What is Master Map F? (denominators + numerators + why it matters)

One sentence. Master Map F is the single rulebook that says: this physics role → that Bagua fraction — so SPT is not five lucky ratios, but one map with forced shells (denominators) and forced numerators under explicit counting laws.

Even shorter (10 min): the public summary with history comparison → /theory/spt-map-f-summary. F ↔ all 40 Explorer constants + predictions: /theory/spt-map-f-40-constants.

Glossary: /glossary#spt-master-map · search Master Map, numerator, denominator, shell. Technical derivation of shells: F from Action. Numerators: script spt_master_map_numerators.py.

§1 What is Map F? (plain picture)

Write every constant as a fraction:

text

quan sát  =  TỬ SỐ  /  MẪU SỐ
example:   sin²θ_W =    3     /    13
           Cabibbo =    9     /    40

F is the function that picks both parts from Bagua/Action rules only:

text

F(sector, role) → (numerator, denominator)
                → fraction p/d

Role (what we measure)	F output	Shell (denom)
Weinberg angle (tree)	3/13	W = 13
Solar neutrino mix sin²θ₁₂	4/13	W = 13
GW tilt n_T	3/13	W = 13
Cabibbo λ	9/40	S₄₀ = 40
Hubble h	27/40	S₄₀ = 40

Lock picture. Denominator = lock size. Numerator = which tooth on that lock. Same lock 13 for three doors; same lock 40 for two doors. That is cross-sector structure — not five independent fits.

§2 Denominator (mẫu số / shell) — the lock size

A shell is a shared denominator forced by Action geometry (Direction 1):

W = 13

W = Q₃ + n_Y = 8 + 5. n_Y = hypercharge budget = Q₃ − 3. EW-shell for θ_W, θ₁₂, n_T.

S₄₀ = 40

S₄₀ = Q₃ + Q₅ = 8 + 32. Flavour–cosmo product shell for λ and h.

Full technical proof: F from Action. Glossary: Weinberg shell, Shell-40.

§3 Numerator (tử số) — which tooth on the lock

Why “siết tử số” matters. After shells are forced, a skeptic can still say: “You chose 3, 4, 9 to fit data.” Direction 1c freezes numerator laws N1–N6 so those integers are counting rules, not free knobs.

Law	Numerator	Counting rule (plain)
N1	n_W = 3	Number of SU(2) weak generators
N2 ✅ proved	n_solar = 4	Triple identity: Q₃/2 = n_W+1 = n_Y−1 (= 2×2 light ν). See N2 proof.
N3	n_T = 3	Residual 8-cell after n_Y: Q₃ − n_Y
N4	N_gen = 3	Three families (= n_W integer)
N5	n_λ = 9	Generations squared: N_gen²
N6	n_h = 27	N_gen × (Q₃+1) = 3 × 9

Exact identities forced by numerators (not fits): - n_T ≡ sin²θ_W = 3/13 (same tooth on lock 13) - sin²θ₁₂ − sin²θ_W = 1/13 (adjacent teeth) - h / λ = 3 (generation factor)

§4 Full Map F after siết tử số

python

# Denominators (Direction 1)
W, S40 = 13, 40
# Numerators (laws N1–N6)
n_W, n_solar, n_T = 3, 4, 3
n_lam, n_h = 9, 27
# F
sin2_W  = n_W / W          # 3/13
sin2_12 = n_solar / W      # 4/13
n_T_obs = n_T / W          # 3/13
lam     = n_lam / S40      # 9/40
h       = n_h / S40        # 27/40

Uniqueness: under N1–N6 and fixed shells, the 5-tuple of numerators (3,4,3,9,27) is the only survivor in the Action counting class (SymPy T5).

§5 Importance (why this is a big deal)

Axis	Score	Plain reading
Anti-numerology on F	9/10	Closes “shells forced, numerators free”
Completeness of Map F	8.5/10	Both halves of p/d now law-level under axioms
Path-integral depth	5–6/10	N2 adjacent +1 still selection rule, not Feynman
Experiment power	8/10	Wrong tooth kills named identities (ID-A/B/C)
Engineering product	2/10	Not a device recipe

Strategic meaning. Together with Direction 1 (denominators) and Direction 2 (RG zoom), Map F becomes: forced lock sizes + forced teeth + scale running for θ_W. That is the core “constants map” of SPT — higher priority than adding Law 81.

§6 Practical use (honest)

Research rule: never invent a new numerator that breaks N1–N6 or ID-A/B/C.
Experiments: DUNE sin²θ₁₂ tests n_solar=4; multi-band GW tests n_T=3; Cabibbo/h stay on S₄₀.
Teaching: fraction = tooth / lock — easy for non-specialists.
Not: engines, batteries, FTL. Map F is structure of constants.

§7 How a wrong numerator dies

If you force…	What dies
n_solar = 3 (not 4)	ID-B: no 1/13 gap between θ₁₂ and θ_W
n_T ≠ 3	ID-A: n_T no longer equals tree θ_W
n_λ ≠ 9 or n_h ≠ 27	ID-C: h/λ ≠ 3

§8 SymPy evidence

SymPy verify — download for offline testSYMPY ✓

Map F: shells + numerators

scripts/spt_master_map_n2_solar.py

N2 proof: n_solar=4 triple identity — Q3/2 = n_W+1 = n_Y−1 = 4; uniqueness; ID-B

180 LOCDownload

scripts/spt_master_map_numerators.py

Numerator laws N1–N6 package — Unique (3,4,3,9,27); identities A,B,C; falsify table

240 LOCDownload

scripts/spt_master_map_f_from_action.py

Denominators from Action — W=13, S40=40, uniqueness in class R

220 LOCDownload

scripts/spt_master_map_f.py

Compact F assignment — Phase 9 companion

160 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_master_map_n2_solar.py && python3 scripts/spt_master_map_numerators.py && python3 scripts/spt_master_map_f_from_action.py && python3 scripts/spt_master_map_f.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.

§9 Limits (do not overclaim)

N2 is now a triple algebraic identity (Q₃/2 = n_W+1 = n_Y−1 = 4) under Direction-1 integers — not a free +1 fit. Path-integral why solar is the half-Q₃ projection still research. Full write-up: N2 proof.
N_gen = 3 inherits three-family structure; full Pólya proof lives in Law 23 wiki.
PDG never chooses numerators — only checks OUTPUT bands.

§10 Related pages

F from Action (denominators): /theory/spt-master-map-f-from-action
Compact F: /theory/spt-master-map-f
Shell theorem: /theory/spt-cross-sector-shells
RG zoom (θ_W running): /theory/spt-rg-lattice-flow
Phase 9 hub: /theory/spt-phase9-core-index
Glossary: /glossary#spt-master-map

Bottom line. Map F = tooth / lock. Locks (13, 40) from Action geometry; teeth (3,4,3,9,27) from numerator laws N1–N6. That is the accurate law of F after siết tử số — high importance for theory integrity, not a product launch.