A registered result in its source context
Original ledger classification: Identity [I] (Clifford dim chain) + Physical [C] (carrier = mode-space spinor)
Recorded statement: RIEMANN-ROCH CARRIER HARDENED FROM g_car TO D5 ITSELF (external-review point 7, 2026-06-14): v189 read g_car=5=h^0(P^1,O(mu4)) and matched rank(D5); this goes one step further -- the 5-dim carrier mode space C_car:=H^0(P^1,O(mu4))…
Statement, assumptions and sources
Paper context: Origin Theory
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ARCH.RRCAR.02
Original ledger classification: Identity [I] (Clifford dim chain) + Physical [C] (carrier = mode-space spinor)
Original claim text
RIEMANN-ROCH CARRIER HARDENED FROM g_car TO D5 ITSELF (external-review point 7, 2026-06-14): v189 read g_car=5=h^0(P^1,O(mu4)) and matched rank(D5); this goes one step further -- the 5-dim carrier mode space C_car:=H^0(P^1,O(mu4)) generates the so(10)=D5 half-spinor by its EVEN CLIFFORD ALGEBRA. (1) [I] dim Lambda^even(C^5) = C(5,0)+C(5,2)+C(5,4) = 1+10+5 = 16 = 2^{g_car-1} = dim S^+ (the repo's dim_Splus). (2) [I] so(10)=D5 (rank 5) has Dirac spinor 2^5=32 and half-spinors 2^4=16 each; S^+ = Lambda^even(C_car), so the 5-dim mode space's Clifford module IS the D5 half-spinor. (3) [I] this closes the chain mu4 -> g_car -> D5 from ONE four-point divisor: cycle side rank H_1=deg-1=3=N_fam (A3); function side h^0=deg+1=5=g_car=dim C_car; Clifford spinor Lambda^even=16=D5 half-spinor=carrier -- D5 is the Clifford spinor of the Riemann-Roch carrier, not a separate input. (4) [C] reading the carrier as Cl^even(C_car) (not merely matching rank(D5)=5) is a geometric hardening, but 'carrier = Clifford spinor of the seam meromorphic mode space' is still a physical [C] step (as in v189); does NOT change the over-determined status of g_car=5. Exact; Wolfram-mirrored
Original classification detail
Identity [I]: Lambda^even(C_car=5-dim)=S^+=16=2^{g_car-1}=the D5 half-spinor, closing mu4->g_car->D5; [C] for the carrier=Clifford-spinor-of-the-mode-space identification (hardens v189, does not change over-determined g_car=5)
Dependencies as recorded
ARCH.RRCAR.01; AX.P2.01; E8.GLU.01; v189_riemann_roch_carrier.py; v1_e8_glue.py
Open the original ledger source, row 263Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-06-14 (three review levers: Marks-via-Lefschetz, E_fail functional, RR [formula in source] D5 — v195–v197)Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
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- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- Sharper still: one anchor [formula in source] , and [formula in source] as a compiler hull ( )TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]Origin Theory
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
- 2026-07-13 ( v474 — the operator level of the entropic-action bridge: the [formula in source] Clifford/spinor structure exhibited on the carrier Fock space, the Hodge fold identified as the [formula in source] conjugation, and the v473 [formula in source] -target decided — supports exactly [formula in source] , the naive pair-block reading killed; no status moves)Research changelog; TFPT — Introduction and reading guide
- 2026-06-17 (The Modular Spectral Closure: Dirac as covariance induction, the modular cutoff, the Kummer/K3 unification, and the assembly certificate, v258– v261)Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v197_rr_carrier_clifford_d5.pyRegistered verification module · registry
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ARCH.RRCAR.02 — hardening the Riemann–Roch carrier from g_car to D5 itself (review point 7): the 5-dim carrier mode space C_car:=H⁰(P¹,O(µ4)) generates the so(10)=D5 half-spinor by its even Clifford algebra. [I] dim Λ^even(C⁵)=C(5,0)+C(5,2)+C(5,4)=1+10+5=16=2^{g_car−1}=dim S⁺; so(10)=D5 half-spinor 2⁴=16=S⁺=Λ^even(C_car); closes the chain µ4→g_car→D5 from one four-point divisor (cycle side rank H_1=3=N_fam(A3); function side h⁰=5=g_car; Clifford spinor=16=D5 half-spinor=carrier). [C] 'carrier = Clifford spinor of the seam mode space' stays physical (as v189), does not change the over-determined g_car=5.
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