Status: candidate ordinary argument with enclosure-carrying numerical steps. Independent Muse and Gemini reviews found no analytic or implementation defect in the complete packet. Model agreement is not an oracle. This document is not a kernel-checked result, and no novelty claim is made.
Candidate result
For phi=(1+sqrt(5))/2 put tau_plus=sqrt(1+phi^2)-phi and tau_minus=-phi-sqrt(1+phi^2), so tau_plus=-1/tau_minus. For either choice tau, use the period-five real coefficients a_tau=(1,tau,-tau,-1,0) mod 5, with D_tau, F_tau, omega_tau, g_tau, H_{plus,t}, H_{minus,t} defined in section 1. In the narrow coordinate s=1/2+iz, the calculation supports
217/200 < Lambda_minus <= 567009/320000 = 1.771903125.
The wide coordinate s=(1+iz)/2 multiplies heat times by four:
217/50 < Lambda_minus_wide <= 567009/80000 = 7.0876125.
In the same narrow frame, the plus function supports Lambda_plus <= 1/2 (section 6), hence Lambda_plus_wide <= 2 in the wide frame by the same unchanged factor of four.
The lower step uses two explicit disks, not a finite scan upgraded into a uniform zero statement. The upper step uses a zero-free half-plane argument and de Bruijn's strip contraction theorem. That theorem allows a complex Hermitian kernel; it does not require a real even kernel.[1]
1. Fourier and Mellin normalization
Fix phi=(1+sqrt(5))/2, tau_plus=sqrt(1+phi^2)-phi, tau_minus=-phi-sqrt(1+phi^2). For either tau, let a_tau=(1,tau,-tau,-1,0) mod 5, and define
omega_tau(x) = sum_{n>=1} n a_tau(n) exp(-pi n^2 x/5), D_tau(s) = sum_{n>=1} a_tau(n)n^(-s), F_tau(s) = (5/pi)^((s+1)/2) Gamma((s+1)/2) D_tau(s).
All of section 1 works in the narrow frame s=1/2+iz. The series for D_tau is absolutely convergent for Re(s)>1. Its expression 5^(-s) sum_{r=1}^4 a_tau(r) zeta(s,r/5) continues it; the residues at 1 cancel. The Mellin transform gives
F_tau(s) = integral_0^infty omega_tau(x) x^((s-1)/2) dx.
To determine the sign, use the finite Fourier transform ahat(b)=sum_{c mod5}a(c) exp(-2pi i bc/5)/5. Put v=sin(2pi/5) and w=sin(4pi/5). Oddness of a gives
ahat(1) = -2i(v+tau*w)/5, ahat(2) = -2i(w-tau*v)/5.
The eigenvector equation is w*tau^2+2v*tau-w=0. Since v/w=phi and v^2+w^2=5/4, its two roots are the stated tau_plus and tau_minus. For the plus root, v+tau_plus*w=+sqrt(5)/2, hence ahat=-i*a/sqrt(5). Poisson summation applied to x exp(-pi t x^2/5) gives
omega_plus(x) = -i sqrt(5) x^(-3/2) sum_{n>=1} n ahat(n) exp(-pi n^2/(5x)), omega_plus(1/x) = +x^(3/2) omega_plus(x).
Consequently g_plus(u)=exp(3u/2)omega_plus(exp(2u)) is real and even. Splitting the Mellin integral at 1 and substituting x=exp(2u) yields
F_plus(1/2+iz) = 4 integral_0^infty g_plus(u) cos(zu) du.
Define the plus heat flow by
H_plus,t(z) = 4 integral_0^infty exp(tu^2) g_plus(u) cos(zu) du.
Thus H_plus,0=F_plus(1/2+iz). Its whole-line Fourier kernel is the real even function K_plus(u)=2 g_plus(u).
For the minus root, v+tau_minus*w=-sqrt(5)/2, hence ahat=+i*a/sqrt(5). Poisson summation gives
omega_minus(x) = i sqrt(5) x^(-3/2) sum_{n>=1} n ahat(n) exp(-pi n^2/(5x)), omega_minus(1/x) = -x^(3/2) omega_minus(x).
Consequently g_minus(u)=exp(3u/2)omega_minus(exp(2u)) is real and odd. Splitting the Mellin integral at 1 and substituting x=exp(2u) yields
F_minus(1/2+iz) = 4i integral_0^infty g_minus(u) sin(zu) du.
Define
H_minus,t(z) = 4 integral_0^infty exp(tu^2) g_minus(u) sin(zu) du.
