Pullback of a closed form is closed. If [ω] ≠ 0 in H²_dR(𝒯), the integral on a homologous two-cycle is invariant. d² = 0 forbids ω = d(dℳ) as a nontrivial class.
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Global charge, local scalar
A closed two-form ω on thermodynamic state space pulls back to F_Q = Φ*ω. Because d_𝒯 ω = 0, d F_Q = 0. The integral on a two-cycle is topological. The pointwise invariant 𝒥_Q = ¼ F_Q² is not. Mixing them is the error the paper exists to prevent.
QRTCH[Σ₂] = ∫Σ₂ Φ*ω
𝒥Q = ¼ FQ μν FQμν
Q / 4π²
0.000
global / topological
⟨𝒥_Q⟩
0.000
local scalar
max 𝒥_Q
0.000
‖H‖
0.000
λ_Q F_Q source
Pullback density on Σ₂
Target T² · winding (n, m)
Winding
n1
m1
Freeze Y
Topological λ_Q
That thermodynamic state space carries such a class, that 𝒥_Q is the physically relevant local invariant, and that A should depend on it, are model assumptions — not theorems.