Reviewed baseline: 1f09dd8a8587506ea83c0c48573d225b7a323bc1 (PR74).
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR72–74.
This is a paper applicability audit, with local algebraic self-review only. No numerical or symbolic scan, new sign candidate, NS row, thaw or manuscript version. The complete 104-node conclusion register and 14 input demands were reviewed; that is not a historical proof/certificate replay. Both original evidence recovery groups remain OPEN.
Question: can a theta addition identity turn the original two-shift pair
into an independently estimable nonnegative transform? The dependency would
have to reach L1[X](r)>=0 for EVERY real r, retaining the complete pair.
This first Laguerre condition alone is not RH. A theta identity without a
signed estimating step does not meet that dependency.
Use the definitions
Theta(x) = sum_(n in Z) exp(-pi*n^2*x), x>0,
h(u) = exp(u/2)*Theta(exp(2u)),
D_u = (partial_u^2-1/4)/4,
phi(u) = D_u h(u),
X(r) = integral_R phi(u)*exp(i*r*u) du = xi(1/2+i*r)/2.
Theta reciprocity makes h even. The complete phi is positive, smooth, even
and rapidly decaying with all derivatives. For the ordinary two-variable
Jacobi function, Theta(x)=theta_3(0 | i*x). Hence our pair has
tau_+ = i*exp(2s+2t), tau_- = i*exp(2s-2t),
elliptic arguments both zero, tau_+/tau_-=exp(4t).
Kharchev–Zabrodin I, arXiv:1502.04603v2, Sections 3.1–3.3 and 4.2, distinguishes the elliptic variables from the modulus. Its addition identities use a common modulus; its basic bilinear identities use tau and 2*tau. Substituting s+t and s-t as elliptic variables would change the object, rather than use an identity for our factors.
There are also valid unequal-modulus identities. Kharchev–Zabrodin II, arXiv:1510.02699v1, Proposition 3.1, uses moduli n1tau and n2tau, positive integers n1,n2, and retains a sum over residue classes modulo n1+n2. These hypotheses match exp(4t)=n1/n2. The common-modulus mismatch therefore does NOT exclude all addition methods. Rational ratios are dense, but an estimate on those ratios would still need the same sign or a controlled limiting error; the identities alone provide neither. No uniform complexity bound or derivative transfer in t is imported.
Special loci also do not settle the problem. At t=0 the two moduli agree; at s=0 they are reciprocal. The latter is exactly a slice of the sort for which NS100 already prohibits an unconditional positive-slice proof for the actual original kernel. Neither locus estimates the complete s integral.
For arbitrary real s,t let x=exp(2s). Absolute convergence of the TWO Gaussian series at fixed s,t permits the bijective reindexing
k=n+m, l=n-m,
(n,m) in Z^2 <=> (k,l) in Z^2 with k=l (mod 2).
Theta(x*exp(2t))*Theta(x*exp(-2t))
= sum_(k=l mod 2) exp[-pi*x/2 *
{cosh(2t)*(k^2+l^2)+2*sinh(2t)*k*l}]. (A)
Indeed n=(k+l)/2 and m=(k-l)/2, which gives the quadratic form printed in (A). No factor of 2 is lost: the map is a bijection onto the parity sublattice, not onto all of Z^2. Both Gaussian factors, all lattice points, and both parity cosets remain.
The real quadratic matrix
Q_t = [[cosh(2t), sinh(2t)],
[sinh(2t), cosh(2t)]]
has eigenvalues exp(2t), exp(-2t), and determinant 1. Thus the series
converges for every fixed real s,t, including t=0. Splitting into
(k,l)=2(p,q)+epsilon*(1,1), epsilon=0,1, expresses (A) as the sum of two
genus-two theta constants with Riemann matrix 2*i*x*Q_t and upper
characteristics (epsilon/2,epsilon/2), lower characteristic zero.
This uses the standard multidimensional series definition in the second
source; it introduces no positivity theorem in the parameter t.
At t=0 the cross term vanishes and the formula reduces to the check
Theta(x)^2 = theta_3(0 | 2*i*x)^2 + theta_2(0 | 2*i*x)^2. (B)
For t!=0 the coupling cannot be dropped from (A). For example, k=l=1
corresponds to the original point (n,m)=(1,0); its exponent is
-pi*x*exp(2t), not -pi*x*cosh(2t). This is an exact one-term check,
not a sampled computation. Diagonalizing Q_t over the reals recovers the
original unequal-modulus factors and must also transport the lattice.
Positive definiteness of Q_t concerns the lattice variables k,l. The required positive definiteness of C concerns matrices formed from arbitrary real translation differences t. These are different properties. Neither the dimension, the determinant, nor positivity of each real Gaussian term supplies the latter.
