REFEREE_W1: a code-disjoint check of the scalar Euler identity

Hunt #136 (below_seven_eighths), referee lane b78-referee, wave 1, 2026-10-21; repaired the same day after an adversarial review (section 7). A second route, sharing no module or function with the first-wave lanes (note N3), for the claim in OpenAI's quasi-Riemann paper that carries the Euler product: Lemma 7.1 (lem:local-euler) and its Gauss-sum inputs, Lemma 4.2 and Equation (4.7). Nothing here is a result until the case log in hunts/README.md says how this hunt ended.

Grades. Every check below is derived: an exact equality in a cyclotomic number field, computed by PARI/GP, at the stated primes and inside the stated truncation boxes. None is a proof of the lemma for all primes, none is kernel-checked, and nothing here bears on real part 1/2. Where a statement is a reading of the paper rather than a computation it is marked inferred.

What is committed and what is only a file. The four files of this directory (REFEREE_W1.md, local_identity_pari.py, test_local_identity_pari.py, disagreements.json) were committed by this lane in b6c4972a8 on the branch hunt/below-seven-eighths. The branch was later pushed to origin: the local reflog records the push at 2026-10-21 21:23:47 UTC, to baa41597e, which contains b6c4972a8. This lane did not push it. The repair after the adversarial review (section 7) is a further local commit on the same branch, and this lane does not push it either. The first-wave lane files the task names (euler_step.py, probe_table.py) were still untracked on the branch at the repair, so no comparison with them was made (section 5).

0. Source and method

The source is paper.tex of github.com/openai/math at adc7f124, read from the campaign cache; its SHA-256 is checked against 42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3 before any line is quoted, and every cited line number is pinned by test_cited_lines_say_what_the_report_says.

Engine: PARI/GP through cypari2 (on the campaign box the PARI library is version 2.17.2 and the cypari2 package version 2.2.4; environment facts, not pinned). Residue symbols come from PARI's nfmodpr reduction of O = Z[omega] to the residue field followed by the (Q-1)/6 power there (quartic symbols in Z[i] the same way). Gauss sums, Jacobi sums and every coefficient are exact PARI polmods modulo polcyclo(N), compared by exact equality; one complex embedding is used only to choose the positive square root of a norm. The first-wave lane uses python-flint convolutions; nothing here imports python-flint.

Test primes (pinned by test_primary_generators_and_symbols): the primary generators 1 + 3w (norm 7), 4 + 3w (13), -2 + 3w (19), -5 (the inert prime 5, norm 25), and 1 + 6w (31, the verifier's prime), with omega_p = chi_p(-1) equal to -1, 1, -1, 1, -1. The primes above 6 lie in S by line 591; there only the claims the paper makes at S are checked (section 1.6). Lemma and equation numbers are the task's; the tests pin the LaTeX labels at the cited lines and that lem:local-euler is the only numbered environment of section 7 (so Lemma 7.1 is an inferred numbering, as is Lemma 4.2 as the second lemma of section 4).

1. Lemma 7.1 and its Gauss-sum inputs

1.1 Lemma 4.2 and the identities of its proof

All 21 identities of lemma_4_2_checks hold exactly at norms 7, 13, 19 and 25 (and 31): gamma_2^3 = -alpha(p), gamma_1 gamma_2 = conj(H(4)) gamma_3 gamma_2^3, |gamma_r| = 1 for r = 1..5, G_0 = -1, tau(A) tau(conj A) = A(-1) P for the five nontrivial powers, J(H, H^3) = H(4) J(H, H), J(H^2, H^2) = -p, tau_2^3 = -pP, and the five facts the proof of Lemma 7.1 quotes (G_1 G_{-1} = omega_p Q, gamma_3^2 = omega_p, gamma_2 gamma_4 = 1, gamma_1 gamma_2 = -alpha(p) conj(chi_p(4)) gamma_3, gamma_4/gamma_1 = -conj(alpha(p))/G(p), lines 3886, 4079, 4117-4121 and 4124; the first version cited 4117-4122, which stops before line 4124, where gamma_2 gamma_4 = 1 is stated). Normalized statements are cleared of square roots before comparison (for example gamma_2^3 = -alpha is tested as G_2^3 = -pP).

