CV076 — composition tail bound

Exact named statement

forall k N. exists d ab ac q wb wc w0 f dn pb pc qb qc ahb ahc x p v rb rc sb sc vb vc zb zc s ub uc. (~d=0) /\ ((NatAtanhPrefix(d,ab,ac,S k)) /\ ((NatZeroBefore(ab,ac,S k,1)) /\ ((BetaAt(ab,ac,0,0)) /\ ((NatBoundedPrefix(ab,ac,2*d,S k)) /\ ((NatHornerValue(ab,ac,2,S k,x)) /\ ((Pow(2,k,q)) /\ ((Le(x+2*d,(2*d)*q)) /\ ((NatExpWeightTable(d,N,wb,wc)) /\ ((forall j a. Le(j,N) ->
  BetaAt(wb,wc,j,a) ->
  ~d=0 ->
  ~a=0) /\ ((BetaAt(wb,wc,0,w0)) /\ ((Factorial(N,f)) /\ ((Pow(d,N,dn)) /\ ((w0=f*dn) /\ ((forall j w f p. Le(j,N) ->
  BetaAt(wb,wc,j,w) ->
  Factorial(j,f) ->
  Pow(d,j,p) ->
  (w*f)*p=w0) /\ ((~w0=0) /\ ((NatPowerTrace(ab,ac,S k,S ((S N)*(S k)),pb,pc,qb,qc,S N)) /\ ((NatScaledHornerTrace(ab,ac,2,S k,ahb,ahc)) /\ ((BetaAt(ahb,ahc,S k,x)) /\ ((Pow(2,S k,p)) /\ ((Pow(2,((S N)*(S k)),v)) /\ ((NatWeightedSeriesTrace(S ((S N)*(S k)),wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,S N)) /\ ((NatRowHornerPrefix(pb,pc,qb,qc,2,S ((S N)*(S k)),vb,vc,S N)) /\ ((NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,S N)) /\ ((Sum(zb,zc,S N,s)) /\ ((forall ub uc. NatPowerRow(rb,rc,sb,sc,S N,ub,uc) ->
  NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\ ((forall k y a b c. Lt(k,S N) ->
  BetaAt(vb,vc,k,y) ->
  Pow(p,k,a) ->
  Pow(2,k,b) ->
  Pow(x,k,c) ->
  a*y=(v*b)*c) /\ ((~p=0) /\ ((~v=0) /\ ((~(d*p)=0) /\ ((~(w0*v)=0) /\ ((p=q*2) /\ ((Le(x,d*p)) /\ ((Le(2*x,2*(d*p))) /\ ((BetaAt(zb,zc,0,w0*v)) /\ ((NatScaledExpRecurrence(d*p,2*x,zb,zc,N)) /\ ((Le(s,9*(w0*v))) /\ ((NatPowerRow(rb,rc,sb,sc,S N,ub,uc)) /\ ((NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\ (forall M. Le(M,((S N)*(S k))) ->
  exists r tb tc u p0 q0 h. (((S N)*(S k))=M+r) /\ ((NatAffineSlice(ub,uc,M,1,tb,tc,S r)) /\ ((NatHornerValue(tb,tc,3,S r,u)) /\ ((Pow(2,M,p0)) /\ ((Pow(3,M,q0)) /\ ((Pow(3,((S N)*(S k)),h)) /\ ((~p0=0) /\ ((~q0=0) /\ ((~h=0) /\ ((~(w0*h)=0) /\ ((Le(u*q0,(9*p0)*(w0*h))) /\ (IRatLe(u,0,w0*h,9*p0,0,q0)))))))))))))))))))))))))))))))))))))))))))))))))))

PD0001 — Le · PD0002 — Lt · PD0013 — BetaAt · PD0015 — Sum · ND0411 — NatAffineSlice · ND0423 — NatExpWeightTable · ND0427 — NatScaledHornerTrace · PD0020 — Pow · PD0023 — Factorial · ND0405 — NatPowerRow · ND0407 — NatPowerTrace · ND0437 — NatHornerValue · ND0438 — NatRowHornerPrefix · ND0421 — NatWeightedSeriesTrace · ND0439 — NatPointwiseProductPrefix · ND0440 — NatBoundedPrefix · ND0408 — NatZeroBefore · ND0425 — NatAtanhPrefix · ND0443 — NatScaledExpRecurrence · ND0444 — IRatLe

Fully expanded target AST

Complete certificate and independent checks

336 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: 2baa8df68b336dc9d80d78e7f1d9797fdeb2859efc39377e00c1c0a18de9f83f
Bundle SHA-256: cf5c84b108e2db80fcaf846e709839e44402011c309e596c4b9c90c74560289e

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.

CV004 · CV008 · CV009 · CV011 · CV019 · CV022 · CV025 · CV030 · CV032 · CV037 · CV038 · CV039 · CV043 · CV044 · CV046 · CV047 · CV048 · CV049 · CV056 · CV057 · CV058 · CV059 · CV060 · CV061 · CV062 · CV063 · CV064 · CV065 · CV066 · CV072 · CV073 · CV074 · CV075

Scope

all actual composition majorant data retained with one decoded terminal row and every in range actual tail certified below nine times two over three to cutoff in valid rational order not analytic limit

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