TFPT scientific source excerpt
Original repository path: verification/v197_rr_carrier_clifford_d5.py
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000001 | """v197 -- ARCH.RRCAR.02: hardening the Riemann-Roch carrier reading from g_car to
000002 | D5 itself (external-review point 7). v189 read g_car = 5 = h^0(P^1, O(mu4)) and
000003 | matched it to rank(D5). This module goes one step further: the 5-dimensional
000004 | carrier mode space C_car := H^0(P^1, O(mu4)) generates the so(10) = D5 half-spinor
000005 | by its even Clifford algebra,
000006 | 
000007 |     Lambda^even(C_car) = S^+ ,    dim S^+ = 2^{g_car-1} = 16 ,
000008 | 
000009 | so the carrier is not merely "rank 5 like D5" but IS the Clifford spinor of the
000010 | 5-dim mode space. The dimensional chain is exact [I]; the physical identification
000011 | "carrier := the Clifford spinor of the seam mode space" stays [C] (as in v189).
000012 | 
000013 |   [I] 1. THE EVEN CLIFFORD DIMENSION.  For a 5-dim space, the even exterior
000014 |         algebra has dim Lambda^even(C^5) = C(5,0)+C(5,2)+C(5,4) = 1+10+5 = 16 =
000015 |         2^{5-1} = 2^{g_car-1} = dim S^+ (the repo's dim_Splus).
000016 |   [I] 2. THE SO(10) HALF-SPINOR.  so(10) = D5 (rank 5) has Dirac spinor 2^5 = 32
000017 |         and half-spinors 2^4 = 16 each; S^+ = Lambda^even(C_car). So the 5-dim
000018 |         mode space, doubled to its Clifford/spinor module, IS the D5 half-spinor.
000019 |   [I] 3. CLOSES THE CHAIN mu4 -> g_car -> D5.  From ONE four-point divisor:
000020 |         cycle side rank H_1 = deg-1 = 3 = N_fam (A3); function side
000021 |         h^0(O(mu4)) = deg+1 = 5 = g_car = dim C_car; and now
000022 |         Lambda^even(C_car) = 16 = the D5 half-spinor = the carrier. So D5 is not a
000023 |         separate input -- it is the Clifford spinor of the Riemann-Roch carrier.
000024 |   [C] 4. THE PHYSICAL IDENTIFICATION STAYS [C].  Reading the carrier as
000025 |         Cl^even(C_car) (rather than merely matching rank(D5)=5) is a geometric
000026 |         hardening, but the identification "carrier = Clifford spinor of the seam
000027 |         meromorphic mode space" is still a physical [C] step (as in v189); it does
000028 |         NOT change the over-determined status of g_car=5.
000029 | 
000030 |   Exact (binomial / Clifford dimensions); Wolfram-mirrored.
000031 | """
000032 | from math import comb
000033 | 
000034 | from tfpt_constants import g_car, N_fam, dim_Splus, check, summary, reset
000035 | 
000036 | MU4 = 4
000037 | 
000038 | 
000039 | def run():
000040 |     reset()
000041 |     print("v197 ARCH.RRCAR.02: C_car = H^0(O(mu4)) (5-dim) generates the D5 half-spinor S^+ via Clifford")
000042 | 
000043 |     h0 = MU4 + 1                                       # h^0(P^1,O(mu4)) = deg+1 = 5 = g_car
000044 |     lam_even = sum(comb(h0, k) for k in range(0, h0 + 1, 2))   # dim Lambda^even(C^5)
000045 |     check("EVEN CLIFFORD DIM [I]: dim Lambda^even(C^%d) = C(5,0)+C(5,2)+C(5,4) = "
000046 |           "%d+%d+%d = %d = 2^{g_car-1} = dim S^+ = %d"
000047 |           % (h0, comb(h0, 0), comb(h0, 2), comb(h0, 4), lam_even, dim_Splus),
000048 |           h0 == g_car and lam_even == 2**(g_car - 1) == dim_Splus == 16)
000049 | 
000050 |     check("SO(10) HALF-SPINOR [I]: so(10)=D5 (rank 5) has Dirac spinor 2^5=%d and "
000051 |           "half-spinors 2^4=%d each; S^+ = Lambda^even(C_car). The 5-dim mode "
000052 |           "space's Clifford module IS the D5 half-spinor"
000053 |           % (2**5, 2**4),
000054 |           2**5 == 32 and 2**4 == dim_Splus)
000055 | 
000056 |     check("CHAIN mu4 -> g_car -> D5 [I]: from ONE four-point divisor -- cycle side "
000057 |           "rank H_1 = deg-1 = %d = N_fam (A3); function side h^0 = deg+1 = %d = "
000058 |           "g_car = dim C_car; Lambda^even(C_car) = %d = the D5 half-spinor = the "
000059 |           "carrier. D5 is the Clifford spinor of the Riemann-Roch carrier, not a "
000060 |           "separate input" % (MU4 - 1, h0, lam_even),
000061 |           (MU4 - 1) == N_fam and h0 == g_car and lam_even == dim_Splus)
000062 | 
000063 |     check("PHYSICAL IDENTIFICATION [C]: reading the carrier as Cl^even(C_car) (not "
000064 |           "merely matching rank(D5)=5) is a geometric hardening of v189, but "
000065 |           "'carrier = Clifford spinor of the seam mode space' is still a physical "
000066 |           "[C] step; it does NOT change the over-determined status of g_car=5", True)
000067 | 
000068 |     return summary("v197 ARCH.RRCAR.02: Lambda^even(C_car)=S^+=16=D5 half-spinor; mu4->g_car->D5 chain [I]/[C]")
000069 | 
000070 | 
000071 | if __name__ == "__main__":
000072 |     raise SystemExit(1 if run() else 0)