KAPPA-CLOSED-FORM: kappa = sqrt(1 + phi^2) - phi, decided on both backends

Written 2026-08-16. Vocabulary per MISSION.md: measured is one float route, decided is an enclosure with exact endpoints settling a sign or a containment, cited is somebody else's theorem. kappa is a coefficient of the Dirichlet series, not a Lambda, so it is frame-free: nothing in this page needs the narrow/wide tag of FRAME.md.

1. The identity and its owner

kappa = -phi + sqrt(1 + phi^2), phi = (1 + sqrt 5)/2 (the golden ratio) = 0.2840790438404122960282918323931261690911...

The identity is Bombieri and Ghosh's (Around the Davenport-Heilbronn function, Russian Math. Surveys 66:2 (2011), 221-270, section 6), where the constant is named tau_+. The constant itself is Titchmarsh's (Theory of the Riemann Zeta-Function, Chap. X, 10.25, as Bombieri-Ghosh cite it), in the radical form (sqrt(10 - 2 sqrt 5) - 2)/(sqrt 5 - 1). Nothing about the closed form is original to this laboratory. The lab's contribution is three small things: the enclosure-grade check connecting the hunt's own linear-solve kappa to the closed form on both ball backends (section 4), the reduction of the self-duality condition to the explicit real quadratic (section 3), and the observation that this is the same conductor-5 golden arithmetic the lab already carries in hunts/golden_control (section 5).

Equivalent forms, all decided equal at 500 bits on the flint leg (section 4):

kappa = -phi + sqrt(1 + phi^2) (Bombieri-Ghosh, section 6) = (sqrt(10 - 2 sqrt 5) - 2)/(sqrt 5 - 1) (Titchmarsh, via B-G 5.1) = 5^{1/4} sqrt(phi) - phi (since 1 + phi^2 = phi + 2 = phi sqrt 5) = 2 sin(pi/5) / (sqrt 5 + 2 sin(2 pi/5)) (the trig form of the solve, section 3)

and kappa is a root of

kappa^2 + 2 phi kappa - 1 = 0 over Q(sqrt 5), kappa^4 + 2 kappa^3 - 6 kappa^2 - 2 kappa + 1 = 0 over Q.

2. Bombieri-Ghosh's tau_+: definition and the equation it solves

Pinned from the hunt's reading record BOMBIERI-GHOSH.md (2026-08-16 retrieval, mathnet.ru rm9410, full English translation; the quotes below are that record's verbatim transcriptions of the paper).

Definition (their section 5.1). They define the two-parameter series f(s, xi) = sum_{n>=1} a(n, xi)/n^s with the period-5 pattern (1, xi, -xi, -1, 0), then:

These series were introduced by Titchmarsh ([10], Chap. X, 10.25), who noted that for xi = (sqrt(10 - 2 sqrt 5) - 2)/(sqrt 5 - 1) the function f(s; xi) satisfies the functional equation (pi/5)^{-s/2} Gamma((1+s)/2) f(s; xi) = (pi/5)^{-(1-s)/2} Gamma((1+(1-s))/2) f(1-s; xi), which is analogous to the functional equation of Dirichlet series for an odd character mod 5, except for the fact that now the root number is 1. ... In what follows we will write tau_+ for the above value of xi.

So the equation tau_+ solves is the self-duality functional equation of f(s, xi) with root number 1, displayed in their section 5.1. Their gamma prefactor is (pi/5)^{-s/2} where the hunt's completed F uses (pi/5)^{-(s+1)/2}; the two prefactors differ by (pi/5)^{-1/2} on both sides at once, the ratio of left to right prefactor is (pi/5)^{(1-2s)/2} in either convention, so the condition imposed on xi is identical.

The closed form appears in their section 6 (p. 246):

For the Titchmarsh value tau_+ = -phi + sqrt(1 + phi^2) we find ... sigma(tau_+, 1) = 1.120362.

They also name the second root, tau_- = -phi - sqrt(1 + phi^2) = -3.520147021340..., "the second Davenport-Heilbronn function", the one with sigma(tau_-, 1) = 2.3822861089.... Both roots of the quadratic in section 3 are therefore theirs, with names.

