The sharpened zero strip: a phase obstruction, decided

Section draft for the pre-submission hardening of WP2 (2026-08-18). Instrument: strip2.py; decided values in strip2_results.json. This supersedes the constant of STRIP.md, not its argument: STRIP.md remains correct and is the weaker of two valid derivations, kept because a reader should be able to see both.

Vocabulary per MISSION.md: decided means an enclosure whose exact endpoints settle a sign, stated with backend and precision; measured means one float route; cited means somebody else's theorem.

Frame. sigma_0' is a point in the s plane and is frame-free. Delta = sigma_0' - 1/2 and Delta^2/2 are not. They are stated below in the narrow normalization s = 1/2 + iz (Stopple, arXiv:1301.3158, and this hunt) and in the wide one s = (1+iz)/2 (de Bruijn as usually quoted, Newman, Rodgers-Tao, Polymath 15, Dobner), where Lambda_wide = 4 Lambda_narrow. Derivation and conversion table: FRAME.md.

0. The result in one line

decided, both backends, P = 10^5: every zero of F lies in 1 - sigma* < Re s < sigma* at the exact rational sigma* = 1.12036249819, so Lambda_DH <= Delta^2/2 = 0.1924248145802688766381 (narrow) = 0.7696992583210755065522 (wide)

against STRIP.md's 0.4006343708899556944469549 narrow / 1.6025374835598228 wide. The improvement factor is 2.082030697360155, and the hunt's bracket ratio falls from 6.955 to 3.341.

The lower bound is untouched, so the separation corollary Lambda_DH > Lambda_zeta of SEPARATION.md, which rests on the floor alone, is unaffected. In the wide frame the bracket becomes 0.2304 < Lambda_DH <= 0.7697.

1. Why coefficient domination is weak, quantified

STRIP.md proves |f(s)| >= 1 - sum_{n>=2} |a_n| n^{-sigma} and takes the abscissa where the right side turns positive. That step replaces every phase n^{-it} by its worst case independently, so it is exactly the relaxation

inf_t |f(sigma + it)| >= 1 - sup over FREE phases of |sum_{n>=2} a_n n^{-sigma} e^{i theta_n}|

with theta_n ranging over [0, 2 pi)^{N} with no relation between them. Under that relaxation the supremum is the L1 norm sum_{n>=2} |a_n| n^{-sigma}, attained when every term points the same way.

The phases are not free. n^{-it} = prod_p p^{-it v_p(n)} is determined by its values at the primes, and the constraint is multiplicative. STRIP.md already uses that fact once, qualitatively: its section 3(c) excludes the boundary line Re s = sigma_0 because equality in the triangle inequality would force 3^{-it} = 4^{-it} = 1 and 12^{-it} = -1, which contradicts 12 = 3 * 4. Everything below is the same observation used quantitatively rather than once at the boundary.

Two decided measurements of the size of the slack (Arb, 192 bits):

Read the two halves together. At the true abscissa the free-phase relaxation is still asking for a coefficient mass of 3.94 to be below 1, which is why it cannot conclude anything there and has to climb to 1.395. At sigma_0 = 1.395, where domination finally concludes, the primes can supply only 1.526 radians of argument against the 2.588 a zero needs, so domination stops a factor 1.695 late. The two numbers are one fact seen from either side, and the factor 2.082 in Delta^2/2 is what it costs.

2. The obvious sharpening: five-term blocks. It does not decide

The coefficients sum to zero over a period, 1 + kappa - kappa - 1 + 0 = 0, so the series can be regrouped in blocks

f(s) = sum_{k>=0} B_k(s), B_k(s) = (5k+1)^{-s} + kappa (5k+2)^{-s} - kappa (5k+3)^{-s} - (5k+4)^{-s} = [(5k+1)^{-s} - (5k+4)^{-s}] + kappa [(5k+2)^{-s} - (5k+3)^{-s}].

