Decision: no candidate admitted. An explicit Volterra construction does produce a nonlocal residual with the complete boundary cancellation. The proposed positive-mixture estimate for that residual fails a necessary condition from the original theta tail. This is a paper admission rejection, not a new NS claim or a closure of general nonlocal certificates.
Reviewed remote baseline: 670864385024041d24612cf620054c1975473928;
local scope commit: b914ac47e8006951a22d030e920d592243e7e0fe. Read AGENTS,
the current admission brief/record, applicable frozen-input and missing-evidence
entries, and PR77 reviews 01, 02, 06. The coordinator’s complete-register
review is inherited, not independently replayed. Both missing-original
recovery groups remain OPEN. No external theorem is imported to establish
the rejection below.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–77. The specific changed hypothesis assessed is a positive mixture for the original pair’s conditional second moment, rather than the failed score Gram or local drift residual. The displayed asymptotic rejects this mixture class. It does not decide positive definiteness of that conditional moment itself.
Use the original complete even theta kernel, with all coefficients one:
phi(u)=sum_(n>=1) [2*pi^2*n^4*exp(9u/2)-3*pi*n^2*exp(5u/2)]
*exp(-pi*n^2*exp(2u)), u>=0;
phi(-u)=phi(u).
w(s,t)=phi(s+t)*phi(s-t),
M(t)=int_R w(s,t) ds > 0,
C(t)=int_R s^2*w(s,t) ds,
k(t)=C(t)/M(t).
There is no prefix, coefficient choice, clock deformation or transform
division. In particular M(t)>0 everywhere. Write
s=(u+v)/2, t=(u-v)/2, b=phi'/phi, and define
V(s,t)=int_(-infinity)^s [a^2-k(t)]*w(a,t) da,
h(u,v)=V(s,t)/w(s,t).
This is an actual construction, not an unspecified correction. Since
(partial_u+partial_v)=partial_s, it gives exactly
R_h(u,v)=(u+v)^2/4-(partial_u+partial_v)h-[b(u)+b(v)]h
=k((u-v)/2).
The defining centering gives V(-infinity,t)=V(+infinity,t)=0 for
every fixed real t. These are precisely the flux boundaries in PR77’s
complete integration-by-parts bridge, including arbitrary fixed translates.
The relevant integrals are absolutely convergent by the complete theta
tails. Symmetry in u,v follows from evenness in t. Thus an independent proof
that k is positive definite would supply this nonlocal certificate. Equivalently,
M is already an autocorrelation, and the pointwise product C=M*k is positive
definite whenever k is. The exact existing transfer is
L1[X](r)=4*int_R C(t)*cos(2*r*t)dt >= 0,
J(r)=16*L1[X](r)>=0, every real r.
Zeros and the low-height polynomial correction are retained automatically. This proves only the first Laguerre target conditionally, not RH or G2. The construction alone estimates no sign: k positive definite is the additional unproved input, not a consequence of conditional centering.
The proposed sufficient arithmetic lemma was
k(t)=int_[0,infinity) sech(2*t)^a dnu(a), all real t,
nu a finite nonnegative measure, nu([0,infinity))=k(0).
Each power with a>0 is positive definite: substitution x=exp(4t)
in its Fourier integral gives
int_R sech(2t)^a*exp(i*r*t)dt
=2^(a-2)*abs(Gamma(a/2+i*r/4))^2/Gamma(a) > 0.
The a=0 member is the constant kernel. Positive mixtures therefore give
the required matrix inequality and the complete transfer above.
The planned estimating method was a positive Laplace representation of the exact original-lattice quotient
Q(z)=k(0.5*arcosh(exp(z))), z>=0;
Q(z)=int exp(-a*z) dnu(a).
This is the positive-power/positive-Laplace class, not arbitrary positive definite k. The first original-coefficient feasibility check is its large-z asymptotic. That check fails before a positive density or derivative-sign argument can be attempted. Thus no unsupported claim that such a method will work is being made.
