AP082 — log exp refinement — irrationality research

Exact named statement

forall d e ab ac bb bc k l x z p q m N cb cc eb ec a0 b0 u v P. ~d=0 ->
  ~e=0 ->
  NatAtanhPrefix(d,ab,ac,S k) ->
  NatAtanhPrefix(e,bb,bc,(S k)+l) ->
  NatHornerValue(ab,ac,3,S k,x) ->
  NatHornerValue(bb,bc,3,(S k)+l,z) ->
  Pow(3,S k,p) ->
  Pow(3,(S k)+l,q) ->
  NatScaledExpRecurrence(d*p,3*x,cb,cc,m) ->
  NatScaledExpRecurrence(e*q,3*z,eb,ec,(S m)+N) ->
  BetaAt(cb,cc,0,a0) ->
  BetaAt(eb,ec,0,b0) ->
  Sum(cb,cc,S m,u) ->
  Sum(eb,ec,S((S m)+N),v) ->
  Pow(2,m,P) ->
  ~p=0 /\ (~q=0 /\ (~P=0 /\ (~(d*p)=0 /\ (~(e*q)=0 /\ (Le(b0*u,a0*v) /\ Le((p*P)*(a0*v),(p*P)*(b0*u)+((9*3)*P+2*p)*(a0*b0)))))))

PD0001 — Le · PD0013 — BetaAt · ND0425 — NatAtanhPrefix · ND0437 — NatHornerValue · PD0020 — Pow · ND0443 — NatScaledExpRecurrence · PD0015 — Sum

Fully expanded target AST

Complete certificate and independent checks

229 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: bccfdfa515c324f3b34a5a205185f1a81d590d8d27d1b41dcba91fbbd65883cc
Bundle SHA-256: 0261db5cc15244a1345941dd847fc8bf283af92dbb74cf19ff6a9f22168eff88

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.

CV009 · CV032 · CV062 · CV068 · CV086 · CV087 · CV088 · CV089 · CV090 · CV092 · CV093 · CV094 · CV095 · CV096 · CV097 · CV098 · CV100 · CV101

Scope

prove supplied logarithm and exponential refinement across independent scales codes and horizons with explicit 27 over ternary power plus 2 over binary power error without limit or code equality assumptions

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