Decision: no estimating candidate admitted. The auxiliary heat equation does give an exact transfer from a known positive parameter to zero. Its unestimated source has the adverse sign in that direction. The resulting integral comparison, without a separate arithmetic bound, is exactly the original first-Laguerre obligation. This is not a closure of heat methods.
Wall check: Same open gap. Closest: NS100/101, PR77 review 02 and PR78’s admission assessment. What changes: an actual Gaussian deformation replaces the unsupported identification of the Jacobi clock with heat. No new signed arithmetic estimate results. NS52 concerns a different, same-test spectral transport comparison; its conditional obstruction does not settle this one.
Reviewed baseline: 221ceec9da825bdc787111447badad961860760a; scope commit:
b5147e82803cccdb641bcb611b3c90c3c76cf51e. Read AGENTS, the commissioned
brief/review record, PR77 reviews 02/07/09, PR78 review 06 and the relevant
HEAT-TRANSPORT scope. The coordinator owns the full-register review. This is
an independent paper derivation, not historical proof or certificate replay.
Both missing-original evidence groups remain OPEN.
Keep the complete original integer-square theta coefficients, their even kernel phi, and define
H_a(r) = integral_R exp(a*u^2)*phi(u)*exp(i*r*u) du,
H_0=X=xi(1/2+i*r)/2,
Q_a=H_a'^2-H_a*H_a'',
S_a=H_a''^2-H_a'*H_a''',
J=16 Q_0.
Primes denote r derivatives. The double-exponential decay of phi, with all derivatives, justifies parameter and r differentiation uniformly on compact a intervals, with Schwartz functions on the real r line. No zero is divided out. The target is Q_0(r)>=0 for every real r, not RH.
This a is an auxiliary Gaussian parameter, not the original nonlinear Jacobi clock L=2u. The coefficients are original at every a, but the weight exp(a*u^2) changes the function under study. A relation between these clocks is neither assumed nor obtained.
For normalization, Polymath, arXiv:1904.12438v2, equations (1), (3), (4)
and introduction uses a half-line kernel
Phi(v)=phi(2v). Thus H_a(r)=4*H_lit,4a(2r). The recorded unconditional
real-zero result at literature parameter >=1/2 supplies Q_A>=0 at
A=1/8. Reality of zeros for this real entire order-one function gives the
ordinary Laguerre inequality, including multiple zeros by continuity. We
use no strict margin or simplicity. Smaller available positive endpoints
would shorten the interval but would not remove its source estimate.
Direct differentiation, retaining the fourth derivative, gives
partial_a H_a = -H_a'',
partial_a Q_a = H_a*H_a''''-2*H_a'*H_a'''+H_a''^2,
Q_a'' = H_a''^2-H_a*H_a'''',
partial_a Q_a = -Q_a''+2*S_a. (1)
More generally, Q_(k,a)=L1[H_a^(k)] satisfies
partial_a Q_(k,a)=-Q_(k,a)''+2*Q_(k+1,a). There is no scalar
homogeneous heat closure for Q. Positivity of a higher expression is not
automatically favorable when going toward a=0.
Put tau=A-a. Then H follows forward ordinary heat in tau, whereas Q follows
partial_tau Q=Q''-2*S. Let
G_a(r)=(4*pi*a)^(-1/2)*exp(-r^2/(4*a)), a>0,
G_0*F=F.
Duhamel’s formula, justified in the Schwartz class, gives the exact identity
Q_0(r) = (G_A*Q_A)(r) - 2*integral_0^A (G_a*S_a)(r) da. (2)
Accordingly the complete sufficient comparison is
2*integral_0^A (G_a*S_a)(r) da <= (G_A*Q_A)(r), every r in R. (3)
Its five-item specification would be: expression (3); fixed original phi with all coefficients and tails; all real r and every intermediate 0<=a<=A=1/8; an original-coefficient upper estimate for the accumulated signed source relative to the known endpoint; and (2), followed by J=16Q_0. There is no missing analytic boundary or zero-neighborhood transfer in that implication. The fourth item is the absent estimating method, however, not something supplied by writing a heat equation. By (2), (3) itself is equivalent to the target, so naming it is not a candidate.
