SilverSight/docs/reviews/COLD_REVIEWER_PROTOCOL.md

Cold Reviewer Protocol

A formal verification protocol for independently checking mathematical claims using only discrete, calculator-verifiable computation.

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1. Goals

Primary goal

Enable a reviewer with no domain expertise (no Lean, no braid theory, no Temperley–Lieb algebras, no differential geometry) to independently determine whether the core claims of the Unified Covariant Field Theory are mathematically sound.

Secondary goals

1. Isolate domain knowledge. All expert-level claims (geometric conjectures, spectral theory, tensor categories) are explicitly gated behind the discrete arithmetic layer. A reviewer who passes the Arithmetic Gate has already validated the foundation without needing to understand the superstructure.

2. Eliminate tautologies. Every invariant must be checked against independent calculation, not against another statement in the same manuscript. No invariant may depend on another invariant.

3. Produce a bounded receipt. A reviewer's final pass/fail verdict is a binary decision: all four invariants hold and no structural red flag is present ⟶ PASS. Otherwise ⟶ FAIL.

4. Provide a formal upgrade path. Each invariant maps directly to a Lean theorem verified by norm_num or dec_trivial. A reviewer who trusts the Lean kernel can skip manual calculation and run lake build SilverSight (3307 jobs, 0 errors).

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2. The Two-Gate Structure

┌────────────────────────────────────────────────────────────────────┐
│                    MANUSCRIPT UNDER REVIEW                          │
│  UnifiedCovariant.lean — Unified Covariant Field Theory            │
├────────────────────────────────────────────────────────────────────┤
│                                                                    │
│  GATE A: ARITHMETIC GATE                                           │
│  ────────────────────────                                           │
│  Verify I₁, I₂, I₃, I₄ independently.                              │
│  Reject immediately if any fails.                                  │
│                                                                    │
│  ↓  (all pass)                                                     │
│                                                                    │
│  GATE B: STRUCTURAL GATE                                            │
│  ────────────────────────                                           │
│  Check that none of the three red-flag claims appear anywhere      │
│  in the manuscript.  Reject if any is present.                     │
│                                                                    │
│  ↓  (no red flags)                                                 │
│                                                                    │
│  GATE C: BUILD GATE                                                 │
│  ────────────────────────                                           │
│  (Optional, for Lean-literate reviewers)                            │
│  Run `lake build SilverSight`.  Must report 0 errors.              │
│                                                                    │
│  ↓  (0 errors)                                                     │
│                                                                    │
│                    ✅ VERDICT: PASS                                 │
│  Core discrete claims are sound.  Higher-level geometric claims     │
│  (Layer 3) are deferred but structurally harmless.                 │
└────────────────────────────────────────────────────────────────────┘

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3. Gate A — Arithmetic Gate

Instructions

You need: a calculator (or pen and paper, or a Lean kernel).

For each invariant, perform the required computation. The answer is predetermined. If your result matches, the invariant passes. If your result differs, reject the manuscript — the foundation is unsound.

DO NOT skip any invariant. DO NOT accept "by symmetry" or "by a similar argument" as a substitute for independent calculation.

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I₁. Golden-Ratio Identity

Claim: \phi = \frac{1 + \sqrt{5}}{2} satisfies \phi^2 - \phi - 1 = 0.

How to verify:

1. Compute \sqrt{5} \approx 2.236067977\ldots
2. Compute 1 + \sqrt{5} = 3.236067977\ldots
3. Divide by 2: \phi = 1.618033988\ldots
4. Compute \phi^2 = (1.618033988)^2 = 2.618033988\ldots
5. Compute \phi^2 - \phi - 1 = 2.618033988 - 1.618033988 - 1 = 0.

Alternative (exact symbolic): [ \phi^2 - \phi - 1 = \frac{(1+\sqrt{5})^2}{4} - \frac{1+\sqrt{5}}{2} - 1 = \frac{1 + 2\sqrt{5} + 5}{4} - \frac{1+\sqrt{5}}{2} - 1 = 0. ]

Lean reference: golden_identity (line 86).

