A registered result in its source context
Original ledger classification: Synthesis/inventory [E] (arithmetic + {a;pi} reduction) + [O] (seam realisation)
Recorded statement: THE CAPSTONE: THE SM STRUCTURAL SECTOR IS Q(zeta30) + GALOIS mu4 x Z2, COMPLETE INPUT {a, pi, v_geo} (2026-06-21, synthesis of the v313-v317 arc). [E] ARITHMETIC SKELETON: the SM STRUCTURAL sector (3 generations, 2 CP phases, the…
Statement, assumptions and sources
Paper context: Origin Theory
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ARITH.CAPSTONE.01
Original ledger classification: Synthesis/inventory [E] (arithmetic + {a;pi} reduction) + [O] (seam realisation)
Original claim text
THE CAPSTONE: THE SM STRUCTURAL SECTOR IS Q(zeta30) + GALOIS mu4 x Z2, COMPLETE INPUT {a, pi, v_geo} (2026-06-21, synthesis of the v313-v317 arc). [E] ARITHMETIC SKELETON: the SM STRUCTURAL sector (3 generations, 2 CP phases, the orbit/hierarchy ordering) lives in the cyclotomic field Q(zeta30): [Q(zeta30):Q]=phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=h(E8), Galois (Z/30)^x=mu4 x Z2 (order 8) -- carrier mu4=(Z/5)^x (golden sqrt5, v313/v315), family/sheet Z2=(Z/3)^x (zeta3=CP conjugation, v316); the 3 generations = the mu3 cube-root orbit (v317), the 2 CP phases = the zeta6 family-factor data (v316). [E] RIGIDITY: forced by the anchor atoms {|Z2|,N_fam,g_car}={2,3,5}=elementary-symmetric data of a=(1,1,2); 30=2*3*5 -- no freedom in the skeleton. [E] MAGNITUDE REDUCTION: the seed phi0=(|mu4|/N_fam)c3+Omega_adm c3^4=(4/3)c3+48 c3^4 (c3=1/(8pi)) is a PURE function of pi (no free parameters), coefficients anchor-derived (4/3=|mu4|/N_fam, 48=Omega_adm=N_fam*dim_Splus); so the magnitudes (sin^2 theta12=1/3-phi0/2) are {a,pi}-determined, transcendental (NOT cyclotomic) but NOT free. [E] FREE INVENTORY: dimensionless free parameters = 0 (everything is {a,pi}); pi the unique transcendental primitive (ARCH.CORE.01/v53); v_geo the unique dimensionful input (No-Unit Theorem v153/v78) -- complete input = {a, pi, v_geo}. [O] HONEST RESIDUAL: this rigidity holds GIVEN the axioms; it does NOT prove the raw seam REALIZES the structure -- the open bedrock QGEO.SYM.01/SEAM.EQUIV.01 stays [O]. VERDICT: SM structural sector = Q(zeta30)+Galois mu4 x Z2 (rigid, anchor-forced, degree=rank E8); magnitudes = {a,pi}-transcendental (phi0); the only genuinely free datum is v_geo; dimensionless free parameters: 0.
Original classification detail
[E] the Q(zeta30)/Galois=mu4 x Z2 skeleton (degree=rank E8) + the anchor-rigidity + the phi0={a,pi} reduction (pure pi, anchor-derived coefficients) + the 0-dimensionless-free-parameter inventory ({a,pi,v_geo}); [O] the seam realisation. A synthesis/inventory capstone of the v313-v317 arc; does NOT close the open bedrock and does NOT add a physical readout.
Dependencies as recorded
GALOIS.FAMILY.01; GALOIS.READOUT.01; COX.COUPLE.01; ARCH.CORE.01; NOUNIT.01; v313; v314; v315; v316; v317
Open the original ledger source, row 415Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-21 (capstone: the SM structural sector is [formula in source] Galois [formula in source] , and the complete input is [formula in source] — zero dimensionless free parameters: v318)Research changelog; TFPT — Introduction and reading guide
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- 2026-06-22 (Wolfram mirror of the arithmetic arc [formula in source] three sharpening steps: the CP-lock made quantitative, the Bisognano–Wichmann keystone, the minimal hypergraph fibre: v322, v323, v324)Research changelog; TFPT — Introduction and reading guide
- 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
- Front matter / supportOrigin Theory
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- 2026-06-22 (the translation clock: the discrete [formula in source] dynamic bridge is the order- [formula in source] Coxeter element, a clock with two coprime hands [formula in source] , read law-inclusive ( [formula in source] ) or live-only ( [formula in source] ): v319)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]origin theory
Registered verification modules
- v318_arithmetic_capstone.pyRegistered verification module · core
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The capstone: the SM structural sector is Q(zeta30) + Galois mu4 x Z2, and the complete input is {a, pi, v_geo} (synthesis of the v313-v317 arc). The structural sector (3 generations, 2 CP phases, orbit/hierarchy) lives in Q(zeta30): [Q(zeta30):Q]=phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=h(E8), Galois (Z/30)^x=mu4 x Z2. Rigidity: forced by the anchor atoms {2,3,5}=e(a). Magnitude reduction: phi0=(4/3)c3+48 c3^4 is a pure function of pi (no free parameters), so the magnitudes are {a,pi}-determined, not free. Free inventory: 0 dimensionless free parameters; pi the unique transcendental; v_geo the one dimensionful unit. Residual: the raw-seam realisation stays open (QGEO.SYM.01/SEAM.EQUIV.01)
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