Let t -> +infinity and A=2*pi*exp(2t). In abs(s)<=t/2, both
positive kernel arguments t+s,t-s tend uniformly to infinity. Their
complete series gives the dominant product
w(s,t)=4*pi^4*exp(9t)*exp(-A*cosh(2s))
*[1-(6/A)*cosh(2s)+9/A^2]
*[1+O(exp(-c*exp(t)))]. (1)
Here c is some fixed positive constant for sufficiently large t. To justify
the uniform relative remainder, factor out the n=1 term in each original
series: the n>=2 terms have exponent gap at least
3*pi*exp(2*(t-abs(s)))>=3*pi*exp(t); their polynomial coefficients are
absorbed by a smaller positive c. The n=1 polynomial is bounded away from
zero throughout this region for large t.
Outside abs(s)<=t/2, the complete bound
phi(x)<=K*exp(-c0*exp(2*abs(x))), for fixed positive K,c0, bounds
the zeroth and second moment tails by a polynomial factor times
exp(-c1*exp(3t)). This is negligible relative to the main integral,
whose size is a positive constant times exp(9t-A)*A^(-1/2).
The absolute tails of the model in (1) obey the same needed negligibility.
Consequently its moments can be extended to the full s-line with errors
smaller than every inverse power of A. This uses no forbidden termwise
full-line unfolding of the original theta series.
Put y=2s, followed by y=x/sqrt(A). Taylor expansion at the unique
minimum of cosh, with the usual Gaussian-integrable remainder and an
exponentially small outer region, gives
I_j(A)=int_R y^j*exp(-A*cosh(y))
*[1-6*cosh(y)/A+9/A^2]dy, j=0,2;
I_0=exp(-A)*sqrt(2*pi/A)*[1-49/(8A)+O(A^-2)],
I_2=exp(-A)*sqrt(2*pi)*A^(-3/2)*[1-53/(8A)+O(A^-2)].
The coefficients are directly checkable: the order A^-1 correction is
-6-x^4/24; Gaussian moments E[x^4]=3, E[x^6]=15
produce -49/8 and -53/8. The nonconstant part of the polynomial
reweight first enters at the next order. Hence the complete original ratio is
k(t)=I_2/(4I_0)+negligible
=1/(4A)-1/(8A^2)+O(A^-3).
Since z=log cosh(2t) and
exp(2t)=2*exp(z)-(1/2)*exp(-z)+O(exp(-3z)), this becomes
Q(z)=exp(-z)/(16*pi)-exp(-2z)/(128*pi^2)+O(exp(-3z)). (2)
Equation (2) contradicts the proposed positive Laplace representation.
Indeed finite nonnegative nu and exp(z)Q(z)->c=1/(16*pi) force
support in [1,infinity): any positive mass below 1 would make that
limit infinite. Dominated convergence then forces nu({1})=c.
It follows that exp(z)Q(z)>=c for every z>=0, whereas (2) makes it
strictly smaller than c for all sufficiently large z. No onset height is
computed or claimed.
NS100 excludes individual slice positivity, which this construction never uses. NS101 shares positive complete kernels and the Volterra centering identity, so those generic facts cannot estimate the target. Its translated mixture changes the original one-lattice tail and Jacobi IVP; no mismatch is advertised as a control pass. The decisive screen here is a necessary condition on the original kernel itself. Davenport–Heilbronn has different coefficients/completion and is not an instance of the exact proposed original-lattice formula. No off-line-zero inference is made. NB target and rate controls do not match this first-Laguerre assertion.
Closest result: PR77 review 06 excludes positive Gaussian mixtures for C. That does not already exclude the present positive sech-power mixtures for the different, exponentially decaying quotient k. PR77’s score-Gram and local-residual failures likewise do not imply (2). The negative second tail coefficient supplies this candidate-specific rejection.
Success would have changed: a positive representation would have supplied an independently sufficient nonlocal certificate for complete J, with no further sign estimate. Actual failure changes: stop this precise positive mixture ansatz. General positive definiteness of k, other h, and the original J target remain open; no adjacent repair is proposed.
Budget: this bounded paper feasibility pass is complete. Zero scan or certification budget; one independent algebra/tail review is appropriate. No research row, thaw, manuscript version, computation run, commit, outreach or agent spawn. Only this assigned internal note was written.
Final wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR76–77. The specified nonlocal positive-mixture candidate is rejected on paper; the full original arithmetic sign remains unestimated.