Hunt #148, step 3. Verdict: no. In F_13[T] the coefficients of the biquadratic (n = 4) theta function at squarefree indices are computable, and they do not vanish identically beyond small degree; but at three primes their modulus depends on the primes themselves, not on their degrees, so no choice of the component at infinity and no normalization by degree makes them multiplicative up to phases. A probe built on them with quartic averaging therefore has no scalar local Euler identity: its principal row is not zeta(4z) L(w) / L(x, eta) times an Euler product. The first failing term is the coefficient of X^3 in the principal row. Exact computation at the stated parameters, not reviewed by any person; files produced on branch hunt/ff-probe, not landed. Every number here is read from results_step3.json and pinned by test_ffprobe.py.
1. The coefficients, and the checks at n = 4
Over F_q(T) with q = 1 mod n, the theta function's Fourier coefficients at an index r are Patterson's rho_0(r, eps, i), one for each of the n components i of the infinite place, computed from the sums C(r, eps, i') of n-th order Gauss sums over the monic moduli of degree i' (SOURCES.md, formulas 5 to 7). Here q = 13, n = 4, the quartic symbol is (a/c)_4 = chi(Res(c, a)), and eps sends g^3 to i for g = 2, the least primitive root. Since 13 = 5 mod 8, eps(chi(-1)) = -1, so the two additive characters of SOURCES.md give different signs, and both are checked.
- rho_0(1, eps, i) = 1, 13 tau(eps chi), 0, 0 for i = 0, 1, 2, 3, and at a linear prime P, rho_0(P, eps, i) = 1, 0, 13^2 tau(eps chi^2), 0: Patterson's values for n = 4 (2007, page 252), with his c_o = (q-1)/n dropped.
- The finite Hecke relation holds in all 36 checked cases, the Hecke relation at infinity in all 20 checked cases in each additive character (with the signs of
SOURCES.md), and rho_0(r b^4) = rho_0(r) at the checked r. - Patterson's four relations for n = 4 (2007, page 252): C(r, eps, 2) = tau(eps) q C(r, eps, 1) eps(chi(-1)) and C(r, eps, 3) = tau(eps^3) q^3 at degree 2, and C(r, eps, 3) = tau(eps^2) q^2 C(r, eps, 1) at degree 3, hold in the 2014 character at every checked r (3, 3 and 2 of them). The degree-4 relation, C(r, eps, 3) = tau(eps) q^2 C(r, eps, 2), fails at r = (T-1)(T-2)(T-3)(T-4); the relation that the Hecke relation at infinity gives there, C(r, eps, 3) = tau(eps) q C(r, eps, 2), holds (a misprint or an extraction error, q^2 for q).
2. Components and forced zeros
For a squarefree index r of degree d and a component i: whether rho_0 vanishes at every sampled r, whether it must (R < 0, or i = i' in the Hecke relation at infinity), and the moduli of the values at the sampled r (component_table; at degree 3 the samples are (T-1)(T-2)(T-3), (T-1)(T-2)(T-5) and (T-1)(T^2+2)).
| d \ i | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 1 | 46.87 = 13^1.5 | 0 (R < 0) | 0 (R < 0) |
| 1 | 1 | 0 (i = i') | 609.34 = 13^2.5 | 0 (R < 0) |
| 2 | 1 | 13 | 609.34 | 7921.40 = 13^3.5 |
| 3 | 0 (i = i') | 0, 26, 18.38 | 0 (i = i') | 0, 15842.79, 11202.55 |
| 4 | 0.5547, 0 | 26, 0 | 338, 0 | 15842.79, 0 |
Every component family vanishes on an odd degree class. A family that reads the coefficient at r in the component i(r) = (b - deg r) mod 4 must vanish, by R < 0 or by i = i', at every degree d = 2b + 1 (mod 4) (derived from the Hecke relation at infinity, checked in the table and by test_ffprobe.py through degree 11). For n = 3 the same computation leaves two families (b = 0, 1) with no forced zero and kills the third, and for n = 2 it recovers the classical theta's support on even degrees. With b = 1, for instance, rho_0 = 1 at every linear prime and rho_0 = 0 at every squarefree index of degree 3, which no function multiplicative up to phases can do.
