SPT substrate — Tier-B evidence: what math can prove (and what only Nature can)

§1 What we are trying to support: H0

Hypothesis H0 (SPT substrate):

• Reality is a discrete Bagua hypercube (Q_n) (nodes + yao), not a smooth continuum down to (a=0).
• The lattice spacing is fixed at (a = \ell_{\mathrm{Pl}}) (Planck length) — a physical cutoff.
• At human/particle scales (L \gg a), physics looks continuum effectively (tiny breaking).

§2 Logic of a Tier-B “proof of substrate”

NOT:   Math alone ⇒ “Nature is Q7”     ✗ impossible

YES:   Structural axes (3+1, SM gauge, 137, 4 forces, Map F)
         ⇒ unique Bagua package = Q7 + a=ℓ_Pl     [B-EXACT uniqueness]
       + lab bounds / cross-domain same a          [B-EXACT + B-PASS]
       + falsifiers F1–F6                          [how Nature can kill H0]

Ontology:  data must prefer H0 over competitors.

So “Tier-B helps SPT substrate correct” means: internal necessity + consistency + kill list — the strongest math can give before experiment.

§3 The evidence chain (easy table)

§4 Plain stories for each pillar

U1 — Why seven, not six or eight?

Want 3 space + 1 time + 3 internal → total 7. Want 1/α integer 137 = 2ⁿ+8+1 → 2ⁿ=128 → n=7. Two independent doors, same key. That is strong uniqueness for (Q_7).

U2 — Why that split of seven?

Among all 36 ways to write 7 as s+t+i, only (3,1,3) survives: stable 3D orbits, one time (causality), three internal for SM-like gauge. That is Law 59 distilled.

U3 — Why four forces?

Q₇ has rotation capacity 14 (=2×7). SM uses 12 (8+3+1); gravity uses 2 (spin-2 frame). No room for a fifth SU(2)-or-bigger force. Substrate predicts the count.

U4 — Why Map F exists on the substrate

Weak generators 3 and hypercharge budget 5 make shell 13. Flavor 8 + cosmo shell 32 make 40. Fractions are not free fits — they are seat counts on the lattice.

U5 — Why fixed a does not contradict “smooth” lab physics

Errors from discreteness fall as ((a/L)^2). At particle scales that is ~(10^{-34}) — far below EW precision. So H0 is compatible with continuum-looking data.

U6 — Smoking gun structure: one spacing

Light speed (membrane flip rate) and cascade slope (d_0=\sqrt{7}/4) share the same membrane spacing. No classical continuum theory forces that link. If experiments force two different spacings, H0 dies.

§5 How to kill the substrate (F1–F6)

Also see Map F pre-reg FC-MF1..4 (θ₁₂, δ_CP, n_T, h/λ) on Action↔F easy and Explorer falsifiers.

§6 What Tier-B proved for you (clear list)

• Proved (B-EXACT): uniqueness of n=7 and (3,1,3); four-force capacity; Map F shells 13 & 40; lab continuum bound form; d₀=√7/4.
• Supported (B-PASS): Ω_DM=35/128 ballpark; (a/L)² OOM; a_Pl inside dispersion bounds.
• Not proved: “Only possible ontology of the universe is Q₇.” That needs Nature via F1–F6 / FC-MF.
• Not proved: Clay strict continuum on ℝ⁴. Different problem; Clay page.

§7 Analogy

Proving a building code unique (must be 7-story, 3+1 rooms, 14 wiring channels…) is Tier-B uniqueness. Proving this city actually built that building is experiment. SPT now has a locked code. Witnesses (experiments) still have to visit the city.

§8 Reproduce offline

pip install sympy numpy && python3 scripts/spt_substrate_tier_b.py && python3 scripts/spt_why_q7_phase9.py && python3 scripts/spt_spacetime_uniqueness.py && python3 scripts/spt_map_f_direction_a_next.py

§9 Bottom line