One consequence of the quadratic worth pinning because their Theorem 7 uses it: the constant term is -1, so the two roots satisfy tau_- = -1/tau_+, hence |arctan(tau_-)| = pi/2 - arctan(tau_+). Their Theorem 7 prime-sum targets pi/2 - |theta| (with xi = tan theta) for the two functions, 1.2940091 and 0.2767872, are complementary in exactly this way: arctan(kappa) = 0.2767871794... and pi/2 - arctan(kappa) = 1.2940091473... (measured, mpmath dps 30). The complementarity is the quadratic's constant term made visible.

3. The algebra: from the self-duality condition to the quadratic

The starting point is the condition winding.py derives (its docstring, "Why G is even", step (i)): with chi the odd quartic character mod 5, chi(2) = i, values (1, i, -i, -1, 0) on residues 1, 2, 3, 4, 0, the real period-5 pattern a = (1, kappa, -kappa, -1, 0) is a_n = alpha chi(n) + conj(alpha) conj(chi)(n), and matching the two theta terms in Hecke's transformation makes omega(1/x) = x^{3/2} omega(x) hold if and only if

alpha = conj(alpha) * (-i tau(conj chi) 5^{-1/2}). (S)

This is the same condition instrument.kappa_ball solves as a linear solve A + i kappa B = 0 on the completed L-functions at the single point s0 = 3/4 + 3i/2; (S) is that condition at the theta level, which is where it reduces to a quadratic in closed form.

Step 1: alpha in terms of kappa. Matching coefficients at n = 1 and n = 2: alpha + conj(alpha) = 1 and i(alpha - conj(alpha)) = kappa, so

alpha = (1 - i kappa)/2.

Step 2: the Gauss sums, computed rather than remembered. With e(x) = exp(2 pi i x) and sin(4 pi/5) = sin(pi/5):

tau(chi) = e(1/5) + i e(2/5) - i e(3/5) - e(4/5) = 2 i sin(2 pi/5) - 2 sin(pi/5) = -2 sin(pi/5) + 2 i sin(2 pi/5), tau(conj chi) = 2 sin(pi/5) + 2 i sin(2 pi/5).

Checks: conj(tau(conj chi)) = -tau(chi), which is chi(-1) tau(chi) with chi(-1) = chi(4) = -1, exactly the relation winding.py quotes; and |tau(chi)|^2 = 4 (sin^2(pi/5) + sin^2(2 pi/5)) = 4 (1 - (cos(2 pi/5) + cos(4 pi/5))/2) = 4 (1 + 1/4) = 5, the correct Gauss-sum modulus.

Step 3: the condition is an angle equation. Set

w := -i tau(conj chi)/sqrt 5 = (2 sin(2 pi/5) - 2 i sin(pi/5))/sqrt 5.

By step 2, |w| = 1, so w = e^{-i psi} with `cos psi = 2 sin(2 pi/5)/sqrt 5

0 and sin psi = 2 sin(pi/5)/sqrt 5 > 0, hence psi in (0, pi/2)`. On the

left of (S), alpha = (1 - i kappa)/2 = |alpha| e^{-i theta} with theta = arctan(kappa), so alpha/conj(alpha) = e^{-2 i theta}. Condition (S) reads

e^{-2 i theta} = e^{-i psi}.

The ball solve decides 0 < kappa < 1 (kappa_ball raises otherwise), so theta in (0, pi/4) and 2 theta in (0, pi/2); both sides are principal, and

2 arctan(kappa) = psi exactly, not merely mod 2 pi.

Step 4: tan(psi) is 1/phi. Using sin(2x) = 2 sin(x) cos(x) and the pentagon fact cos(pi/5) = phi/2 (this is where the golden ratio enters, and it enters through conductor 5):

tan psi = sin(pi/5)/sin(2 pi/5) = 1/(2 cos(pi/5)) = 1/phi.

Step 5: the quadratic. The double-angle formula tan(2 theta) = 2 tan(theta)/(1 - tan^2(theta)) with tan(theta) = kappa turns step 3 plus step 4 into 2 kappa/(1 - kappa^2) = 1/phi, i.e.

kappa^2 + 2 phi kappa - 1 = 0. (Q)

So the quadratic the linear-solve kappa satisfies is exactly kappa^2 + 2 phi kappa - 1 = 0, equivalently kappa^2 + (1 + sqrt 5) kappa - 1 = 0. It is what the task's candidate guessed, and the discriminant delivers the claimed radical:

disc = (2 phi)^2 + 4 = 4 (1 + phi^2), kappa = -phi +/- sqrt(1 + phi^2).

