Direction 1 — Master Map F from the SPT Action (easy guide + uniqueness)

What this page is (one minute). Phase 9 showed that several constants share denominators 13 and 40. This page answers: why those denominators, from the SPT Action? We write 6 short axioms (A0–A5) that package the Action + lattice, then prove four theorems (T1–T4) with a downloadable SymPy script. No advanced math required to follow the story — formulas are boxed, words come first.

Start here if terms are new. Full plain guide (what F is, denominator vs numerator, importance): /theory/spt-master-map-f-explained. Glossary: /glossary#spt-master-map. This page focuses on forcing the shells (denominators) from Action. Numerators: laws N1–N6 on the explained page + spt_master_map_numerators.py.

Honest scope. We prove F under explicit Action axioms (Tier-B EXACT algebra). We do not claim a full path-integral derivation of every numerator from continuum QFT. Direction A thickens the bridge (mode-space W, solar 4, S₄₀ sum-shell, force capacity): /theory/spt-map-f-direction-a — continuum path-integral still OPEN.

§0 Terminology table (click → glossary)

Read this table first. Each row links to the live glossary (searchable). Full group: SPT — Master Map & shells.

Term	Plain meaning	Glossary
Master Map F	One rule: (sector, role) → a fraction built only from Bagua integers	Open
SPT Action (S)	The single master formula nature extremises; all laws come from S	Open
Q₇ / Q₃ / Q₅	Hypercubes: 128 / 8 / 32 vertices — the integer alphabet of SPT	Open
DANode / DAbit	Basic node of reality; 7 binary bits (Âm/Dương) per node	Open
Weinberg shell W = 13	Shared denominator for θ_W, solar θ₁₂, and GW tilt n_T	Open
Shell-40 (S₄₀)	Shared denominator Q₃+Q₅=40 for Cabibbo λ and Hubble h	Open
n_weak = 3	Three SU(2) weak generators — numerator of tree sin²θ_W	Open
n_Y = 5	Hypercharge budget = Q₃−3; enters the shell as W=Q₃+n_Y	Open
Ansatz class R	All fractions p/d with Bagua denominators {8,13,16,32,40,…}	Open
Tier-B EXACT	Exact algebraic identity under stated axioms (Δ ≡ 0)	Open
Ab initio	Built from first principles / axioms — PDG only for checking	Open
Blind prediction	A number frozen before using data to build F	Open

§0 — Learn terms once, then the theorems read like English.

§1 The story in plain language

Imagine the SPT Action as a single recipe book for the universe. The recipe says: matter lives on a fixed grid of DANodes, each with 7 DAbit. Three bits are “internal” (forces), three are space, one is time. The force part of the recipe uses the internal three bits and produces the Standard Model gauges — strong, weak, hypercharge.

From that force content we count: 3 weak generators, and a residual hypercharge budget of 5 on the 8-cell Bagua. The shell (shared denominator) is not “3+5=8” — it is “whole 8-cell + hypercharge budget 5” → 13. Every angle that only “talks through” this electroweak shell must use denominator 13. That is why θ_W, solar mixing, and a GW tilt lock together.

A second shell, 40 = 8+32, ties the Cabibbo angle to the Hubble number h. Two shells, five observables, zero free fit parameters on the right-hand side.

§1b Picture: keys vs one lock

Before Direction 1 (easy to attack): SPT looked like picking five lucky keys that each fit five different locks (θ_W, θ₁₂, n_T, λ, h). A skeptic says: “You chose fractions after seeing the data.” After Direction 1 (harder to dismiss): The Action’s force-counting cuts one lock cylinder of size 13. Three doors (electroweak angle, solar neutrino mix, primordial GW tilt) must use that same lock. A second cylinder of size 40 links Cabibbo to Hubble h. Failing one door without the others becomes difficult — that correlated failure mode is how real theories live or die.

