# Technical Report IV — The Symmetric-Product Origin of the Jacobian Counterexample

Reproducibility bundle. One self-contained exact-arithmetic verifier; no floating
point is used in any assertion. Total replay ≈ a few seconds.

## Contents
- `verify_sym3.py`   — the verifier. Proves checks A–G below and writes `certificate.json`.
- `certificate.json` — machine-generated result ledger (produced by the verifier).
- `requirements.txt` — `sympy` only.
- `SHA256SUMS.txt`   — SHA-256 of every file in this bundle.
- `index.html`       — artifact browser.

## One-command reproduction
```
pip install -r requirements.txt
python verify_sym3.py
```
Expected: `"status": "PASS"` with checks A_keller_det … G_k1_row.

## What is proved (all exact / symbolic)
- **A** Φ₃ = `construct(1,2)` : 𝔸³ → 𝔸³ has det J ≡ −1/2 (nonzero constant ⇒ Keller).
- **B** Φ₃ is equivariant for the torus x↦tx, y↦t⁻¹y, z↦t⁻²z with target weights (−2,−1,1).
- **C** The generic fiber is 3 points whose x-coordinates are the roots of an explicit
  **depressed** (trace-zero) cubic x³ + C₁x + C₀ attached to the image (a,b,c), with
  D = 108a²c² − 36abc + 8a + 4b³c − b², C₁ = (1−3bc)/D, C₀ = −c/D. The vanishing of the
  x² coefficient is the fingerprint of "unordered triple of roots": Φ₃ is the affine
  insert map on Sym³(ℙ¹).
- **D** Explicit rational non-injectivity witness:
  Φ₃(−1/2, 7, 25) = Φ₃(1/2, 1, 1) = (15/2, 8, 0), two distinct rational points (a ±1/2 fold).
- **E** Alpöge's July 20 2026 map (det −2) is generic-degree-3, torus-(1,−1,−2)-equivariant
  and non-injective — the same graded-gauge class as Φ₃ (Report II, Theorem 6). Hence
  Alpöge's map **is** the affine Sym³ insert map, up to gauge.
- **F** Dimension-two firewall: the construction has a (d−k) pole; demanding a
  dimension-two / generic-degree-2 member forces d=k, giving coefficient (k+1)/(d−k)=∞
  and `construct(k,k)` fails. No n=2 member exists — consistent with JC₂ remaining open.
- **G** The k=1 row (generic degrees 3…7): det −1/2 and torus-equivariant throughout.
  **Honest scope:** these are maps 𝔸³→𝔸³; only n=3 (dim = degree = 3) is the genuine
  Sym³ insert map. Higher members are dimension-3 power-weighted lifts (Report III),
  NOT Sym^{d+1} insert maps.

## Provenance & honesty
The base counterexample is due to L. Alpöge with Claude Fable 5 (July 20, 2026). This
report supplies a geometric *why* (the missing "story"), building on AGNT Labs Reports
I–III. No expert human review yet; published to solicit it. The general-dimension Sym^n
insert-Keller construction (n ≥ 4, genuine dimension n) is an explicitly stated open
frontier, not a claimed result.