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I₂. Fixed-Point Gap

Claim: [ \sigma = \frac{9984}{65536} = \frac{39}{256},\qquad \tau = \frac{1}{7},\qquad \sigma - \tau = \frac{17}{1792} > 0. ]

How to verify:

1. Reduce \frac{9984}{65536}: divide numerator and denominator by 256. [ \frac{9984}{65536} = \frac{9984 \div 256}{65536 \div 256} = \frac{39}{256}. ] (Check: 256 × 39 = 9984, 256 × 256 = 65536. ✓)

2. Compute the difference with common denominator 1792: [ \frac{39}{256} - \frac{1}{7} = \frac{39 \times 7}{256 \times 7} - \frac{1 \times 256}{7 \times 256} = \frac{273}{1792} - \frac{256}{1792} = \frac{17}{1792}. ]

3. Check positivity: 17 > 0. ✓

Also verify: \sigma > \tau (since 17 > 0).

Lean reference: spectral_gap_positive (line 99).

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I₃. Fibonacci Values

Claim: F_7 = 13 and F_8 = 21.

How to verify: Run the recurrence from the definition.

n	F_n	Calculation
0	0	(by definition)
1	1	(by definition)
2	1	F_0 + F_1 = 0 + 1
3	2	F_1 + F_2 = 1 + 1
4	3	F_2 + F_3 = 1 + 2
5	5	F_3 + F_4 = 2 + 3
6	8	F_4 + F_5 = 3 + 5
7	13	F_5 + F_6 = 5 + 8
8	21	F_6 + F_7 = 8 + 13

Both values match the claim.

Lean reference: fibonacci_dims (line 106), fib7_is_13 (line 232).

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I₄. Sidon Uniqueness

Claim: For a,b,c,d \in \{0,1,2,3,4,5,6,7\},

[ 2^a + 2^b = 2^c + 2^d ;\Longrightarrow; {a,b} = {c,d}. ]

How to verify — method 1 (binary expansion):

1. Write each integer 2^k in binary: it is a 1 followed by k zeros. [ 2^0 = 1_2,; 2^1 = 10_2,; 2^2 = 100_2,; \ldots,; 2^7 = 10000000_2. ]

2. The sum 2^a + 2^b in binary has:

   • Case a = b: a single 1 in position a+1 (carry).
   • Case a \neq b: exactly two 1 bits, at positions a and b.

3. Binary representation is unique. Therefore if two sums are equal, the sets of bit positions must be identical.

How to verify — method 2 (exhaustion, 8⁴ = 4096 cases):

Check all quadruples (a,b,c,d). If 2^a + 2^b = 2^c + 2^d then \{a,b\} = \{c,d\}. This is a finite computation. With a computer:

for a in range(8):
  for b in range(8):
    for c in range(8):
      for d in range(8):
        if 2**a + 2**b == 2**c + 2**d:
          assert {a,b} == {c,d}

No counterexample exists.

Lean reference: sidon_unique (line 110), verified by dec_trivial.

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4. Gate B — Structural Gate

Instructions

Read through the entire manuscript. If any of the following three claims appears verbatim or in spirit, reject.

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Red Flag 1: J^2 = -I

✗ WRONG	✓ CORRECT
J^2 = -I	J^2 = J + I

Why it matters: J = \phi \cdot \mathrm{id}_V satisfies the golden-ratio polynomial x^2 - x - 1 = 0. Its eigenvalues are \phi and -1/\phi, not \pm i. It is not an almost-complex structure. Claiming J^2 = -I would make J a complex structure on a real vector space, which changes the entire geometric interpretation.

How to check: Find the definition of J (or goldenEndomorphism) and verify its defining relation. The correct relation is J^2 = J + I.

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Red Flag 2: \Delta_7 is Kähler

✗ WRONG	✓ CORRECT
\Delta_7 is Kähler	\mathbb{CP}^7 is Kähler (simplex is 7-real-dimensional, odd)

Why it matters: A Kähler manifold must have even real dimension. The open simplex \Delta_7 = \{p \in \mathbb{R}_{>0}^8 \mid \sum p_i = 1\} has dimension 8 - 1 = 7 (odd). Kähler on \Delta_7 is impossible.

How to check: Find any claim that a Kähler structure exists on \Delta_7 or on a 7-dimensional (or odd-dimensional) manifold. If the manuscript refers to \mathbb{CP}^7 (real dimension 14) instead, this red flag is avoided.