§1c Before vs after (what changed)

Aspect	Before Direction 1	After Direction 1
What is 13?	A lucky denominator that fits several numbers	Counted from Action gauge content: W = Q₃ + n_Y
Cross-sector 13	Observed pattern (suggestive)	Theorem under axioms A0–A5 (T1–T2)
Is F unique?	Best-looking assignment	Unique in finite class R once shells fixed (T4)
SPT looks like…	A catalogue of nice ratios	A map forced by the recipe book (Action)

§2 The six Action axioms (A0–A5)

A0 Substrate

Each DANode has N_yao = 7 DAbit on Q₇.

A1 Partition

(space, time, internal) = (3,1,3) — unique for 7 (Law 59).

A2 Action form

S = kinetic + fermion + gauge·rotation + V(φ). Gauge saturates internal yao.

A3 Gauge count

SU(3)×SU(2)×U(1) → 8+3+1. n_weak=3, n_Y=Q₃−3=5.

A4 EW shell

W := Q₃ + n_Y = 8+5 = 13. Shared by all pure-EW-shell observables.

A5 Product shell

S₄₀ := Q₃ + Q₅ = 40 for flavour↔cosmo product structure.

§3 Four theorems (what SymPy checks)

T1 (shell W): W = Q₃ + n_Y = 13 and sin²θ_W^tree = n_weak/W = 3/13.
T2 (shared W): sin²θ₁₂ = 4/13, n_T = 3/13; identities n_T ≡ sin²θ_W and sin²θ₁₂ − sin²θ_W = 1/13.
T3 (shell-40): S₄₀ = 40, λ = 9/40, h = 27/40, h/λ = 3 exact.
T4 (uniqueness): Inside finite class R of Bagua rationals, once Action fixes (W,S₄₀)=(13,40), the map F with numerators (3,4,3,9,27) is the unique solution.

python

n_weak, n_Y = 3, 5          # from SU(2) + residual on Q_3
W = 8 + n_Y                 # Q_3 + n_Y  →  13
assert Rational(n_weak, W) == Rational(3, 13)
S40 = 8 + 32                # Q_3 + Q_5  →  40
assert Rational(9, S40) == Rational(9, 40)
assert Rational(27, S40) / Rational(9, S40) == 3

Key lines from spt_master_map_f_from_action.py

§4 SymPy evidence (run yourself)

SymPy verify — download for offline testSYMPY ✓

Direction-1 verification scripts

Run: python scripts/spt_master_map_f_from_action.py — expect OVERALL: PASS.

scripts/spt_master_map_f_from_action.py

F from Action + uniqueness — A0–A5 → T1–T4; PDG output check; blind list frozen

220 LOCDownload

scripts/spt_master_map_f.py

Phase 9 Master Map F (assignment) — Earlier assignment form of F (companion)

160 LOCDownload

scripts/spt_cross_sector_shells.py

Cross-sector shell theorem — Standalone shell-13 / shell-40 identities

120 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_master_map_f_from_action.py && python3 scripts/spt_master_map_f.py && python3 scripts/spt_cross_sector_shells.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.

§5 Precision vs nature (OUTPUT only)

Observable	F form	Δ vs PDG	Tier
sin²θ_W tree	3/13	0.195%	B-PASS
sin²θ₁₂	4/13	0.226%	B-PASS
n_T	3/13	identity with θ_W	B-EXACT
λ_Cabibbo	9/40	0.000%	B-EXACT match
h_Planck	27/40	0.148%	B-PASS

PDG numbers never enter the right-hand side. Blind predictions (δ_CP=270°, f_NL=3/2, r=1/300) are frozen and not used to choose shells — see blind predictions.

§6 Why this discovery matters

Before: F looked like a clever fit of Bagua fractions to data.
After Direction 1: W=13 is counted from Action gauge content; sharing W is a theorem; uniqueness holds inside class R once shells are fixed.
Scientific meaning: fail θ_W without failing n_T becomes hard — cross-sector lock is how real theories die or live.

