A registered result in its source context
Original ledger classification: Identity/Lattice [E] (number-field split) + [C]/[O] (functor verdict)
Recorded statement: DO THE DISCRETE-KERNEL AND DYNAMIC RATES TRANSLATE? AN EXACT NUMBER-FIELD SPLIT (2026-06-21, v312/v313 follow-up). [E] FIELD SPLIT: the affine-E8 adjacency spectrum is the rational part {0,+-1,+-2} in Q plus the golden part {+-phi,+-1/phi}…
Statement, assumptions and sources
Paper context: Origin Theory
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RATE.TRANSLATE.01
Original ledger classification: Identity/Lattice [E] (number-field split) + [C]/[O] (functor verdict)
Original claim text
DO THE DISCRETE-KERNEL AND DYNAMIC RATES TRANSLATE? AN EXACT NUMBER-FIELD SPLIT (2026-06-21, v312/v313 follow-up). [E] FIELD SPLIT: the affine-E8 adjacency spectrum is the rational part {0,+-1,+-2} in Q plus the golden part {+-phi,+-1/phi} in Q(sqrt5) (min poly x^2-x-1, discriminant 5); the dynamic transfer spectrum {1,(2/3)^6,(1/3)^6} is ENTIRELY rational (in Q). [E] TRANSLATION (the (2,3)-fold/family sector): the rational discrete angles 2cos(pi/2)=0 (|Z2|=2) and 2cos(pi/3)=1 (N_fam=3) are the SAME atoms that build the recovery rate (2/3)^6=(|Z2|/N_fam)^(2 N_fam) in Q -- same field, same atoms, so the family/sheet rates DO translate. [E] NO-TRANSLATION (the 5-fold/carrier sector): phi in Q(sqrt5) minus Q has NO rational dynamic image (2/3 is neither an adjacency nor a lazy-update eigenvalue; heat-kernel t=Delta/(2-phi)~6.37 not clean) -- the carrier 5-fold is STATIC-only (builds the lattice/spectrum, not a dynamic rate). [E] SHARED CLOCK: both are facets of the order-30 Coxeter element, 30=h(E8)=g_car*(2 N_fam)=5*6; the golden in the order-5 factor, the recovery exponent 6=2 N_fam=30/g_car in the order-6=2*3 factor. [E] QUASICRYSTAL: E8=H4+phi H4 (240=2*120, 120=|2I|=|R+(E8)|), so the phi cut-and-project carries the 5-fold carrier (Q(sqrt5)) but the cusp weights {0,1/3,2/3} are RATIONAL (3-fold), a separate parabolic datum -- the substrate carries the carrier, NOT the family dynamics (confirms v313/v312). [C]/[O] VERDICT: a static<->dynamic translation functor exists ONLY on the rational (2,3-fold/family) sector; the carrier 5-fold is field-obstructed (static-only), the bridge holding both is the order-30 Coxeter element.
Original classification detail
[E] the Q vs Q(sqrt5) field split of the discrete spectrum + the dynamic-in-Q fact + the (2,3)-sector translation + the 5-fold field obstruction (heat-kernel + eigenvalue negatives) + the order-30 shared clock + the quasicrystal counts; [C]/[O] the translation-functor verdict. Answers the discrete<->dynamic translation question exactly (a field split); does NOT close the substrate question.
Dependencies as recorded
GOLD.ATOMS.01; SEAM.EQUIV.GAP.01; v313; v54; v56
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- Origin Theory
Sections that explicitly cite this claim
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- 2026-06-21 (do the discrete-kernel and dynamic rates translate? An exact number-field split [formula in source] vs [formula in source] : v314)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
- 2026-06-21 (couple or factorize? the order-30 Coxeter element couples the carrier and family sectors as a cyclotomic compositum, Galois [formula in source] : v315)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- The unique attractor: a gapped transport selects ONE fixed pointorigin theory
- The unique attractor: a gapped transport selects ONE fixed pointinput 4 origin theory
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- 2026-06-21 (does the Galois [formula in source] organize the readouts? Yes — the CP phases are the family-factor cyclotomic data, Galois [formula in source] CP conjugation: v316)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]Origin Theory
All 18 linked sections
- 2026-06-24 (Harvesting the QFT/gravity machinery: v385 the UV-branch kill-test surface, v386 the graviton-exchange amplitude, v387 the gap-driven correction calculus)
- Front matter / support
- Outcome
- 2026-06-23 (Bird's-eye synthesis: two new structural patterns — v383 the universal spectral-gap principle and v384 the residual is certification, not construction)
- The unique attractor: a gapped transport selects ONE fixed point
- Front matter / support
- 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)
- 2026-06-22 (external-review pass: a registry bug fix, the NA62 2026 rare-kaon update, CP-phase prose hygiene, and three firewall/visualisation enhancements: v202, v203, v305, v307, v321)
- The whole integer skeleton from one pair [formula in source]
- The unique attractor: a gapped transport selects ONE fixed point
Registered verification modules
- v314_rate_translation.pyRegistered verification module · frontier
Read the description recorded in the module registry
Do the discrete-kernel and dynamic rates translate? An exact number-field split. The affine-E8 adjacency spectrum splits into the rational part {0,+-1,+-2} in Q and the golden part {+-phi,+-1/phi} in Q(sqrt5); the dynamic transfer spectrum {1,(2/3)^6,(1/3)^6} is entirely rational. So the family rates DO translate (the rational discrete angles 2cos(pi/2)=0=|Z2|, 2cos(pi/3)=1=N_fam build the recovery rate (2/3)^6=(|Z2|/N_fam)^(2 N_fam) in Q), but the carrier 5-fold (golden, Q(sqrt5)) has NO rational dynamic image -- a field obstruction, so it is static-only. The shared object is the order-30 Coxeter element (30=g_car*(2 N_fam)=5*6); the phi quasicrystal carries the carrier, not the cusp-weight family dynamics
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