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Red Flag 3: \dim(\mathrm{TL}_7) = 13

✗ WRONG	✓ CORRECT
\dim(\mathrm{TL}_7) = 13	\dim(\mathrm{TL}_7) = C_7 = 429

Why it matters: The (n)-th Catalan number is [ C_n = \frac{1}{n+1}\binom{2n}{n}, \qquad C_7 = \frac{1}{8}\binom{14}{7} = \frac{3432}{8} = 429. ] The value 13 is the Fibonacci integer F_7, which arises only in the specialized Temperley–Lieb quotient at q = e^{i\pi/5} (Fibonacci anyon model). Confusing 13 with 429 is a dimension error of factor ~33×, which invalidates any spectral or geometric argument that depends on it.

How to check: Find any claim about \dim(\mathrm{TL}_7), the dimension of the Temperley–Lieb algebra on 7 strands. If it is 13, reject. If it is 429 (or the Fibonacci quotient is explicitly named), this red flag is avoided.

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5. Gate C — Build Gate (Optional)

For reviewers with access to Lean 4 and Mathlib:

cd /home/allaun/SilverSight
lake build SilverSight

Expected result: 3307 jobs, 0 errors.

If the build fails, the manuscript has a formalization error.

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6. Extending the Protocol

Adding a new invariant to Gate A

Every new Layer-1 invariant must satisfy:

1. Independence. No invariant may reference another invariant's conclusion. Each must be checkable from first principles.

2. Finiteness. The verification must be a finite computation: rational arithmetic, integer arithmetic, or finite case analysis (dec_trivial). No limits, no infinite series, no analysis.

3. Lean correspondence. Each invariant must have a corresponding Lean theorem verified by norm_num or dec_trivial.

4. Documentation in this document. Add a new subsection with:

   • The exact mathematical claim
   • A step-by-step calculator verification procedure
   • The Lean reference

Adding a new red flag to Gate B

Every new structural red flag must:

1. Be an unambiguously wrong statement that a non-expert could plausibly write.
2. Have a clear correction and a brief explanation of why the wrong version is harmful.
3. Be listed in the Structural Gate table.

Adding a new layer

The protocol supports exactly three layers:

Layer	Content	Gate	Standard
1	Discrete foundations	Gate A	0 sorries
2	Mechanical theorems	Gate B (transitively)	0 sorries
3	Geometric conjectures	Deferred	sorry permitted

A new layer must be assigned to one of these three. No "Layer 1.5" or "Layer 2b" may bypass Gate A. If a claim is not discrete and finite, it must be Layer 3 or be restated in discrete form.

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7. Decision Procedure

For each invariant I₁–I₄:
    verify independently
    if any fails → REJECT (Arithmetic Gate fail)

For each red flag R₁–R₃:
    check manuscript
    if any is present → REJECT (Structural Gate fail)

Optionally:
    run `lake build SilverSight`
    if errors → REJECT (Build Gate fail)

Otherwise → PASS

What PASS means

The four discrete invariants are mathematically sound. No obvious structural error is present. The manuscript is ready for expert review of the Layer 3 geometric conjectures.

What FAIL means

The manuscript has a foundational error. Corrections must be made to the offending invariant or red flag before any higher-level claims can be evaluated.

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8. Quick Reference

Arithmetic Gate (I₁–I₄)

ID	Claim	Verification	Lean
I₁	\phi^2 - \phi - 1 = 0	Expand (1+\sqrt{5})^2/4	golden_identity
I₂	\sigma - \tau = 17/1792 > 0	Common denominator 1792	spectral_gap_positive
I₃	F_7 = 13,\; F_8 = 21	Run recurrence to term 8	fibonacci_dims
I₄	Sidon uniqueness	Binary expansion uniqueness	sidon_unique

Structural Gate (Red Flags)

ID	Wrong claim	Correct
R₁	J^2 = -I	J^2 = J + I
R₂	\Delta_7 is Kähler	\mathbb{CP}^7 is Kähler
R₃	\dim(\mathrm{TL}_7) = 13	\dim(\mathrm{TL}_7) = C_7 = 429