A registered result in its source context
Original ledger classification: Atom trichotomy [E]/[C]: {2,3,5} ram/inert/split in Z[i] (2 ram, 3 inert, 5 split) vs Z[omega] (3 ram, 2,5 inert); the ramified prime of each ring = its atom; each atom is the unique ramified prime of one facet of Q(i,sqrt-3,sqrt5), product 30=h(E8); seam/flavor/golden reading [C]
Recorded statement: The atom trichotomy (2026-06-25): {2,3,5} are ramified/inert/split in the two exceptional CM rings, and each atom is the unique ramified prime of one quadratic facet of K=Q(i,sqrt-3,sqrt5) (v403). Sharpens v222/v230 (atoms read only as CM…
Statement, assumptions and sources
Paper context: Origin Theory
The exact identifier, original classification and full claim are available below.
The source wording and classification below are preserved as recorded. A paper link, shared script or authored topic relation does not upgrade the claim or report a new proof run. Read later reviewed findings separately.
Read the exact registered claim and its dependencies
ARITH.TRICHOTOMY.01
Original ledger classification: Atom trichotomy [E]/[C]: {2,3,5} ram/inert/split in Z[i] (2 ram, 3 inert, 5 split) vs Z[omega] (3 ram, 2,5 inert); the ramified prime of each ring = its atom; each atom is the unique ramified prime of one facet of Q(i,sqrt-3,sqrt5), product 30=h(E8); seam/flavor/golden reading [C]
Original claim text
The atom trichotomy (2026-06-25): {2,3,5} are ramified/inert/split in the two exceptional CM rings, and each atom is the unique ramified prime of one quadratic facet of K=Q(i,sqrt-3,sqrt5) (v403). Sharpens v222/v230 (atoms read only as CM norms) and v390/v394/v403 (the three facets) into one prime-splitting table. [E] Z[i] (square/seam, tau=i): 2=|Z2| RAMIFIES (2=-i(1+i)^2), 3=N_fam INERT (3 mod 4, norm 9=N_fam^2), 5=g_car SPLITS ((2+i)(2-i), 1 mod 4) -- the carrier rank g_car=5 FACTORS on the seam. [E] Z[omega] (hex/flavor, tau=omega): 3=N_fam RAMIFIES (N(1-omega)=3), 2=|Z2| INERT (norm 4), 5=g_car INERT (norm 25=g_car^2). [E] the ramified prime of each ring IS its atom: Z[i] ramifies at 2 (sheet/seam), Z[omega] at 3 (family/flavor). [E] the three facets Q(i)/Q(sqrt-3)/Q(sqrt5) (disc -4,-3,5) ramify over exactly one atom each (2,3,5), product 2*3*5=30=h(E8); 2 imaginary (CM/dynamic) + 1 real (RM/static) = the v403 reading at the prime level. [C] the seam/flavor/golden reading. Exact number theory composing existing facts; no new number.
Original classification detail
Identity
Dependencies as recorded
ARITH.HULL.01; CMNORM.DUAL.01; COXETER.ATOMS.01; COXETER.PRIME2.01; v403; v222; v390; v394; v313
Open the original ledger source, row 484Location recorded in the ledger
These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.
- Origin Theory
Sections that explicitly cite this claim
A literal claim identifier is present in these paper sections. The section and its premises determine what the citation supports.
- 2026-06-25 (The Gaussian operator [formula in source] the atom trichotomy: v415, v416)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.
- 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
- 2026-06-25 (The cyclotomic norm triple [formula in source] the carrier-5 clock: v418)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]Origin Theory
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- The whole integer skeleton from one pair [formula in source]origin theory
Registered verification modules
- v416_atom_trichotomy.pyRegistered verification module · core
Read the description recorded in the module registry
The atom trichotomy: {2,3,5} are ramified/inert/split in the two exceptional CM rings, and each atom is the unique ramified prime of one quadratic facet of K=Q(i,sqrt-3,sqrt5) (v403). Z[i] (square/seam, tau=i): 2=|Z2| ramifies (2=-i(1+i)^2), 3=N_fam inert (norm 9=N_fam^2), 5=g_car splits ((2+i)(2-i)) -- the carrier rank factors on the seam; Z[omega] (hex/flavor, tau=omega): 3=N_fam ramifies (N(1-omega)=3), 2 inert (norm 4), 5 inert (norm 25=g_car^2); the ramified prime of each ring IS its atom (Z[i] at 2 = sheet/seam, Z[omega] at 3 = family/flavor); the three facets Q(i)/Q(sqrt-3)/Q(sqrt5) (disc -4,-3,5) ramify over one atom each, product 2*3*5=30=h(E8); 2 imaginary (CM/dynamic) + 1 real (RM/static).
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.