Direction A — Thickening the Action → Map F bridge (A1–A4)

One sentence. Direction 1 packaged Action content as axioms A0–A5 and forced shells 13 & 40. Direction A asks whether those shells have effective Action stories (mode space, light sector, sum-shell, force capacity) — and locks them in SymPy — without claiming a full continuum path-integral proof.

Easy guide (15 min): /theory/spt-map-f-action-easy. Depends on: F from Action · N2 · F ↔ 40. Scripts: spt_map_f_direction_a.py · _plus.py · _next.py.

Honest scope. CLOSED = discrete algebra + Action-integer stories. OPEN = continuum Wilsonian path-integral, Feynman form factors, dynamical product rule for S₄₀, full Prop_X(r) from S. Same honesty class as Direction 1.

§1 Why Direction A (after Map F)

Map F is already useful: shared locks, family kills, Explorer columns. The scientific gap peers will ask first is: is shell 13 only a counting slogan, or an effective structure read off the Action? Direction A attacks that gap in four sub-tracks without inventing Law 81.

Track	Question	Status after this page
A1	Why denominator = Q₃+n_Y?	CLOSED toy (mode space ⊕)
A2	Why solar tooth = 4 from S?	CLOSED algebra + light 2×2
A3	Why S₄₀ = Q₃+Q₅ (flavor×cosmo)?	CLOSED sum-shell uniqueness
A4	Tr(J·Ṙ) forces ↔ F locks?	CLOSED capacity + assignment

§2 A1 — Mode-space / Wilsonian toy for W

Idea. After integrating out UV yao not in the EW projection, the IR counting space of mixing channels is a direct sum:

text

H_EW  ≅  H_trigram  ⊕  H_Y
dim H_trigram = Q₃ = 8
dim H_Y       = n_Y = Q₃ − n_W = 5
dim H_EW      = W = 13

sin²θ_W^tree = n_W / dim(H_EW) = 3/13

Why not a pure sublattice of Q₇? W=13 does not divide Q₇=128, so the shell is not “every 13th vertex.” That supports the ⊕ sum of two budgets reading over a pure quotient. False shells {8, 12, n_W+n_Y=8, 16} fail the Action filter d=Q₃+n_Y.

Still open for A1. Continuum Wilsonian block-spin with a rigorous fixed-point proof that the IR measure lands on dim=13. The toy makes A4 natural; it does not replace constructive QFT.

§3 A2 — Solar tooth 4 from fermion / light sector

Four routes coincide at 4 (N2 theorem + light sector):

R-A Q₃/2 = 4 — half of trigram cell (solar projects on half Q₃).
R-B n_Y−1 = 4 — hypercharge budget minus vacuum pole.
R-C n_W+1 = 4 — next ordered role after weak generators on W.
R-D 2×2 = 4 — with m_ν1=0, effective solar 2ν block: 2 massive light states × 2 mixing slots (fermion-term story of S).

Identity I1: sin²θ₁₂ − sin²θ_W = 1/13 still holds. Full Feynman derivation of “why half Q₃” remains open (same honesty as N2 wiki).

§4 A3 — S₄₀ = Q₃ ⊕ Q₅ (flavor × cosmo sum-shell)

Reading (not product Q₃·Q₅). Two orthogonal counting spaces on the cascade ladder:

text

Flavor geometry   ~  Q₃ = 8
Cosmo ladder step ~  Q₅ = 32
S₄₀ = dim(H_flavor ⊕ H_cosmo) = 8 + 32 = 40

λ = N_gen² / S₄₀ = 9/40
h = N_gen·(Q₃+1) / S₄₀ = 27/40
h/λ = 3 EXACT

Among power-of-two sum-shells Q_i+Q_j that host (9/d,27/d) with h/λ=3, the Action-motivated pair is the adjacent ladder rungs Q₃+Q₅ (flavor shell + next cosmo shell). Dynamical “why sum not product” from full cascade EOM remains open.

§5 A4 — Rotation term, four kernels, F locks

Action term ½Tr(J·Ṙ) + unified-force Law 42: generator capacity on Q₇ is 14 (yao-pair). SM uses 12 (8+3+1); gravity uses spin-2 frame (+2 geometric) — no fifth SU(N).

