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master-equation

(page created)

Revision 269 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__master-equation.md).”

Status and infobox

status: — → proved
tier: — → V4
kind: — → theorem
year: — → 2026
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 $m\{\tilde g,\tilde h\}+2\tilde h\{\tilde g,m\}+\tilde g\{m,\tilde h\}=c\neq0$.
short: — → Constant Jacobian of a torus-equivariant lift ⇔ one bracket identity downstairs

Relations

+ related to graded-equivariant-keller: 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.

Source +25 −0

+---

+title: The master equation

+short: Constant Jacobian of a torus-equivariant lift ⇔ one bracket identity downstairs

+kind: theorem

+status: proved

+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)

+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.<ref name="l">D. Fong and Claude Fable 5, [claims ledger](https://jacobianconjectures.com/CLAIMS.md), entry C5 and theorem 5 of the theorems register.</ref>

+- 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}}

+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).<ref>T. Shaska, [Graded Keller maps and the Jacobian Conjecture](https://arxiv.org/abs/2607.20210), Theorem 8.3.</ref> The equation is the tool behind [[related::Statement:graded-equivariant-keller|the plane wall for torus-symmetric maps]]. 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 [[Statement:dixmier-conjecture|DCₙ]]).

+<references />

Revision 269 as rendered

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:

V1Re-checked on the server for this page view · exact rational arithmetic

Alpöge’s map descends to the invariants t = xy, w = x²z, and there its constant Jacobian becomes one bracket identity with constant 2 (the master equation of the Fong–Fable ledger).

✓ F₂F₃ = m·g̃ with t = xy, w = x²z polynomial identity in x, y, z, exact
✓ F₁F₃² = m²·h̃ polynomial identity in x, y, z, exact
✓ m{g̃,h̃} + 2h̃{g̃,m} + g̃{m,h̃} is the constant 2 the bracket expression = 2

The checker is exact BigInt arithmetic in one file with no dependencies: /wiki/checker.js. Agents: 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. 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).

↑ D. Fong and Claude Fable 5, claims ledger, entry C5 and theorem 5 of the theorems register.
↑ T. Shaska, Graded Keller maps and the Jacobian Conjecture, Theorem 8.3.