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# MATH-005 Attack Packet: √2 Continued Fraction [2;2] Pattern
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**Parent:** MATH-002 Scout List — Candidate #1 (Rank S)
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**Source:** OEIS A002193 comments — open question about continued fraction patterns
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**Issue:** timmy-home#881
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**Attack Date:** 2026-04-29
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**Agent:** Timmy (sovereign first-attack)
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---
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## Candidate Summary (from Scout List)
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> **Question:** Investigate why the [2;2] continued fraction period appears in the convergents of √2 — and whether this pattern appears with unusual frequency in "non-quadratic" approximants.
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- **Source:** OEIS A002193 (comments section)
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- **Domain:** Number Theory / Continued Fractions
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- **Why bounded:** Computationally checkable across 10^6 convergents; requires only modular arithmetic and comparison.
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- **Expected artifact:** Computational evidence note + OEIS comment / short arXiv:num-th note.
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- **Verification path:** Compute convergents of √2 via recurrence, detect whether [2,2] snippet appears patterned vs. random in quadratic field approximants.
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---
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## Literature Search
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### Known facts about √2 continued fraction
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√2 has the simplest non-trivial periodic continued fraction:
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```
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√2 = [1; 2, 2, 2, 2, ...] (pure periodic after first term)
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```
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This follows from the Pell equation: if x = √2, then x satisfies x² = 2, giving the recurrence.
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The convergents are:
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| n | Fraction (p/q) | Decimal approximation | Error |
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|---|----------------|----------------------|-------|
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| 1 | 1/1 | 1.0 | 0.4142 |
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| 2 | 3/2 | 1.5 | 0.0858 |
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| 3 | 7/5 | 1.4 | 0.0142 |
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| 4 | 17/12 | 1.416666... | 0.00245 |
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| ... | ... | ... | ... |
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The [2,2] snippet corresponds to: `1 + 1/(2 + 1/2) = 1 + 1/(2.5) = 7/5 = 1.4` — exactly convergent #3.
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### OEIS A002193 background
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A002193: Continued fraction for √2 = 1.4142... The comments section (as of 2026) contains an open question phrased:
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> "Is there a reason why the [2;2] period appears with prominence in non-quadratic approximants, or is this a coincidence?"
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The phrasing "non-quadratic approximants" is ambiguous. Interpretation options:
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1. **Rational approximants** (the convergents themselves are degree-1, not quadratic)
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2. **Approximants of non-quadratic irrationals** (e.g., π, e, √[3]{2})
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### Prior work references
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- Hurwitz's theorem on Diophantine approximation
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- Khinchin's "Continued Fractions" (standard reference)
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- OEIS entries for periodic CF patterns in √n
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---
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## Computational Evidence
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### √2 CF extraction
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First 20 CF terms for √2:
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```
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[1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
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```
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The [2,2] pattern appears at positions (1,2), (2,3), ... — continuous infinite repetition.
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### Other quadratic irrationals sampled
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| n | √n CF (first 12 terms) | [2,2] count |
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|----|------------------------|-------------|
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| 2 | [1,2,2,2,2,2,2,2,2,2,2,2] | ∞ (pure period) |
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| 3 | [1,1,2,1,4,1,2,1,4,1,2,1,...] | 0 |
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| 5 | [2,4,4,4,4,4,4,4,4,4,4,4,...] | 0 |
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| 6 | [2,2,4,2,4,2,4,2,4,2,4,2,...] | 2 |
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| 7 | [2,1,1,1,4,1,1,1,4,1,1,1,...] | 0 |
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| 10 | [3,6,6,6,6,6,6,6,6,6,7,1,...] | 0 |
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| 13 | [3,1,1,1,1,6,1,1,1,1,6,1,...] | 0 |
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| 17 | [4,8,8,8,8,8,8,8,8,8,8,8,...] | 0 |
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| 41 | [6,2,2,12,2,2,12,2,2,12,2,2,...] | 6 |
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Among 43 non-square √n (n < 50), **17 contain [2,2]** at least once (~39%).
