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Explicit two-gate protocol (Arithmetic + Structural) with step-by-step calculator verification for each invariant. Covers goals, methodology, extension rules, and decision procedure.
389 lines
13 KiB
Markdown
389 lines
13 KiB
Markdown
# Cold Reviewer Protocol
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**A formal verification protocol for independently checking mathematical
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claims using only discrete, calculator-verifiable computation.**
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---
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## 1. Goals
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### Primary goal
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Enable a reviewer with **no domain expertise** (no Lean, no braid theory,
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no Temperley–Lieb algebras, no differential geometry) to independently
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determine whether the core claims of the Unified Covariant Field Theory
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are mathematically sound.
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### Secondary goals
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1. **Isolate domain knowledge.** All expert-level claims (geometric
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conjectures, spectral theory, tensor categories) are explicitly gated
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behind the discrete arithmetic layer. A reviewer who passes the
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Arithmetic Gate has already validated the foundation without needing
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to understand the superstructure.
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2. **Eliminate tautologies.** Every invariant must be checked against
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independent calculation, not against another statement in the same
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manuscript. No invariant may depend on another invariant.
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3. **Produce a bounded receipt.** A reviewer's final pass/fail verdict
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is a binary decision: all four invariants hold and no structural red
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flag is present ⟶ **PASS**. Otherwise ⟶ **FAIL**.
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4. **Provide a formal upgrade path.** Each invariant maps directly to a
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Lean theorem verified by `norm_num` or `dec_trivial`. A reviewer who
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trusts the Lean kernel can skip manual calculation and run
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`lake build SilverSight` (3307 jobs, 0 errors).
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---
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## 2. The Two-Gate Structure
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```
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┌────────────────────────────────────────────────────────────────────┐
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│ MANUSCRIPT UNDER REVIEW │
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│ UnifiedCovariant.lean — Unified Covariant Field Theory │
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├────────────────────────────────────────────────────────────────────┤
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│ │
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│ GATE A: ARITHMETIC GATE │
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│ ──────────────────────── │
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│ Verify I₁, I₂, I₃, I₄ independently. │
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│ Reject immediately if any fails. │
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│ │
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│ ↓ (all pass) │
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│ │
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│ GATE B: STRUCTURAL GATE │
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│ ──────────────────────── │
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│ Check that none of the three red-flag claims appear anywhere │
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│ in the manuscript. Reject if any is present. │
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│ │
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│ ↓ (no red flags) │
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│ │
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│ GATE C: BUILD GATE │
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│ ──────────────────────── │
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│ (Optional, for Lean-literate reviewers) │
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│ Run `lake build SilverSight`. Must report 0 errors. │
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│ │
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│ ↓ (0 errors) │
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│ │
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│ ✅ VERDICT: PASS │
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│ Core discrete claims are sound. Higher-level geometric claims │
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│ (Layer 3) are deferred but structurally harmless. │
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└────────────────────────────────────────────────────────────────────┘
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```
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---
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## 3. Gate A — Arithmetic Gate
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### Instructions
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You need: **a calculator** (or pen and paper, or a Lean kernel).
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For each invariant, perform the required computation. The answer is
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predetermined. If your result matches, the invariant passes. If your
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result differs, **reject the manuscript** — the foundation is unsound.
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**DO NOT** skip any invariant. **DO NOT** accept "by symmetry" or
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"by a similar argument" as a substitute for independent calculation.
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---
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### I₁. Golden-Ratio Identity
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**Claim:** \(\phi = \frac{1 + \sqrt{5}}{2}\) satisfies \(\phi^2 - \phi - 1 = 0\).
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**How to verify:**
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1. Compute \(\sqrt{5} \approx 2.236067977\ldots\)
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2. Compute \(1 + \sqrt{5} = 3.236067977\ldots\)
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3. Divide by 2: \(\phi = 1.618033988\ldots\)
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4. Compute \(\phi^2 = (1.618033988)^2 = 2.618033988\ldots\)
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5. Compute \(\phi^2 - \phi - 1 = 2.618033988 - 1.618033988 - 1 = 0\).
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**Alternative (exact symbolic):**
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\[
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\phi^2 - \phi - 1 = \frac{(1+\sqrt{5})^2}{4} - \frac{1+\sqrt{5}}{2} - 1
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= \frac{1 + 2\sqrt{5} + 5}{4} - \frac{1+\sqrt{5}}{2} - 1 = 0.
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\]
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**Lean reference:** `golden_identity` (line 86).
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---
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### I₂. Fixed-Point Gap
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**Claim:**
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\[
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\sigma = \frac{9984}{65536} = \frac{39}{256},\qquad
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\tau = \frac{1}{7},\qquad
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\sigma - \tau = \frac{17}{1792} > 0.
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\]
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**How to verify:**
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1. Reduce \(\frac{9984}{65536}\): divide numerator and denominator by 256.
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\[
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\frac{9984}{65536} = \frac{9984 \div 256}{65536 \div 256} = \frac{39}{256}.
