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The complete original theta-tail response

Abstract

The complete original theta-tail response.

Theorem 1.1 (A relative rational response and the original source-node debits).

Lean statement: D5/S3/Zeros/Jensen/OriginalPrimalTailResponse.original_primal_tail_response

Proof. Machine-checked in Lean as D5/S3/Zeros/Jensen/OriginalPrimalTailResponse.original_primal_tail_response (✓ std3). ∎

Source. Repository-derived.

Commentary.

The original polynomial is B_(n,t)(x)=sum from j=0 to n of (-1)^j*(n)_jt^jx^(2j)/(n^j(2j)!). Put alpha=t/(2n), A=pi*(N+1)^2, Q_N=sum over m>N of q_m, and H_N=sum over 1<=m<=N of q_m, with q_m and the full cubic origin flux kappa_N as in the original primal tail. The denominator is exactly Z=Re xi(1/2), which is positive unconditionally. The density has total mass one. For every n>=1, the full-density expectation J_n(t) is the real source polynomial at -t, its complex coercion is the canonical complex Jensen evaluation, and J_n(t)=(2/Z)(integral BH_N+integral B*Q_N). The complex polynomial uses shift zero and scale 1/n, including n=1.

There is an absolute K>0, chosen before a,N,n,t. For every a>0, one C_W(a)>0 works for all N,n>=1 and t>0 with alpha<=a. All full-density and half-line integrals in the formula are integrable. If I=integral over x>0 of BQ_N, the original bound is |tI/kappa_N-1|<=C_W(a)(N+1)^4/t for every N>=1. At t>=2C_W(a)(N+1)^4, every real or canonical complex source node has (2/Z)integral BH_N<=-kappa_N/(Zt).

For A>=max(12,a), the sharper relative response is |(t+4A^2)I/kappa_N-1|<=K/A. If also A>=2K, then I>=kappa_N/(2(t+4A^2))>0. Every actual real or complex source node consequently has (2/Z)integral BH_N <=-kappa_N/(Z(t+4*A^2))<0. No positive lower bound on alpha, rescaled t, or degree is required. Common and multiple nodes are retained; no derivative or cofactor denominator occurs.

The small residual keeps 4A^2Q_N-Q_N’’ grouped. With s=pim^2 and z=sexp(2x), its leading polynomial difference is bounded by 16z^2*(z^2-A^2)+151z^3. The leading term splits into s^4exp(4x)(exp(4x)-1) and (s^2-A^2)s^2exp(4x). The first has the essential x factor after exp(4x)-1<=4xexp(4x); the second vanishes exactly at the first omitted mode. Weighted mode integration gives 230s^2exp(-s)+16*(s^2-A^2)sexp(-s). All later modes are summed with their scale gaps. If S_k=shiftedBudget(k)-1, L=230*(1+S_2)+(8/pi)S_3 and K=(2/11)L. The entire weighted residual is at most LA^2exp(-A), whereas the literal full flux is at least (11/2)A^3exp(-A).

The literal finite coefficient recurrence proves B’‘-alphaxB’+tB=0, B(0)=1 and B’(0)=0. The derivative of exp(-alphax^2)(B^2+B’^2/t) is -2alphaxexp(-alphax^2)B^2, giving |B|<=exp(alphax^2/2). The full Green bracket BF’-B’F+alphaxBF has derivative BA_alpha(F)+tB*F. Polynomial weighted jet bounds pay each infinity product for both F=Q_N and F=A_alpha(Q_N); both origin terms remain. The second Green application and the original full-tail budget give the all-cutoff P bound with the same C_W(a). The first Green application and the grouped residual give the relative response. Exact raw moments and the complete series split supply the source correspondence and node debits. These conclusions assert no node existence, cofactor sign, full source coupling, or Riemann hypothesis.

References