Adversary 3: attack on the upper bound Lambda_DH <= 0.4006343708899557

Scratch analysis, 2026-08-16. Remit: break the upper side. Nothing in this file is an enclosure; every number below is measured on one route unless it says otherwise, and the routes are my own scratch code, not the hunt's instruments.

Verdict: the inequality survives. One major defect found, in the normalization dictionary, which does not move the number but does move what the number means.

1. The defect: the hunt's Lambda is not Dobner's Lambda

THEOREM13.md section 6 says:

Dobner's deformation xi_t^F((1+iz)/2) = int e^{tu^2} Phi_F(u) e^{izu} du is the hunt's H_t up to the constant factor 2 (Phi even turns the full-line integral into 2 int_0^inf ... cos(zu) du), so the two share zeros and share Lambda exactly.

That is false. Dobner's argument satisfies s = (1 + iz)/2 = 1/2 + iz/2; the hunt's satisfies s = 1/2 + iz. The two differ by a dilation in z, not by a constant. Measured (mpmath dps 30, my own quadrature):

So Phi_F(u) = Phi_DH(2u), G_0(z) = H_0(z/2), and G_t(z) = H_{t/4}(z/2). Hence

Lambda^Dobner = 4 * Lambda^hunt.

Why the inequality still stands

de Bruijn's Theorem 13 is applied entirely inside the hunt's own normalization, on the hunt's own Delta, so no factor of 4 enters the derivation. Dobner is used only for the half-line structure and for Lambda >= 0, and both properties are invariant under t -> t/4. The claim, as stated with its own definitions of Phi_DH, H_t and Lambda_DH, is correct.

Why it matters anyway

NOVELTY.md calibrates the result against "the zeta record 0 <= Lambda_zeta <= 0.22 (Rodgers-Tao; Polymath 15), Lambda_zeta < 1/2 (Ki-Kim-Lee 2009)". Those numbers live in the standard convention, the one Dobner uses, the one Rodgers-Tao use, and the one zeta/heatflow.py uses: H_0(z) = (1/8) Xi(z/2), kernel exp(-pi n^2 e^{4u}). The hunt's Phi_DH uses exp(-pi n^2 e^{2u}), which is the a = 1 convention. Converting the hunt's headline into the convention its own calibration is quoted in:

0.2304 < Lambda_DH <= 1.6025374836 (standard convention)

and conversely zeta's classical 1/2 becomes 1/8 in the hunt's convention. A reader who sets "Lambda_DH <= 0.4006" beside "Lambda_zeta < 1/2" concludes the DH constant is comfortably below zeta's classical bound. In a common normalization the DH upper bound is 3.2 times zeta's classical 1/2 and 7.3 times Polymath's 0.22.

Fix, either way round: state Phi_DH(u) = 8 e^{3u} sum_n n a_n exp(-pi n^2 e^{4u} / 5), which puts H_0(z) = Xi_DH(z/2) and matches zeta/heatflow.py exactly, or keep the present convention and print both numbers side by side. In both cases delete the "share Lambda exactly" sentence.

2. Attacks that failed

2.1 Theorem 13 read from the original scan

Fetched https://pure.tue.nl/ws/files/1769368/597490.pdf and read pages 203 to 205 as images. Theorems 10, 11, 12, 13 and 14 match THEOREM13.md word for word. In particular Theorem 13 (p. 205) reads

If F(t) satisfies the conditions of Theorem 10, and if all the roots of (3.6) lie in the strip |Im z| <= Delta, then all the roots of g(z) = int_{-inf}^{inf} F(t) e^{(1/2)lambda^2 t^2} e^{izt} dt lie in the strip (3.8) |Im z| <= {Max(Delta^2 - lambda^2, 0)}^{1/2}.

with no "(all but a finite number of the roots)" parenthetical in either hypothesis or conclusion, unlike Theorems 11 and 12 on the same and facing pages, which do carry it. Theorem 10 (p. 204) gives the class conditions as exactly: b > 2, F integrable over the real line, (3.4) F(t) = (F(-t))*, and (3.5) F(t) = O(e^{-|t|^b}). No positivity, no monotonicity, F may be complex. The multiplier is e^{(1/2)lambda^2 t^2}, so t = lambda^2/2 and the all-real threshold is t >= Delta^2/2. Kill condition 1 of MISSION.md is genuinely not triggered.

2.2 The triangle equality case on Re s = sigma_0

Correct as written, and airtight. Every term a_n n^{-s} with n not divisible by 5 is nonzero, so equality in the triangle inequality forces all of them onto a common ray. a_3 = -kappa < 0 forces 3^{-it} = 1, a_4 = -1 < 0 forces 4^{-it} = 1, a_12 = kappa > 0 forces 12^{-it} = -1, and 12 = 3 * 4.

