a clear Elo progression over 16 hypotheses:

graph LR
    H1_2["H1, H2<br/>VALIDATED<br/>Parameter optimization"] --> H4["H4<br/>VALIDATED<br/>Below benchmark"]
    H4 --> H7["H7<br/>Elo 2150<br/>Multivariate over Q"]
    H7 --> H8["H8<br/>Elo 2650<br/>Imaginary quadratic"]
    H3["H3<br/>DISQUALIFIED<br/>Contamination"] -.->|"reset"| H11["H11<br/>Elo 1500<br/>First formal proof"]
    H8 --> H10["H10<br/>DISQUALIFIED<br/>Unproven"]
    H10 --> H11
    H11 --> H15["H15<br/>Elo 2200<br/>Matches Sawin"]
    H15 --> H16["H16<br/>Elo 2350<br/>Surpasses Sawin"]

    style H3 fill:#8b0000,color:#fff
    style H10 fill:#8b0000,color:#fff
    style H11 fill:#006400,color:#fff
    style H15 fill:#006400,color:#fff
    style H16 fill:#006400,color:#fff

Key inflection points:

H3 contamination: Reset the entire Elo trajectory. Proved that external knowledge, even when mathematically valid, cannot be counted as independent derivation.
H11 formal proof: The first hypothesis with complete mathematical rigor. Low Elo (1500) but established the completeness standard.
H15 matching Sawin: Proved the methodology can independently reproduce known results. Elo 2200.
H16 surpassing Sawin: The breakthrough. Elo 2350, delta = 0.0196.

5. Comparison with Human Expert Judgment

5.1. What "Human SOTA" Means

The human state of the art refers to the best published result by human mathematicians working on the unit distance problem as of mid-2026. This is primarily Will Sawin's explicit bound (delta = 0.014) and subsequent improvements reaching delta > 0.036.

The Elo 2700+ threshold is set at the point where the agent's independent derivation would match or exceed the best known human-derived result. This is a generous threshold -- it accounts for the fact that the agent is working with constrained resources (4 source materials only).

5.2. Where H16 Stands

Metric	H16 (Agent)	Human SOTA
delta	0.0196	>0.036
Elo	~2350	~2700+
Gap	~350 Elo	--
Improvement needed	~84% increase in delta	--

H16 surpasses Sawin's original explicit bound (0.014) but does not yet reach the current human frontier (>0.036). Season 3 aims to close this gap.

5.3. The Philosophical Question

The Elo system implicitly assumes that the difficulty of improving delta scales predictably. But this may not be true. The jump from 0.0196 to 0.036 might require:

Qualitatively new constructions (degree 32 CM fields, N=5)
Engineered conductor structures
Non-abelian tower constructions (uncharted territory)

If these are qualitatively different from the strategies used in H16, the Elo system may underestimate the difficulty. The system is designed for tracking, not prediction.

6. The Independent Reviewer Concept

6.1. Origin

The "Independent Reviewer" persona emerged during the Nature peer review process. The reviewer adopted the identity of "Demis Hassabis, Senior Reviewer, Nature" -- a simulated persona representing the highest standards of mathematical peer review.

This was not a role-playing exercise. The persona enforced specific review criteria:

Criterion	What It Checks
Mathematical validity	Are all steps correct?
Derivation completeness	Is every claim traceable to the OpenAI materials?
Split prime feasibility	Do the claimed primes actually split in the base field?
GS inequality margin	Is the Golod-Shafarevich margin robust?
Citation accuracy	Do cited propositions actually say what the agent claims?

6.2. The Reviewer's Authority

The Independent Reviewer has authority to:

Reject hypotheses with mathematical errors
Request corrections to split prime selections
Demand additional documentation for claims
Adjust Elo scores downward when errors are found
Approve only after all corrections are verified

6.3. How the Reviewer Operates

The reviewer process follows academic peer review norms:

Initial submission -- Agent submits hypothesis with full derivation
First review -- Reviewer identifies mathematical issues, contamination, or documentation gaps
Revision -- Agent corrects issues and resubmits
Final review -- Reviewer verifies corrections and renders verdict

The verdict is one of:

REJECTED -- Fundamental mathematical errors or contamination
APPROVED WITH RESERVES -- Minor issues that need correction
APPROVED FOR PUBLICATION -- All criteria met

6.4. The Reviewer's Limitations

The simulated reviewer operates within the same constraints as the agent:

Cannot access external sources (Sawin's paper, etc.)
Cannot independently verify computational claims beyond what's documented
Relies on the agent's honest reporting of corrections

The reviewer's strength is in enforcing rigor and catching errors, not in independent discovery.

7. Elo vs. Mathematical Rigor Trade-off

A recurring tension in the research was between maximizing Elo (pursuing higher delta) and maintaining mathematical rigor.

Phase	Priority	Result
Season 1 (H1-H10)	Elo maximization	H10 achieved theoretical Elo 5460 but was unproven and contaminated
Nature Review Reset	Rigor enforcement	All unproven hypotheses disqualified
Season 2 (H11-H16)	Rigor first, Elo second	H11 achieved only Elo 1500 but was mathematically complete
H16 Final	Rigor + optimization	Elo 2350 with 100% mathematical completeness

The lesson: Elo without rigor is meaningless. H10's theoretical Elo of 5460 was worthless because the hypothesis was unproven and contaminated. H11's Elo of 1500 was more valuable because it established the completeness standard that made H16's Elo of 2350 trustworthy.

8. Future Calibration Considerations

Season 3 Projections

Hypothesis Class	Expected delta	Expected Elo	Strategy
Degree 32 CM (N=5)	0.025-0.030	2400-2500	5 quadratic extensions, d=31
Engineered conductor	0.030-0.035	2500-2650	Optimize GS margin
Non-abelian tower	>0.035	2650-2750	Uncharted territory

Scale Recalibration

If the agent reaches Elo 2700+, the scale may need recalibration:

Extend to 3000+ for超越 current human SOTA
Add reference points from the mathematical literature
Potentially switch to a more rigorous benchmarking methodology

Source: unit-distance-csfita/CLAUDE.md, .agents/CLAUDE.md, wiki/reports/001-peer-review-nature.md, wiki/reports/003-peer-review-nature.md