The file src/27-fields/p11.py of the validation package of the paper 27-fld, rendered with an anchor per line; the file itself is beside this page.
1# framed-rational status: EXACT -- integer / modular-F_p / cyclotomic arithmetic; no float in any asserted check (float, if present, only formats an exact rational for display).2"""3P11: the substrate residue Omega = 5 (mod 12).45(a) Arithmetic: symmetry-completeness needs 4 | Omega-1 (the quarter-turn, in the6 split torus of order Omega-1, complex QM); colour triality needs 3 | Omega+17 (the Z_3 centre, in the non-split torus of order Omega+1). By CRT,8 Omega = 1 (mod 4) AND Omega = 2 (mod 3) <=> Omega = 5 (mod 12).9 The two conditions sit in the two non-trivial frame tori.10(b) Status: =1 mod 4 is DEFINITIONAL (the corpus's symmetry-complete shell, complex11 QM); =2 mod 3 is the colour-centre / stable-matter condition.12(c) Forced vs selection -- relational resolution: the Carrier is the totality and the13 Subject is within it (no external multiverse), so the residue is FORCED by the14 self-consistency of an observer-bearing substrate, not selected among cosmoi.15(d) Reduction: the residue is the arithmetic shadow of the four-fold (Q_4, mod 4)16 and the cubic (colour/generations, mod 3) structures.17"""18from sympy import isprime, primerange1920print("="*68)21print("P11 the substrate residue Omega = 5 (mod 12)")22print("="*68)2324# ---- (a) CRT: the two conditions <=> 5 mod 12 ----25print("\n(a) CRT over residues mod 12:")26for r in range(12):27 c4 = (r % 4 == 1) # quarter-turn / complex QM (4 | Omega-1)28 c3 = (r % 3 == 2) # colour Z_3 centre (3 | Omega+1)29 if c4 and c3:30 print(f" Omega = {r} mod 12 : (=1 mod4: {c4}) AND (=2 mod3: {c3}) <-- the unique residue")31print(" => Omega = 5 (mod 12) is the unique residue with BOTH; 5 mod4=1, 5 mod3=2.")32print(" quarter-turn lives in the Omega-1 (split) torus; colour Z_3 in the Omega+1 (non-split) torus.")3334# ---- (c) density among primes: Dirichlet 1/phi(12) = 1/4 ----35import collections36cnt = collections.Counter(p % 12 for p in primerange(5, 200000))37tot = sum(cnt[r] for r in (1,5,7,11))38print("\n density of large primes by residue mod 12 (Dirichlet 1/4 each):")39for r in (1,5,7,11):40 print(f" p = {r:2d} mod 12 : {cnt[r]/tot:.3f}")41print(f" => Omega = 5 mod 12 (admissible substrate) has density 1/4 among primes;")42print(f" admissible Omega near 10^122 are abundant (Dirichlet).")4344# ---- corpus example shells ----45print("\n corpus example shells mod 12 (all symmetry-complete, =1 mod 4):")46for p in (13, 53, 61, 157, 421):47 print(f" F_{p}: {p} mod 12 = {p%12} ({'=5: has colour Z_3' if p%12==5 else '=1: cube roots in base field, no colour centre'})")48print(" (small shells illustrate the structure; the cosmic Omega ~ 10^122 needs")49print(" the extra colour condition =2 mod 3, hence =5 mod 12.)")5051print("\n(c) RELATIONAL resolution (forced, not selected):")52print(" The Carrier is the totality; the Subject (observer, composite matter) is a")53print(" nested shell WITHIN it. There is no external multiverse to select among.")54print(" A self-consistent observer-bearing substrate must host its own observers,")55print(" which requires complex QM (=1 mod 4) and colour-confined matter (=2 mod 3).")56print(" Hence Omega = 5 mod 12 is FORCED by self-consistency, not anthropically chosen.")57print("\n(d) The residue is the arithmetic shadow of the four-fold (Q_4, mod 4) and the")58print(" cubic (colour/generations, mod 3); not an independent free parameter.")59print("="*68)60