Decision: no candidate admitted. The standard parity/character duplication identities give an exact complete recurrence, but no independent estimating step. In particular, an unrestricted positive-semidefinite matrix of the completed three-sector Laguerre blocks is impossible, even for the original theta function. This is a scoped rejection of one possible cone, not a closure of nonlinear duplication methods or of the original target.
Wall check: Same open gap — ZERO-GEOMETRY, NS100/101, PR75/77/78. What changes: a paper check of completed parity-sector recurrence and its candidate matrix order; no new arithmetic sign input survives.
Reviewed baseline 221ceec9da825bdc787111447badad961860760a; current AGENTS,
BRIEF, checkpoints, relevant register/frozen-input scopes, MISSING, PR75’s
addition derivation, PR77 reviews 01–03 and PR78 stops. The coordinator’s
full-register review is inherited, not a historical proof replay. Both
original-evidence recovery groups remain OPEN. No scan was run.
Let a=log(2), D=(d_u^2-1/4)/4, and, for j=2,3,4, define
h_j(u)=exp(u/2)*theta_j(0 | i*exp(2u)),
phi_j=D h_j, X_j(r)=int_R phi_j(u)*exp(i*r*u)du.
phi_3=phi, X_3=X=xi(1/2+i*r)/2.
Use the exact integer, half-integer and alternating integer coefficients of
these theta constants. The fourth half-characteristic has
theta_1(0|tau)=0, so its completed sector is identically zero; it is not a
discarded nonzero contribution. All moduli have positive imaginary part.
The defining series and modular reflection conventions are in
Kharchev–Zabrodin I, sections 2.1, 2.2 and 2.6.
The identities below follow directly by partitioning even and odd integers,
not by applying a common-modulus addition theorem to unequal moduli.
theta_3(0|i*x)=theta_3(0|4*i*x)+theta_2(0|4*i*x),
theta_4(0|i*x)=theta_3(0|4*i*x)-theta_2(0|4*i*x), x>0.
phi_3(u)=[phi_3(u+a)+phi_2(u+a)]/sqrt(2),
phi_4(u)=[phi_3(u+a)-phi_2(u+a)]/sqrt(2). (1)
Apply (1) separately at u=s+t and u=s-t. Their modulus ratio remains
exp(4t); it has not been set to one. The operator D commutes with these
translations. At either real end the growing leading exponential of each
h_j is annihilated by D; modular reflection exchanges the 2 and 4 sectors.
Alternatively, solving (1) gives the exact completed identities
phi_2(u)=sqrt(2)*phi(u-a)-phi(u),
phi_4(u)=sqrt(2)*phi(u+a)-phi(u). (2)
Thus all completed sectors and polynomially weighted derivatives decay at both ends. All integrations below are ordinary complete integrals with vanishing boundaries. No transform of an uncompleted growing h_j is used. The characteristic sectors are signed; their positivity is not assumed.
For j=0,1,2 set
K_bc,j(t)=int_R s^j*phi_b(s+t)*phi_c(s-t)ds.
Applying (1) to both factors and translating s gives
K_33,j(t)=(1/2)*sum_(b,c in {3,2}) sum_(l=0)^j
binom(j,l)*(-a)^(j-l)*K_bc,l(t). (3)
All four pair sectors and all lower moments remain. In particular the
second-moment recurrence contains K_bc,2-2*a*K_bc,1+a^2*K_bc,0.
Deleting the first or zeroth moment is not a positive recurrence for the
actual target C=K_33,2.
Fourier transforming (2), with the displayed positive-exponent convention, gives, at every real r,
v=(X_3,X_2,X_4)^T=A*X,
A=(1, sqrt(2)*exp(i*a*r)-1, sqrt(2)*exp(-i*a*r)-1)^T. (4)
The complete pair Fourier matrix is Hermitian:
S_bc=4*int_R K_bc,2(t)*exp(2*i*r*t)dt
=X_b'*conj(X_c')
-[X_b''*conj(X_c)+X_b*conj(X_c'')]/2.
The factors follow from u=s+t, v=s-t, whose Jacobian is 1/2.