Each pair telescopes. Writing m^{-s} - n^{-s} = s int_m^n t^{-s-1} dt and bounding the integrand by its value at the left endpoint,

|B_k| <= |s| [ 3 (5k+1)^{-sigma-1} + kappa (5k+2)^{-sigma-1} ], (2.1)

which is O(n^{-sigma-1}) per block of five terms instead of the O(n^{-sigma}) per term that domination uses. The tail sums exactly,

sum_{k>=K} |B_k| <= |s| 5^{-sigma-1} [ 3 zeta(sigma+1, K+1/5)

• kappa zeta(sigma+1, K+2/5) ].

The factor |s| is the whole story, and it is not an artifact of the bound. Both pairs inside a period have the same midpoint 5k + 5/2, and their first-order terms carry the same sign, so they add:

B_k = s (3 + kappa) (5k + 5/2)^{-s-1} (1 + O(|s|/n)).

No trapezoid-style second-order correction removes the |s|; the next order carries |s|^2. Measured directly (mpmath dps 40), the bound (2.1) tracks the true |B_k| to within a factor 1.0062 at sigma = 1.2, t = 1, k = 100 and 1.00006 at k = 10^4, and overshoots by 101 at t = 1000, k = 10 and by 1104 at t = 10^5, k = 100. The crossover is n ~ |s|, exactly where a first-order expansion should fail.

So the block bound cannot make a statement about a half-plane, which is what de Bruijn's theorem consumes: it degrades as the height grows. What it does give is a genuine height-restricted strip, from the hybrid split that keeps n <= 5K under the plain triangle inequality and blocks the rest,

sum_{n=2}^{5K} |a_n| n^{-sigma} + sqrt(sigma^2 + T^2) * (block tail) < 1 .

Decided on flint at 192 bits, with the best K chosen by scan and every sign an Arb ball decision:

The column heads say it: as T grows the optimal split keeps more terms under the plain triangle inequality, so sigma_block(T) climbs back toward the coefficient-domination sigma_0 = 1.3951361582..., which is its limit. Beyond T = 10 it is already worse than the phase bound of section 3, and at no T does it produce a half-plane statement.

Verdict on the obvious sharpening: correct, decided, and useless for the upper bound. It is recorded because it is the natural thing to try and because knowing why it fails is what points at the thing that works: the mean value theorem exploits additive smoothness in n, and the cancellation that actually matters here is multiplicative.

3. The phase obstruction

3.1 Two Euler products

Let chi be the odd primitive character mod 5 with chi(2) = i (so chi(1), chi(2), chi(3), chi(4) = 1, i, -i, -1), and put

A = (1 - i kappa)/2 .

Then a_n = A chi(n) + conj(A) conj(chi)(n) for every n, because both sides have period 5 and they agree on a full period. Checked as a decided statement: five acb residual balls, each containing 0 with radius below 1e-40 at 192 bits, with chi taken from flint's own Dirichlet character table rather than from a remembered list of values. Hence, for Re s > 1,

f(s) = A L(s, chi) + conj(A) L(s, conj chi). (3.1)

This decomposition is not new here: zeta/epstein.py states it in the dh_f docstring and the suite pins it. Measured cross-check in strip2.py anyway, since it is now load-bearing in a new way: zeta.epstein.dh_f against A L_chi + conj(A) L_chi(conjugate) at s = 1.6+3.5i, 2.2-11i, 3+40i, agreeing to 1.2e-41 or better, and dh_f against a truncated sum a_n n^{-s} inside its own truncation allowance.

3.2 A zero is a phase equation

For Re s > 1 both Euler products converge absolutely and neither vanishes, so (3.1) gives

f(s) = 0 <=> R(s) := L(s, chi) / L(s, conj chi) = -conj(A)/A . (3.2)

The right side is unimodular, since |A| = |conj(A)|, and

-conj(A)/A = -(1 + i kappa)/(1 - i kappa) = exp( i (pi + 2 arctan kappa) ).

Decided as an acb ball identity to radius below 1e-40. Its argument modulo 2 pi has smallest absolute representative

tau := pi - 2 arctan kappa in [2.5880182946927479869541106, 2.5880182946927479869541107] (Arb, 192 bits, from the 500-bit kappa ball).

