PKG-SIDE24-001
Peer-review capsule
Reviewed fixed-side scope

Bargmann–Fock · 3D torus · H₀

Side-24 Persistence Coefficient Explorer

Live map of the closed theorem: short-lifetime density ν ∼ c · ℓ⁻¹ᐟ³, the periodization correction P₃(24) e⁻²⁸⁸, and the exact scope firewall from your Drive package.

Planar coefficient c₍₃,∞₎

0.041775932

Relative @ L=24

-1.486e-118

Remainder bound

|ε₂₄| < 10⁻¹⁸⁰

Closed mathematical statement

From 00_START_HERE and LS-CLS-077 — absolute coefficient chain at reviewed finite-side scope.

PKG-SIDE24-001
2026-07-28
d = 3 · L = 24 · H₀

Main theorem

For the normalized cubic-periodized three-dimensional Bargmann–Fock field on the flat torus of side L = 24, the first-moment density of finite nonessential superlevel H₀ persistence lifetimes satisfies

ν(3,24)(ℓ) = c(3,24) · ℓ−1/3 (1 + o(1)),   ℓ → 0⁺

c₍₃,∞₎ planar

0.04177593184

0.0417759318405983…

relative @ L=24

-1.4862e-118

P₃(24) e⁻²⁸⁸

c₍₃,₂₄₎

0.04177593184

from relative enclosure

Finite-side relative correction

c(3,24) / c(3,∞) − 1 = P₃(24) e−288 + ε₂₄

P₃(24) = -620813376/35 ,   |ε₂₄| < 10−180

The factor e−288 is the leading nearest-image weight: with covariance images exp(−|x + nL|²/2), the six faces at distance L = 24 contribute exp(−L²/2) = exp(−288). The next shell is O(e−576), which sits well under the remainder bound.

Externally closed components

  • Local coefficient reduction
  • Elder-rule selected / all-typed ID
  • Selection rate 1 − pᵣ = O(r³)
  • Absolute Kac–Rice + lifetime norms
  • Ordered-pair + directed-sphere multiplicity
  • Finite-side relative correction

Planar closed form

c(3,∞) = 21/6 31/3 (29√6 − 36) Γ(1/6) / (432 π5/2)

Status: Mathematics closed at reviewed fixed-side scope. Sources linked from Drive package folder PKG-SIDE24-001.