The mixed terms are included. In matrix notation,
S=v'*v'^*-(v''*v^*+v*v''^*)/2 = V*B*V^*,
V=[v,v',v''],
B=[[0,0,-1/2],[0,1,0],[-1/2,0,0]]. (5)
This tests one precise sufficient candidate: S(r)>=0 in Hermitian matrix
order for every real r, with the exact original coefficients. Its (3,3)
sector would give L1[X]>=0, hence J>=0. A hoped-for method was to
propagate this matrix order through duplication/reflection, retaining all
sectors. That candidate already fails on the original data:
det[A,A',A'']=-4*i*a^3,
det V=-4*i*a^3*X^3.
Whenever X(r)!=0, V is invertible. B has eigenvalues 1,1/2,-1/2,
so congruence in (5) gives S two positive and one negative eigenvalue.
At a zero of X, S=|X'|^2*A*A^*>=0, including a multiple zero. No
division by X was needed, and the proposed all-height matrix condition
fails away from zeros regardless of the unknown sign of its original
diagonal entry. This is not a negative value of L1[X].
Subtracting the known phase correction does not supply an alternative estimate. Direct expansion of (4) yields
S=L1[X]*A*A^*+X^2*B_A,
B_A=A'*A'^*-(A''*A^*+A*A''^*)/2.
The corrected matrix is positive semidefinite exactly when L1[X]>=0,
since A’s first component is one. Calling that corrected matrix positive
just repeats the scalar target. No finite constant-matrix closure is
proposed or used.
Keeping finer sectors does not justify dropping their interactions. For
m=2^k, 0<=b<m, define the complete residue series
T_b,m(x)=sum_(n=b mod m) exp(-pi*n^2*x),
Phi_b,m=D[exp(u/2)*T_b,m(exp(2u))].
Phi_b,m=Phi_b,2m+Phi_(b+m),2m,
sum_(b=0)^(m-1) Phi_b,m=phi. (6)
The full additive-character representation is
T_b,m(x)=(1/m)*sum_(l=0)^(m-1) exp(-2*pi*i*l*b/m)
*sum_n exp(2*pi*i*l*n/m)*exp(-pi*n^2*x).
No character is removed. For each fixed m, Poisson summation gives the
small-x leading term 1/(m*sqrt(x)); D kills its corresponding exponential.
Together with the large-x series this justifies the completed transforms
at each finite level. It does not give a uniform-in-m norm bound: the dual
Gaussian scale includes m^2*x.
Writing F_b,m=Fourier(Phi_b,m), (6) yields exact block aggregation of
the Hermitian quadratic S(F) from (5). The original scalar is the sum
of all entries. Positive diagonal bounds or separate sector tail bounds
leave the cross entries unestimated. An infinite-level argument would
need a specified cone or signed quadratic estimate, an invariant refinement
inequality in that order, and a uniform convergence bound in a norm
controlling two r derivatives. None has been supplied here. These are
additional open inputs, not consequences of the partition identity.
Likewise nonlinear Landen identities for squares of constants may define
a different state and cone; the rejection (5) does not exclude them.
Their differentiated complete pair comparison is currently unspecified.
The unequal-modulus identity in
Kharchev–Zabrodin II, Proposition 3.1
requires positive-integer modulus multiples and positive imaginary part;
it matches exp(4t)=n1/n2, not all shifts via an unproved derivative limit.
It supplies no estimating cone in this audit.
NS100 excludes a per-slice sign requirement; (3) retains the full integral. NS101 changes the literal original lattice/IVP. However (2), (4) and (5) can be formed from any smooth real even base kernel, including its mixture: their algebra alone cannot distinguish it. The claimed sector-matrix positivity fails directly on the original, so no numerical control command is needed to reject it. Davenport–Heilbronn does not have the original coefficient/completion hypotheses; NS74/83 concern different NB targets. These mismatches are not passes of a sign screen.
A successful independent signed estimate for the complete recurrence would
give L1[X]>=0, equivalently
J=(r^2+1/4)^2*(Y'^2-Y*Y'')+2*(r^2-1/4)*Y^2>=0, at every real r.
First Laguerre alone is not RH. Here the unrestricted matrix estimate is
false and the corrected scalar estimate has no method; no candidate
survives. Budget: one bounded paper review, zero computation. Stop without
a row, thaw or version. The broader nonlinear arithmetic route stays open.
Final wall check: Same open gap; the specific all-sector matrix cone is rejected. Closest: PR75/77/78 and NS100/101.