# Single-u-Mode Cyclic Rigidity Theorem

**Date:** 2026-07-21
**Status:** proved algebraically; exact checker PASS; external expert review not yet obtained.

Let `k` be a characteristic-zero field, let integers `m,K >= 1`, and let

`g(u,y) = a0 + u^K a(y)`, with `a0 in k*` and `a(y) in k[y]`.

Suppose `H in k[u,y]` and `c in k*` satisfy

`m u H_u g_y - H_y(g + m u g_u) = c`.                         (1)

Then `a(y)=0`. Hence `g=a0` is constant and the corresponding cyclic-equivariant map

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

is a polynomial automorphism.

## Proof

Write

`H(u,y)=sum_{n=0}^M h_n(y)u^n`, extending `h_n=0` for `n>M`.

The coefficient of `u^0` in (1) is

`-a0 h_0'(y)=c`,

so `h_0` is affine linear and `h_0'=-c/a0 != 0`.

For every `n>=1`, the coefficient of `u^n` is

`-a0 h_n' + 1_{n>=K}[m(n-K)h_{n-K}a'-(1+mK)a h_{n-K}']=0`.    (2)

If `n` is not divisible by `K`, induction from (2) gives `h_n'=0`; these coefficients are constants and do not affect the argument. On the residue-zero chain define `f_l=h_{lK}`. Then

`a0 f_l' = mK(l-1)a'f_{l-1}-(1+mK)a f_{l-1}'`.               (3)

Assume `a != 0` and put `d=deg a`. We claim

`deg f_l = l d + 1` for every `l>=0`.

It is true for `l=0` because `f_0=h_0` is affine linear with nonzero slope. Suppose it holds for `l-1`, and write leading terms

`a=A y^d`,  `f_{l-1}=B y^e`,  where `e=(l-1)d+1` and `A B !=0`.

The leading coefficient on the right of (3) is

`A B [mK(l-1)d-(1+mK)e]`

`= -A B [(l-1)d+1+mK]`,

which is nonzero in characteristic zero. Therefore

`deg f_l' = l d`, hence `deg f_l = l d+1` and `f_l !=0`.

Thus infinitely many coefficients `h_{lK}` are nonzero, contradicting `H in k[u,y]`. Hence `a=0`.

With `g=a0`, equation (1) gives `H_y=-c/a0`, so

`H(u,y)=-(c/a0)y+p(u)`.

The map has explicit inverse

`x=F_2/a0`,

`y=-(a0/c)[F_1-p((F_2/a0)^m)]`.

QED.

## Verification

Run:

`python verify_single_u_mode_theorem.py`

The verifier checks the coefficient recurrence for 35 `(m,K)` pairs and stress-tests the degree law on 210 exact rational cases. Current certificate SHA-256:

`64eda00925854422decb5b17249ae08bff94552cdb6614b886e62683af708a24`

## Scope

This is an infinite-class rigidity theorem, not a proof of `JC_2`. It covers arbitrary `m`, arbitrary active u-exponent `K`, arbitrary polynomial `a(y)`, and unbounded degree/support for `H`, but only `g` with one nonconstant u-mode beyond its forced constant boundary term.