# FRW triangle (one-loop three-site cosmological correlator, a = 2) — ε-form campaign: the evaluable objects

Everything on this page is computed at runtime from **exact inputs only** — the exact
bivariate-rational connection, the exact symbolic gauge chain, the exact word lists, and
closed-form periods. No finite-precision constant enters any computation
(`cosmo-evaluate.py` enforces this; see the honest-scope block for the boundary-vector
cache/recompute contract).

## 1. Closed L₂ periods (certified, both kinematic regions)

The elliptic Picard–Fuchs operator L₂ of the maximal cut has the closed period basis

```
m(λ)   = (λ² − 1) / (9λ² − 1)                      (Legendre modulus, a = 2)
ϖ₀(λ)  = K(m) / sqrt(1 − 9λ²)
ϖ₁(λ)  = K(1 − m) / sqrt(1 − 9λ²)
W      = ϖ₀ ϖ₁' − ϖ₁ ϖ₀' = π / (2 λ D₄),   D₄ = (λ² − 1)(9λ² − 1)
dτ/dλ  = π / (2 λ D₄ ϖ₀²)
```

with mpmath/arb principal branches; the continuation to the transport region
λ ∈ [1/2, 7/10] is pinned by L₂-annihilation and the Wronskian identity (both
branch-sensitive; certified > 100 d in the campaign, re-checked live by the script
against a direct quadrature of the defining K-integral).

## 2. The symbolic-ε connection A16 and the two elliptic curves

The full 16×16 connection ∂_λ M = A(λ, ε) M is exact: 66 nonzero entries, bivariate
rationals in (λ, ε), max deg_ε = 2 (artifact sha256 `de035018b116c7…`, held-out-prime
and held-out-ε audited). Two structural facts are re-verified at runtime:

- **Exact** (Fraction arithmetic on a full bidegree grid — a rigorous
  polynomial-identity proof): `A16[z5,z5] = ε · dlog D₄`.
- **The proven √Q4 twist** (the second elliptic curve): at each root of
  `Q4 = 20λ⁴ + 39λ³ + 29λ² + 9λ + 1` the residue matrix of A16 has poles only in the
  e₆ row, rank 1, with sole nonzero eigenvalue **ε − 1/2** — the signature that forces
  the sqrt(Q4)-twisted line bundle. The FRW triangle thus lives on **two** curves: the
  L₂ Legendre curve through the period frame, and y² = Q4(λ) through the e₆ twist and
  the (E, z₅) third-kind tail.

## 3. Terminal ε-form and the rigidity theorem (campaign result, frozen)

After the exact gauge chain (τ-frame ∘ Eichler strictification U₁ ∘ grading ∘
z-sector pre-gauge) the connection reaches the terminal normal form

```
conn_G3 = gauge_Wt⁻¹ (ε · Ã_final) + S
```

with S letter-free and **rigid** (7 slots at ε²/ε³): connection-level full strictness
over *any* finite letter alphabet with unipotent letter-poly gauges is refuted
(5 lemmas). The irreducible ε² core is 1 log generator + 2 second-kind integrals on
y² = Q4.

## 4. Boundary-free per-ε-order holonomy (the headline gate — live)

Ψ(x; ε) solves the rotated ε-form system with Ψ(1/2) = **I** exactly — the boundary is
the identity, no boundary constants enter. The campaign banked Ψ as explicit
iterated-integral **word lists** per matrix entry per ε-order:

```
order        ε⁰    ε¹     ε²      ε³
words        10    254    3437    47077     (272 dressed letters + Eichler n_k composites)
```

`cosmo-evaluate.py` evaluates these words as iterated integrals over the closed periods
(composite Chebyshev–Lobatto spectral quadrature, acb ball arithmetic) and gates them
against a **live independent oracle**: arb Taylor-ODE transport of the *original* SUB9
block of A16 at scalar ε nodes, rotated by the exact gauge chain T₉ (Laurent in ε, so
per-order references are extracted by an ε-Vandermonde solve — the campaign's own gate
design, rebuilt from scratch at runtime). Measured on the reference box at `--dps 40`
(bars are minima over all nonzero matrix entries):

```
eps^0: 63.0 d    eps^1: 61.2 d    eps^2: 61.5 d    eps^3: 60.6 d     (all PASS, bar 30)
dps-scaling: 63.0/61.2/61.5 -> 75.0/73.3/72.2 at --dps 52 (PASS)
mutation control (one word x (1+1e-30)): gate collapses to 30.0 d (PASS)
```

Sample value (an actual banked word-sum, computed live):

```
Ψ₁[Js0, Js1](7/10) = −11.29475492713311728138875163354820796124 i   (≥ 61 d vs oracle)
```

The ε→0 mechanism is visible in the graded words: naive diag(ε) regrading fails
(A[e₇,{e₈,e₉}](ε=0) ≠ 0 exactly); the limit resolves via the kernel condition on the
(e₈,e₉) 1/ε-pole vector.

## Honest scope

- Correlator **values** at a kinematic point transport from the λ = 1/2 boundary
  9-vector, which the evaluator now prints (Point 4) from a sha-pinned cache, each
  component clamped to its own two-leg certificate (42.9–46.8 d). The cache is a
  speed layer, not the definition: `--boundary-recompute <comps>` regenerates any
  component from its written integrals at the requested precision (measured costs
  in `--help`), and any cache/sha/value-integrity miss raises `CertError` fail-closed.
- The holonomy path is the campaign's frozen gate path [1/2, 7/10] (word lists are
  path-specific by convention).
- Certified λ domains for the period branches: (0, 1/3) and [1/2, 7/10).

Interface: `python3 cosmo-evaluate.py [--dps D] [--point 'lam=3/5,eps=1/7'] [--boundary-recompute COMPS]
[--orders K] [--check] [--skip-holonomy]` (requires mpmath, sympy; python-flint for the
holonomy point).