CV031 — normalized composition — irrationality research

Exact named statement

forall N. Lt(1,N) ->
  exists d ab ac wb wc db dc pb pc qb qc tb tc vb vc cb cc z fn pn. ((~d=0 /\ (NatAtanhPrefix(d,ab,ac,N) /\ (NatExpWeightTable(d,N,wb,wc) /\ ((forall j a. Le(j,N) ->
  BetaAt(wb,wc,j,a) ->
  ~d=0 ->
  ~a=0) /\ (NatEulerPrefix(ab,ac,db,dc,N) /\ (NatPowerTrace(ab,ac,N,N,pb,pc,qb,qc,N) /\ (NatWeightedSeriesTrace(N,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\ (NatPowerRow(tb,tc,vb,vc,S N,cb,cc) /\ ((forall j x y. Lt(j,N) ->
  BetaAt(cb,cc,j,x) ->
  NatConvolutionCoefficient(db,dc,N,cb,cc,N,j,y) ->
  d*(j*x)=y) /\ ((forall i a b c. Lt(S(S i),N) ->
  BetaAt(cb,cc,i,a) ->
  BetaAt(cb,cc,S i,b) ->
  BetaAt(cb,cc,S(S i),c) ->
  (S(S i))*c=i*a+2*b))))))))))) /\ (BetaAt(wb,wc,0,z) /\ (~z=0 /\ (NatExpAtanhRecurrence(cb,cc,N,z) /\ (forall i a. Lt(i,N) ->
  BetaAt(cb,cc,i,a) ->
  ((i=0 /\ a=z) \/ (exists k. i=S k /\ a=2*z))))))) /\ (Factorial(N,fn) /\ (Pow(d,N,pn) /\ (z=fn*pn /\ (forall j w f p. Le(j,N) ->
  BetaAt(wb,wc,j,w) ->
  Factorial(j,f) ->
  Pow(d,j,p) ->
  (w*f)*p=z))))

PD0001 — Le · PD0002 — Lt · PD0013 — BetaAt · ND0401 — NatConvolutionCoefficient · ND0405 — NatPowerRow · ND0407 — NatPowerTrace · ND0421 — NatWeightedSeriesTrace · ND0410 — NatEulerPrefix · ND0423 — NatExpWeightTable · ND0425 — NatAtanhPrefix · ND0400 — NatExpAtanhRecurrence · PD0020 — Pow · PD0023 — Factorial

Fully expanded target AST

Complete certificate and independent checks

272 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: 255502b20ed9e685e90c18c8a121a614c14ffe23ae181d749f51ced7a170f19e
Bundle SHA-256: 37a4fbb48c282439ac816d0efea5198bc6905ad57e7736aad23242cd96f8be71

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.

CV001 · CV002 · CV004 · CV008 · CV009 · CV011 · CV012 · CV019 · CV020 · CV021 · CV022 · CV024 · CV025 · CV027 · CV028 · CV029

Scope

same actual composition weights normalized by factorial powers not rational evaluation or analytic identity

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