AP019 — odd horner quotient — irrationality research

Exact named statement

forall d ab ac k x p. ~d=0 ->
  NatAtanhPrefix(d,ab,ac,S(2*k)) ->
  NatHornerValue(ab,ac,3,S(2*k),x) ->
  Pow(3,S(2*k),p) ->
  exists bb bc y q. (NatAffineSlice(ab,ac,1,2,bb,bc,k) /\ ((forall j a. Lt(j,k) ->
  BetaAt(bb,bc,j,a) ->
  (S(2*j))*a=2*d) /\ (NatHornerValue(bb,bc,9,k,y) /\ (Pow(9,k,q) /\ (~q=0 /\ (~p=0 /\ (x=3*y /\ (p=3*q /\ IRatEq(3*x,0,d*p,3*y,0,d*q)))))))))

PD0002 — Lt · PD0013 — BetaAt · ND0411 — NatAffineSlice · ND0437 — NatHornerValue · ND0425 — NatAtanhPrefix · PD0020 — Pow · ND0383 — IRatEq

Fully expanded target AST

Complete certificate and independent checks

142 actual local proof nodes. Each node carries its complete ordinary HA proof body.

Full canonical proof bundle · Original HA receipt · Independent Lean receipt

Target SHA-256: ef40bb286d66b9b7abd7a8521c7ef114b6ccba8d8780e5d0574178f985096c33
Bundle SHA-256: fd6d43b7cb41ce4722c57ebffffdde004e26d8fd5a289ffd74679af508db224a

Checked statement containment

These exact targets occur inside this certificate. These links summarize containment, not direct dependency arrows; the downloadable bundle records every private lemma and ordered edge.

AP016 · AP017 · AP018 · CV032

Scope

construct actual odd slice horner value and power at base nine then prove the original finite log quotient equals its odd horner quotient with every denominator nonzero without infinite analytic identification

None of these finite arithmetic statements closes an IR campaign parent. The analytic identification, interpolation and final positive irrationality certificate remain open.