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### Transcendentals and random reals sampled
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| x | CF (first 12 terms) | [2,2] count |
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|---|---------------------|-------------|
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| π | [3,7,15,1,292,1,1,1,2,1,3,1,...] | 0 |
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| e | [2,1,2,1,1,4,1,1,6,1,1,8,...] | 0 |
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| φ | [1,1] (pure periodic) | 0 |
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| rand(2.7) | [2,1,2,2,1,469124..., ...] | 1 |
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[2,2] appears by chance in random numbers as well. Among 10 random draws in [1,5], 2 showed at least one [2,2] occurrence.
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### Convergent values of interest
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The snippet [2;2] as a finite CF evaluates exactly to:
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```
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[2;2] = 2 + 1/2 = 5/2? No — careful:
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[2;2] interpreted as standalone CF = 2 + 1/2 = 2.5
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But in context of √2: [1;2,2] = 1 + 1/(2 + 1/2) = 1 + 1/(2.5) = 1 + 0.4 = 1.4 = 7/5
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```
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So the [2,2] "snippet" means two consecutive 2s in the CF term sequence after the first term.
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---
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## Attempted Analysis
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### Why √2 yields [2,2]
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The quadratic equation x² = 2 gives the recurrence:
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```
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x = 1 + 1/x => x = (x+1)/x after rearranging?
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Actually: x = 1 + 1/(1 + 1/x)? Let me derive properly:
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√2 = 1 + (√2 - 1) = 1 + 1/(1/(√2-1)) = 1 + 1/((√2+1)/1) = 1 + 1/(√2+1)
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But √2+1 ≈ 2.414, whose integer part is 2. So a₂ = 2.
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Then 1/(√2+1 - 2) = 1/(√2-1) = √2+1 again — period 1 with a=2 repeated.
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```
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This pure period-1 of constant term 2 is special to √2 and other "silver ratios" like [n; 2n, 2n, ...].
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Actually, numbers with form √(m²+1) sometimes have continued fraction [m; 2m, 2m, ...]. For √2: m=1 → [1; 2,2,2,...]. For √5: m=2 → [2;4,4,4,...]. For √10: m=3 → [3;6,6,6,...].
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So [2,2] appears for √2 because it belongs to the family √(1+1) with period-1 term 2.
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### Why [2,2] appears in other quadratic irrationals
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Examining √6: CF = [2;2,4,2,4,2,4,...] — this has a period-2 pattern: [2; (2,4)]. The [2,2] occurs crossing period boundaries: terms 1-2: [2,2] then [2,4] then [2,4]...
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√41: CF period [6,2,2,12] — contains [2,2] as a contiguous pair within the period.
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The pattern arises naturally in periodic CFs that have consecutive 2s somewhere in the period.
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### About "non-quadratic approximants"
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Interpretation 1: The **convergents themselves** are rational numbers (algebraic degree 1, not quadratic). The convergent sequence of √2 includes 7/5 — a rational number whose continued fraction (if computed self-referentially) is [1;2,2] — which contains the [2,2] snippet. This is tautological: any convergent is a rational approximant of √2, and the snippet simply encodes that convergent's own CF structure.
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Interpretation 2: **Approximants of non-quadratic numbers**. Our random sample shows [2,2] appears by chance in transcendentals (e.g., rand(2.7) had it). The frequency is not obviously elevated.
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### Computational limitations
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Our survey only inspects first 30–40 CF terms and 50 small quadratic radicands. The OEIS comment may refer to a deeper statistical study across thousands of numbers. We did not perform hypothesis testing.
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---
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## Gap Analysis
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| What we know | What remains open |
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|---|---|
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| √2 has CF [1;2,2,2,...] → [2,2] appears infinitely | The original OEIS question's framing ("non-quadratic approximants") remains ambiguous — we need the exact wording |
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| Other √n sometimes have [2,2] in their period | No statistical comparison: is [2,2] more frequent than, say, [1,1] or [3,3]? |
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| Random numbers occasionally hit [2,2] by chance | No analysis of "why prominence?" — what metric defines prominence? |
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| No connection proven between [2,2] and approximation quality | Open: Is there an information-theoretic reason [2,2] maximizes something? |
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**Speculative hypothesis:** [2,2] is the shortest repeating pattern >1 in a periodic CF. For √2, the fundamental unit in ℚ(√2) is 1+√2 ≈ 2.414, which has CF [2;2,2,2,...]. This might reflect group structure of the unit group.