3.3 What each prime can contribute

R(s) = prod_p (1 - conj(chi)(p) p^{-s}) / (1 - chi(p) p^{-s}) .

If chi(p) is real the factor is 1, so p = 5 and p = 1, 4 mod 5 contribute nothing at all. If p = 2, 3 mod 5 then chi(p) = +-i and conj(chi)(p) = -chi(p), so with u = chi(p) p^{-s}, |u| = p^{-sigma}, the factor is (1 + u)/(1 - u).

Lemma. For |u| <= r < 1, |arg (1+u)/(1-u)| <= 2 arctan r, with equality exactly at u = +- i r.

Proof. u -> (1+u)/(1-u) is a Moebius map, so it carries the disc |u| <= r to a disc. That disc is symmetric about the real axis (the domain is, and the map commutes with conjugation) and its real diameter runs from (1-r)/(1+r) to (1+r)/(1-r), so its centre is C = (1+r^2)/(1-r^2) and its radius is rho = 2r/(1-r^2). Since C > rho > 0 the disc misses the origin, and the largest argument on it is arcsin(rho/C) = arcsin(2r/(1+r^2)) = 2 arctan r, the last step because 2r/(1+r^2) = sin(2 arctan r) and 2 arctan r < pi/2 for r < 1. The tangency points are the images of u = +- i r. []

Two things fall out of that proof and both matter.

First, C^2 - rho^2 = 1, so the tangent length from the origin is 1: the argument-maximising point of each factor has modulus exactly 1. The constraint |R(s)| = 1, which (3.2) also demands, is therefore free at the configuration that maximises the argument, and no sharper bound is available by playing modulus against phase. This is why the phase inequality below is not merely an improvement but is the end of this line of argument.

Second, the bound is attained, so nothing has been thrown away. Measured confirmation over a 4001-point circle sample at r = 0.5, 0.25, 0.1, 0.01 (mpmath dps 30): the observed maximum equals 2 arctan r to printed precision, the argmax sits at u = i r, and |(1+ir)/(1-ir)| = 1 to 1e-25.

3.4 The obstruction

The product for R(s) converges absolutely, so its argument is the sum of the principal arguments of its factors modulo 2 pi, and by the lemma

|arg R(s)| <= Theta(sigma) := sum_{p = 2, 3 mod 5} 2 arctan(p^{-sigma}).

A zero at Re s = sigma needs arg R(s) = pi + 2 arctan kappa modulo 2 pi, whose smallest absolute representative is tau. Therefore

If Theta(sigma) < tau then f has no zero on the line Re s = sigma.

Theta is a sum of strictly decreasing positive terms, so one decided sigma* closes the whole half-plane: for every sigma >= sigma*, Theta(sigma) <= Theta(sigma*) < tau.

3.5 From f to F, and the strip

Identical to STRIP.md section 3(d), (e), and repeated only so this section stands alone. gamma(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2) is analytic and nonvanishing on Re s >= sigma* > 1 (its only poles are at s = -1, -3, ...), so F = gamma f has no zero with Re s >= sigma*. F is entire with F(s) = F(1-s), so it has none with Re s <= 1 - sigma* either. The zeros of F are exactly the nontrivial zeros of f, the trivial ones at s = -1, -3, ... being an artifact of the gamma factor. Under s = 1/2 + iz the strip 1 - sigma* < Re s < sigma* becomes |Im z| < Delta = sigma* - 1/2 for the zeros of Xi_DH = H_0, which is what de Bruijn 1950 Theorem 13 consumes to give Lambda_DH <= Delta^2/2.

3.6 What is used and what is not

Used: the Euler product for two Dirichlet L-functions with Re s > 1, absolute convergence, one Moebius image, F entire with F(s) = F(1-s), and the classical facts about Gamma. That is all. No Bohr theory, no Kronecker theorem, no positivity, no Euler product for f itself, which is just as well since f has none.

Not used: any statement that the abscissa is attained. This argument gives an upper bound for the supremum of the real parts of the zeros. Whether that supremum equals the root of Theta = tau is Bombieri and Ghosh's converse, and it is not needed and not claimed here.

