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INDEPENDENT SYMBOLIC MODE-SIGN DERIVATION
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Line 1 — Matsubara mixed derivative:
  Δτ = τ'−τ
  ∂_τ ∂_τ' exp[i n κ Δτ] = kappa**2*n**2
  MATSUBARA_SIGN = PLUS

Line 2 — Euclidean raising / parallel propagator:
  lim g^{ττ'} = +1/f
  TIME_PARALLEL_PROPAGATOR_SIGN = PLUS

Line 3 — Mode assembly:
  Analytic: +1/(4*pi**2) * sum_l (2*l+1) * sum_{n>=1} (n**2 * kappa**2 / f) * g_nl
  Printed:  -1/(4*pi**2) * sum_l (2*l+1) * sum_{n>=1} (n**2 * kappa**2 / f) * mathsf{g}_nl  - 1/(4*pi**2) * {geo brace}

INDEPENDENT_SYMBOLIC_MODE_SIGN = DERIVED

Independent symbolic derivation yields DERIVED: analytic [g^{tau tau'} W_,tau tau']_mode = PLUS structure; printed formula carries a global MINUS relative to that.