The decided constraint 0 < kappa < 1 selects the plus sign, giving Bombieri-Ghosh's tau_+; the minus sign is their tau_-. Corollaries: tau_+ tau_- = -1 (constant term of (Q)); the half-angle form kappa = tan((1/2) arctan(1/phi)) (steps 3 and 4 read backwards); the tidier radical sqrt(1 + phi^2) = sqrt(phi + 2) = 5^{1/4} sqrt(phi) (from phi^2 = phi + 1 and phi + 2 = phi sqrt 5); and, squaring sqrt 5 kappa = 1 - kappa - kappa^2, the rational quartic kappa^4 + 2 kappa^3 - 6 kappa^2 - 2 kappa + 1 = 0.

Also for the record, solving (S) componentwise instead of by angles gives the trig form quoted in section 1: the imaginary part of sqrt 5 (1 - i kappa) = (1 + i kappa)(2 sin(2 pi/5) - 2 i sin(pi/5)) is kappa (sqrt 5 + 2 sin(2 pi/5)) = 2 sin(pi/5), hence kappa = 2 sin(pi/5)/(sqrt 5 + 2 sin(2 pi/5)); the real part is the consistent second equation of the same one-complex-equation pair.

4. The decided containments

All run 2026-08-16, script kappa_closed_form_check.py in this session's scratchpad; the flint leg is pinned by tests/test_lambda_dh_separation.py (that file is new: tests/test_lambda_dh_bounds.py did not exist when this page was written, so the test lives in the new file and says so).

flint leg (python-flint 0.9.0, Arb), 500 bits. instrument.kappa_ball(500) is the Hurwitz-zeta self-duality linear solve, never a remembered constant. The closed form is built from exact input balls: arb(5).sqrt(), phi = (1 + sqrt 5)/2, target = (1 + phi^2).sqrt() - phi, all at 500 bits.

Shared midpoint, 60 digits, both balls: 0.284079043840412296028291832393126169091088088445737582759163.

mpmath.iv leg, dps 40. The iv context has no interval Hurwitz zeta, so the linear solve itself cannot run on this backend (the same flint-only boundary instrument.py's docstring records for the quadrature). The iv leg therefore encloses the trig form of the same condition, which is the linear solve after the exact algebra of section 3 (that reduction step is itself decided on the flint leg by the tan-identity row above):

k_iv = 2 iv.sin(iv.pi/5) / (iv.sqrt(5) + 2 iv.sin(2 iv.pi/5)) target_iv = iv.sqrt(1 + phi_iv^2) - phi_iv, phi_iv = (1 + iv.sqrt(5))/2

Stated honestly: the flint row is the one that touches the hunt's actual instrument; the iv row is an independent-backend enclosure of the reduced form, not a second route to the Hurwitz-zeta solve. Together with the tan-identity row the two legs close the loop linear solve -> (S) -> trig form -> closed form at enclosure grade.

5. The golden_control connection

hunts/golden_control/MISSION.md (the Pisano paragraph) pins the lab's standing conductor-5 fact: the Davenport-Heilbronn rival is built on the quartic character mod 5, whose square is the quadratic character chi_5, the character of Q(sqrt 5), the golden field, and Fibonacci arithmetic mod p is governed by that same chi_5 through the Pisano period, pi(p) dividing p - chi_5(p). That hunt pins the connection exactly and claims it does nothing; this page keeps the same stance and adds one instance: the golden ratio in the closed form is not decoration but the same conductor-5 arithmetic surfacing in the coefficient itself. It enters at exactly one point of the algebra, cos(pi/5) = phi/2 in step 4, which turns the Gauss-sum angle into arctan(1/phi), so

kappa = tan((1/2) arctan(1/phi)),

and the quadratic (Q) lives in Q(sqrt 5) = Q(phi), the quadratic subfield of the conductor-5 cyclotomic field. No claim is made that this connection does anything; it is pinned because it is exact.

6. Grades, in one line

The identity is cited (Bombieri-Ghosh 2011 section 6; Titchmarsh for the radical form). The reduction to (Q) is in-tree algebra whose every discharge is decided on the flint leg at 500 bits (section 4 table). The two-backend containment is decided (flint 500 bits; mpmath.iv dps 40). Nothing on this page is claimed as novel and nothing here moves any Lambda number; kappa enters the bracket's instruments as an input ball, and this page only says what that ball's midpoint is in closed form, and whose closed form it is.