1.2 Equation (4.7), Lemma 4.3 and Equation (4.6)

Equation (4.7) holds exactly at the squarefree indices c = 1 + 3w, 7 (both primes above 7), (1 + 3w)(4 + 3w), (4 + 3w)(-5), 19 (both primes above 19) and (1 + 3w)(-5), and in the slow tier at the three-prime index (1 + 3w)(4 + 3w)(-2 + 3w): gamma_2(c)^3 = mu(c) alpha(c), gamma_1 gamma_2 = mu alpha G, G = conj(chi_c(4)) gamma_3 (that is, the quadratic Gauss sum equals gamma_3) and |G| = 1. At each of these c the direct quadratic sum also equals Lemma 4.3's four-term value. At c = p^t the four-term value matches direct summation for t <= 3 (norms 7 and 25) and t <= 2 (13, 19), and Equation (4.6) (G(p^3) = gamma_3(p), R(p, p) = gamma_3(p)^2 = chi_p(-1)) and G(p^2) = chi_p(4) (line 953) hold.

1.3 The local scalar C_p

The lift of lines 3877-3883 (g_{chi^r}(p^t, p^{j'})) equals direct summation over O/p^t for every r and j' <= t + 1, for t <= 3 at norm 7 and t <= 2 at 13, 19, 25 (slow tier: t <= 4 at 7, t <= 3 at 13 and 25). Then C_p(t, k, j'), computed from its definition (local-scalar-definition, the sum over d mod p^k done term by term with its congruence), equals the paper's table (local-scalar-values) at every key of the boxes t <= 13, k <= 3, j' <= 29 (norm 7, 1680 keys, inner Gauss sums summed directly for t <= 2) and t <= 13, k <= 2, j' <= 29 (13, 19, 25, 1260 keys each, direct for t <= 1). Above those depths the inner sums use the lift, which is the one input taken from the paper beyond the directly summed range.

1.4 The closed form

(local-P) equals Lemma 7.1's closed form (old-eq:5.13a) as an exact identity of truncated local series in X = Q^{-x}, B = Q^{-w}, V = Q^{-6z}, through the box t <= 13, k <= 2, m <= 4, for all six rows j = v_p(u), all six unit phases rho, at norms 7, 13, 19 and 25 (144 identities). The comparison multiplies the series by (1-R)(1-V)(1-W_loc) and matches the closed form's polynomial numerator, so each of the four families of the table (eq:local-family-table) is tested through r = 1, and the family with t = 6r + 1 through r = 2. It is not vacuous: at norm 7 the six rows carry 28, 24, 26, 28, 26, 26 nonzero series coefficients in the box (at rho = zeta_6^2) and the numerator of row 0 has 11 monomials. Every monomial X^t carries eta(p)^t, so eta = 1 suffices; eta = zeta_12^5 and eta = 2 were also run and agree.

Two further derivations agree with the same summands. They share this module's Gauss sums, C_p and field arithmetic, so they are distinct derivations, not independent implementations:

• The phase bookkeeping (lines 3953-3996). The raw coefficient at an index supported on one prime, gamma_2(p)^{e0} conj(alpha)^t eta^t conj(G(p^t)) R(p^t, p^k) rho^{k-t} C_p, with gamma_2 summed directly and G, R from the four-term value, equals the summand of (local-P) through t <= 13, k <= 2, m <= 3, at all four primes.
• The global coefficient. K_eta(c, n, s, a; u) built from local-full-coefficient itself, with S = {(2), (lambda)}, b_* = 2 lambda, xi of order three at 2 and quadratic at lambda, tau, Xi, and F(s, A, H) as a sum of complete Gauss sums modulo b_* A (lines 3742-3745), equals Q^{e0/2+l} times the summand at single-prime indices (11 cases with t <= 3, k <= 2, m <= 1 at norm 7 for rows 0, 1, 2 and two units; 6 cases at norms 13 and 25 for rows 0 and 1; rows 3 and 4 at t = 4 in the slow tier). It separates into prime factors (line 3994): the coefficient at an index supported on two primes is the product of its two one-prime coefficients for the pairs (7, 7bar), (7, 13) and (7, 25), each for eta = 1 and eta = chi_{-4} o N, and for u = 1, zeta_6, 1 + 3w.

