A registered result in its source context
Original ledger classification: Construction/proof [E]/[O] (cusp weight from a minimal rewrite)
Recorded statement: DERIVING THE CUSP WEIGHT 2/3 FROM A MINIMAL REWRITE (2026-06-22, sharpening HYP.FIBER.01/v324 + HYP.INJECT.01/v312). A minimal local rule -- a family channel with N_fam=3 slots, one absorbing attractor (w=0) and |Z2|=2 surviving (the Z2…
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
HYP.REWRITE.01
Original ledger classification: Construction/proof [E]/[O] (cusp weight from a minimal rewrite)
Original claim text
DERIVING THE CUSP WEIGHT 2/3 FROM A MINIMAL REWRITE (2026-06-22, sharpening HYP.FIBER.01/v324 + HYP.INJECT.01/v312). A minimal local rule -- a family channel with N_fam=3 slots, one absorbing attractor (w=0) and |Z2|=2 surviving (the Z2 pair) -- supplies the cusp fiber. [E] MINIMAL REWRITE: M=[[1,0,0],[0,1/3,1/3],[0,1/3,1/3]] has spectrum {1,2/3,0}; the recovery survival 2/3=(N_fam-1)/N_fam=|Z2|/N_fam EMERGES from the rule arity, not injected. [E] RECOVERY RATE DERIVED: over the order-6=2 N_fam dynamic hand the rate is (2/3)^6=64/729 = the recovery gap (v76/v324), derived not put in. [E] NON-GRAPH-SPECTRAL (proof, sharpening v312): 2/3 is NOT a root of the affine-E8 charpoly x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1) (p(2/3)=-3520/19683 != 0), so the recovery rate cannot be an adjacency eigenvalue -- necessarily the branching channel. [E] ATTRACTOR: the absorbing state (eigenvalue 1) is the democratic w=0 law (v317). [O] HONEST RESIDUAL: the rewrite DERIVES 2/3 from the arity {|Z2|,N_fam}={2,3} (the anchor atoms, still the input P2/v23) -- v324's injected datum REDUCED to the anchor arity, not derived from nothing. UPDATE 2026-07-15 (v486/HYP.REWRITE.02): the uniform rule is the DEGENERATE case (spectrum {2/3, 0}); the unique lazy Z2-pair walk (stay 1/2, hop 1/6, leak 1/3) generates the FULL physical spectrum {1, (2/3)^6, (1/3)^6} over the same order-6 hand -- the (1/3)^6 mode now has a local-rule origin; superseded as full-spectrum generator, the 2/3 arity story unchanged.
Original classification detail
[E] the rule spectrum {1,2/3,0} + the recovery-rate derivation (2/3)^6 + the non-graph-spectral PROOF p(2/3)!=0; [O] the arity {2,3} is still the anchor input. Sharpens v324/v312 (the injected datum reduced to the anchor arity); not a derivation of the atoms.
Dependencies as recorded
HYP.FIBER.01; HYP.INJECT.01; v312; v324; v317; v76
Open the original ledger source, row 426Location 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.
- 2026-06-22 (next-steps round: the keystone collapsed to one shared premise (Lean [formula in source] one citable theorem), the [formula in source] solver suite, the cusp weight derived from a minimal rewrite, and the [formula in source] pressure point: FORM.QGEO.BW.01, v325, v326, v327, v328)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
- 2026-07-15 (II) ( v487 — the lazy split FORCED by the resummed clock: the walk's one-step spectrum is the complete v124 clock ladder below the wall, deck parity [formula in source] rung index, and the two comb tones are one bend apart — the v486 \ ``split selection'' closes onto established structure; no gate closes, no marker moves)Research changelog; TFPT — Introduction and reading guide
- The compiler: why [formula in source]TFPT — Introduction and reading guide
- 2026-07-15 ( v486 — generator hunt: the FULL transfer spectrum [formula in source] from ONE forced local rule — the unique lazy [formula in source] -pair walk [formula in source] ; the previously generator-less [formula in source] mode gets its origin; v327's uniform rule superseded as full-spectrum generator — no gate closes, no marker moves)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]Origin Theory
- 2026-06-24 ( [formula in source] leading term from icosahedral combinatorics: v396)Research changelog; TFPT — Introduction and reading guide
- 2026-06-25 (F_pole external: the Koide source [formula in source] pole transfer is one-loop EM: v420)Research changelog; TFPT — Introduction and reading guide
Registered verification modules
- v327_hypergraph_rewrite.pyRegistered verification module · frontier
Read the description recorded in the module registry
Deriving the cusp weight 2/3 from a minimal rewrite (sharpening v324/v312). A minimal local rule -- a family channel with N_fam=3 slots, one absorbing attractor (w=0) and |Z2|=2 surviving -- supplies the cusp fiber. M=[[1,0,0],[0,1/3,1/3],[0,1/3,1/3]] has spectrum {1,2/3,0}; 2/3=(N_fam-1)/N_fam=|Z2|/N_fam EMERGES from the rule arity, not injected. Over the order-6=2 N_fam hand the rate is (2/3)^6=64/729 = the recovery gap (v76/v324), derived. NON-GRAPH-SPECTRAL (proof, sharpening v312): 2/3 is NOT a root of the affine-E8 charpoly (p(2/3)=-3520/19683 != 0), so the recovery rate cannot be an adjacency eigenvalue. The absorbing state (eigenvalue 1) is the democratic w=0 law (v317). Honest residual: the arity {2,3} (the anchor atoms) is still the input (P2) -- v324's injected datum reduced to the anchor arity. Substrate = (2,3,5)-adjacency (carrier) x this minimal branching rule (family).
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.