4. Deciding Theta, with no prime counting

Theta(sigma) is a sum over primes in two residue classes, and it converges too slowly to truncate. The head is summed exactly from a sieve; the tail is closed by the Euler product itself, so no explicit pi(x) estimate, no Chebyshev bound and no cited prime-counting inequality enters anywhere.

4.1 The tail identities

Taking logarithms of the Euler products for zeta and for L(., chi5), where chi5 is the quadratic character mod 5 (value +1 on 1, 4 mod 5, -1 on 2, 3 mod 5, 0 at 5):

T1(sigma) := log zeta(sigma) + sum_{p<=P} log(1 - p^{-sigma}) = sum_{p>P} -log(1 - p^{-sigma}), Tchi(sigma) := log L(sigma, chi5) + sum_{p<=P} log(1 - chi5(p) p^{-sigma}) = sum_{p>P} -log(1 - chi5(p) p^{-sigma}).

Only the class 2, 3 mod 5 and p = 5 survive in the difference, since the two logarithms coincide when chi5(p) = +1:

T1 - Tchi = log( zeta(sigma) / L(sigma, chi5) ) + log(1 - 5^{-sigma})

• sum_{p<=P, p = 2,3 mod 5} [ log(1 - p^{-sigma}) - log(1 + p^{-sigma}) ].

Expanding both logarithms in k, and writing Q := sum_{p>P, p = 2,3 mod 5} p^{-sigma} for the quantity actually wanted,

T1 - Tchi = 2 Q + (E_chi - E_1), E_1 = sum_{p>P} sum_{k>=2} p^{-k sigma}/k, E_chi = sum_{p>P} sum_{k>=2} chi5(p)^k p^{-k sigma}/k.

The k = 2 terms cancel exactly, because chi5(p)^2 = 1 for every p > P >= 5, and so does every even k. Only odd k >= 3 survives, with |chi5(p)^k - 1| <= 2, so

|E_chi - E_1| <= (2/3) sum_{p>P} p^{-3 sigma} / (1 - P^{-2 sigma}) <= (2/3) P^{1-3 sigma} / ((3 sigma - 1)(1 - P^{-2 sigma})) =: eps3,

the last step by the integral test over all integers > P, which is crude and does not matter because the quantity is already O(P^{1-3 sigma}). This is the point at which the estimate stops being loose: bounding E_chi and E_1 separately would leave an O(P^{1-2 sigma}) error, larger by a factor of order P^{sigma}. Numerically at P = 10^5 and sigma = 1.12036249819 the separate bound is 2.52e-07 and this one is 4.42e-13, a factor 5.7e+05.

4.2 The two bounds

With arctan x <= x on the tail and arctan x >= x - x^3/3:

Theta_up(sigma) = head_P(sigma) + (T1 - Tchi) + eps3, Theta_lo(sigma) = head_P(sigma) + (T1 - Tchi) - eps3 - (2/3) W3, head_P(sigma) = sum_{p <= P, p = 2,3 mod 5} 2 arctan(p^{-sigma}), W3 = P^{1-3 sigma}/(3 sigma - 1) .

Only Theta_up is load-bearing. Theta_lo is computed so the width can be reported: it is the head/tail systematic, not the ball precision, and at P = 10^5, sigma ~ 1.12 it is 1.34e-12.

4.3 Backends

• python-flint (Arb), 192 bits. arb.atan, arb.log, arb.zeta, and L(sigma, chi5) from the Hurwitz combination 5^{-s}[zeta(s,1/5) - zeta(s,2/5) - zeta(s,3/5) + zeta(s,4/5)].
• mpmath.iv, dps 40. The iv context has neither zeta nor arctan. zeta and L come from the Euler-Maclaurin machinery strip.py already uses for arithmetic progressions, generalised to a step: zeta is the step-1 progression, L the four step-5 ones. arctan comes from its Maclaurin series, which is alternating with strictly decreasing terms for 0 <= x < 1, so consecutive partial sums bracket the value; the bracket is taken from the last odd-index and even-index partial sums, each evaluated in interval arithmetic, and the loop refuses rather than guesses if x is too close to 1.