Write H(s,t)=h(s+t)h(s-t). The original pair is obtained by differentiating the complete object, not by using H as a substitute for phi_+phi_-:
phi(s+t)*phi(s-t) = (1/256)*P H(s,t),
P = partial_s^4 - 2*partial_s^2*partial_t^2 + partial_t^4
-2*partial_s^2 -2*partial_t^2 +1. (C)
To check the factor, each u derivative becomes
partial_(u+)=(partial_s+partial_t)/2 or
partial_(u-)=(partial_s-partial_t)/2. The two original operators are
((partial_s+partial_t)^2-1)/16 and
((partial_s-partial_t)^2-1)/16; their product is P/256.
As a second check, testing on exp(alpha(s+t)+beta(s-t)) gives multiplier
(alpha^2-1/4)*(beta^2-1/4)/16 in either representation.
The unregularized integral of s^2 H diverges, because H~exp(|s|) at both s ends for fixed t. Instead define the exact smooth subtraction
R(s,t)=H(s,t)-2*cosh(s),
M_j(t)=integral_R s^j*R(s,t) ds, j=0,2. (D)
P annihilates 2*cosh(s), since on a t-independent function it is
(partial_s^2-1)^2. This is a smooth subtraction, not exp(|s|) with its
distributional derivatives. For t in each compact interval, complete
theta reciprocity and its Gaussian tails give
R(s,t) = -exp(-|s|) + O_T(exp(c_T*|s|-d_T*exp(2*|s|))),
for suitable finite c_T and positive d_T as |s|->infinity, and the needed derivatives have the same integrable character. Thus (D), differentiation under its integral, and the following s integrations by parts are valid:
integral_R s^2*R_ss ds = 2*M_0,
integral_R s^2*R_ssss ds = 0.
Consequently the COMPLETE associated kernel is
C(t) = integral_R s^2*phi(s+t)*phi(s-t) ds
= [(partial_t^2-1)^2*M_2(t)
-4*(partial_t^2+1)*M_0(t)]/256. (E)
These are supporting identities and an explicit boundary treatment, not a lower bound. The original single-lattice coefficients enter H and its tail estimates; the differential calculus itself is generic.
There is a second boundary issue if (E) is Fourier transformed. Even after the s regularization, M_0 and M_2 must not be Fourier transformed separately as ordinary integrable functions without a new justification. For example, h(u)>=exp(|u|/2) follows from reciprocity and Theta(x)>=1 at positive x. For t>=0 this yields H(s,t)>=exp(max(|s|,t)). Integrating the resulting lower bound for R first on |s|<=t and then on |s|>t proves
M_0(t) >= 2*(t-1)*exp(t), t>=0. (F)
Both lower integrals converge at each fixed t: the exterior is -exp(-|s|).
In detail their sum is 2*t*exp(t)-4*sinh(t)-2*exp(-t), which equals (F).
Thus M_0 grows rather than decays at large t. The COMPLETE bracket in (E),
by its original definition, equals 256*C and is rapidly decaying. Its
cancellations cannot be replaced by ordinary Fourier transforms of the
separate growing moments. A distributional transform, if chosen, would
require its own domain/boundary accounting and still no sign is automatic.
The subtraction is also signed: at (s,t)=(0,0),
R=Theta(1)^2-2<0, since Theta(1)<=1+2/(exp(pi)-1)<6/5.
At s=0, R(0,t)>=exp(t)-2>0 for t>log(2). This does not assert a negative
C or L1; it prevents simply carrying termwise positivity through the
regularization.
The target in these coordinates is still
integral_R [(partial_t^2-1)^2*M_2
-4*(partial_t^2+1)*M_0]*cos(2*r*t) dt >= 0,
for every real r. (G)
The complete bracket is kept inside the integral. By (E) and PR73 this integral is 64*L1X. There is no division by a zero, no parity/source restriction on the translates, and no assertion that (G) is an independent estimate. No method controlling the signed cancellation in (G) was obtained.
No sign candidate survives admission. The checked addition formulas do not supply a comparison for the actual two moduli. The valid lattice identity and completed moment formula retain exactly the signed comparison already open. Do not run a lattice, residue-class, dimension or height scan on the strength of these representations.
Success of a future original-arithmetic estimating inequality for (G), or the original pair directly, would establish the first-Laguerre target only. Failure of the literal common-modulus substitution says nothing against every unequal-modulus or nonlinear method. No broad no-go result is claimed. Budget used: one bounded two-source applicability and paper-boundary audit, one findings PR; zero further computation budget without a concrete method. No external outreach. RH, G2 and the cofinal signed-arithmetic lower bound remain open.