<!--
Sections 1-2 and the validation constants were copied verbatim from the
verification records of this work (2026-06). Section 3 (the closed t-tail) was
derived 2026-07-03/04; its numbers are quoted from the executed
zgamma-evaluate.py gate log of 2026-07-04 (round 3, derived constants); the
script recomputes them live.
-->

# gg → Zγ exact-m_t elliptic masters (sectors E125/E127) — final result

Two-loop gg → Zγ with exact top mass, deformation M = m_Z²/m_t² = 7/25 (m_t² = 1).
The hard sector E125 holds the canonical UT pair {g31 = R_a (holomorphic-period
master), g32 = R_b (quasi-period master)} on the deformed Γ₀(4) elliptic curve
y² = P(P+s−M)(P²+(s−M)P−4s). **No compact one-line closed form exists for the full
(s,t)-dependent masters**; the deliverable is (1) a named-Eichler-integral
spine, (2) 25 cusp boundary constants in a named conductor-4 ring, (3) the
t-tail in closed form: exact polynomials in t over derived boundary constants
(boundary-constant provenance, the exact endpoint split CLOSED by the
2026-07-04 seed derivation: the collinear branch and all corner masters are FULLY ANALYTIC,
and all SEVEN analytic-branch seeds {G5..G11} are derived — uniform Γ-sum
closed forms gated 61.1–65.9 d, worst over 13 ε-orders each, against the
dedicated independent AMFlow oracle, independently re-gated at a second
quadrature config; the provisional AMFlow seed set — the seeds' source
until 2026-07-04 — is retired).

## 1. The elliptic spine (named closed form, s-dependent part)

    g_spine(s) = s^(-2ε) * ((s−r₂)/(s−r₁))^((10√93/93) ε)
                 × [ c₁ + c₂·τ(s) + Eich[B_{2,4}](s) ],
    where  625 r² + 9650 r + 49 = 0   (r₁, r₂ exact algebraic),
    B_{2,4}(τ) = E₂(τ) − 4 E₂(4τ)   (Γ₀(4) weight-2 Eisenstein kernel, a₀ = 1/8),
    Eich[B_{2,4}](s) = ∫_{i∞}^{τ(s)} B_{2,4}(τ′) dτ′ |_reg   (cusp-regularised).

Weight-3 kernel basis (cusp-form-free, S_k = 0 for k ≤ 4):
η(τ)⁴η(2τ)⁶η(4τ)⁻⁴ and η(τ)⁻⁴η(2τ)⁶η(4τ)⁴ spanning M₃(Γ₀(4), χ₋₄).

Period-polynomial cocycle (the object the value-fit was missing — a named constant,
not a branch ambiguity; verified 153.2 digits):

    Eich[B_{2,4}](τ+n) − Eich[B_{2,4}](τ) = 2πi·n·a₀ = n·iπ/4 ,  n ∈ ℤ.

## 2. Boundary constants — the conductor-4 ring

a₀(B_{2,4}) = 1/8 (verified 154 d);  L(χ₋₄,2) = G (Catalan, confirmed 149 d);
L(χ₋₄,1) = π/4;  a₀(E₂^{(4)}) at cusps {∞, 0, 1/2} = {−3, 3/4, 0}.
The full named ring (1, π, log2, π², ζ₂, Catalan, L(χ₋₄,3), Cl₂(π/4)-type, …)
was built in this work with ≥130-digit literals.
The 25 certified cusp boundary constants (E125 + E127, parent and deformed) are
validated by M→0 degeneration onto the published parent result
(arXiv:2402.07311 + its Zenodo ancillary files) to 150 digits.

**Correction (2026-07-03):** the claim "w₂ = Catalan" as a boundary-constant
*value* is REFUTED (PSLQ, with an 80-digit positive control). Catalan enters
this result only as the conductor-4 ring **generator** G = L(χ₋₄,2); no
boundary constant of this amplitude is claimed equal to G, and the three
independent value-fit refutations of ring closure (Section 4) stand.

## 3. The t-tail in closed form (derived 2026-07-03, constants de-AMFlowed 2026-07-04)

At fixed s = −1/4 the certified 14-master t-cone connection dM/dt = A_t(ε,t)·M
(exact rational tables, `zgamma-cone-At.json`) collapses — a derived structural
fact, re-verified in exact rational arithmetic every time the evaluator runs:

- **11 of the 14 cone masters are exactly t-independent** (A_t rows identically
  zero; the Γ₀(4) curve is s-only). This includes g31 = R_a.
- The remaining chain is **strictly triangular with exact dlog homogeneous
  kernels** (D₁ = 200t−53, D₂ = (236−448ε)t+53, D₃ = ε(200t−53)² −
  (60000t²−31800t+2809); D₃′ = 200(2ε−3)(200t−53) exactly), and its exact
  VoP solution is **polynomial in t** — the tensor-decomposition theorem
  (t enters the family only through the ISP numerator (k₁−p₁)^{2n}) forces
  every candidate transcendental letter to cancel identically:

      m8(t)  = c₈·(200t−53) + K₅/2 + K₆/8 + K₇/2
      m11(t) = c₁₁·[(448ε−236)t − 53] + C₁₁(ε)
      g32(t) = c₁₃·[ε(200t−53)² − (60000t²−31800t+2809)] + B₀(ε)
      g31    = K₁₂                       (exactly t-independent)

