Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Tests and predictions
The published foundation in this area
20 authored claim references lead into the original ledger and paper sections. This source record remains available even where no separate reviewed research packet is attached.
The ledger is the authority for claim status. Graph edges and model annotations aid navigation, while algebraic identities, computations, external reports, and physical identifications carry different evidential weight; shared inputs correlate readouts.
Starting structure in the Atlas orientation
The ledger distinguishes claim status; sources, contracts, and checks document results within their stated scope.
Which connection is sought
The open work is independent testing, complete premises, holdouts, and correlated-output accounting without status inflation.
How it connects
Validation reconnects each topic edge to original sources and ledger status; check counts are not independent evidence.
The concrete element/orbit and arithmetic statements are available. No fresh kernel run, nor a newly checked single Lean theorem about all abstract subgroups or lattice isomorphisms. The general subgroup conclusion also uses finite group theory; concrete E8 basis embedding and physical carrier selection are separate.
Read the exact result, assumptions and evidence
Result
In Fin4 x Fin4 with qZ=(5x^2+3y^2) mod8, named Lean declarations classify all six isotropic elements and four odd-coordinate isotropic generators. Explicit orbit identities give H1 and H2; klein_not_isotropic gives the nonisotropic Klein witness, spinor_swap_exchanges swaps the two orbits, and unique_halfway_stage classifies the only nonzero isotropic order-two element (2,2). Index and sector arithmetic are additional separate declarations.
Required assumptions
Specified finite carrier discriminant form
The model is El=Fin4 x Fin4 with componentwise addition modulo4 and qZ(x,y)=(5x^2+3y^2) mod8. Its identification with a concrete physical carrier is not inferred without premises.
Algebraic uniqueness under the stipulated Pascal/half-spinor exhaustion condition. Original source and proof text reviewed; no fresh Lean kernel run. AX.P2.01 remains a declared physical interface; the equation does not select itself from an original source.
Read the exact result, assumptions and evidence
Result
CarrierRankUniqueness.carrier_rank_pascal_unique states and proves for every natural g that 2^g=g^2+g+2 iff g=5. Its proof combines the small cases with pascal_growth for all g>=6. This is unbounded; PascalLadder has a separate bounded K=2 statement. For g=5, N_fam=3 and g+N_fam=8 are computed.
Required assumptions
Pascal exhaustion condition as input
g is natural; the stipulated equation is 2^g=g^2+g+2. The equation itself is the additional carrier exhaustion rule.