Direction 2 — Lattice RG Flow (easy guide): zoom from the DANode grid to everyday scales

What this page is (one minute). RG = Renormalization Group = how “constants” change when you zoom energy/distance. SPT starts from a fixed lattice of DANodes. Direction 2 builds a pipeline: lattice hop → UV scale → tree boundary from Direction 1 → Bagua log window → structural IR shift for sin²θ_W — without hand-pasting 0.00042. Script PASS: spt_rg_lattice_flow.py.

Glossary first. /glossary?q=RG · group /glossary#spt-rg-lattice. Terms: RG, β-function, running, μ scale, hopping, δ_RG.

Honest scope. We do deliver a Bagua-clean structural RG pipeline and closed forms (Tier-B EXACT algebra). We do not claim a full derivation of SM β-coefficients b₁,b₂,b₃ from the SPT Action, nor that 137 is predicted by dynamical integration alone. Full Action β remains open (Tier-A research).

§0 Terminology table (→ glossary)

Term	Plain meaning	Glossary
RG	How couplings change when you zoom scale	Open
β-function	β(g) = dg / d ln μ — the zoom speed of a coupling	Open
Running	A “constant” that depends on energy scale μ	Open
μ (scale)	Energy/resolution dial you are using	Open
Hopping / c	Jump rate between DANodes; c = 2Ja/ℏ	Open
μ_UV	Highest lattice frequency ∼ c/a = 2J/ℏ	Open
t_ = 32*	Bagua log window 4·Q₃ of ln(μ_high/μ_low)	Open
δ_RG	Structural one-loop shift 1/(1024π) ≈ 3.1×10⁻⁴	Open
1/α = 137	IR combinatorial spine — not a Planck miracle	Open
Tree 3/13	High-scale sin²θ_W from Direction 1 (boundary)	Open

§1 Picture: zooming the DANode fabric

Imagine the universe as a fixed mesh of DANodes (like a fabric of knots). Up close (Planck), you see every knot and every hop. Far away (atoms, labs), the fabric looks smooth. RG is the math of that zoom: the effective strength of forces can look different at each zoom level.

SPT’s speed of light is not an extra postulate: excitations hop to neighbours at c = 2Ja/ℏ. The highest “note” the lattice can play is μ_UV ∼ c/a. That is the natural UV end of the RG pipe. The IR end still has the combinatorial fact 1/α = 137 (count of Q₇ structure) — kept separate from dynamical running.

Keys vs zoom. Direction 1 fixed the lock size (shell W=13) for electroweak angles. Direction 2 asks: when we zoom from the high-scale lock setting 3/13 down to lab energies, how much does sin²θ_W move — using only Bagua integers + π, not a hand-typed 0.00042.

§2 The RG pipeline (six steps)

1. Hop

c = 2Ja/ℏ — signal front on the grid

2. UV dial

μ_UV = c/a = 2J/ℏ — highest lattice scale

3. IR spine

1/α_IR = Q₇+Q₃+1 = 137 combinatorial

4. Boundary

sin²θ_W(μ_high) = 3/13 from Direction 1

5. Zoom window

t_* = 4·Q₃ = 32 exact e-folds of ln μ

6. Structural shift

δ_RG = t_*/(2π·Q₇²) = 1/(1024π); add to 3/13

python

t_star = 4 * 8                          # 32
delta_RG = t_star / (2 * pi * 128**2)   # 1/(1024*pi)
sin2_IR = Rational(3, 13) + delta_RG    # ≈ 0.23108
# PDG 0.23122 → Δ ≈ 0.06% (OUTPUT check only)

Key lines from spt_rg_lattice_flow.py

§3 SymPy evidence

SymPy verify — download for offline testSYMPY ✓

Direction-2 lattice RG scripts

scripts/spt_rg_lattice_flow.py

Lattice RG flow (main) — R1–R7: hopping, 137, 3/13, t_*=32, δ_RG, separation

200 LOCDownload

scripts/spt_rg_substrate_framework.py

Phase 9 RG honesty frame — Earlier separation 137 IR vs open UV

100 LOCDownload

scripts/spt_master_map_f_from_action.py

Direction 1 boundary 3/13 — Tree sin²θ_W from Action gauge content

220 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_rg_lattice_flow.py && python3 scripts/spt_rg_substrate_framework.py && python3 scripts/spt_master_map_f_from_action.py

Or quick-verify with AI (Grok / Claude / ChatGPT)

Don't want to install Python? Paste the prompt straight into Grok / Claude / ChatGPT / Gemini — the AI fetches the public script URL below and independently verifies each assertion in ~30 s. Open grok.com or claude.ai , paste, send.

⚠️ AI can be wrong — running the Python above is the only 100% certain check. Full AI guide →

Inputs: Bagua integers + π/√ only — no CODATA, no PDG, no calibration (Tier B). SymPy-verified as exact fractions (not floating-point). See full context at /theory/sympy-breakthrough-2026.

§4 Numbers (OUTPUT vs PDG)

Quantity	SPT closed form	Value	Tier
1/α_IR	Q₇+Q₃+1	137 exact	B-EXACT
sin²θ_W tree	3/13	0.230769	B-EXACT
t_*	4·Q₃	32 exact	B-EXACT
δ_RG	1/(1024π)	≈ 3.11×10⁻⁴	B-EXACT form
sin²θ_W IR struct	3/13 + 1/(1024π)	≈ 0.23108 vs PDG 0.23122 (Δ 0.06%)	B-PASS band

§5 Why it matters + practical use

Before: people could confuse “137” with a Planck-scale magic number, and θ_W running was a hand correction.
After Direction 2: explicit lattice→IR pipeline; δ_RG is a closed form; PDG only checks the result.
Practical: peer narrative (“we have a structural RG sketch”); experiment weighting still with Direction-1 shells; not a device recipe.
Importance: strategic ~7.5/10 on the theory path (connects combinatorics to running); engineering ~2/10; full SM β from Action still the long climb.

§6 What stayed open — and Direction 2b progress

Đợt 55 / Direction 2b advances all three former open doors on /theory/spt-rg-open-closure (script spt_rg_action_beta_alpha.py): 1. b₁,b₂,b₃ = Bagua EXACT forms 41/10, −19/6, −7 (path-integral why still research). 2. α₁,α₂ 1-loop flow over t_*=32; 137 spine kept combinatorial. 3. Substrate vs continuum separated; equivalence E to Clay a→0 still open. Precision θ_W channel remains δ_RG on this page — not naive long absolute α-flow.

§7 Related pages

Phase 9 hub: /theory/spt-phase9-core-index
Older RG honesty frame: /theory/spt-rg-from-substrate
Direction 1 (boundary 3/13): /theory/spt-master-map-f-from-action
⭐ Direction 2b — open gaps: /theory/spt-rg-open-closure
Glossary RG group: /glossary#spt-rg-lattice

Bottom line. RG from the lattice = zoom math on the DANode fabric. Direction 2 ships a Bagua-clean pipeline and δ_RG = 1/(1024π) so θ_W running is no longer a magic paste — while full Action β stays an open mountain.