Ordinary derivation, self-reviewed, pending independent reproduction. Source checked on 2026-09-28: Zhu, arXiv:2608.24827v2, <https://arxiv.org/html/2608.24827v2>, equations (6), (12), (13), Lemma 6.1 and section 6.2. Only this source and the hunt's existing theory were checked; no novelty search. No source copy is retained in this restricted write area. The source's odd benchmark is a measured matrix value 9.1183e-15 and a full-space lower bound 8.2065e-15 at L=4/5, T#=150, H=0, N=200.
1. Cross terms and the reduction
The theory file RESULTS.md, pole calculation at lines 182 onward and its section 1.5 parity row, supports the odd application. Here are both missing cross-term calculations explicitly, without editing the theory lane.
Use F(t)=integral f(x) exp(itx) dx. For real f=e+o, F_e is real even and F_o imaginary odd, so Re(F_e conj(F_o))=0 pointwise. The pole is 2(c^2-s^2), c=integral e cosh(x/2), s=integral o sinh(x/2), and contains no mixed term. Thus Q(e+o)=Q(e)+Q(o). For complex f=u+iv with u,v real, the Fourier cross term is odd in t, because F_u(-t)=conj(F_u(t)) and likewise for v. It integrates to zero against the real even symbol. The pole 2|c|^2-2|s|^2 also splits into real and imaginary parts. Consequently Q(f)=Q(u)+Q(v), and orthogonality splits the squared norms. For an infinite positive archimedean integral, interpret Q as an extended closed form; alternatively first prove this on its finite form domain.
The sine:16 H is a real even sum of out-of-band cosines. Its full-line integral against |F|^2 vanishes by Fourier support, independently of parity. With S bounding P_L-H globally, T>=15/4 and beta=log(T/(2pi))-1/T-S>0, the increasing lower envelope gives
Q_odd >= R_odd = beta I + C - 2 s s^T, C_ij = (1/pi) integral_0^T (Psi_L+H-beta) v_i v_j dt.
This uses the existing envelope result; it is not a new verification of H.
2. Basis and pole from Rodrigues' formula
For n=2i+1 put nu=n+1/2 and T_i(x)=sqrt(nu/L) P_n(x/L). Legendre orthogonality makes these an orthonormal basis of the real odd subspace. The general Fourier identity is
integral_-1^1 P_n(y) exp(izy) dy = 2 i^n j_n(z).
Hence F_i(t)=i v_i(t), v_i(t)=(-1)^i 2 sqrt(L nu) j_(2i+1)(Lt). The matrix uses F_i conj(F_j)=v_i v_j. Multiplying F_i F_j would introduce the wrong minus sign in the entire multiplier matrix.
Let a=L/2. The odd integral against cosh(ay) vanishes, so the sinh moment equals the exp(ay) moment. Rodrigues and n integrations by parts give
integral_-1^1 P_n(y) exp(ay) dy = a^n/(2^n n!) integral_-1^1 (1-y^2)^n exp(ay) dy = 2 sqrt(pi/(2a)) I_(n+1/2)(a). s_i = 2 sqrt(L nu) sqrt(pi/(2a)) I_nu(a).
There is no (-1)^i in s_i. The head pole is -2 s_i s_j.
3. Odd tail sequences and indices
The same Rodrigues integral at z=x real gives |j_n(x)| <= x^n/(2n+1)!!, since the beta integral of (1-y^2)^n is 2^(n+1) n!/(2n+1)!!. For the real exponential, exp(ay)<=exp(a) gives |i_n(a)| <= exp(a) a^n/(2n+1)!!. Thus for odd integer n,
a_n = 2 sqrt(L(n+1/2)) (LT)^n/(2n+1)!!, pi_n = 2 sqrt(L(n+1/2)) exp(L/2) (L/2)^n/(2n+1)!!.
These majorize |v_n(t)| for 0<=t<=T and |s_n|, respectively. For i=0,...,N-1 the orders are 1,...,2N-1. The first discarded order is n0=2N+1, whose denominator is (4N+3)!!. The helper dfact_log(n)=lgamma(2n+2)-n log(2)-lgamma(n+1) computes log((2n+1)!!), with n the polynomial order, not the basis index.
The step-two ratios are, with x=LT or L/2 respectively,
r_x(n) = x^2/((2n+3)(2n+5)) sqrt((n+5/2)/(n+1/2)).
Both factors decrease with n>=0, so checking r_x(n0)<1 in Arb licenses alpha=sum a_n <= a_n0/(1-r_LT(n0)) and pis=sum pi_n <= pi_n0/(1-r_(L/2)(n0)), sums over odd n>=n0. This is also why a_n0 and pi_n0 bound each tail coefficient.
Let K=(T/pi) sup_[0,T]|Psi_L+H-beta|, b_i=2 sqrt(L nu_i), b_max=max b_i, b_sum=sum b_i, s_max=max |s_i| and s_sum=sum |s_i|. All maxima use ball upper bounds. Absolute entry bounds, including the negative pole diagonal, give
eps_D = K a_n0 alpha + 2 pi_n0 pis, B1 = K b_max alpha + 2 s_max pis, Binf = K a_n0 b_sum + 2 pi_n0 s_sum, eps_B = sqrt(B1 Binf).
The tail satisfies D>= (beta-eps_D)I and the coupling norm is <=eps_B. The pole sign disappears only inside these absolute-value estimates. It remains negative in the retained matrix.
4. Quadrature and head positivity
The Poisson integral yields |v_i(z)|<=2 sqrt(L nu_i) exp(L|Im z|) for both odd and even n. Therefore the existing ellipse and node-shift derivation applies after substituting the odd nu_i. With h=1/2 and rho=1+sqrt(2), the rectangle has half-width h/sqrt(2), height h/2. The radius-0.1 disk around each real node lies in this rectangle, so the Cauchy derivative bound used for snapping nodes remains valid.
The reducer encloses the head, subtracts a proposed shift, constructs an exact dyadic lower-triangular factor and bounds its Arb residual by the maximum absolute row sum r. The head radius plus quadrature has operator norm at most e. Therefore
lambda_min(R_odd) >= min(lambda0-r-e, beta-eps_D)-eps_B.
The reported decimal is rounded downward from the exact Arb lower endpoint. The inverse-iteration value and sign-flip comparison are measurements. For a failed Cholesky pivot, a reconstructed Schur vector is evaluated against the ball matrix plus quadrature error, distinguishing a numerical refusal from an enclosed negative Rayleigh quotient.
If both sectors succeed, the weaker endpoint bounds all complex f by section 1. This remains a fixed-window candidate, with the ordinary reduction and shared numerical implementation pending independent review.