Thus H_minus,0=-i F_minus(1/2+iz). Its whole-line Fourier kernel is K_minus(u)=-2i g_minus(u): K_minus(u)=conj(K_minus(-u)). The rest of this packet writes H_t, g, omega, F, D for the minus objects H_minus,t, g_minus, omega_minus, F_minus, D_minus. The sign and factor are pinned against an independent Hurwitz-zeta evaluation and by the odd modular identity; the plus sign and cosine identity are pinned by the even_theta_transform and plus_zero_time_cosine_identity checks in verify.py and the two new tests in test_heat.py. The minus coefficients are bounded by M=abs(tau_minus), which is greater than one; the plus coefficients satisfy abs(a_n)<=1 since 0<tau_plus<1.
At positive infinity, g(u) decays as a polynomial exponential times exp(-pi exp(2u)/5); at negative infinity use oddness. This proves the required super-Gaussian decay, integrability, entire dependence on z, and locally uniform dependence on every real heat parameter t.
2. Two local disks, with the stronger one at time 217/200
The time-one disk used the exact rational values
c = 7.646830873064 + 0.425938287679 i, r = 1/10000000.
The disk avoids the real axis. odd_ball.py encloses the infinite heat integral and its first derivative at c, and an upper bound M2 for abs(H_1''(z)) throughout this disk. The exact dyadic endpoints in rouche.json, rechecked as rational inequalities by verify.py, imply
abs(H_1(c)) < 1/10^14, abs(H_1'(c)) > 15/1000, M2 < 235/100.
Taylor's integral remainder on each straight segment from c gives
abs(H_1(z)-H_1(c)-H_1'(c)(z-c)) <= M2*r^2/2.
On the disk boundary, compare H_1 with H_1'(c)(z-c). Their difference has magnitude at most abs(H_1(c))+M2*r^2/2, strictly less than r*abs(H_1'(c)). The coarse rational lower bound on this margin is
5999913/4000000000000000 > 0.
Rouche's theorem therefore gives exactly one zero counted with multiplicity in the disk. It is simple and non-real. A rounded root-finder residual alone would not establish this statement.
The stronger disk uses
t = 217/200, c = 7.543145542463 + 0.072646330251 i, r = 1/10000000.
It remains disjoint from the real axis. Four enclosure configurations of the same Arb calculation, at 96, 128 and 160 bits, imply the coarse bounds
abs(H_t(c)) < 1/10^14, abs(H_t'(c)) > 28/10000, M2 < 19/10.
The corresponding exact coarse margin is
559961/2000000000000000 > 0.
The same Taylor/Rouche argument therefore places exactly one simple non-real zero in this later disk. This proves only survival at t=217/200; the measured approach to the real axis between this time and 1.09 is not promoted to a collision-time statement.
3. Error bounds actually consumed
For derivative order k=0,1,2, put y=abs(Im(c))+r when bounding the disk, and y=abs(Im(c)) at the centre. On a real integration path, the wave factor and its k-th derivative are bounded by u^k exp(yu).
For the theta tail after n=N, set rho=exp(-pi(N+1)/5). Since n^2>=n(N+1) for n>N, the omitted omega terms on u>=0 have absolute sum at most
M (N+1) rho^(N+1)/(1-rho)^2.
On [0,U] multiply this by 4 U^(k+1) exp(tU^2+(3/2+y)U). The calculation uses t>=0.
For the integral after U, assume U>=1 and set
V=exp(2U), C=pi/5-(tU^2+(3/2+y)U)/V > 0.
The code also checks exp(-pi V/5)<29/100, giving sum n exp(-pi n^2 exp(2u)/5)<=2exp(-pi exp(2u)/5). The decreasing functions u exp(-2u) and u^2 exp(-2u) give the integrand bound 8M u^k exp(-C exp(2u)). Substitution v=exp(2u) and the fact that (log v)^k/v decreases for log v>=k bound the tail by
4 M U^k exp(-CV)/(CV).
The finite integrand is entire. Arb's complex integrator encloses its finite integral; both explicit errors are added to both complex components. The second-derivative majorant replaces every coefficient by M and the sine by exp(yu), a nonnegative real integrand. It bounds the whole disk, not sampled boundary points. An insufficient theta cutoff returns an inconclusive local inequality. It does not silently discard a tail.
4. Upper strip from prime phases
Let chi be the primitive quartic character modulo 5 with chi(2)=i, and A=(1-i*tau_minus)/2. The coefficient identity gives
D(s)=A L(s,chi)+conj(A)L(s,conj(chi)).
