Bilinearity of the traced mixed coordinate mixedB #
The degree-one pairing mixedB t x y = (heisMarking t x y).tameValue.z + (…).wildValue.z is
bilinear in the offsets (x, y). Via bridge_tame/bridge_wild this reduces to bilinearity
of (stokesEval c x y r).z for an arbitrary free-group word r, which is an induction on r
using the HeisLift coordinate cocycle rules: the .a-coordinate depends only on x, the
.l-coordinate only on y, the .g-coordinate on neither, and the .z-coordinate is the
bilinear cross-term Σ λ_left(a_right).
This is the general-offset toolkit consumed by the trivial-module Gram matrix (the Prop. 5.15 proof part i) and the ramified mixed Hessian (the §5 proof layer).
The .a-coordinate of a Stokes evaluation is independent of the dual offsets y.
The .l-coordinate of a Stokes evaluation is independent of the primal offsets x.
Canonical form of stokesEval_a_indep (dual offsets set to 0).
Canonical form of stokesEval_l_indep (primal offsets set to 0).
The .a-coordinate is additive in the primal offsets x.
The .l-coordinate is additive in the dual offsets y.
.z is additive in the primal offsets x.
.z is additive in the dual offsets y.
The tame .z in closed form (trivial action) #
For a trivial C-action, fgTame = g₀⁻¹ g₁ g₀ g₁⁻² evaluates (untwisted Heisenberg) to the
bilinear form below. Crucially every term carries an index-1 (τ) factor, so it vanishes on
the split cocycles {x₁ = 0} — i.e. the trivial-module degree-one pairing is carried entirely
by the wild relator, not the tame one.
The tame .z (trivial action) vanishes on the split cocycles x₁ = 0, y₁ = 0.
Conjugation by an a=l-slice element g (g.a = g.l = 0) fixes the central coordinate, even
when g.z ≠ 0 and the base acts nontrivially: (conjP p g).z = p.z. (The two g.z contributions
cancel in ZMod 2.) Strengthens conjP_z_of_slice by dropping g.z = 0 — needed because on
general offsets g₀ = σ₂² has g₀.z = y₀(x₀) ≠ 0.
Wild .z, piece 1: the x₁^σ = σ⁻¹x₁σ factor (trivial action) #
One factor of the wild relator wildValue = h₀·u₁⁻¹·x₁^σ·c₀. Its central coordinate is the
symplectic pairing of the σ- and x₁-slots, y₃(x₀) − y₀(x₃) — the (0,3)/(3,0) Gram entries.
Wild .z, piece 2: c₀ = [d₀,z₀] ↦ 0 on cocycles. The symplectic commutator vanishes
because d₀.a = d₀.l = 0 there (liftMarking_d0_u = x₁ = 0). Same argument as heisMarking_c0_z,
with x₁ = y₁ = 0 in place of x₀-support.
Wild .z, piece 3: h₀ ↦ y₂(x₂) on cocycles — the main term, giving the (2,2) Gram
entry. Mirrors heisMarking_h0_z (the x₀-supported ↦ λ(c)) with x₁=y₁=0 in place of
x₀-support: the d₀-derived leaf coords still vanish (liftMarking_d0_u = x₁ = 0), the ω₂ in
d₀.z cancels via the dg·d₀ pair in char 2, and g₀ = σ₂² is a=l-slice (char-2 doubling) so
conjP_z_of_alzero handles its nonzero .z.
Wild .z assembly on cocycles: peeling wildValue = h₀·u₁⁻¹·(x₁^σ)·c₀ keeps the sum of the
four factor .z's plus one cross-term. The u₁.l-terms — from inv_z
(u₁⁻¹.z = u₁.z + u₁.l(u₁.a)) and from the (x₁^σ)-cross ((h₀u₁⁻¹).l = −u₁.l) — cancel
because u₁.a = (x₁^σ).a = x₃ (both
are the same primal Fox derivative), leaving the opaque u₁.z (the ω₂ scalar) confined to the
(3,3) slot.
The trivial-module degree-one pairing on cocycles:
mixedB t x y = y₂(x₂) + y₃(x₀) − y₀(x₃) + u₁.z, the tame part vanishing
(stokesEval_tame_z_trivial_cocycle) and the wild part from the peel. The opaque u₁.z is the
ω₂ scalar, confined to the (3,3) slot.
u₁.z is confined to the (3,3) slot #
u₁.z = 0 when y₃ = 0 (on cocycles): u₁ = powOmega2(x₁τ) and x₁τ has l = 0, z = 0, so
its powers do too.
u₁.z = 0 when x₃ = 0 (on cocycles), dually.