Note this step is not load-bearing: Theorem 13 wants a closed strip, and step (b) evaluated at the decided rational sigma* = 1.3951361582351097210613589375 already delivers |Im z| <= sigma* - 1/2 without it.

2.3 The functional equation and the gamma factor

Rebuilt from scratch, sharing no code with the hunt. Linear solve for kappa at s = 2.3 + 1.7i (mpmath dps 50) gives

kappa = 0.2840790438404122960282918323931261690911 imaginary residue 7.08e-51 |kappa - KAPPA_REF| = 1.19e-41

and |F(s) - F(1-s)| <= 5.42e-51 at s = 1.1+0.3i, 0.2+5i, -1.5+2.2i. The completing factor is (pi/5)^{-(s+1)/2} Gamma((s+1)/2), the odd-character shape. Structurally kappa is forced: with F = c Lambda(s, chi) + conj(c) Lambda(s, chibar) and c = (1 - i kappa)/2, the requirement F(s) = F(1-s) is exactly eps(chi) = conj(c)/c, a unit modulus condition with a unique real solution. So kappa is the constant making it hold, not a constant chosen to nearly make it hold.

2.4 The trap: trivial zeros and the gamma poles

Worked out and it holds. Measured at dps 50:

So f has simple zeros at s = -1, -3, -5, ..., each exactly cancelling a simple pole of Gamma((s+1)/2), and F is finite and nonzero there. Xi_DH inherits no zero from the gamma factor, because neither (pi/5)^{-(s+1)/2} nor Gamma has any zero at all. The trivial zeros are zeros of f and not of F, and STRIP.md section 3(e) states this correctly.

This is a check with teeth rather than a formality. If f had a double zero at any s_m = -(2m+1), F would vanish at Re s = -(2m+1), which is outside the strip, and the upper bound would collapse. It does not happen at the four points tested, and it cannot happen anywhere, because F(s_m) = F(1 - s_m) and 1 - s_m = 2m + 2 lies in the half-plane Re s >= 2 where g(2) < 0 already puts f away from zero.

2.5 Uniformity in the height

The domination |f(s)| >= 1 - sum_{n>=2} |a_n| n^{-Re s} depends on Re s only: |n^{-s}| = n^{-Re s} exactly. There is no height-dependent term anywhere in the argument, so no large |Im s| escape route exists. This is a property of the construction, not a claim needing a check.

2.6 H_0 really is Xi_DH, with the stated Phi_DH

Derived independently. F(s) = int_0^inf Theta(x) x^{(s+1)/2} dx/x with Theta(x) = sum_n n a_n exp(-pi n^2 x / 5); substituting x = e^{2u} gives F(1/2 + iz) = int_{-inf}^{inf} 2 e^{3u/2} Theta(e^{2u}) e^{izu} du, and the half-line cosine form doubles the prefactor to 4. So the constant 4 is forced, not fitted. Measured agreement of H_0 against F(1/2 + iz):

Fitting H_0(z) = c Xi_DH(a z) gives c = 1 exactly and a = 1 - 7.4e-42 i. Phi evenness measured at |u| = 0.1, 0.4, 0.9, 1.5: relative differences 2.0e-41, 0, 5.0e-41, 2.9e-38.

2.7 The strip constant, recomputed

Independent route (mpmath dps 50, Hurwitz zeta, my own kappa):

sigma_0 = 1.3951361582351097210613588973265388 Delta = 0.895136158235109721061358897327 Delta^2/2 = 0.400634370889955694446954776081

inside the hunt's flint bracket [1.3951361582351097210613588712, 1.3951361582351097210613589375]. A brute-force check that does not use the residue-class identity at all (direct float64 sum of |a_n| n^{-sigma} over n <= 2e6 plus an integral-test tail) brackets g(sigma_0) in [-4.21e-3, +3.99e-3], consistent with 0.

The headline decimal is a rounding in the safe direction: the exact Delta*^2/2 at the decided rational sigma* is 0.4006343708899556944469548120 (outward), and the claimed 0.4006343708899557 exceeds it by 5.553e-18.

2.8 The strip-to-time dictionary, recalibrated

The operator was derived rather than recalled: multiplying by u^2 under int Phi e^{izu} du is -d^2/dz^2 on the transform, so H_t = exp(-t d^2/dz^2) H_0. Verified on the actual DH object by central differences at (t, z) = (0.4, 2.3), dps 30: dH/dt = 1.95104693155701e-4 versus -d^2H/dz^2 = 1.95104693155765e-4, residual 6.43e-17.

Bare conjugate pair p(z) = z^2 + D^2 flows to z^2 + D^2 - 2t, so it lands at exactly D^2/2. Measured by 200-step bisection on the exact finite series:

D^2/8 and 2D^2 are both refuted by a factor of 4. Distant real spectators only accelerate the landing, approaching the bound from below: (z^2 + D^2)(z^2 - A^2) at D = 0.895136158235 gives 2t*/D^2 = 0.8738, 0.9846, 0.99936, 0.99998 for A = 3, 10, 50, 300. Two pairs at unequal heights land strictly early, so the bound reads off the deepest pair, as it should.

