A History-Tagged Wheel: Complete Axiomatization with Retrocausal Structure, Celestial Soft Charge, and Ramanujan Grading — Definitions and Elementary Proofs

We define RH = R x H where H = Sigma* is free monoid of derivation histories, and extended division: a / b = (1, h1) if r1=r2=0 and h1=h2, else bot if h1 != h2. This gives same-zero vs different-zero theorem: E=(0, [[e^{i pi}+1]]) and T=(0, [[1-1]]) satisfy E/E=1_H, T/T=1_H, E/T=bot, T/E=bot, despite pi1(E)=pi1(T)=0. This preserves same-origin cancellation while flagging different-origin 0/0 as bot, unlike Carlstrom wheel where (x-x)/(x-x)=bot always. We then extend H to H_ext = H_past x H_future x Q_soft x P where Q_soft is abelian group of supertranslation charges (Pasterski) and P is partition grading with Ramanujan grading |p|. Operation (h_p h'_p, h'_f h_f, Q+Q', p+p') and involution T(h_p,h_f,Q,p)=(h_f,h_p,-Q,p). We prove: (4) full + is associative, coherent restriction +_coh is associative iff retrocausal chain matches, (5) Q_soft: ker pi1 -> Q_soft is homomorphism and E/T=bot iff Q_soft differs, (6) deg: H_ext -> N preserves Ramanujan class mod5 for same-zero and gives bot for different-zero. This construction is new as a combined object, though each component is built from known ideas (wheels, provenance semirings, soft charges, partitions). No physical law for lambda is claimed. Formula A_protected = A e^{-lambda V S_loop} with S_loop=Omega remains conjecture requiring measured lambda. Pure math stops at Theorem 6. Keywords: wheel algebra, division by zero, provenance, retrocausal, celestial holography, Ramanujan partitions