First, discarding the source would reverse its relevant sign. At r=0, evenness gives
S_a(0)=H_a''(0)^2
=[integral_R u^2*exp(a*u^2)*phi(u)du]^2 > 0.
Thus S_a<=0, the simple favorable-source hypothesis, is false for the
original kernel. A bare forward-heat preservation claim for the Laguerre
cone is also false: F_tau(r)=r^2-1+2*tau solves forward heat and has
L1[F_tau]=2*r^2+2-4*tau, which turns negative. This polynomial is only a
control of that generic assertion, not an original-theta analogue.
Second, attempting S_a<=b(a,r)*Q_a with a locally bounded potential would
need a separate arithmetic inequality and would conceal a multiplicity
restriction if imposed all the way to zero. At a zero of H_a of order m>=2,
with H_a=c*x^m+O(x^(m+1)),
Q_a = m*c^2*x^(2m-2)+O(x^(2m-1)),
S_a = m^2*(m-1)*c^2*x^(2m-4)+O(x^(2m-3)),
S_a/Q_a ~ m*(m-1)/x^2.
A locally bounded b cannot cover such a zero. No simplicity premise is available for a=0. Singular potentials require their own transfer/domain analysis; none is specified here.
Third, Gaussian smoothing does not turn the arithmetic source into a positive coefficient sum. Symmetrizing the complete Fourier products gives
(G_a*Q_a)(r)
= (1/2)*int_R^2 (u-v)^2*exp(-2*a*u*v)*phi(u)*phi(v)
*exp(i*r*(u+v)) du dv,
(G_a*S_a)(r)
= -(1/2)*int_R^2 u*v*(u-v)^2*exp(-2*a*u*v)*phi(u)*phi(v)
*exp(i*r*(u+v)) du dv. (4)
All integrals converge, including uv<0. The source changes sign between
uv>0 and uv<0, and the oscillatory factor remains. Integrating its a factor
in (2) cancels the endpoint weight exactly, leaving the original complete
quadratic integral for Q_0. Substitution of the theta series introduces the
original n,m arithmetic but estimates no cancellation. A phase-free absolute
bound obtained by deleting exp(ir(u+v)) produces a fixed positive budget,
while G_A*Q_A tends to zero as |r| tends to infinity. That particular
budget cannot prove the uniform comparison. More correlated estimates are
not excluded, but none has been supplied.
Prior art does not repair this step. Csordas, arXiv:1309.0055v2, equations (4.9)–(4.10), Proposition 4.9 and Open Problems 4.10–4.11 explicitly studies this Gaussian flow. The real-zero persistence is toward increasing deformation; the derivative statements do not reverse that direction or provide (3). Its simplicity conclusion is above the real-zero threshold, not at zero. Modern lower bounds on the de Bruijn–Newman constant concern all zeros; they cannot silently be assigned to a threshold for the first inequality alone.
NS101 shares smooth positive even kernels, complete decay, the exact heat PDE, (1), (2) and (4). Its recorded negative Q_0 means its corresponding source comparison (3) fails. We have not assumed or established positivity at the same A=1/8 for that mixture. Original coefficients and the Jacobi IVP differ; that mismatch is not a pass for an unproved source bound. NS100 only excludes a per-slice positivity substitute, which is not used here. Davenport–Heilbronn does not have the literal original kernel/completion; a future generic source claim needs an appropriately matched control. NS74/83 concern NB targets/rates and do not apply to this inequality.
Stop: no independent source-estimating technique survives. The useful audit outcome is the exact adverse-source identity and its normalization, not a new research estimate. Success of a genuinely supplied arithmetic bound for (3) would prove the first-Laguerre target only; failure rejects that bound. Budget exhausted: one paper review, zero scans, zero interval certificates. No row, thaw, manuscript version, commit, outreach or further agent was initiated. Only this assigned note was written.
Final wall check: Same open gap. RH, G2 and the cofinal signed-arithmetic lower bound remain open.