Direction 2b — Closing the open RG gaps (β, α-flow, continuum frame)

§0 Terms (→ glossary)

§1 Picture: three open doors

Direction 2 built the zoom pipe (hop → window t_=32 → δ_RG). Three questions remained: (1) What are the zoom speeds b_i? (2) Can α itself flow toward 137? (3) Is this the same as continuum Clay RG?* This page answers with forms + flow + a hard separation — not marketing.

§2 OPEN-1 — b₁, b₂, b₃ as Bagua rationals

The textbook SM one-loop coefficients are exactly writable from Q₃ alone (plus fixed 6 and 10):

b1 = (5*Q3 + 1)/10   # 41/10
b2 = -(2*Q3 + 3)/6   # -19/6
b3 = -(Q3 - 1)       # -7

Meaning. We no longer “import mysterious decimals.” We state: these β-spines are Bagua integers. Remaining: derive why the prefactors 10 and 6 from a path integral on S (still research).

§3 OPEN-2 — α₁, α₂ flow over t_*=32

• Boundary: sin²θ_W=3/13 (Direction 1) + UV seed 1/α_em = Q₇ = 128.
• Flow: 1/α_i(low) = 1/α_i(high) + (b_i/2π)·t_* with Bagua b_i.
• Reconstruct 1/α_em(low) = (5/3)/α₁ + 1/α₂ (Tier-A band vs PDG M_Z).
• Combinatorial 137 spine remains exact — dynamical flow sits nearby; spine not replaced.

§4 OPEN-3 — Substrate RG ≠ continuum Clay

L_sub: a = ℓ_Pl fixed, V→∞, μ_UV = 2J/ℏ (a cancels). L_cont: a→0 on ℝ⁴ (Clay). Direction 2+2b live in L_sub. Equivalence map E: L_sub ↔ L_cont remains open — same honesty as the Clay wiki retraction. Closing substrate RG does not win the Millennium prize by itself.

§5 SymPy

pip install sympy numpy && python3 scripts/spt_rg_action_beta_alpha.py && python3 scripts/spt_rg_lattice_flow.py

§6 Practical meaning

• Research: β coefficients are no longer foreign SM decimals — they are Q₃ rationals you can audit.
• Honesty: know which channel predicts θ_W (δ_RG) vs which only bookkeeps α flow.
• Clay boundary: do not claim continuum prize from substrate RG papers.
• Not hardware. Still no engine/battery from β-functions.

§7 Still open after 2b

• Path-integral why for b_i prefactors (10, 6, …).
• Sub-percent absolute 1/α_em(M_Z) from first principles alone.
• Equivalence E: substrate ↔ classical continuum (Clay).