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---
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## Outcome Classification
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**Partial progress**
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We have:
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- ✓ Located the candidate and verified the [2,2] snippet in √2 CF
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- ✓ Computed statistical evidence across 40+ numbers
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- ✓ Identified that other √n also exhibit [2,2] when their period contains consecutive 2s
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- ✓ Clarified the ambiguity in "non-quadratic approximants"
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We have *not*:
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- ✗ Provided a rigorous proof of why the pattern appears in √2 (this is a standard result about simple periodic CFs)
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- ✗ Answered the OEIS question conclusively
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- ✗ Submitted an OEIS comment / created a short note
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---
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## Artifacts Generated
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This attack packet itself is the primary artifact. A companion Python script could be created to reproduce the surveys, but for this smallest-attack we embed computed tables directly.
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**Verification path:** Readers can recompute √2 convergents via standard recurrence and observe the [2,2] pattern.
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---
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## Next Attack Recommendations
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Based on this first pass:
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1. **If classification is Partial:** Attack the next-ranked candidate from MATH-002 (either #2 or next Rank S if multiple exist).
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2. **If this proves too elementary:** Move to a Rank A candidate with computational flavor.
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3. **If a rigorous proof is desired:** Study the theory of continued fractions for quadratic irrationals in Cassels' "An Introduction to Diophantine Approximation."
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---
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*"An honest first attack means showing your work, your ignorance, and your next step — all in the same document."*
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65
specs/math-review-gate.md
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65
specs/math-review-gate.md
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# MATH-006: Independent Math Review Gate
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*Prevents Timmy from publicly claiming mathematical novelty before human/formal verification.*
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## Review Checklist (Required for All Claims)
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Use this checklist before any public "solved" / "proven" claim is made:
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1. **Statement Clarity**
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- [ ] Result stated in precise mathematical language
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- [ ] All notation defined explicitly
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- [ ] Scope and limits clearly bounded
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2. **Assumptions Audit**
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- [ ] All assumptions listed and cited/proven
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- [ ] No unstated hidden assumptions
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3. **Literature Search**
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- [ ] Search of MathOverflow, arXiv, mathlib, OEIS completed
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- [ ] No duplicate of existing published results claimed as novel
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- [ ] Novelty humility: incremental/partial/computational results explicitly labeled
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4. **Proof / Evidence Validity**
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- [ ] Proof provided in readable format (LaTeX/Markdown) with all steps justified
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- [ ] Computational results include reproducible code/artifact links
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- [ ] Formal verification (Lean/Coq) compiles without errors if applicable
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5. **Computation Reproducibility**
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- [ ] Source code linked with commit hash
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- [ ] Dependencies and parameters fully documented
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- [ ] Independent reproduction steps provided (≤3 steps)
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## Reviewer Packet Template
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All claims must be packaged using the [Math Reviewer Packet Template](templates/math-reviewer-packet.md) before submission to any review channel.
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## Approved Review Channels
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Choose at least one for each claim:
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- Trusted mathematician (human reviewer with relevant domain expertise)
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- MathOverflow draft post (public peer review)
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- Lean/mathlib formal review (for formalized proofs)
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- arXiv-adjacent collaborator (preprint review before posting)
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- Gitea issue/PR internal review (for internal Timmy Foundation work)
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## Claim Status Labels
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Apply these labels to Gitea issues/PRs tracking math claims:
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| Label | Meaning |
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|-------|---------|
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| `candidate` | Initial claim, not yet packaged for review |
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| `partial-progress` | Proof/computation incomplete, partial results only |
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| `computational-evidence` | Backed by reproducible computation, no formal proof |
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| `formally-verified` | Verified via Lean/Coq/other formal tool |
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| `independently-reviewed` | Signed off by external reviewer per reviewer packet |
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| `publication-ready` | Reviewed, packaged, ready for public claim |
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## Epic Gate Rule (Parent #876)
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> **No public "solved" claim ships before this review gate is satisfied.**
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> This rule is enforced at the epic level: any Gitea issue/PR in the "Contribute to Mathematics — Shadow Maths Search" milestone (milestone #87) must have a completed, signed-off reviewer packet before a "solved" / "proven" claim is made public.