1.5 H_p and the analytic statements

The defect formula local-H-defect holds exactly in the box at all four primes, rows and phases, and H_p - 1 has no X^1 term: the -D entry of J_0 cancels the Euler factor D = eta(p) conj(chi_p(u)) Q^{-x}, which is what places 1/L^S(x, eta conj(chi_.(u))) in eq:probe-high.

The convergence statements were read as exponent bookkeeping over the monomials of the polynomial (1-D)(1-R)(1-V)(H_p - 1) (symbolic, in sympy, exact rationals). In region one, at eps0 = 1/1000 the largest monomial is Q^{-1001/1000} in rows 0 and 1 and Q^{-1/1000} in rows 2 to 5; at eps0 = 1/10 it is Q^{-11/10} in rows 0 and 1 and Q^{-3/100}, Q^{-1/25}, Q^{-1/10}, Q^{-3/50} in rows 2 to 5. At those values and at eps0 = 1/50 every row is inside the stated O(Q^{-1-eps_H}) (row 0, p not dividing u) and O(Q^{-eps_H}) (rows 1 to 5, p | u), eps_H = min(eps0, 1/50). In region two the maxima are exactly -363/200 and -33/40, attained by the terms the paper names (Q^{1-x_r-w_r-6z_r} and Q^{1-x_r-w_r}), and the tail exponent 1 - 363/200 = -163/200 gives c_H = 4/5. This checks the bookkeeping of lines 4148-4185 given the closed form; it is not a proof of normal convergence.

1.6 At the primes above 6

Only what the paper claims there was checked, with S = {(2), (lambda)} and with the inert prime 5 added: the six units are distinct modulo 3 (line 580), xi is a character of (O/b_*)^x of order three at 2 and quadratic at lambda (lines 3349-3355), |tau| = 1 (line 3367), g_xi(b_*, H) = 0 when H meets S (line 3755), and g_xi(b_*, h) = conj(xi(h)) g_xi(b_*, 1) for h prime to S. Lemma 4.2 and Lemma 7.1 are not claimed at S and were not run there.

Verdict on (1). No defect was found. At norms 7, 13, 19 and 25, inside the stated boxes, Lemma 7.1's closed form, its scalar factorization at the level of single-prime and two-prime coefficients, and its Gauss-sum inputs hold exactly. Derived, at those primes and boxes only.

2. Rivals: the identity fails, and where

Placement. The identity reads the target only through eta(A) in local-full-coefficient, linearly. The paper places a Dirichlet character there as a -> chi(N a) (line 6725). This lane extends that rule, which the paper states only for characters, to a rival's Dirichlet coefficients a(n): f(a) = a(N a) goes in the eta slot, and the rival's bad primes join S (S = {(2), (lambda), (5)} for both rivals).

• Davenport-Heilbronn. f(a) = c_1 chi(N a) + c_2 conj(chi)(N a), chi mod 5 with chi(2) = i, c_{1,2} = (1 -+ i kappa)/2: a linear combination of two Hecke characters of F. kappa is built exactly in Q(zeta_20) and agrees with zeta/epstein.py's pinned value (0.2840790438404123 to fifteen places). The coefficients are 1, kappa, -kappa, -1, 0 by n mod 5.
• Epstein, D = -15 (hunt #120). f(a) = r(N a)/2 for Q0 = x^2 + xy + 4y^2, with r = (1 * chi_-15) + (chi_-3 * chi_5), hunt #120's identity in the paper's vocabulary; the formula matches a direct lattice count at the 17 values of n tested.

