A registered result in its source context
Original ledger classification: Construction [E]/[C] (fibered substrate)
Recorded statement: THE MINIMAL HYPERGRAPH SUBSTRATE THAT WORKS -- A (2,3,5)-NETWORK FIBERED BY A 3-FOLD CUSP CHANNEL (2026-06-22, the constructive v312 follow-up): reproduces BOTH the E8 skeleton AND the recovery gap. [E] NETWORK FACTOR (carrier/5-fold):…
Statement, assumptions and sources
Paper context: Origin Theory
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Read the exact registered claim and its dependencies
HYP.FIBER.01
Original ledger classification: Construction [E]/[C] (fibered substrate)
Original claim text
THE MINIMAL HYPERGRAPH SUBSTRATE THAT WORKS -- A (2,3,5)-NETWORK FIBERED BY A 3-FOLD CUSP CHANNEL (2026-06-22, the constructive v312 follow-up): reproduces BOTH the E8 skeleton AND the recovery gap. [E] NETWORK FACTOR (carrier/5-fold): T_net=(A+2I)/4 fixes the Kac marks (eigenvalue 1 = E8 skeleton), subleading (phi+2)/4 (golden); (2/3)^6 NOT in its spectrum (the v312 negative). [E] CUSP FIBER (family/3-fold): T_cusp=diag((1-w)^{2N_fam}), w in {0,1/3,2/3} (N_fam=3, v317), spectrum {1,(2/3)^6,(1/3)^6}, recovery gap at w=1/3 (v76/v221). [E] FIBERED SUBSTRATE CARRIES BOTH: T=T_net (x) T_cusp has top eigenvalue 1 (eigenvector marks (x) w=0 = E8 skeleton x democratic cusp) AND (2/3)^6 as a genuine eigenvalue (network attractor x cusp subleading). [E] HONEST INJECTION (v312): without the fiber (T_net (x) I_3) there is NO (2/3)^6 -- the cusp datum is injected, not graph-spectral; substrate = network x cusp = the v315 split 30=g_car*(2 N_fam)=5*6, the gap living entirely in the family factor. [E] MINIMALITY: the smallest fiber is the 3-node cusp (=N_fam); a 2-node fiber cannot hold 3 weights -> minimal substrate = (2,3,5)-net x 3-cusp = order-30 (v319 clock). [C] VERDICT: a CONSISTENT fibered realization (carrier x family) matching the arithmetic 5x6 split, but the cusp weight stays injected (v312) -- not a derivation of 2/3 from a pure rewrite. Sharpens v312; the open rewrite question is unchanged.
Original classification detail
[E] the fibration construction (T_net (x) T_cusp carries the skeleton + the gap) + the v312 negative (no gap without the fiber); [C] the 'minimal substrate = carrier x family' reading matching the v315 5x6 split; [O] deriving the cusp weight from a pure rewrite stays open. Sharpens v312; not a closure.
Dependencies as recorded
HYP.INJECT.01; COX.COUPLE.01; GALOIS.FAMILY.01; v312; v315; v317; v319; v221; v76
Open the original ledger source, row 422Location recorded in the ledger
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- Origin Theory
Sections that explicitly cite this claim
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- 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
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
- The compiler: why [formula in source]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
- 2026-06-22 (consistency + de-accretion editorial pass: current-state alignment across papers, website and ledger; no new modules)Research changelog; TFPT — Introduction and reading guide
- 2026-06-24 (Coupled hypergraph rewrite: v395 unifies carrier [formula in source] family in one local rule)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]origin theory
Registered verification modules
- v324_hypergraph_fiber.pyRegistered verification module · frontier
Read the description recorded in the module registry
The minimal hypergraph substrate that works: a (2,3,5)-network FIBERED by a 3-fold cusp channel reproduces both the E8 skeleton and the recovery gap (the constructive v312 follow-up). Network factor (carrier/5-fold): T_net=(A+2I)/4 fixes the Kac marks (eigenvalue 1=E8 skeleton), subleading (phi+2)/4 (golden); (2/3)^6 NOT in its spectrum (v312 negative). Cusp fiber (family/3-fold): T_cusp=diag((1-w)^{2N_fam}), w in {0,1/3,2/3} (N_fam=3, v317), spectrum {1,(2/3)^6,(1/3)^6}. Fibered substrate carries both: T=T_net (x) T_cusp has top eigenvalue 1 (eigenvector marks (x) w=0 = skeleton x democratic cusp) AND (2/3)^6 as a genuine eigenvalue (network attractor x cusp subleading). Honest injection (v312): without the fiber (T_net (x) I_3) there is no (2/3)^6 -- the cusp datum is injected, not graph-spectral; substrate = network x cusp = the v315 split 30=g_car*(2 N_fam)=5*6. Minimality: the smallest fiber is the 3-node cusp (=N_fam) = order-30 (v319 clock). Verdict: a consistent fibered realization, not a derivation of 2/3 from a pure rewrite
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