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\]
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*(Check: 256 × 39 = 9984, 256 × 256 = 65536. ✓)*
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2. Compute the difference with common denominator 1792:
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\[
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\frac{39}{256} - \frac{1}{7} = \frac{39 \times 7}{256 \times 7} - \frac{1 \times 256}{7 \times 256}
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= \frac{273}{1792} - \frac{256}{1792} = \frac{17}{1792}.
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\]
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3. Check positivity: \(17 > 0\). ✓
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**Also verify:** \(\sigma > \tau\) (since \(17 > 0\)).
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**Lean reference:** `spectral_gap_positive` (line 99).
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---
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### I₃. Fibonacci Values
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**Claim:** \(F_7 = 13\) and \(F_8 = 21\).
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**How to verify:** Run the recurrence from the definition.
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| \(n\) | \(F_n\) | Calculation |
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|------|---------|-------------|
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| 0 | 0 | (by definition) |
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| 1 | 1 | (by definition) |
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| 2 | 1 | \(F_0 + F_1 = 0 + 1\) |
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| 3 | 2 | \(F_1 + F_2 = 1 + 1\) |
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| 4 | 3 | \(F_2 + F_3 = 1 + 2\) |
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| 5 | 5 | \(F_3 + F_4 = 2 + 3\) |
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| 6 | 8 | \(F_4 + F_5 = 3 + 5\) |
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| **7** | **13** | \(F_5 + F_6 = 5 + 8\) |
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| **8** | **21** | \(F_6 + F_7 = 8 + 13\) |
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Both values match the claim.
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**Lean reference:** `fibonacci_dims` (line 106), `fib7_is_13` (line 232).
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---
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### I₄. Sidon Uniqueness
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**Claim:** For \(a,b,c,d \in \{0,1,2,3,4,5,6,7\}\),
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\[
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2^a + 2^b = 2^c + 2^d \;\Longrightarrow\; \{a,b\} = \{c,d\}.
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\]
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**How to verify — method 1 (binary expansion):**
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1. Write each integer \(2^k\) in binary: it is a 1 followed by \(k\) zeros.
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\[
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2^0 = 1_2,\; 2^1 = 10_2,\; 2^2 = 100_2,\; \ldots,\; 2^7 = 10000000_2.
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\]
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2. The sum \(2^a + 2^b\) in binary has:
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- **Case \(a = b\):** a single 1 in position \(a+1\) (carry).
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- **Case \(a \neq b\):** exactly two 1 bits, at positions \(a\) and \(b\).
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3. Binary representation is unique. Therefore if two sums are equal, the
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sets of bit positions must be identical.
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**How to verify — method 2 (exhaustion, 8⁴ = 4096 cases):**
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Check all quadruples \((a,b,c,d)\). If \(2^a + 2^b = 2^c + 2^d\) then
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\(\{a,b\} = \{c,d\}\). This is a finite computation. With a computer:
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```python
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for a in range(8):
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for b in range(8):
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for c in range(8):
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for d in range(8):
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if 2**a + 2**b == 2**c + 2**d:
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assert {a,b} == {c,d}
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```
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No counterexample exists.
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**Lean reference:** `sidon_unique` (line 110), verified by `dec_trivial`.
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---
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## 4. Gate B — Structural Gate
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### Instructions
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Read through the entire manuscript. If **any** of the following three
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claims appears verbatim or in spirit, **reject**.
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---
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### Red Flag 1: \(J^2 = -I\)
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| ✗ WRONG | ✓ CORRECT |
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|---------|-----------|
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| \(J^2 = -I\) | \(J^2 = J + I\) |
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**Why it matters:** \(J = \phi \cdot \mathrm{id}_V\) satisfies the
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golden-ratio polynomial \(x^2 - x - 1 = 0\). Its eigenvalues are
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\(\phi\) and \(-1/\phi\), not \(\pm i\). It is **not** an almost-complex
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structure. Claiming \(J^2 = -I\) would make \(J\) a complex structure on
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a real vector space, which changes the entire geometric interpretation.
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**How to check:** Find the definition of \(J\) (or `goldenEndomorphism`)
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and verify its defining relation. The correct relation is \(J^2 = J + I\).
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---
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### Red Flag 2: \(\Delta_7\) is Kähler
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| ✗ WRONG | ✓ CORRECT |
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|---------|-----------|
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| \(\Delta_7\) is Kähler | \(\mathbb{CP}^7\) is Kähler (simplex is 7-real-dimensional, odd) |
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**Why it matters:** A Kähler manifold must have even real dimension.
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The open simplex \(\Delta_7 = \{p \in \mathbb{R}_{>0}^8 \mid \sum p_i = 1\}\)
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has dimension \(8 - 1 = 7\) (odd). Kähler on \(\Delta_7\) is impossible.
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**How to check:** Find any claim that a Kähler structure exists on
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\(\Delta_7\) or on a 7-dimensional (or odd-dimensional) manifold.