For Re(s)>1, both Euler products are nonzero. A zero of D would force their ratio to equal -conj(A)/A. Writing kappa=tau_plus=-1/tau_minus, its smallest absolute phase is 2 atan(kappa).
The primes congruent to 1 or 4 modulo 5 contribute ratio one, as does 5. Each remaining prime contributes (1+u)/(1-u) with abs(u)=p^(-sigma). Its phase has magnitude at most 2 atan(p^(-sigma)). Thus a zero is excluded if
Theta(sigma)=2 sum_{p=2,3 mod5} atan(p^(-sigma)) < 2 atan(kappa).
This is the necessary Euler-factor argument recorded in ../lambda_dh_bounds/STRIP2.md, not a new phase-obstruction theorem. Here a simpler tail replaces that instrument: above the integer cutoff P,
2 sum_{p>P} atan(p^(-sigma)) <= 2 P^(1-sigma)/(sigma-1).
An exact sieve through P=10000 and two independent interval backends decide the inequality at sigma=953/400. The lower endpoint of the phase margin is positive on both backends. The mpmath interval implementation uses an arctangent Taylor series with an explicit remainder, not a float arctangent. Monotonicity excludes every Re(s)>=953/400. The functional equation reflects this exclusion to the left. Since the gamma factor has no zeros or poles in the right region, every zero of the entire completed F lies in abs(Im(z))<753/400.
De Bruijn's contraction therefore makes all zeros real by (753/400)^2/2=567009/320000.[1] The sharper sibling abscissa recorded in the older hunt is not required and has not been rerun here.
5. Threshold and strict lower bound
Let S be the real times at which all zeros of H_t are real. It is nonempty by section 4. De Bruijn's theorem makes S upward closed: for a real-rooted H_t, apply the strip theorem with arbitrarily small positive strip widths to its Hermitian kernel exp(tu^2)K(u).[1]
S is closed. Indeed, H_t varies locally uniformly with t, and no H_t is identically zero, by injectivity of the Fourier transform of the nonzero integrable kernel. If t_n in S tends to t and H_t had a non-real zero, choose a small disk avoiding the real axis and with a zero-free boundary. Uniform convergence and Rouche would put a non-real zero of H_(t_n) there, a contradiction. The stronger zero in section 2 implies 217/200 notin S. Upward closure then excludes all times at most 217/200 from S, and closedness gives
S=[Lambda_minus,infinity), 217/200<Lambda_minus<=567009/320000.
This argument uses the Hermitian strip theorem directly. It does not need an even-kernel restriction, a proof that all zero trajectories have been tracked, or the assumption that no pair enters from infinity.
6. Controls and exact scope
verify.py repeats both local enclosures at 96, 128 and 160 bits, with theta cutoffs 24, 26 and 28 and integration cutoffs 4 and 7/2. Direct mpmath quadrature at 55 digits agrees with the Arb values of H and H' at both times. This is a floating-point independent check, not a second enclosure backend for the quadrature. The prime-strip sign is enclosed on both Arb and mpmath.iv. Moving the centre outside the tiny disk and underresolving the theta series both prevent the local decision. Normalization tests reject a factor-of-two error and the wrong modular sign.
The subject is itself the second Davenport-Heilbronn rival. No implication from shared symmetries to RH is proposed, so a generic rival-passing test would not strengthen the analytic argument. The relevant comparison is the first DH function with the opposite theta parity, in the same heat frame.
That comparison does not need the inherited first-function bound. At sigma=3/2, sum the plus-function coefficients through N=20 and use abs(a_n)<=1 for the rest:
sum_{n>=2} abs(a_n)n^(-3/2) <= sum_{n=2}^20 abs(a_n)n^(-3/2)
- integral_20^infinity x^(-3/2) dx = head + 1/sqrt(5) < 1.
Both Arb and mpmath.iv enclose this strict inequality in first_comparison.json. First-term domination and the plus functional equation put all zeros of F_plus, hence of H_plus,0, in a narrow z-strip of half-width one. The same de Bruijn contraction applied to the plus cosine flow H_plus,t gives
Lambda_plus <= 1/2 < 217/200 < Lambda_minus,
in the narrow frame, i.e. Lambda_plus_wide <= 2 by the same unchanged factor of four. Lambda_plus here is the threshold of H_plus,t, not of the generic or minus flow.
Thus the two conductor-five heat constants are separated by this packet if the ordinary analytic argument and enclosed computations survive review. The sharper inherited upper bound for the first function is not needed.
The claim concerns a heat threshold for this explicitly defined function, not zeta's RH, the location of every second-DH zero at time zero, the exact landing time of the observed pair, or novelty in the literature.
