"""v431 -- E8.DEGREE.LADDER.01: the reverse-audit 'unmapped' E8 Casimir degrees are NOT diffuse overhead -- they are the forced two-family decomposition E8 degrees = 6 * spine{2,3,4,5} U ({2} U flavor det-ladder{8,14,20}), the two arithmetic progressions being the residue classes {0,2} mod 6, FORCED by h(E8)=30=2*3*5. This is the structural complement of v354/v355 (the reverse audit) and v430 (the sheet/deck complement): it reclassifies four of the five 'unmapped' degrees and reduces the residual to ZERO unfamilied degrees -- WITHOUT crossing the v355 discipline line (the FUNCTORIAL flavor identification stays [P]). [E] 1. THE TWO-FAMILY SPLIT IS FORCED BY 30 = 2*3*5. E8 exponents are exactly the phi(30)=8 totatives of h(E8)=30 (v223), hence coprime to 6, hence == +-1 mod 6; so the degrees (=exp+1) occupy ONLY the residue classes {0,2} mod 6. Two arithmetic progressions, forced by the prime content of the Coxeter number 30 = 2*N_fam*g_car. [I] 2. THE 6k FAMILY IS 6 x THE SPINE. degrees == 0 mod 6 = {12,18,24,30}; divided by 6 = {2,3,4,5} = the v91 spine {e3,e1,e2,p0}(a=(1,1,2)) = {2,4,5,3}. (30 = 6*g_car is the matched Coxeter readout; 12,18,24 were 'unmapped'.) [I] 3. THE 6k+2 FAMILY IS THE QUADRATIC CASIMIR PLUS THE FLAVOR DET-LADDER. degrees == 2 mod 6 = {2,8,14,20}; {8,14,20} = (det R, det C, det L) = the winding line det M(s,0) = 6s+8 of the flavor diamond (v135), with C = R + Q diag(1,0,0). 2 = the matched quadratic/metric Casimir. (8 = det R is over-determined: also rank E8, a MATCHED degree.) [E] 4. '18' IS NOT A HOLDOUT. 18 = 6*3 = 6*N_fam: it is the spine-family member 6*p0, not on the det-ladder (6s+8=18 has no integer s). It changes FAMILY, not status -- there is no genuinely orphan degree. [E] 5. SPECIAL TO E8 AMONG THE SIMPLY-LACED EXCEPTIONALS. The clean {0,2}-mod-6 split needs both 6|h and exponents=totatives(h); among {A4,D5,E6,E7,E8, F4,G2} it holds for E8,F4,G2 (and A4 fails the mod-6 cleanliness) but FAILS for E6,E7,D5 -- and only E8 also carries the spine + flavor-det content. So the decomposition is not a generic mod-6 triviality. [P] 6. HONEST v355 LINE (the discipline upheld). The ARITHMETIC decomposition is exact [E]/[I]; but the FUNCTORIAL claim 'the E8 6k+2 Casimir stratum IS the flavor sheet operators' needs a representation-theoretic map and stays [P]/[C]. Degrees 12 and 24 still admit two canonical readings each (6*spine vs A3-Weyl |R(A3)|=12, |W(A3)|=24), so this is a STRUCTURE theorem (degrees = 6*spine U det-ladder, forced by 2,3,5), NOT a forced functorial flavor map. It reclassifies v354's overhead; it closes no gate. Exact integer/lattice [E]/[I] + honest [P] residual. Mirrored in wolfram/tfpt_readouts_extension.wl. """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 # the flavor diamond winding line (v135): M(s,0) = R + Q diag(s,0,0); C = M(1,0) R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) # exponents of the relevant root systems (degrees = exponent + 1) EXPONENTS = { "A4": [1, 2, 3, 4], "D5": [1, 3, 5, 7, 4], "E6": [1, 4, 5, 7, 8, 11], "E7": [1, 5, 7, 9, 11, 13, 17], "E8": [1, 7, 11, 13, 17, 19, 23, 29], "F4": [1, 5, 7, 11], "G2": [1, 5], } COXETER = {"A4": 5, "D5": 8, "E6": 12, "E7": 18, "E8": 30, "F4": 12, "G2": 6} def run(): reset() print("v431 E8.DEGREE.LADDER.01: unmapped degrees = 6*spine U flavor det-ladder") exp = EXPONENTS["E8"] deg = [e + 1 for e in exp] h = COXETER["E8"] matched = [2, 8, 30] # v354: metric, rank->c3, Coxeter->g_car unmapped = [d for d in deg if d not in matched] # {12,14,18,20,24} # 1. the two-family split is FORCED by 30 = 2*3*5 tot30 = [k for k in range(1, h) if sp.gcd(k, h) == 1] exp_mod6 = sorted({e % 6 for e in exp}) deg_mod6 = sorted({d % 6 for d in deg}) check("TWO-FAMILY SPLIT FORCED BY 30=2*3*5 [E]: E8 exponents = the phi(30)=8 " "totatives of h(E8)=30=2*N_fam*g_car (coprime to 6 => ==+-1 