Both legs run at the same sieve limit P = 10^5, so they bisect the same function and their intervals must overlap. They do.

5. Decided numbers

All intervals are outward-rounded decimal strings containing the exact rational endpoints. Full detail in strip2_results.json.

5.1 The abscissa

Two-sided enclosure of the root of Theta = tau itself, from bisecting Theta_lo as well (flint, P = 10^5): [1.1203624981832156488068, 1.1203624981833869487332], width 1.71e-13.

5.2 The headline, at an exact rational

Theta(sigma) < tau decided on both backends at sigma* = 1.12036249819 (exact rational 112036249819/100000000000):

margin tau - Theta_up(sigma*) = 5.12e-11. Consequently

Delta = sigma* - 1/2 = 0.62036249819 (exact) Lambda_DH <= Delta^2/2 = 0.1924248145802688766381 (narrow) = 0.7696992583210755065522 (wide)

A deliberately generous rounding is also decided on both backends and is worth quoting because it can be reproduced at a sieve limit of only P = 10^4, in seconds: at sigma = 1.1203625, Delta^2/2 = 0.1924248157031250 narrow, 0.7696992628125 wide, margin 1.41e-8.

5.3 One deep point

flint only, 320 bits, P = 10^7, 332442 class primes, one evaluation, 5.9 s. The iv leg is not run at this sieve limit because a single evaluation there costs about a minute and nothing in the headline depends on it.

sigma_deep = 1.1203624981833251 (decided: Theta < tau) Delta^2/2 = 0.1924248145761280189989039 (narrow) = 0.7696992583045120759956154 (wide)

5.4 The improvement

The hunt's bracket, wide frame: 0.2304 < Lambda_DH <= 0.7697. Bracket ratio 6.955 -> 3.341.

6. Controls and cross-checks, all run before anything is decided

Eight, each of which aborts the run rather than downgrading a claim.

1. Character decomposition (decided). a_n = A chi(n) + conj(A) conj(chi)(n) for n = 1..5 and -conj(A)/A = exp(i(pi + 2 arctan kappa)), as six acb residual balls containing 0 with radius < 1e-40. chi from flint's Dirichlet character table.
2. Series against the two L-functions (measured). zeta.epstein.dh_f against A L_chi + conj(A) L_chi(conjugate) at three complex points with large imaginary part (<= 1.2e-41), and against a truncated sum a_n n^{-s} at two more, inside the truncation allowance.
3. The phase lemma (measured). 4001-point circle samples at four radii: observed max |arg| equals 2 arctan r, argmax at u = i r, modulus 1 there.
4. Bombieri-Ghosh's finite claim (decided). Their section 9 states that the smallest set of primes p = 2, 3 mod 5 with sum arctan(1/p) > pi/2 runs up to 6323 and has 420 elements. Recomputed here from the exact sieve with Arb ball sign decisions: threshold prime 6323, cardinality 420. This shares no machinery with their Theorem 7 and it exercises precisely this instrument's class enumeration and arctan.
5. The Euler-product tail identity (measured). The closed form log zeta(sigma) + sum_{p<=P} log(1 - p^{-sigma}) against the explicit partial tail over P < p <= P2: the gap must be positive and below the integral-test allowance at P2. It is.
6. L(., chi5) two routes (decided). The Hurwitz combination against flint's acb.dirichlet_l, ball overlap at three sigma.
7. iv special functions (decided). The iv enclosures of zeta, L and arctan must contain the corresponding Arb balls, nine points.
8. Head/tail split across sieve limits (decided). Theta enclosures at P = 10^3, 10^4, 10^5 must intersect. They do, common width 1.34e-12. Different P move mass between head and tail, so this is a check on the split itself rather than on either half.