2.9 The same machinery on zeta reproduces the literature

In zeta/heatflow.py's convention H_0(z) = (1/8) Xi(z/2), the classical zero-free half-planes give zeros of Xi in |Im| <= 1/2, hence Delta = 1 after the z/2, hence Delta^2/2 = 1/2. That is de Bruijn's published bound exactly.

The same run also prices the elementary route. Feed zeta through the DH argument's own domination (root of zeta(sigma) = 2):

sigma_0(zeta) = 1.7286472389981836181 Delta in the standard convention = 2 * 1.228647239 = 2.457294478 Delta^2/2 = 3.0191480758

which is 6.04 times worse than the true 1/2. The Euler product is what buys zeta that factor. DH has none, so a comparable slack should be assumed here and is not visible from inside the argument.

3. Sharpness

0.4006 is not close to tight, and all the slack is in the strip constant, not in de Bruijn's 1/2, which section 2.8 shows is attained on the extremal family.

Where the zeros actually are. The repo's pinned off-line zero has Re rho = 0.808517, so |Im z| = 0.3085. The census's deepest new pair at gamma = 531.2797 has beta = 0.846954, so |Im z| = 0.3470. Against Delta = 0.8951 that is a factor of 2.6 in the strip and about 6.7 in the bound. DH is known to have zeros with Re s > 1, so the true supremum is at least 0.5, but that only lifts the honest range: if the supremum were 0.5 the de Bruijn bound would be 0.125, and at 0.6 it would be 0.18. With the measured floor at 0.0577, the truth sits somewhere in (0.0577, 0.4006] and nothing on disk narrows that.

The strip constant is improvable now, with an argument the hunt already has. STRIP.md step 3(c) proves the triangle bound is never attained, via 12 = 3 * 4, and then discards the quantity. Keep it. On Re s = sigma the values n^{-it} are, by Kronecker and the Q-independence of the log p, exactly free independent phases on the primes, so the closure of the head's value set is the set of sums with arbitrary prime phases. Define

B_M(sigma) = min over prime phases of |sum_{n<=M} a_n n^{-sigma} e^{i <v(n), theta>}| - sum_{n>M} |a_n| n^{-sigma}

Any sigma with B_M(sigma) > 0 is a valid strip constant, and M = 1 is the hunt's present bound. Measured (float grade, L-BFGS with 25 restarts, tail by the Hurwitz identity):

M = 4 buying nothing and M = 12 buying the first gain is exactly the hunt's own n = 12 witness: on n <= 4 the choice 2^{-it} = -1, 3^{-it} = 1 attains the triangle bound, and 12 is the first n that cannot be aligned with it. That is a 3.4 percent improvement in the bound for eleven extra terms, and the sequence is monotone non-increasing in M by construction.

This is not a float-only route. The head is a trigonometric polynomial in finitely many phases with an explicit gradient bound, so its minimum over the torus can be enclosed by a Lipschitz grid using the same interval machinery strip.py already carries, and the tail is the same Hurwitz expression. The improvement is available at the grade the hunt already works at.

The literature offers no better lever that I can find. Ki-Kim-Lee 2009's Lambda_zeta < 1/2 is a zeta-specific strictness result, and the all-but-finitely-many statements (de Bruijn Theorems 11 and 12, Ki-Kim 2003 Theorem 2.2) have weaker conclusions at the same constant. Theorem 13's 1/2 is sharp. The strip is the only place to push.

4. Minor items

• THEOREM13.md section 5.3 asserts that g(u) = (pi/5) e^{2u} - (3/2)u - u^3 is increasing on [2, 30] from a 201-point grid. A grid does not show monotonicity. One line does: g'(u) = (2 pi/5) e^{2u} - 3/2 - 3u^2, and at u = 2 that is 68.6 against 13.5, with the exponential dominating 3u^2 thereafter. Replace the grid claim with the derivative.
• The headline 0.4006343708899557 is a rounded-up form of Delta*^2/2 and reads as if exact. Say so, or quote the outward endpoint.

5. Scratch code

/tmp/claude-0/-home-user-zeta-lab/ea2c50e5-c6c1-5be9-ac30-eda79b8fac85/scratchpad/: a3_strip.py (kappa, sigma_0, trivial zeros), a3_transform.py (H_0 versus Xi_DH, scale fit), a3_dict.py (polynomial flow calibration), a3_zeta.py (backward heat on the DH object, zeta convention), a3_dobner.py (the normalization defect), a3_sharp.py and a3_sharp2.py (Bohr value set and the improved strip constant), plus the de Bruijn scan debruijn1950.pdf. No hunt artifact was modified.