There is a degree-32 Lehmer equality target with a point over \(\mathbb Z_2\) inside one explicit odd-transvection family.
Not merely solutions modulo many powers of two. An actual 2-adic point.
The construction starts with a product \(T(a)\) of \(32\) elementary transvections, where every parameter is odd, and forms
$$ B(a)=T(a)^{\mathsf T}T(a). $$
For every integral specialization, \(B(a)\) is symmetric, positive definite, and unimodular. The characteristic polynomial of the associated anti-symplectic matrix is controlled by \(16\) independent coefficient equations:
$$ \Phi:\mathbb Z_2^{32}\longrightarrow\mathbb Z_2^{16}. $$
Earlier lifting found exact odd solutions through modulus \(2^{30}\), but finite compatibility by itself does not produce a point in \(\mathbb Z_2\). The missing datum was the singular Jacobian.
The Jacobian has total 2-adic defect eight#
At the retained solution \(a\), reconstruct the \(16\times32\) Jacobian modulo \(2^{30}\) by division-free finite differences. The 2-adic Smith valuations of its nonzero invariant factors are
$$ \boxed{ (0^{13},1,3,4). } $$
So thirteen directions are units, and the remaining three singular directions contribute total determinantal valuation
$$ s=1+3+4=8. $$
This is not inferred from the Smith form alone. The retained columns
$$ 0,1,5,6,8,11,13,15,17,18,19,21,22,23,25,29 $$
give an explicit \(16\times16\) minor with valuation exactly eight.
The residual coefficient error vanishes modulo \(2^{30}\). Multivariate Hensel lifting needs
$$ v_2(\Phi(a)-q)>2v_2(\det J). $$
Here that is the tiny inequality
$$ \boxed{30>2\cdot8.} $$
Therefore the finite solution lifts to
$$ \widetilde a\in\mathbb Z_2^{32}, \qquad \Phi(\widetilde a)=q. $$
The correction is 2-adically smaller than the starting vector, so every parameter remains odd.
What this does and does not say#
The old sign-restricted family, with parameters only in \(\{-1,1\}\), misses the target already modulo \(16\). Allowing arbitrary odd 2-adic parameters completely changes the local picture: the equality fiber is not empty at two.
That removes every finite 2-adic obstruction for this target in this chart. It does not produce a rational or integral parameter vector, and it does not settle the degree-32 comparison. Any remaining failure must come from global arithmetic descent, another prime, the archimedean condition, or a compatibility obstruction not visible in \(\mathbb Z_2\).
I like how little is needed at the final step. Thirty binary digits looked like a long computational ladder. Once the Jacobian defect was measured, they became a theorem because thirty is simply bigger than sixteen.
Notebook references: R-0391, R-0442