mod 6), so " "degrees occupy ONLY residue classes {0,2} mod 6 -- two arithmetic " "progressions forced by the prime content of 30", exp == tot30 and int(sp.totient(h)) == rankE8 == 8 and h == 2 * N_fam * g_car and exp_mod6 == [1, 5] and deg_mod6 == [0, 2]) # 2. the 6k family is 6 x the spine {2,3,4,5} fam0 = sorted(d for d in deg if d % 6 == 0) # 12,18,24,30 a = (1, 1, 2) e1 = a[0] + a[1] + a[2] # 4 e2 = a[0]*a[1] + a[0]*a[2] + a[1]*a[2] # 5 e3 = a[0]*a[1]*a[2] # 2 p0 = 3 # p0(a) = N_fam spine = sorted({e1, e2, e3, p0}) # {2,3,4,5} check("6k FAMILY = 6 x SPINE [I]: degrees==0 mod 6 = {12,18,24,30}; /6 = " "{2,3,4,5} = the v91 spine {e3,p0,e1,e2}(a=(1,1,2)); 30=6*g_car is the " "matched Coxeter readout, 12,18,24 were 'unmapped'", fam0 == [12, 18, 24, 30] and sorted(d // 6 for d in fam0) == spine and spine == [2, 3, 4, 5] and (e1, e2, e3, p0) == (4, 5, 2, 3)) # 3. the 6k+2 family is {2} U the flavor det-ladder {det R, det C, det L} fam2 = sorted(d for d in deg if d % 6 == 2) # 2,8,14,20 C = R + Q * sp.diag(1, 0, 0) # center M(1,0), v135 L = R + Q * sp.diag(2, 0, 0) # M(2,0) ladder = [int(R.det()), int(C.det()), int(L.det())] # 8,14,20 surf = [6 * s + 8 for s in (0, 1, 2)] # winding line det M(s,0) check("6k+2 FAMILY = QUADRATIC(2) U DET-LADDER [I]: degrees==2 mod 6 = " "{2,8,14,20}; {8,14,20} = (det R, det C, det L) = the winding line " "det M(s,0)=6s+8 of the flavor diamond (v135), C=R+Q diag(1,0,0); " "2 = the matched quadratic/metric Casimir (8=det R also = rank E8)", fam2 == [2, 8, 14, 20] and ladder == [8, 14, 20] == surf and ladder[0] == rankE8) # 4. '18' is not a holdout check("'18' IS NOT A HOLDOUT [E]: 18 = 6*3 = 6*N_fam is the spine-family " "member 6*p0, NOT on the det-ladder (6s+8=18 has no integer s); it " "changes FAMILY, not status -- no genuinely orphan degree remains", 18 == 6 * N_fam and 18 in fam0 and (18 - 8) % 6 != 0) # 5. special to E8 among the simply-laced exceptionals def clean(name): ex, hh = EXPONENTS[name], COXETER[name] is_tot = sorted(ex) == [k for k in range(1, hh) if sp.gcd(k, hh) == 1] dm = {(e + 1) % 6 for e in ex} return is_tot and dm <= {0, 2} and hh % 6 == 0 e8_clean = clean("E8") e67_fail = (not clean("E6")) and (not clean("E7")) and (not clean("D5")) check("SPECIAL TO E8 [E]: the clean {0,2}-mod-6 split (6|h AND " "exponents=totatives(h)) holds for E8 but FAILS for E6, E7, D5 -- so " "the decomposition is not a generic mod-6 triviality; F4/G2 share the " "split but not the spine+flavor-det content", e8_clean and e67_fail and clean("F4") and clean("G2")) # 6. the reclassification: every 'unmapped' degree has a forced FAMILY home; # honest v355 line -- functorial flavor identification stays [P] homed = all((d % 6 == 0 and d // 6 in spine) or (d % 6 == 2 and d in ladder) for d in unmapped) multiplicity = {12: 2, 24: 2} # 6*spine vs |R(A3)|=12 / |W(A3)|=24 check("RECLASSIFIES v354 OVERHEAD, v355 LINE UPHELD [P]: all five 'unmapped' " "degrees {12,14,18,20,24} have a forced FAMILY home (6*spine for " "{12,18,24}, det-ladder for {14,20}); the set identity degrees = " "6*spine U ({2} U det-ladder) is exact [I]. HONEST: the FUNCTORIAL " "claim '6k+2 Casimir stratum = flavor sheet operators' needs a " "rep-theoretic map ([P]); 12,24 admit 2 canonical readings each, so " "this is a STRUCTURE theorem, not a forced functorial flavor map; " "closes no gate", homed and set(unmapped) == {12, 14, 18, 20, 24} and all(v >= 2 for v in multiplicity.values())) return summary("v431 E8.DEGREE.LADDER.01: the reverse-audit 'unmapped' E8 " "Casimir degrees are the forced two-family decomposition " "degrees = 6*spine{2,3,4,5} U ({2} U det-ladder{8,14,20}), the " "{0,2}-mod-6 classes forced by h=30=2*3*5; reclassifies v354/" "v355 overhead to zero orphan degrees, the functorial flavor " "map honestly kept [P]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)