A bounded search for second-DH heat bounds did not locate a directly relevant prior bracket. That does not establish absence of prior art. The finite Fourier calculation, Mellin representation, Rouche theorem and strip contraction are existing mathematics; any original contribution here is the explicit new application and reproducible numerical packet, subject to review.
7. Residue-orbit corollary (exact algebra)
Let phi=(1+sqrt(5))/2, tau_plus=sqrt(1+phi^2)-phi, tau_minus=-phi-sqrt(1+phi^2), and a_tau=(0,1,tau,-tau,-1) on residues mod 5. The following hold exactly, by simplification plus complete enumeration over Z/5Z (see test_residue_orbit_is_exact_nonresidue_permutation; no float assigns this identity):
tau_plus*tau_minus=-1, since withs=sqrt(1+phi^2)the product is-(s^2-phi^2)=-1.- For every
n mod 5,tau_plus*a_minus(n)=a_plus(2n mod 5). - The full unit action
r in (Z/5Z)^*ona_plusis:r=1givesa_plus(n);r=2givestau_plus*a_minus(n);r=3gives-tau_plus*a_minus(n)(because3=-2 mod 5andais odd);r=4gives-a_plus(n)(because4=-1 mod 5andais odd). - Nonzero global scaling multiplies
D_tau,F_tau, andH_tau,tpointwise but moves no zero and changes no threshold, so the unit orbit has exactly two zero/heat classes up to scale: residues{1,4}give the plus class, nonresidues{2,3}give the minus class. - With the reviewed enclosed separation
Lambda_plus<=1/2<217/200<Lambda_minus, the de Bruijn-Newman threshold is therefore not invariant under the nonresidue permutationn->2n mod 5, even though the coefficient multiset is unchanged up to the global scaletau_plus. This is scoped to the conductor-five pair. It does not prove root-number sign alone causes the gap, a general statement for other conductors, novelty, or anything about RH. - This matches the finite-Fourier signs already proved in section 1: plus eigenvalue
-i/sqrt(5)(even cosine class) versus minus+i/sqrt(5)(odd sine class). The nonresidue action switches the modular/even-cosine versus odd-sine class for this pair, without a general causal claim beyond it.
Verification level: ordinary algebraic corollary of the reviewed and enclosed separation. Not kernel-checked; no novelty claim.
Sources: [1] https://arxiv.org/html/2005.05142v2 - Dobner, extended Selberg class, version 2
The doors
Active constraints at the optimum
There is no numerical optimizer in this hunt, so “optimum” means the binding inequalities in each endpoint of the bracket. For the lower endpoint, the active constraint is the local Taylor/Rouche margin. The linear boundary term is about 2.86e-10 at the later disk; the centre and remainder contributions total about 1.37e-14. The radius is therefore not close to its mathematical limit, but the derivative is shrinking as the conjugate pair approaches the real axis. For the upper endpoint, the active constraint is the Euler-phase budget at sigma=953/400. Its enclosed margin is about 9.89e-5; the simple all-integer tail above 10000, rather than the finite prime head, is the part most open to sharpening. Neither constraint says the current bracket is exact.
Frozen-constant inventory
The frozen constants are the narrow frame s=1/2+iz, heat multiplier exp(tu^2), tau_plus=sqrt(1+phi^2)-phi, tau_minus=-phi-sqrt(1+phi^2), the time-one centre and radius, the coarse Rouche bounds 1e-14, 0.015, and 2.35, theta cutoffs 24, 26, and 28, integration cutoffs 4 and 7/2, phase cutoff 10000, and strip abscissa 953/400. The first-function comparison freezes sigma=3/2 and N=20. Changing any of these requires rerunning the exact checker rather than carrying the displayed conclusion forward. The wide-frame values are derived by the fixed factor four and are not an independent computation.
Information class of each door
The first door is a later rational heat time for the same conjugate pair. Its information class is local analytic data: Arb enclosures of one value, one derivative, and a disk-wide second-derivative majorant. A successful later disk raises the lower bound; a failed disk leaves the attempt unresolved. The second door is a sharper zero strip. Its information class is the quartic mod-five Euler products plus an explicit prime-phase tail. It can improve the upper endpoint but cannot determine the heat constant by itself. The third door is structural: compare the even and odd conductor-five kernels under the same deformation and seek a theorem explaining their separated thresholds. Its information class must include the theta-transform sign, not merely shared gamma factors or finite zero counts. A zero-counting experiment may scout that door, but no asymptotic counting law is asserted here. Prior-art search remains a separate door before any novelty language.