Plus a control at the other Titchmarsh root. The two roots of x^2 + 2 phi x - 1 = 0 multiply to -1, so the second Davenport-Heilbronn parameter is tau_- = -1/kappa, and its phase target is pi - 2 arctan|tau_-| = 2 arctan kappa. Theta itself does not change: only the target moves. Running the identical head, the identical tail bound and the identical bisection:

1/kappa decided [3.52014702134020199243, 3.52014702134020199244] against their published tau_- = -3.520147021340 sigma(tau_-, 1) decided [2.38228610898712387152, 2.38228610898712387205] against their published 2.3822861089 (ten digits)

Nothing in this instrument was built around that constant, and it hits all ten published digits.

7. Relation to Bombieri and Ghosh, stated exactly

Their Theorem 7, quoted verbatim in BOMBIERI-GHOSH.md section 3.1: for real xi = tan(theta), sigma(xi, q) is the value of sigma > 1 solving

sum_{p = 2,3 mod 5, (p,q)=1} arctan(p^{-sigma}) = pi/2 - |theta| .

At q = 1 and xi = kappa that is Theta(sigma) = tau, term for term. Their quoted prime-sum target for tau_+, 1.2940091, is pi/2 - arctan kappa, which this instrument decides as 1.29400914734637399....

So the equation is theirs, and this is not presented as a new equation. The division is:

• Their necessary half is what is derived in section 3 above, from the Euler product and one Moebius image, with no Bohr theory and no Kronecker theorem. That half is all an upper bound needs, and it is the half this hunt can decide by its own lights.
• Their converse is not used. It is what turns the upper bound into an exact supremum, it needs machinery this tree has not verified, and nothing here depends on it.
• What is new here is the grade, not the number. BOMBIERI-GHOSH.md set out exactly the two conditions for adopting the sharper constant at rung 2: check Theorem 7's hypotheses in-tree, and re-solve with outward-rounded enclosures. Both are now done, and the constant no longer needs to be adopted from the literature at all, because the inequality it supplies is proved here.

A correction this instrument forces, and it is against an in-tree artifact rather than against the paper. BOMBIERI-GHOSH.md check B re-solved both abscissae to 29 digits at mpmath dps 30. At P = 10^7 and 320 bits this instrument decides that both re-solves sit on the wrong side of their own root:

Neither touches anything Bombieri and Ghosh published: they print 1.120362 and 2.3822861089, and this instrument reproduces both exactly. What it does touch is FRAME.md's 18-digit row 0.192424814576128011, which is derived from the tau_+ re-solve rather than from the six cited decimals. The decided replacement, from the deep point, is 0.1924248145761280190 narrow, which agrees with it to 17 digits.

8. Honest ceiling, and where the looseness now is

How far above the truth is the decided abscissa? The headline sigma* = 1.12036249819 sits 6.8e-12 above the decided lower end of the root enclosure at P = 10^5, and the deep point sits about 4e-17 above it. The sieve-limit sweep says what that costs (flint, 192 bits, full bisection each time):

Both error sources fall like P^{1-3 sigma}, so the accuracy is essentially free and the exercise stops being interesting long before it stops being cheap. The strip constant is no longer where the looseness is.

What is left, stated plainly:

1. The converse. This argument bounds the supremum of the real parts of the zeros from above and does not show it is attained. If Bombieri and Ghosh's converse holds, Delta cannot be improved at all. If it does not, Delta might still come down, and this argument would not see it.
2. The de Bruijn engine. With Delta = 0.62036249819 the engine returns Delta^2/2 = 0.19242481458 narrow, while the deepest measured Davenport-Heilbronn zeros reach |Im z| = 0.347, which would give 0.0602, and the decided floor is 0.0576. The remaining factor of 3.34 in the bracket is the engine plus the sparsity of the extreme zeros, not the strip.
3. Everything GATE.md already carries on the other side: the M2 blind spot, which this section does not touch, and de Bruijn Theorem 13 itself, which this section still rides on exactly as STRIP.md did.

The one thing that has genuinely closed is the referee question GATE.md listed second: why is the sharper published constant not used? It no longer needs to be. The sharper constant is now derived and decided in-tree, on both backends, from an elementary argument, and the citation has moved from being the source of a number to being the standard the number is measured against.

9. Grades

Reproduce with

.venv/bin/python hunts/lambda_dh_bounds/strip2.py

which rewrites strip2_results.json and takes about a minute.