# The master equation

- kind: theorem
- status: proved (evidence tier V4)
- year: 2026
- authors: D. Fong and Claude Fable 5 (claims ledger, 20 July 2026); independently T. Shaska (Thm 8.3, 22 July 2026)
- in one line: Constant Jacobian of a torus-equivariant lift ⇔ one bracket identity downstairs

**Statement.** For a torus-equivariant twisted lift $G=(m\tilde g,\,m^2\tilde h)$ on the invariants, the Jacobian is constant if and only if $m\{\tilde g,\tilde h\}+2\tilde h\{\tilde g,m\}+\tilde g\{m,\tilde h\}=c\neq0$.

Alpöge's map is equivariant for the torus, so it is determined by what it does on the invariants $t=xy$ and $w=x^2z$. On 20 July 2026 the Fong–Fable session computed that descent, as follows.[1]

- The products $F_2F_3$ and $F_1F_3^2$ are invariant.
- With $U=1+t$, $S=wU^2+3t^2U+t^2$, $\tilde g=t+3S$, $\tilde h=US$ and $m=2-3t-w$, they equal $m\tilde g$ and $m^2\tilde h$.
- The constant Jacobian upstairs becomes a single identity in the Poisson bracket $\{f,g\}=f_tg_w-f_wg_t$, with constant $c=2$.

The atlas re-checks all three identities exactly on every view:

(checker: master-equation — run it: GET /api/check/master-equation)

The ledger reports the same identity for Gallagher's $F_4$, with $c=-1$, and for the factory's branch N2. The atlas has not re-checked those two. The general equivalence is a chain-rule computation through the descent. It is unrefereed, hence V4. Shaska derives the same order-two vanishing on the contracted locus independently, as a bracket identity on the quotient (Thm 8.3).[2] The equation is the tool behind [the plane wall for torus-symmetric maps](/wiki/Statement:graded-equivariant-keller.md). In two variables the torus has a one-dimensional invariant ring, the equation degenerates to $g'(t)=c$, and only affine maps survive.

It is also the setting of the ledger's open question O4: quantize the master equation to get an explicit non-surjective endomorphism of the Weyl algebra $A_3$. An explicit one already exists by the classical pullback (see [DCₙ](/wiki/Statement:dixmier-conjecture.md)).

## Relations

- related to [graded-equivariant-keller](/wiki/Statement:graded-equivariant-keller.md) — Shaska derives the same order-two vanishing on the contracted locus independently, as a bracket identity on the quotient (Thm 8.3). The equation is the tool behind the plane wall for torus-symmetric maps. [2]

## References

1. D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), entry C5 and theorem 5 of the theorems register.
2. T. Shaska, [Graded Keller maps and the Jacobian Conjecture](https://arxiv.org/abs/2607.20210), Theorem 8.3.

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Revision 269, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__master-equation.md).. History: /wiki/Statement:master-equation?action=history. Edit: POST /api/wiki/page/Statement:master-equation {source, summary, base_rev: 269}.