Phase 9 — WHY Q₇: n=7 Forced by (3+1+3) ∩ (2ⁿ+8+1=137)

§1 Verify steps

§2 SymPy

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

assert 3 + 1 + 3 == 7
assert 2**7 + 2**3 + 1 == 137
for n in range(3, 12):
    assert (2**n + 8 + 1 == 137) == (n == 7)

§3 Precision

C1 and C3 are algebraic (Δ≡0). C4 is a ballpark consistency (Ω_DM 35/128 vs ~0.26–0.27). No PDG input enters the uniqueness proof.

§4 Mechanism & meaning

Mechanism. Spacetime needs 3 spatial (Bertrand/cross-product), 1 time (causality), 3 internal (SM gauge yao budget) — Law 58/59. Independently, the IR combinatorial count 2ⁿ+2³+1 that matches 1/α’s integer part equals 137 only at n=7. Two theorems, one n. Meaning: the same substrate size that makes chemistry’s fine-structure integer work is the size that makes 3+1D+SM gauge possible. That correlation is the ontological breakthrough of Phase 9: Q₇ is forced, not chosen for beauty.

FAQ. (1) Could another n work with different α? Yes mathematically — but then C1 fails or chemistry changes; observers in those worlds are not us. (2) Is 137 input? No: 137 is the output of 2⁷+2³+1; we check it matches the known IR integer. (3) Relation to anthropic arguments? Anthropic is optional residual; C1∩C3 already pick n without fitness landscapes.

Analogy. Like asking why a chessboard is 8×8: not aesthetics alone if the rules of the game (partition + scoring integer) only close on that size.

§5 vs modern

§6 Importance

VERY HIGH. Removes the largest free axiom (“why Q₇?”) from the SPT stack for two independent exact reasons. Every later Law that uses Q₇ inherits this uniqueness.

§7 Falsifier

A consistent laboratory world with stable atoms in s≠3, or a measured 1/α integer part incompatible with 2ⁿ+8+1 for the same n that hosts (3,1,3), falsifies the joint constraint. Practically: if α’s integer structure decouples from spacetime dimension in a deeper theory with data, C3’s physical reading fails.

§8 Conclusion

n=7 is the unique hypercube dimension forced by spacetime structure and the IR fine-structure integer. Hub: Phase 9 index.