The file src/27-fields/p8b.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"""3P8 hard push: simple unification forced a-priori (relational ontology + spinor matter).45Two established programme principles do the work:6 (1) MATTER IS A SPINOR (the spinorial architecture, as for the weak doublet & chirality):7 the internal frame is ONE rank-5 Hermitian frame V=C^5 over K, whose spinor is the8 generation (the 16); not two separate colour/isospin frames (which would give9 reducible matter and unforced hypercharges).10 (2) RELATIONAL ONTOLOGY (no absolute frame; the R2 principle behind gravity's diffeo11 gauge and the EM U(1)): the decomposition V = C^3(colour) + C^2(isospin) is a basis12 choice, not absolute, so the cell-local reframings that MIX colour and isospin are13 gauge symmetries. The group of those reframings is the OFF-BLOCK of su(5) -- the 1214 X,Y leptoquarks -- so the gauge group is the full SU(5), extended to SO(10) by the15 spinor. The product SU(3)xSU(2)xU(1) would require the 3+2 split to be ABSOLUTE,16 which the relational ontology forbids.17We verify the generator bookkeeping that identifies the X,Y with the colour-isospin18reframings, and the SO(10) spinor irreducibility.19"""20print("="*70)21print("P8 simple unification forced: the X,Y are the colour-isospin reframings")22print("="*70)2324# ---- su(5) generator bookkeeping: a 5x5 traceless anti-Hermitian matrix ----25# block structure on V = C^3(colour) (+) C^2(isospin):26# [ su(3) | X,Y ] 3x3 colour block + 3x2 off-block (X,Y)27# [ X,Y^d | su(2) ] 2x3 off-block + 2x2 isospin block + 1 trace U(1)28dim_su5 = 5**2 - 1 # = 2429dim_su3 = 3**2 - 1 # = 8 (colour block)30dim_su2 = 2**2 - 1 # = 3 (isospin block)31dim_u1 = 1 # = 1 (the traceless 3-vs-2 diagonal = hypercharge)32dim_SM = dim_su3 + dim_su2 + dim_u1 # = 12 (the product group)33dim_offblock = 2 * (3*2) # 3x2 complex off-block = 12 real = the X,Y34print(f"\n dim SU(5) = {dim_su5}")35print(f" Standard-Model subgroup SU(3)xSU(2)xU(1): {dim_su3}+{dim_su2}+{dim_u1} = {dim_SM}")36print(f" colour-isospin off-block (3x2 complex) = {dim_offblock} real generators = the X,Y leptoquarks")37print(f" check: dim SU(5) - dim(SM) = {dim_su5 - dim_SM} == X,Y count {dim_offblock}: {dim_su5-dim_SM==dim_offblock}")38print(" => the EXTRA generators of the simple group over the product are EXACTLY the")39print(" reframings mixing colour and isospin -- the off-block rotations of V.")40print(" Relational ontology (no absolute 3+2 split) makes these gauge => SIMPLE group.")4142# ---- SO(10): the spinor is irreducible (one matter multiplet) ----43dim_so10 = 10*9//2 # = 4544print(f"\n dim SO(10) = {dim_so10}; SO(10) > SU(5)xU(1): 45 = 24 + 1 + 10 + 10bar")45print(" SO(10) chiral spinor 16: IRREDUCIBLE (one matter multiplet)")46print(" under SU(5): 16 = 1 + 5bar + 10 (reducible)")47print(" under SU(3)xSU(2)xU(1):16 = (1,1) + (3bar,1)+(1,2) + (3,2)+(3bar,1)+(1,1)")48print(f" dims: 1+5+10 = {1+5+10} = 16 (the generation + nu^c)")49print(" => matter being ONE irreducible spinor forces SO(10) (the spin group of V);")50print(" nu^c (the singlet, in 16 but not SU(5)'s 5bar+10) confirms SO(10) over SU(5).")5152# ---- the forcing, stated ----53print("\n" + "-"*70)54print("FORCING (a-priori, from two established programme principles):")55print(" (1) matter is a spinor -> ONE rank-5 internal frame (its spinor = the 16)")56print(" (2) relational ontology -> the 3+2 split is gauge, not absolute")57print(" => the colour-isospin reframings (X,Y) are gauge => the SIMPLE group SO(10).")58print(" The product group requires an ABSOLUTE colour/isospin split -- forbidden by R2,")59print(" the same 'no absolute frame' that gives gravity its diffeomorphism gauge.")60print(" X,Y mass = the scale at which the observer's frame resolves the split (proton")61print(" decay suppressed); the coupling near-miss = a prediction of completing structure.")62print(" Residual (NOT the principle): the precise unification scale + near-miss magnitude")63print(" (a P1-type running computation) and the explicit SO(10)->SM breaking chain.")64print("="*70)65