One-sentence meaning. Direction 1 upgrades SPT from “a catalogue of beautiful numbers” to “a map with two forced locks cut from the Action (under axioms A0–A5).” That is a structural step toward a real theory — not a product launch, and not “TOE finished.”

§6b Practical meaning (honest — what it is / is not for)

Not an engineering product. Direction 1 does not give engines, batteries, FTL, or chips. Those live in separate internal vision tracks and must not be sold as “proved by F.”

Practical use	What it means for the project
Research compass	Every new constant must ask: which shell? Forced by Action or only fitted?
Weighted experiments	DUNE (θ₁₂, δ_CP), precision EW, PTA/LISA (n_T) are correlated bets — one bad shell can kill the cluster.
Peer / paper narrative	Can say: shells from gauge content + uniqueness in class R + script — stronger than “I fitted 3/13.”
Filter bad ideas	“Change Q₇ to fix one constant” breaks 137 and 3+1+3 together — save years of dead ends.
Teaching	Count generators → shell 13 → several phenomena share one lock — non-experts can follow.

§6c Importance (how big is this, really?)

Axis	Score (1–10)	Plain reading
Must-do step on a real TOE path	9	Yes — this is the right next science after Phase 9
Full ab-initio depth	6–7	Strong under axioms; not full path-integral yet
Anti-numerology credit	8	Uniqueness + derived shell help a lot
Near-term engineering	2	No product — do not oversell
Paper / peer value	7–8	If written short + honest scope
“TOE is finished”?	~4	Only one core piece — keep calm

Strategic importance ~8/10 for the framework. Product importance ~2/10. Compared with adding five more PDG-fitting Laws: Direction 1 matters more. Compared with “solving Clay”: different track — Direction 1 matters more for SPT as a theory of constants.

§6d Limits in plain words (do not self-deceive)

Axioms A0–A5 still package the Action’s structure — not every detail is derived from a continuum path integral.
The solar numerator “4” is still motivated counting, not as hard as “W=13 from gauge content.”
SymPy PASS = algebra + uniqueness in class R — not experimental confirmation.
“Zero free parameters” still means: no continuous fit knobs on the RHS — discrete axioms remain.

How to feel about this win. Correct strategic direction — not a victory lap for a finished TOE. Real assets: story + script + cross-sector locks + uniqueness. Natural next science: tighten numerators from S, or lattice RG, or pre-registered experiments (DUNE/LZ) — not Law 81–90.

§7 Compared with ordinary particle physics

Question	Standard Model	SPT Direction 1
Where does sin²θ_W come from?	Input (or GUT 3/8 + running)	3/13 from W=Q₃+n_Y under A3–A4
Link to neutrino solar angle?	Independent PMNS fit	Same shell W → 4/13
Link to GW tilt n_T?	Usually unrelated	n_T ≡ 3/13 = tree θ_W
Cabibbo vs Hubble h?	Independent	Same S₄₀; h/λ=3 exact

§8 How to kill this (and what next)

If precision data force sin²θ_W and sin²θ₁₂ off the 3/13–4/13 pair without a shared-shell rescue, T2 dies. If Cabibbo leaves 9/40 while h stays 27/40, T3 dies. Deadlines: DUNE/T2K (θ₁₂, δ_CP), LiteBIRD/PTA (n_T), CODATA/PDG updates. Next science: RG from lattice (substrate RG) and dynamics d₀(t).

§9 Where to go next on this site

Glossary group: /glossary#spt-master-map
Phase 9 hub: /theory/spt-phase9-core-index
⭐ Direction 2 — Lattice RG (zoom 3/13 → lab): /theory/spt-rg-lattice-flow
Earlier assignment page: /theory/spt-master-map-f
Shell theorem only: /theory/spt-cross-sector-shells
One Action: /theory/the-one-spt-action (or search Action in glossary)

Bottom line. Direction 1 upgrades Master Map F from “nice fractions” to “shells counted from the Action + uniqueness in class R.” Learn the §0 terms, run the script, then judge T1–T4 yourself.