Map F — Easy full guide: what it is, the conditions, Tier-B proof

1. The real question (not “pretty numbers”)

SPT writes several measured quantities as Bagua fractions that look beautiful:

sin²θ_W (tree)  ≈  3/13
sin²θ_12         ≈  4/13
n_T              ≈  3/13
Cabibbo λ        ≈  9/40
Hubble proxy h   ≈  27/40

The scientific question is not “do they match PDG roughly?” but:

Is this a forced law under clear rules — or a lucky coincidence of integers?

Map F is the name of the single rulebook that tries to answer that with one map, not five independent fits.

2. What is Map F? (lock and tooth)

Write every constant as a fraction:

quan sát  =  TỬ SỐ  /  MẪU SỐ
            =  “răng”  /  “cỡ ổ khóa”

F is one function:

F(lĩnh vực, vai trò đo)  →  phân số Bagua
(không nhét số PDG vào vế phải; PDG chỉ để kiểm SAU)

3. The conditions — package P (this is the important part)

Think of P as the rulebook of the game. The theorems say: if you accept the whole rulebook, then the fractions lock.

P0 — Bagua integers only (no PDG as input)

Allowed building blocks: (Q_n = 2^n) (8, 16, 32, 64, 128…), binomial counts, and pure math like √7. Forbidden as inputs: CODATA α, PDG masses, hand-tuned knobs to match data.

P1 — Split 7 yao as 3 + 1 + 3

This split is a structural choice (STRUCT), not forced by “only powers of two.” Without it, there is no internal sector to count weak weight from.

P2 — R_axis: n_w = number of internal axes (= 3)

n_w  :=  i  =  3

Spectral reading (Tier-B under P1): on the internal subcube (Q_i), the Laplacian spectral gap has multiplicity (i). So (n_w) = “number of weight-1 flip directions inside” = 3. This avoids typing the letters S-U-(2), but it is still a counting rule, not a free miracle.

P3 — Residual budget n_Y

n_Y  :=  Q_3 − n_w  =  8 − 3  =  5

Inside the 8-cell trigram budget, remove the weak weight 3; 5 channels remain (hypercharge-style residual).

P4 — EW shell W (critical rule)

W  :=  Q_3 + n_Y  =  8 + 5  =  13

// NOT  W = n_w + n_Y = 3 + 5 = 8
// Shell = trigram cell + residual Y
// Weak weight sits in the NUMERATOR only

Tree Weinberg angle:

sin²θ_W^tree  =  n_w / W  =  3/13

P5 — Flavour/cosmo shell S (minimal Q₃+Q_k)

Honest fact: the pair ((9/d,\,27/d)) with (h/\lambda=3) works for every (d>27). That alone does not pick 40. We need the shell rule:

S = Q_3 + Q_k   (k ≥ 3)
    with  0 < 9/S < 1  and  0 < 27/S < 1
    and S minimal

→ only k=5 survives as minimal:  S = 8+32 = 40

P6 — Numerators on shell W

sin²θ_12 = 4/13
n_T      = 3/13
I1: n_T ≡ sin²θ_W
I2: sin²θ_12 − sin²θ_W = 1/13

P7 — Numerators on shell S

λ  =  3² / S           =  9/40
h  =  3·(Q_3+1) / S    =  27/40
h/λ = 3  EXACT

P8 — Finite class R of Bagua denominators

Only denominators built from (Q_a), (Q_a+Q_b), (Q_a+k) in a fixed range. Then impose:

• U_NUM — fix numerators {3,4,3} on one denom and {9,27} on another
• U_SHELL — force denoms to be exactly W=13 and S=40

4. Conditions in one line

BẮT BUỘC (package P)
  Q_n = 2^n
  7 hào = 3 + 1 + 3
  n_w = 3  (trục nội / gap mult)
  n_Y = 5
  W = 8+5 = 13     ← shell EW
  S = 8+32 = 40    ← shell flavour/cosmo (min Q3+Qk)
  tử số 3,4,3 trên W  và  9,27 trên S
  class R hữu hạn + U_NUM ∩ U_SHELL

⇒  Map F duy nhất
    3/13, 4/13, 3/13, 9/40, 27/40   (Tier-B EXACT)

KHÔNG NẰM TRONG ĐIỀU KIỆN
  PDG · continuum Action · “chỉ số đẹp không luật chơi”

5. What “Tier-B EXACT proof” means (and does not mean)

Proved Tier-B under P (scripts): shells (13,40); closed forms; identities I1–I2; N2 lock; unique F in class R; L2_STRICT fails (capacity alone does not pick (n_w)); L2_STRUCT passes (P1+P2 pick (n_w=3)).

• Does mean: if you accept P, the fractions are forced and unique in R — machine-checkable.
• Does not mean: Nature must be Q₇; continuum path-integral derived P; ToE finished; PDG proves uniqueness (PDG is OUTPUT only).

6. Picture: from chaos to one map

  Số Bagua thô          →  nhiều map (hỗn loạn)
       +
  Luật chơi P (STRUCT)  →  đúng 1 map F
       +
  Chứng đại số Tier-B   →  3/13, 4/13, 9/40… Δ=0
       +
  PDG                   →  chỉ soi gương (OUTPUT)

Before the audit: “Map F looks pretty.” After: “Map F is a conditional theorem — and we know exactly what the conditions are.”

7. PDG comparison (OUTPUT only — not part of the proof)

8. Referee-safe claim (use this wording)

9. What is still open (real breakthrough targets)

1. Derive P1–P2 (why 3+1+3 and (n_w=i)) from Action dynamics — not from wanting sin²θ_W ≈ 0.23.
2. RG: run tree (3/13) to IR (\sin^2\theta_W(M_Z)) without hand-inserting textbook numbers.
3. Keep blind predictions frozen (n_T, δ_CP, …) — do not retune after data.

10. Reproduce offline (SymPy)

pip install sympy numpy && python3 scripts/spt_map_f_tier_b_proof.py && python3 scripts/spt_map_f_spectral_shell.py && python3 scripts/spt_map_f_uniqueness_minimal.py && python3 scripts/spt_map_f_uniqueness_enum.py && python3 scripts/spt_master_map_f_from_action.py

Research notes (repo): docs/research/map-f/claim-sheet.md, smuggling-matrix.md, tier-b-proof.md.

11. Bottom line