# Full Cyclic Rigidity Theorem

**Date:** 2026-07-21
**Status:** complete internal proof; exact symbolic and adversarial checks PASS; independent human peer review not yet obtained.

## Theorem

Let `k` be a characteristic-zero field, let `m>=1`, and let `H,g in k[u,y]`. If

`m u H_u g_y - H_y(g+m u g_u)=c`, with `c in k*`,                (1)

then `g in k*`. Consequently every polynomial map

`F(x,y)=(H(x^m,y), x g(x^m,y))`

with nonzero constant Jacobian is a polynomial automorphism.

For `m>=2`, this proves the cyclic-equivariant Jacobian conjecture for the diagonal source and target action `(x,y)->(zeta*x,y)`.

## Proof

### 1. Boundary normalization

Write `g=sum_{j=0}^K a_j(y)u^j`. Evaluating (1) at `u=0` gives

`-H_y(0,y)a_0(y)=c`.

Both factors are polynomials whose product is a nonzero constant. Hence

`a_0 in k*`, and `H(0,y)` is affine linear with nonzero y-slope.

Assume for contradiction that `K>=1`.

### 2. The highest Newton slope of g

Define

`rho=max_{j>=1} deg(a_j)/j`.

Write `rho=p/q` in lowest terms, with `p>=0`, `q>=1`. Give the variables the rational weight

`wt(y)=1`, `wt(u)=-rho`.

The weight-zero part of `g` is a nonconstant polynomial in

`t=u^q y^p`:

`g_[0]=A(t)`,

with `A(0)=a_0 !=0` and `deg A>=1`. Every other term of `g` has strictly negative weight.

Let `sigma` be the largest weight occurring in `H`, and let `H_[sigma]` be the corresponding weighted-homogeneous component. Since `H(0,y)` has a nonzero linear term, `sigma>=1`.

Weighted differentiation gives

`wt(partial_u)=rho`, `wt(partial_y)=-1`.

Therefore the highest-weight component of the left side of (1) has weight `sigma-1`. Since `sigma>=1`, terms of g with negative weight cannot contribute to this component. Because the right side has weight zero:

- if `sigma>1`, the highest component must vanish;
- if `sigma=1`, it must equal the constant `c`.

### 3. Parametrization of the top edge of H

Choose the unique integers `0<=r<q` and `s>=0` for which

`s-rho*r=sigma`.

Then

`H_[sigma]=u^r y^s Phi(t)`

for some nonzero `Phi in k[t]`. Since `sigma>=1`, one has `s>0`.

A direct chain-rule computation gives

`m u (H_[sigma])_u (g_[0])_y
 -(H_[sigma])_y(g_[0]+m u(g_[0])_u)`

`=u^r y^(s-1) B(t)`,

where

`B(t)=-p t A Phi' - s A Phi - m q sigma t A' Phi`.             (2)

### 4. The edge obstruction

If `sigma>1`, the top component must vanish, so `B=0`.

Let `N=deg Phi>=0` and `D=deg A>=1`. The coefficient of `t^(N+D)` in `B` is the nonzero product of the leading coefficients of `A` and `Phi` times

`-(pN+s+m q sigma D)`.

Every term inside the parentheses is nonnegative, and `s>0`. Hence this coefficient cannot vanish in characteristic zero. Contradiction.

If `sigma=1`, the top component must be the constant `c`. But `u^r y^(s-1)B(t)` can be constant only when `r=0`, `s=1`, and `B(t)` is constant. Since `D>=1`, equation (2) has degree `N+D`, and its top coefficient is again

`-(pN+1+m q D)`

times nonzero leading coefficients. It is therefore nonzero, so `B` is nonconstant. Contradiction.

Thus `K=0`, so `g=a_0 in k*`.

### 5. Explicit inverse

Equation (1) now reduces to

`-a_0 H_y=c`,

hence

`H(u,y)=-(c/a_0)y+h(u)`.

Writing target coordinates `(P,Q)=F(x,y)`, the inverse is

`x=Q/a_0`,

`y=-(a_0/c)[P-h((Q/a_0)^m)]`.

Therefore `F` is a polynomial automorphism. QED.

## Corollaries

1. The reflection-equivariant class from Report II is rigid without any bounded-degree assumption.
2. Every odd-order cyclic class is rigid as well; Miyanishi's even-order theorem is not needed for this normal form.
3. The divisibility equation

   `u^(m-1) Jac(H,K)^m = c^m K^(m-1)`, with `K=u g^m`,

   has no nonconstant-Jacobian solutions arising from this cyclic lift.
4. A two-dimensional Keller counterexample, if one exists, cannot commute with any nontrivial linear cyclic pseudoreflection after polynomial conjugacy into this normal form.

## Machine verification

`python verify_full_cyclic_rigidity.py`

Current result:

- 28 exact determinant-identity checks;
- symbolic verification of equation (2);
- 96,840 exact positivity checks for the terminal coefficient;
- 120 arbitrary multimode `g` exclusions through `deg_u H<=7`, `deg_y H<=9`, certified modulo prime 1,000,003.

Certificate SHA-256:

`11a02fb467545f48b65e8c24cd8c5bed2c45341bdbd7816060a0de2534f96460`

## Scope and novelty caution

This is not a proof of the unrestricted two-dimensional Jacobian conjecture. It closes the cyclic-equivariant normal form above. A targeted web search found Miyanishi's theorem for even-order finite-group equivariance but did not find this all-m cyclic Newton-edge proof. A comprehensive priority claim requires specialist literature review and human peer review.