Kernel	Count	Map F lock / form
SU(2) weak	3	W=13 · tooth 3 → 3/13
U(1)_Y → EM	1 (+ n_Y budget 5)	137 = Q₇+Q₃+1; feeds W
SU(3) strong	8	Layer C cascade / α_s
Spin-2 gravity	2 (frame)	Layer B/D hierarchy

Open: full propagator Prop_X(r) derived from S for each kernel (Law 42 quantitative upgrade).

§6 Bridge status matrix

Track	Closed now	Still open
A1	Mode-space ⊕ toy; false-shell filter	Continuum Wilsonian fixed point
A2	N2 four routes = 4; I1	Feynman solar form factor
A3	S₄₀ as Q₃⊕Q₅; h/λ=3	Dynamical sum rule from EOM
A4	14-cap; kernel→lock table	Prop_X(r) from S

§7 What you may claim (and may not)

May: Map F shells have mode-space / sum-shell / capacity stories consistent with Action integers; SymPy PASS; PDG is OUTPUT only.
May: Direction A reduces the “slogan only” critique relative to bare A0–A5 packaging.
May not: “Full path-integral derivation of F from S is complete.”
May not: New free parameters or Law 81.

§8 Direction A+ — Tier-B scoreboard (Đợt 61)

Next-step script spt_map_f_direction_a_plus.py upgrades discrete bridge claims and classifies tiers honestly:

Claim	Tier	Result
A1+ EW-preserving decimation keeps dim H_EW = W = 13	B-EXACT	PASS
A3+ ∂W/∂d₀ = ∂S₄₀/∂d₀ = 0 (shells ⟂ cascade slope)	B-EXACT	PASS
A4+ 137, 3/13, h/λ=3, capacity 14	B-EXACT	PASS
A4+ α_EM, α_W, hierarchy, sin²θ_W vs PDG	B-PASS	PASS (Δ < 1% / scheme band)
Bridge+ δ_RG = 1/(1024π); IR = 3/13+δ_RG	B-EXACT + B-PASS OUTPUT	PASS (IR Δ ~0.06% PDG)
Continuum Wilsonian fixed point / full path-integral	OPEN (not Tier-B)	Not claimed
α_s zero-calibration ab initio	OPEN / A-track	Not claimed here

Verdict A+: discrete Action↔F bridge upgrades PASS Tier-B (EXACT backbone + PASS outputs). Continuum QFT depth remains OPEN — same honesty as Direction 1. OVERALL script: PASS.

Still next (not Tier-B yet)

Continuum block-spin fixed point with full Euclidean invariance.
Path-integral derivation of dim H_Y = 5 from ∫𝒟φ e^{iS}.
Pre-reg FC-MF1–4 as external experimental judges.

§9 Reproduce offline

SymPy verify — download for offline testSYMPY ✓

Direction A + A+ verification

A: six stages PASS. A+: Tier-B scoreboard PASS (EXACT + PASS). Continuum OPEN.

scripts/spt_map_f_direction_a.py

Direction A (A1–A4 bridge) — Mode-space W, false shells, N2+R-D, S40, force capacity

280 LOCDownload

scripts/spt_map_f_direction_a_plus.py

Direction A+ (Tier-B next steps) — A1+ ranks, A3+ d0-decouple, A4+ couplings, δ_RG

320 LOCDownload

scripts/spt_master_map_f_from_action.py

Direction 1 — shells from axioms — A0–A5, T1–T4

220 LOCDownload

scripts/spt_master_map_n2_solar.py

N2 triple identity — n_solar=4

180 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_map_f_direction_a.py && python3 scripts/spt_map_f_direction_a_plus.py && python3 scripts/spt_master_map_f_from_action.py && python3 scripts/spt_master_map_n2_solar.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.

§10 Bottom line

Direction A thickens the Action→F bridge; Direction A+ shows the discrete upgrades pass Tier-B: - B-EXACT: W=13 rank-fixed under decimation; shells ⟂ d₀(t); 137 + 3/13 + h/λ=3 + cap 14; δ_RG=1/(1024π). - B-PASS: coupling OUTPUT bands; IR sin²θ_W ≈ 0.23108 (Δ ~0.06%). - OPEN: continuum path-integral / full Prop_X / α_s zero-calib. 0 new Laws. Judge by scripts spt_map_f_direction_a.py + _plus.py — not scoreboard volume.