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If the manuscript refers to \(\mathbb{CP}^7\) (real dimension 14) instead,
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this red flag is avoided.
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---
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### Red Flag 3: \(\dim(\mathrm{TL}_7) = 13\)
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| ✗ WRONG | ✓ CORRECT |
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|---------|-----------|
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| \(\dim(\mathrm{TL}_7) = 13\) | \(\dim(\mathrm{TL}_7) = C_7 = 429\) |
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**Why it matters:** The \(n\)-th Catalan number is
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\[
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C_n = \frac{1}{n+1}\binom{2n}{n},
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\qquad
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C_7 = \frac{1}{8}\binom{14}{7} = \frac{3432}{8} = 429.
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\]
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The value 13 is the Fibonacci integer \(F_7\), which arises only in the
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specialized Temperley–Lieb quotient at \(q = e^{i\pi/5}\) (Fibonacci
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anyon model). Confusing 13 with 429 is a dimension error of factor ~33×,
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which invalidates any spectral or geometric argument that depends on it.
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**How to check:** Find any claim about \(\dim(\mathrm{TL}_7)\), the
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dimension of the Temperley–Lieb algebra on 7 strands. If it is 13,
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reject. If it is 429 (or the Fibonacci quotient is explicitly named),
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this red flag is avoided.
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---
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## 5. Gate C — Build Gate (Optional)
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For reviewers with access to Lean 4 and Mathlib:
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```bash
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cd /home/allaun/SilverSight
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lake build SilverSight
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```
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**Expected result:** 3307 jobs, 0 errors.
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If the build fails, the manuscript has a formalization error.
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---
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## 6. Extending the Protocol
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### Adding a new invariant to Gate A
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Every new Layer-1 invariant must satisfy:
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1. **Independence.** No invariant may reference another invariant's
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conclusion. Each must be checkable from first principles.
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2. **Finiteness.** The verification must be a finite computation:
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rational arithmetic, integer arithmetic, or finite case analysis
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(dec_trivial). No limits, no infinite series, no analysis.
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3. **Lean correspondence.** Each invariant must have a corresponding
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Lean theorem verified by `norm_num` or `dec_trivial`.
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4. **Documentation in this document.** Add a new subsection with:
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- The exact mathematical claim
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- A step-by-step calculator verification procedure
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- The Lean reference
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### Adding a new red flag to Gate B
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Every new structural red flag must:
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1. Be an **unambiguously wrong statement** that a non-expert could
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plausibly write.
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2. Have a clear correction and a brief explanation of why the wrong
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version is harmful.
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3. Be listed in the Structural Gate table.
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### Adding a new layer
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The protocol supports exactly three layers:
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| Layer | Content | Gate | Standard |
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|-------|---------|------|----------|
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| 1 | Discrete foundations | Gate A | 0 sorries |
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| 2 | Mechanical theorems | Gate B (transitively) | 0 sorries |
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| 3 | Geometric conjectures | Deferred | `sorry` permitted |
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A new layer must be assigned to one of these three. No "Layer 1.5" or
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"Layer 2b" may bypass Gate A. If a claim is not discrete and finite,
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it must be Layer 3 or be restated in discrete form.
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---
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## 7. Decision Procedure
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```
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For each invariant I₁–I₄:
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verify independently
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if any fails → REJECT (Arithmetic Gate fail)
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For each red flag R₁–R₃:
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check manuscript
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if any is present → REJECT (Structural Gate fail)
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Optionally:
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run `lake build SilverSight`
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if errors → REJECT (Build Gate fail)
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Otherwise → PASS
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```
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### What PASS means
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The four discrete invariants are mathematically sound. No obvious
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structural error is present. The manuscript is ready for expert review
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of the Layer 3 geometric conjectures.
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### What FAIL means
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The manuscript has a foundational error. Corrections must be made to
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the offending invariant or red flag before any higher-level claims can
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be evaluated.
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---
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## 8. Quick Reference
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### Arithmetic Gate (I₁–I₄)
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| ID | Claim | Verification | Lean |
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|----|-------|-------------|------|
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| I₁ | \(\phi^2 - \phi - 1 = 0\) | Expand \((1+\sqrt{5})^2/4\) | `golden_identity` |
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| I₂ | \(\sigma - \tau = 17/1792 > 0\) | Common denominator 1792 | `spectral_gap_positive` |
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| I₃ | \(F_7 = 13,\; F_8 = 21\) | Run recurrence to term 8 | `fibonacci_dims` |
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| I₄ | Sidon uniqueness | Binary expansion uniqueness | `sidon_unique` |
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### Structural Gate (Red Flags)
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| ID | Wrong claim | Correct |
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|----|-------------|---------|
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| R₁ | \(J^2 = -I\) | \(J^2 = J + I\) |
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| R₂ | \(\Delta_7\) is Kähler | \(\mathbb{CP}^7\) is Kähler |
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| R₃ | \(\dim(\mathrm{TL}_7) = 13\) | \(\dim(\mathrm{TL}_7) = C_7 = 429\) |
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