Standard · direct manipulation

PT calculators

Three interactive tools let the reader manipulate the arithmetic cascade of the sieve in the browser: anomalous dimensions, branch holonomies and the bare-to-dressed reading of α_EM.

The goal is not to replace the companion scripts, but to make the formulas legible: move μ, change p, observe when a prime becomes active, and see how q+ and q− carry two distinct readings.

Plain JavaScript calculations, with no additional fit parameter.

Calculator 1

γ_p(μ) — anomalous dimensions and activity

Canonical formula (T6): γ_p = 4·q^p−1·(1−δ_p) ÷ (μ·δ_p·(2−δ_p)) with q = 1 − 2/μ and δ_p = (1−q^p)/p. A prime is active if γ_p > s = 1/2.

μ = 15

p	δ_p	γ_p	status

Calculator 2

sin²(θ_p) — holonomy on both branches

Algebraic identity (T6): sin²(θ_p) = δ_p·(2 − δ_p). Branch q₊ = 1 − 2/μ (couplings, leptons, α_EM); branch q₋ = e^−1/μ (geometry, quarks, CKM).

p = 3 μ = 15

Branch q₊ (couplings)

q₊ = —

δ_p = —

sin²θ_p = —

Branch q₋ (geometry)

q₋ = —

δ_p = —

sin²θ_p = —

Calculator 3

α_EM — from bare product to dressed value

Computed at μ* = 15, branch q₊. The BA5 product α_bare = ∏ sin²θ_p over {3, 5, 7} gives 1/136.28. Dressing through the binary channel p = 2 closes the gap to 1/137.036.

sin²θ₃

—

sin²θ₅

—

sin²θ₇

—

Bare product: α_bare = sin²θ₃·sin²θ₅·sin²θ₇ α_bare = — = 1/—

Binary-channel dressing (F(2), R51): F(2) = 0.7583

α_dressed⁻¹ = α_bare⁻¹ + F(2): 1/α ≈ —

Full PT (spiral + echo + 2-loop, ch. 10): 1/α = 137.035 999 083

CODATA 2022: 1/α = 137.035 999 084

PT — CODATA gap: 0.004 ppb. Zero parameters fitted at any step. For the feedback spiral and echo terms, see monograph chapter 10.

Want more depth? Theorem T6 (holonomy), T5 (fixed point), essay on α_EM, verification scripts.