In the identity's terms at a good prime of norm P: eta(p) := f(p) (forced, since the t = 1 coefficient is -eta rho^{-1}, eta or -eta (Q-1) in every row), a_p = conj(alpha)^3 f(p)^3, b_p = conj(alpha)^2 f(p)^2 conj(chi_p(4)), R = a_p^2 Q^{4-6x-6z}, D = f(p) conj(chi_p(u)) Q^{-x}, while the series carries the rival's own f(p^t) where a character has eta(p)^t.

The failing local terms. Both steps of the identity fail for both rivals.

1. The single-prime closed form. With eta(p) = f(p), the first monomial where the rival's series leaves the closed form is, in the u = 1 row, the R(1-Q^{-1}) entry (family (0, 2r+2), r = 0, the monomial X^6 V), where f(p^6) replaces f(p)^6: DH at norms 7 and 13 (f(p^6) = -1 against kappa^6), Epstein at 7, 13 (1 against 0), 19 and 31 (7 against 64). In the other rows the first failure is J_2 (row 2, t = 3), the first entry of J_3 (row 3), J_4 (row 4) and J_5 (row 5) for DH at 7 and 13 and Epstein at 19 and 31; for Epstein at 7 and 13, where f(p^3) = 0 = f(p)^3, row 2 first fails at t = 6 and row 3 at the X^4 entry of J_3. Below t = 6 the u = 1 row has monomials only at t = 0 and t = 1, and its X^6 V coefficient carries C_p(6, 0, 6) = Q^6 - Q^5, which is never zero (test_row_zero_reads_only_t_0_1_below_six, at norms 7, 13, 19 and 25): that row fails at X^6 V exactly when f(p^6) != f(p)^6.
2. Separation into prime factors (line 3994). In the u = 1 row the coefficient at the squarefree index c = 7 = (1 + 3w)(-2 - 3w), which the product of local factors gives as D_p D_pbar (the product of the two J_0 entries -eta rho^{-1} Q^{-x}), comes out as f(7) times the same Gauss-sum phase. The phase itself separates (the trivial character passes), so at a pair separation holds exactly when f(pq) = f(p) f(q). DH: f(7) = a(49) = -1 against f(p) f(pbar) = kappa^2 = 0.0807009. Epstein: 1 against 0. Also failing: DH at (7, 13) (1 against -kappa^2) and Epstein at (19, 19bar) (3 against 4).

Which step fails where: both, on the same primes. No tested index has a rival pass the single-prime closed form and fail only the separation.

• DH. The placed coefficient depends only on N a mod 5. Where chi(N p) is real (N p = 1 or 4 mod 5: among the tested primes those of norm 19 and 31, and every inert prime, whose norm is a square), f(p^t) = f(p)^t for every t and f(pq) = f(p) f(q) for every q: DH agrees with the Hecke character chi o N on every power of p, and neither step can fail at p. Where chi(N p) is not real (N p = 2 or 3 mod 5: norms 7 and 13), the single-prime closed form fails at X^6 V, and separation fails at every pair of two such primes (-1 against kappa^2, or 1 against -kappa^2) and at no other pair. The coefficient statements are exhaustive over the residues mod 5 (test_dh_both_steps_fail_on_the_same_primes); the closed form was computed at norms 7, 13, 19 and 31, and the global coefficient at (7, 7bar), (7, 13), (19, 19bar) and (7, 19). The single-prime identity is therefore a shared lemma for DH only at primes where DH is locally a character, and at those primes separation holds as well.
• Epstein. The single-prime closed form fails at every tested prime (7, 13, 19, 31). At a split prime p | ell, f(p^t) = sum_{i <= t} chi_5(ell)^i, so f(p^6) is 7 against 64 when chi_5(ell) = 1 and 1 against 0 when chi_5(ell) = -1; from the coefficients this holds at all 21 split ell below 200. Separation fails at the conjugate pairs (p, pbar) over one rational prime (computed at (7, 7bar) and (19, 19bar); from the coefficients, f(ell^2) != f(ell)^2 at all 21) and holds across distinct rational primes (computed at (7, 13) and (7, 19), both sides zero; from the coefficients, at all 210 pairs of the 21), as test_epstein_single_prime_fails_everywhere_and_separation_only_at_conjugates pins. For Epstein the step that fails at every tested prime is the single-prime closed form, and separation is shared at pairs over distinct rational primes.