3. The degree-3 coefficients depend on the primes, not on their degrees
Read the coefficients by degree, with any component choice. At degree 0, 1 and 2 the squared moduli are fixed by the component alone: at every linear prime they are 1, 0, 13^5, 0, and at every pair of linear primes 1, 13^2, 13^5, 13^7, for i = 0, 1, 2, 3 (moduli_by_lambda, twelve primes and 21 pairs; the irreducible quadratic of the table agrees). At degree 3 only i = 1 and i = 3 survive, and there rho_0 is a constant times Patterson's C(r, eps, 1) = tau(eps chi) H(r), with
H(r) = sum over a in F_13 with r(a) != 0 of conj(eps(chi(r(a)))),
the classical character sum of a cubic polynomial (with this hunt's additive character e = e_o(a_-1); Patterson 2007, page 253, prints -tau(eps) for the 2007 character). Over all 2028 squarefree monic cubics (degree3_census), the squared modulus abs(H(r))^2 is:
| factorization of r | abs(H)^2 = 0 | nonzero | nonzero value |
|---|---|---|---|
| three linear primes (286) | 234 | 52 | 52 |
| linear times irreducible quadratic (1014) | 546 | 468 | 26 |
| irreducible cubic (728) | 0 | 728 | 13 |
For r = (T-1)(T-2)(T-lambda), lambda = 3, ..., 12, H(r) vanishes except at lambda = 5 and 11, where abs(H)^2 = 52. A function whose modulus is multiplicative over primes, times any factor depending on the degree or even on the factorization type, is constant in modulus on each row of this table. These coefficients are not: on three linear primes the modulus is 0 for 234 triples and 26 for 52 (at i = 1). The ratio of section 4's notation, abs(f(P1 P2 P3))^2 abs(f(P1) f(P2) f(P3))^2 / (abs(f(P1 P2) f(P1 P3) f(P2 P3))^2 abs(f(1))^2), is defined only at lambda = 5 and 11 for the admissible component choices (three_body), and the zeros at the other eight values of lambda are the defect. This is the hypergeometric behaviour Patterson describes for three prime factors (2014, lines 470 to 481).
4. Consequence for the probe, and the named term
In the F_13[T] analogue of the probe, with quartic symbols, quartic averaging and the theta coefficients read by degree, the principal row's coefficient at an index A with s = a = 1 is K(A) = F(1, A, 1) theta(A) eta(A) times phases of modulus one, and F(1, A, 1) = g_{xi chi_A}(T A, 1) is the Gauss sum of a primitive character modulo T A for squarefree A prime to T, of squared modulus 13^(1 + deg A) (the q = 7 analogue is checked in results_ledger.json). So abs(K(A)) is, up to the degree, abs(theta(A)), and section 3 is a statement about K. A scalar local identity would make K multiplicative up to phases, and it fails first at the coefficient of X^3 in the principal row, at A = (T-1)(T-2)(T-lambda), which vanishes for eight of the ten lambda and not for the other two. The squarefree coefficients do not vanish identically beyond small degree (section 2), so the verdict is not the S3 one (a theta supported on squares); it is that the coefficients exist and are not multiplicative.
5. What the template would price, and what that is worth
Hunt #136's order-k template (PROBE.md there, sections 3 and 4, derived and measured there, unreviewed) prices k = 4 at 7/8 for the balanced probe, 13/16 at the per-summand corner and 4/5 at perfect row counts (recomputed here as exact fractions, results_ledger.json), and at 0.8105686 for the endpoint with the paper's row counts carried over (hunt #136's variant A, measured there, quoted and not rerun). All four are conditional on hunt #136's L1 to L3 (closed-form squarefree coefficients with a pairing, a twisted reflection, a scalar Euler identity). In F_13[T], L1 fails in the sense of sections 2 and 3, so the template prices nothing for this probe. Over F_q[T] the truth is 1/2 for every L-function concerned (Weil).
6. What transfers to Q(i), and what does not (inferred)
- The forced zeros of section 2 come from the infinite place of F_q(T), which is non-archimedean with n components; the infinite place of Q(i) is complex and has no such components. That obstruction does not transfer.
- The degree-3 behaviour of section 3 is a statement about the finite places: the coefficient at three primes is not determined by the coefficients at fewer primes. Over number fields the Hecke theory of Kazhdan and Patterson likewise stops determining the squarefree coefficients of degree-n theta functions for n >= 4 (Deligne's non-uniqueness of Whittaker models, as hunt #136 records via David, Dunn, Hamieh and Lin). The function-field computation is evidence, not proof, that S4's L1 fails over Q(i) for the same reason.
7. Limitations
- One q (13) and one embedding eps; components chosen per degree (a rule choosing a component per index would not be the theta function's coefficients in any natural sense); degree-4 data only as samples.
- The verdict is about scalar Euler identities of Lemma 7.1's type. A probe whose principal row is not an Euler product could still isolate 1/L by another mechanism; none is proposed here.
- The quartic probe itself (its Poisson coefficients and local factors at q = 13) is not built: the obstruction is located in its theta input, and the reduction from K to theta in section 4 uses only the Gauss-sum modulus of a primitive character.