Map F ↔ 40 constants — links, layers, and predictions

§1 Map F in one picture (reminder)

Every Map F output is a fraction built only from Bagua/Action counting:

F(sector, role)  →  TỬ / MẪU  =  “răng” / “ổ khóa (shell)”

Two primary locks (Direction 1 + N1–N6):
  W   = Q₃ + n_Y = 8 + 5 = 13     (electroweak family)
  S₄₀ = Q₃ + Q₅  = 8 + 32 = 40    (Cabibbo + Hubble h)

Core assignments:
  sin²θ_W^tree = 3/13     sin²θ₁₂ = 4/13     n_T = 3/13
  λ_Cabibbo    = 9/40     h_Planck = 27/40     (⇒ h/λ = 3 EXACT)
  1/α_IR       = Q₇ + Q₃ + 1 = 137

Wider alphabet (still F-style rationals, other denominators): (Q_n=2^n), binomials (C(7,k)), (d_0=sqrt{7}/4), shell 128, 136, 16, 57, … See F from Action · N1–N6 · N2 proof.

§2 Four layers: how F touches the 40 constants

The Derivation Explorer lists exactly 40 constants (derivationData.ts). Map F does not give each a new formula — it assigns each to a structural role.

                    ┌─ Lock W=13 ── sin²θ_W, θ₁₂, (n_T)
         Map F ────┼─ Lock S₄₀=40 ─ V_us (λ), h (H₀)
         (core)    └─ Lock 137 ──── α
              │
              ▼
    Alphabet Qₙ / C(7,k) / 128 ── Ω_b, Ω_DM, Ω_Λ, n_s, d₀, ε_GW, θ₁₃, θ₂₃, …
              │
              ▼
    Cascade + EW machinery ── 12 masses, m_W/m_Z/m_H, Λ, Δm²_ν, …
              │
              ▼
    Substrate / geometric ── c, ℏ, G, CHSH, Hawking, Bekenstein, chirp, …

§3 Layer A — hard Map F locks (core)

§4 Layer B — same Bagua alphabet (other shells)

Still “F-style” (numerator/denominator from counting), but not forced onto W=13 or S₄₀=40.

§5 Layer C — inherits F via EW / cascade

These constants are not pure p/d on shell 13/40, but their pipelines import F outputs (especially sin²θ_W and d₀) or the same cascade bookkeeping Law 37 uses.

§6 Layer D — same Action, far from shell 13/40

Still SPT-derived in the Explorer, but killing them does not kill Map F shells — it kills substrate / virtual-DA / geometric Laws instead.

• c — membrane flip-rate (speed of light from membrane).
• hbar, G — action/geometry normalisations on the lattice.
• chshBound — Tsirelson 2√2 (Law 46); yao spin algebra, not shell 13.
• hawkingT, bekensteinS — virtual-DA tunneling / Z₂ bit-count (Laws 61, 45).
• Mc150914 — chirp mass channel; gravity sector, not F tooth.

§7 Full roster — all 40 Explorer ids × layer

Rough tally: A ≈ 5–6 · B ≈ 11 · C ≈ 15 · D ≈ 8 — sums to 40. Layer tags are research labels, not extra free parameters.

§8 Cross-sector identities (already forced by F)

These are structural predictions already inside F — no new fit knobs.

§9 Predictions P1–P6 — near term (2026–2035)

§10 Predictions P7–P13 — mid term (cascade / cosmo / DM)

Cascade & EW precision (Layer C)

• P7 — Twelve fermion mass ratios fixed by d_i/d₀ = h_i + C_i/Q₃ with d₀ = √7/4. HL-LHC/FCC systematics that break the cascade ratios kill Law 37 (the cascade leg of F).
• P8 — m_H/v = √(33/128) holds. FCC-ee m_H, v: durable >~0.2% fail kills shell-128 Higgs form.
• P9 — M_W, M_Z in the Law 55 band via F’s θ_W + cascade v. Future ±5 MeV M_W outside band kills EW VEV pipeline.
• P10 — m_ββ ∈ ~[1.5, 3.7] meV (NH band, Law 62) tied to PMNS F angles. nEXO / KamLAND-Zen outside band (after NME systematics) pressures the flavour map.

Cosmology & dark sector (Layer B + Laws 50, 64–66)

• P11 — m_DM ~ 60 GeV band + σ_SI ~ 4×10⁻⁴⁷ cm² (Laws 64, 66). LZ 2025–27 / XLZD: deep null or mass far outside band.
• P12 — Ω_DM shape 35/128 in Bagua closure; DESI/Euclid Ω_DM h² leaving closure pressures Layer B.
• P13 — n_s = 55/57 ≈ 0.9649; r ≈ 12/N_e² with N_e = 60. CMB-S4 / LiteBIRD: n_s fail or large-field r ≫ 0.01.

§11 Meta-predictions P14–P18 — programme rules

These are not new numbers — they are rules Map F imposes on future research and communication.

• P14 — No sixth “lucky” constant may use denominator 13 with a free numerator. Any new tooth on W needs an N-style counting law (like N1–N6).
• P15 — Every new constant claim must declare: which lock, which tooth, which Law N# / script — or admit Class C calibration.
• P16 — Falsifying one member of lock 13 endangers all three channels (θ_W tree, θ₁₂, n_T). Single-experiment “local” excuses are not enough.
• P17 — Falsifying lock 40 endangers Cabibbo and h together (I3).
• P18 — 1/α = 137 stays combinatorial IR; lab running must not “repair” 137 by a free UV knob (RG Direction 2 honesty).

§12 Kill calendar (compact)

Companion registry: Blind predictions · Honest status.

§13 Plain-language analogy

Think of the 40 constants as a full orchestral score SPT has already written note-by-note. Map F is the key signature and bar lines: not every note sits on the same chord, but chord 13 forces EW + solar (+ GW tilt) to stay in one key, chord 40 forces Cabibbo + Hubble together, and 137 / 128 / √7 set the tuning of α, Ω, and the cascade metronome. The scientific claim is not “we have many correct notes” — it is “if one instrument on chord 13 goes out of key, the whole section is wrong.”

§14 What this page does not claim

• Not “all 40 constants are pure fractions on shells 13 and 40.” Only Layer A is hard-locked that way.
• Not a new numbered Law and not a scoreboard Law-count bump (still 80 Laws).
• Not peer-reviewed consensus; predictions are commitments to be killed, not trophies.
• Not a Clay / TOE “finished” claim — see Clay track separately.

§15 Reproduce the F core offline

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

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