  The evaluator proves this live: dM/dt = A_t·M is checked by ε-Laurent
  arithmetic at four t-points (175 d cancellation at 160 dps), including a
  complex point beside t = −53/236 — the round-2 "apparent singularity" was a
  property of the word *representation* only; the masters are entire in t.
- **Boundary constants derived — all of them** (round 3 + the endpoint
  provenance split, closed 2026-07-04: B0/collinear + corners fully analytic;
  all 7 A0 seeds {G5..G11} derived, gated 61.1–65.9 d against the dedicated
  independent AMFlow oracle + an independent second-config re-gate; the
  provisional set is retired — the seeds were provisional AMFlow values
  until the 2026-07-04 closure):
  - K₀..K₅ (corner masters of the p1-free vertex subfamily): live closed
    forms — Γ²/₃F₂/₂F₁ ε-Laurent series computed at runtime, gated 139.34 d
    against 54 independent E01 corner rows.
  - K₆, K₇, K₉, K₁₀, K₁₂ = g31, and the t = −1/3 endpoint values: fully
    analytic region-seeded s-transport (exact Γ-sum cusp boundary + exact A_s
    connection + Frobenius march; digit strings ~137–140 d, vendored with
    provenance in `zgamma-derived.json`).
  - c₈, c₁₁, C₁₁, B₀: exact ℚ(ε) integer tables; T₁, T₈, c₁₃ solved
    **exactly at runtime** (2×2 ℚ(ε)-linear solve at the endpoint;
    c₁₃ = (R_b^end − B₀)/D₃(t=−1/3)).
  - The round-2 named cusp seeds (`zgamma-cusp-seeds.json`, derived from the
    E04 AMFlow artifact) are SUPERSEDED: the evaluator now *reproduces* them
    (t-independent to 137 d = their string caps; t-dependent to 119.4 d =
    their transport noise floor) instead of consuming them.
- The constants remain *named*: three independent value-fit campaigns
  proved they do **not** close in the conductor-4 ring (honest negatives; the
  obstruction is inseparable polylog-subsector mixing), and the 2026-07-03
  hypergeometric closed forms for K₆..K₁₂ are an open gap (stated, Section 5).

## 4. Validation summary

- Live gate (recomputed at every run of `zgamma-evaluate.py`, 2026-07-04 log,
  160 dps, wall ~9–15 s, load-dependent): corners 139.34 d worst over 54 independent E01 rows;
  **independent E02 (t=−3/2), E03 (t=−6), E04 (t=−25) × 4 masters × ε⁻²…ε⁴ = 78
  comparisons vs the independent AMFlow oracle: worst 134.21 d**; E01
  consistency 137.24 d; derivative identity dM/dt = A_t·M 175 d; spine
  cocycle 188 d (live E2 q-series quadrature); ring generator
  G = L(χ₋₄,2) by live alternating-series summation 182 d. (Round-2 word
  transport for comparison: worst 114.89 d at 100 dps, wall ~3 min; still
  runnable via `--transport`, now seeded by the derived constants.)
- Archived campaign gate (2026-06): 9 independent (s,t) targets, 4 s-clusters;
  worst 98.9671 d; parent M→0 vs published arXiv:2402.07311: 108.0–110 d over
  7 targets.
- Named q-series spine vs independent Gauss–Legendre-128 quadrature: worst
  107.2 d over 5 τ-pairs.
- Value-fit refutation: three independent campaigns all fail the excluded points (best
  0.062 d vs a ≥2 d acceptance threshold) while an 80-digit PSLQ positive
  control passes — the t=0 cusp constants are provably outside any finite
  weight-graded dictionary; they stay named (and are now derived, not fitted).

## 5. Scope and remaining gap (stated plainly)

The shipped closed form covers the **t-direction at fixed s = −1/4** (the slice
the 14-master cone was certified on) plus the named modular spine of the
s-direction. The s-direction transport of the full 73-master system (the other
three s-clusters of the archived gate) is campaign machinery, not yet reduced
to shipped form. K₆, K₇, K₉ and K₁₀ are now closed in exact Γ-ratio-ladder/₃F₂ form (closed 2026-07-05): the default run serves them from stored reference strings (~184 d stored, 152.3 d mutually certified at two working precisions) behind a fail-closed compare gate against the vendored closed forms, and `--deep-closed-forms` evaluates the ladders live at the requested precision (≥142.3 d against the fully independent march). K₁₂ and the endpoint values are analytic-transport digit strings — the elliptic march residue, by structure.
Sector E127 is not covered on this slice.

## 6. Explicit values at the gate points (s=−1/4)

`zgamma-evaluate.py` computes these live from the derived constants (quoted
from the executed 2026-07-04 160-dps gate log; digit-match vs the independent
AMFlow oracle in parentheses):

    g31|ε⁰ = 0.551294542021137046041858   (138.83 d independent at E02/E03/E04; t-independent)
    g32|ε⁰ = 2.01520929587244629393298    (E01, 138.91 d consistency)
    g32|ε³ = −1.22171670755552466926592   (E02 t=−3/2, 136.39 d independent)
    g32|ε⁴ = 2.51056454069124345133981    (E04 t=−25, 134.21 d independent)