A History-Tagged Wheel v3: Classification Theorem — Uniqueness of R_H^ret (v1+v2+v3 Unified)
Authors/Creators
Description
v3 NEW - CLASSIFICATION THEOREM - This is new math as classification (Theorem 2.1):
Main Theorem: Let W be associative wheel satisfying
(A1) Provenance: ker pi = {epsilon_h}, epsilon_h / epsilon_h = 1_H, epsilon_{h1}/epsilon_{h2}=bot if h1!=h2, W = R x H(W)
(A2) Supertranslation invariance: Q_soft: ker pi -> Q(W) homomorphism, Q_E !=0 for Euler void
(A3) Ramanujan grading: deg: W -> N detecting p(5k+4)=0 mod5
Then W contains R_H^ret = R[Q_soft][P] \rtimes (H_past x H_future) as canonical subwheel. If W minimal (no proper subwheel satisfies A1-A3), W is isomorphic to R_H^ret.
Corollary: In any model satisfying A1-A3, E/E=1_H, E/T=bot is forced, not conventional. Q_soft(E)!=Q_soft(T) forced by supertranslation Ward identity. This is representation theorem analogous to Cayley.
v1 construction: R_H = R x H, H=Sigma* free monoid, extended division a/b = (1,h1) if r1=r2=0 and h1=h2 else bot if h1!=h2. E=(0,[[e^{i pi}+1]]), T=(0,[[1-1]]) satisfy E/E=1_H, E/T=bot despite pi1(E)=pi1(T)=0. Same-zero 1_H vs different-zero bot.
v2 forced components: (7) H=Sigma* initial among provenance-preserving models, (8) coherence T(h_f)=h_p unique for associativity of coherent product, (9) Q_soft forced by supertranslation invariance - any invariant homomorphism ker pi1 -> Q_soft factors through projection, hence E/T=bot forced by Q_E!=0, (10) deg=|p| forced by Ramanujan grading.
Novelty: Combined object H_ext = H_past x H_future x Q_soft x P with classification is new. Individual components from known ideas: wheels Carlstrom 2004, provenance semirings Green et al 2007, soft charges Pasterski celestial holography, partitions Ramanujan 1919. No physical lambda law claimed. A_protected = A e^{-lambda V S_loop} remains conjecture.
Keywords: wheel algebra, classification theorem, representation theorem, division by zero, provenance, retrocausal, supertranslation, Ramanujan partitions, Euler identity
Previous records:
v1: 10.5281/zenodo.23934709
v1+v2 unified: 10.5281/zenodo.23934989
This record v3: 10.5281/zenodo.23935347 - adds classification theorem