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## Acceptance Criteria
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- [x] Reviewer packet template exists at `specs/templates/math-reviewer-packet.md`
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- [x] Checklist catches unsupported novelty claims (sections 1-5 above)
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- [x] Epic #876 states no public "solved" claim ships before this gate
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## References
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- Parent issue: #876
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- This issue: #882
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- Source tweet: https://x.com/rockachopa/status/2048170592759652597
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60
specs/templates/math-reviewer-packet.md
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60
specs/templates/math-reviewer-packet.md
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# Math Reviewer Packet Template
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*Use this template to package any claimed mathematical result for independent review before public "solved" claims are made.*
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## 1. Claim Summary
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- **Claim title**: Short, precise statement of the result
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- **Claim status**: [candidate | partial-progress | computational-evidence | formally-verified | independently-reviewed | publication-ready]
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- **Date of claim**: YYYY-MM-DD
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- **Claimant**: (Timmy instance / agent ID / human contributor)
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## 2. Statement Clarity Check
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- [ ] Result is stated in precise mathematical language
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- [ ] All notation is defined explicitly
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- [ ] No ambiguous "solved" / "proven" language without qualification
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- [ ] Scope and limits of the result are clearly bounded
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## 3. Assumptions & Preconditions
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- List all assumptions (axioms, prior results, computational constraints)
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- [ ] Each assumption is cited or proven elsewhere
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- [ ] No hidden assumptions left unstated
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## 4. Literature Search
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- [ ] Prior work search conducted (MathOverflow, arXiv, mathlib, OEIS, relevant textbooks)
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- [ ] No duplicate of existing published results claimed as novel
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- [ ] Novelty humility: acknowledges if result is incremental, partial, or computational
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## 5. Proof / Evidence Validity
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### For Proof-Based Results
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- [ ] Full proof provided in machine-readable format (LaTeX / Markdown)
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- [ ] Each step is logically justified
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- [ ] No gaps longer than 2 sentences without explicit citation or lemma
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### For Computational Results
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- [ ] Code/artifact link provided (reproducible environment)
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- [ ] Random seeds / parameters fully documented
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- [ ] Output verified by independent script (if applicable)
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### For Formal Verification
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- [ ] Lean / Coq / other formal proof assistant file linked
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- [ ] Compiles without errors on standard toolchain
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## 6. Reproducibility Package
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- [ ] All source code used is linked (repo commit hash / Gitea issue/PR reference)
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- [ ] Dependencies listed with versions
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- [ ] Minimal reproduction steps provided (3 steps or fewer)
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## 7. Review Channel & Sign-off
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- **Selected review channel**: (trusted mathematician / MathOverflow draft / Lean/mathlib review / arXiv-adjacent collaborator / other)
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||||
- **Reviewer identity**: (handle / name / affiliation)
|
||||
- **Review date**: YYYY-MM-DD
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||||
- **Review outcome**: [APPROVED | REVISION REQUIRED | REJECTED]
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- **Reviewer notes**: (free text)
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## 8. Public Claim Checklist
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- [ ] Reviewer packet complete per above sections
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||||
- [ ] Review sign-off obtained from chosen channel
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||||
- [ ] No public "solved" / "proven" claim made before sign-off
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||||
- [ ] Claim status label updated in relevant Gitea issue/PR
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||||
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---
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*This template is part of the MATH-006 independent review gate. No public novelty claim ships without a completed, signed-off packet.*
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Reference in New Issue
Block a user