# Cold Reviewer Formula

**Protocol for independent verification of the Unified Covariant Field Theory**

A reviewer needs only a calculator and this checklist. No knowledge of Lean,
braid theory, or Temperley–Lieb algebras is required to pass the arithmetic
and structural gates.

---

## Arithmetic Gate

Verify the following four invariants independently.

### I₁. Golden-ratio identity

\[
\phi = \frac{1 + \sqrt{5}}{2}, \qquad \phi^2 - \phi - 1 = 0.
\]

*Check:* Compute \(\phi^2\), subtract \(\phi + 1\), obtain 0.

### I₂. Fixed-point gap

\[
\sigma = \frac{9984}{65536} = \frac{39}{256}, \qquad
\tau = \frac{1}{7},
\]

\[
\sigma - \tau = \frac{39}{256} - \frac{1}{7}
            = \frac{273 - 256}{1792}
            = \frac{17}{1792} > 0.
\]

*Check:* Common denominator 1792.  Numerator \(273 - 256 = 17 > 0\).

### I₃. Fibonacci values

\[
F_7 = 13, \qquad F_8 = 21.
\]

*Check:* Run the recurrence \(F_0=0,\;F_1=1,\;F_{n+2}=F_{n+1}+F_n\) to term 8.

### I₄. Binary / Sidon uniqueness

For \(a,b,c,d \in \{0,\dots,7\}\),

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

*Reason:* Binary expansion of integers is unique.  A sum of two powers of two
has either one bit set (\(a=b\), yielding \(2^{a+1}\)) or exactly two bits set
(\(a \neq b\)).  Equality therefore forces the same exponent multiset.

---

## Structural Gate

Ensure the manuscript does **not** make any of the following claims.

### Red Flag 1 — Wrong endomorphism relation

| ✗ Prohibited | ✓ Correct |
|---|---|
| \(J^2 = -I\) | \(J^2 = J + I\) |

where \(J = \phi \cdot \mathrm{id}_V\).  This operator satisfies the
golden-ratio polynomial \(x^2 - x - 1 = 0\), with eigenvalues
\(\phi\) and \(-1/\phi\).  It is **not** an almost-complex structure.

### Red Flag 2 — Kähler on the simplex

| ✗ Prohibited | ✓ Correct |
|---|---|
| \(\Delta_7\) is Kähler | \(\dim \Delta_7 = 7\) (odd, impossible) |

The open probability simplex

\[
\Delta_7 = \{ p \in \mathbb{R}_{>0}^8 \mid \sum p_i = 1 \}
\]

has dimension \(8 - 1 = 7\).  Every Kähler manifold is even-dimensional,
so \(\Delta_7\) cannot carry a Kähler structure.  If a Kähler example is
desired, work on \(\mathbb{CP}^7\) (real dimension 14) with the standard
Fubini–Study metric.

### Red Flag 3 — Temperley–Lieb dimension

| ✗ Prohibited | ✓ Correct |
|---|---|
| \(\dim(\mathrm{TL}_7) = 13\) | \(\dim(\mathrm{TL}_7) = C_7 = 429\) |

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 arises only in specialized settings: the Fibonacci category or
Fibonacci anyon model at \(q = e^{i\pi/5}\) (quantum dimension \(\phi\)),
where the fusion rule is \(\phi \times \phi = 1 + \phi\) and simple-object
dimensions follow the Fibonacci sequence.  It is **not** the dimension of
the ordinary Temperley–Lieb algebra.

---

## Reviewer Decision Procedure

1. **Verify I₁–I₄.**  Reject immediately if any arithmetic invariant fails.
2. **Check that none of the three red-flag claims appear.**  Reject if any is present.
3. **Only after both gates pass** should higher-level geometric, categorical,
   or dynamical constructions be evaluated.

---

## Source

The reference implementation is:

- **File:** `formal/SilverSight/PIST/UnifiedCovariant.lean`
- **Proof status:** I₁–I₄ verified at compile time by `norm_num`/`dec_trivial`
  (0 sorries, 0 axioms).  Red flags explicitly documented and avoided.
  Layer 3 conjectures tagged `sorry` pending Mathlib's differential geometry API.
- **Build:** `lake build SilverSight` — 3307 jobs, 0 errors.
