PASS on the assigned real-variable boundary audit. OPEN on the sign.
Reviewed scientific baseline: d93aaf6b8dbe1fcbebf6fff4f6d1f71d993f4208
(PR75). Read AGENTS section 0 and project premise, the present review brief
and complete draft, and PR75’s complete derivation.md. The parent supplied
the repository-wide conclusions/dependency review; this review independently
checks the following boundary calculations, not all historical proofs or
certificates. No scans, numerical computation, extra agents or external
outreach were used.
Wall check: Same open gap. Closest: NS100/101 and PR74/75. What changes: a globally smooth decaying primitive permits ordinary transforms of the separate moments. It supplies no sign estimate for the corrected expression.
Equations (1)-(4), draft lines 28-65, are correct with a=1/2.
The definition f=2*cosh(u/2)-h(u) is global. Theta reciprocity gives
h(-u)=h(u), while its series and every derivative converge locally uniformly
on the real line. Thus f is smooth and even, including at zero, and every
odd derivative at zero vanishes. No reflected half-line approximation is
being differentiated.
More explicitly, for each derivative order k there is a finite c_k such
that
abs(f^(k)(u)) <= c_k*exp(-abs(u)/2), every real u.
On the positive half-line, differentiate the exponentially small series in
(1). Each derivative is a sum of polynomial factors in
z_n=pi*n^2*exp(2u) times exp(u/2-z_n), hence is negligible compared with
exp(-u/2) at infinity. Compact intervals are controlled by smoothness;
evenness covers the other tail. This proves the derivative bound, the unit
tail ratio, and every claimed weighted integrability assertion.
The Green factor is correct: G_a=exp(-a*abs(u))/(2*a), whose first
derivative has jump -1, satisfies (a^2-d^2)G_a=delta. At a=1/2 its
denominator is one. Since phi is smooth and has exponentially weighted
integrable tails, g=4*G_a*phi is a classical smooth solution of (2), and
abs(g(u)) <= 4*exp(-abs(u)/2)
* integral_R exp(abs(v)/2)*phi(v) dv.
Consequently g decays at both ends. The smooth difference f-g solves
w''=a^2*w and is a linear combination of exp(a*u) and exp(-a*u);
decay at both endpoints forces both coefficients to vanish. This justifies
uniqueness without treating the cusp in the Green kernel as a cusp in f.
Its distributional delta is precisely the defining source and leaves no
additional atom or boundary term after convolution.
The inherited positivity of phi can also be seen directly at u>=0:
its nth summand is exp(u/2)*(2*z_n^2-3*z_n)*exp(-z_n), positive since
z_n>=pi; reciprocity covers u<0. Thus convolution gives f>0, and
(1) gives the strict upper envelope f<exp(-abs(u)/2), including zero.
Equation (6), draft lines 93-104, has the correct constants. Writing
T=abs(t) and F(s,t)=f(s+t)*f(s-t), the basic envelope is
abs(F(s,t)) <= exp(-max(abs(s),T)).
Direct integration gives
integral_R exp(-max(abs(s),T)) ds = 2*(T+1)*exp(-T),
integral_R s^2*exp(-max(abs(s),T)) ds
= 2*(T^3/3+T^2+2*T+2)*exp(-T).
The same bound with a finite derivative-dependent constant holds for every
mixed derivative partial_s^j partial_t^k F, by the bound on derivatives
of f. This is sufficient simultaneously for:
s integral, with compact-uniform domination;s endpoint terms in (7), since polynomial multiples of derivatives
vanish at both endpoints;t transforms and t integrations by parts, since every
derivative of either A_j obeys a constant times its corresponding
polynomial-exponential envelope in (6).Equivalently f is a Schwartz function, and so is F after the invertible
linear change of coordinates; integrating its weighted s slices preserves
that property. The explicit envelopes are stronger information about the
tails and do not conflict with the Schwartz statement.
Therefore integral s^2*F_ss=2*A_0 and
integral s^2*F_ssss=0 in (7) are legitimate. Equations (5) and (8) are
ordinary Fourier identities on every real frequency. No interpretation by
finite part, analytic regularization or distributional Fourier transform is
needed for them.
An optional strengthening of the exposition is to show explicitly how the
old and new completed products relate. Put c(u)=2*cosh(u/2), so h=c-f.
With the old PR75 notation R=h(s+t)*h(s-t)-2*cosh(s),
R(s,t)-F(s,t)
= 2*cosh(t)-c(s+t)*f(s-t)-c(s-t)*f(s+t).
This is a pointwise equality of smooth functions. The right side must be
kept as a complete expression before integration in s: its three displayed
terms separately include nonintegrable constants at infinity. Their sum
decays at both s endpoints, as follows either from PR75’s proven tail for
R and the new bound for F, or directly from the unit tails of f.
The pair operator P in PR75 annihilates this entire difference. Each mixed
c*f term has one factor killed by d^2-1/4; on 2*cosh(t) the operator
reduces to (partial_t^2-1)^2, also zero. Consequently the differential
moment expressions built from old M_j and new A_j are identical. They
are two complete representations of 256*C; the replacement is not a
discarding of the old growing terms one at a time. No new delta appears at
s=0, t=0, s=t or s=-t, since all underlying functions are smooth.
No correction to (1)-(8), the explicit envelopes or their real boundary justifications is required. The displayed old/new comparison would make the absence of hidden terms easier to verify, but is an explanatory addition rather than a repair. This review does not infer a Laguerre sign from any envelope. Equation (10) remains the full original sign obligation.