The first version of this section called the single-prime identity a shared lemma for DH in the sense of ALIGNMENT.md section 5, and the separation the step DH cannot pay. That reading is withdrawn (section 7); the computed values it rested on are unchanged.

Scope. The placement tests the identity's terms, not the rivals' zeros: the norm-composed series are not the rivals themselves (for DH it is c_1 L(chi) L(chi chi_-3) + c_2 L(chibar) L(chibar chi_-3)), and no zero of either placed series is computed here. The kill condition "a claimed arithmetic step also holds for a function with a zero beyond the claimed abscissa" does not fire on this step.

3. The lower-order probes, one prime each

These are the referee's own constructions from the task's one-line descriptions, built before any lane file was opened (this lane has never read the probe lane's definitions; section 5). The one report about them, from the adversarial review, is attributed as such in the S1 bullet. "Scalar" means in mu_infinity * alpha(p)^Z, the values a Hecke character can take at p; the test is exact (x^12 alpha^{-a} a root of unity), and alpha(p) is checked not to be a root of unity.

• S1, cubic theta with the conjugate cubic twist (k = 3), at norm 7. gamma_2 gamma_4 = 1 exactly, also at the two-prime indices 7, (1 + 3w)(4 + 3w), (1 + 3w)(-2 + 3w), (4 + 3w)(-2 + 3w) and (1 + 3w)(1 + 6w) and at the products of the conjugate primes above 13 and above 19: the product itself carries no Moebius sign and no cross phase. This holds at every squarefree c prime to 6, by an argument the adversarial review supplied: at a prime, tau(H^2) tau(H^4) = H^2(-1) P = P (section 1.1's tau(A) tau(conj A) = A(-1) P with A = H^2), and for c = p_1 ... p_r the CRT cross factors of g_2(c) and g_4(c) multiply to prod_{i != j} chi_{p_i}(p_j)^6 = 1 (derived: a two-line argument, checked exactly at the indices just listed).
• The paper's row normalization. At one prime the u = 1 coefficient of X is +conj(alpha(p) G(p)) eta rho^{-4}: the angular factor conj(alpha) does appear, as the kill says, together with conj(G(p)). Across primes this coefficient separates only where the pair phase is trivial. Its squarefree part is conj(alpha(c) G(c)), and G(pq) = G(p) G(q) R(p, q), so at pq it is conj(R(p, q)) times the product of the two one-prime values. R(p, q) = -1 at (7, 13), (7, 19), (13, 19) and (7, 31), and +1 at the conjugate pairs above 7, 13 and 19 (exact, with Gamma summed directly and compared with Lemma 4.3; test_S1_does_not_separate_in_the_paper_row_normalization). In the same normalization the paper's own probe gives mu(c) at every one of these indices and separates; dropping the conj(G) insertion, a planted lesion, flips both verdicts at (7, 13) (test_S1_pair_phase_lesion_without_the_G_insertion). So in that normalization S1 has no single Euler product L(s, eta conj(alpha)), in either position. Reading line 3416, which expands conj(G(A)) over the characters of T-hat, what remains is a finite combination over T-hat (inferred, not computed here). The first version of this bullet stated the paper-normalization coefficient at one prime only.
• The bare row. With no angular, no G and no R insertion (all three are there to cancel phases that gamma_1 gamma_2 = mu alpha G produces and this product does not) the coefficient of X is exactly +eta rho^{-4}, and it separates. At norms 7 and 13, inside the box t <= 7, k <= 2, m <= 3, the bare-row local factor at u = 1 has the scalar shape (1-V)^{-1} (1-W)^{-1} (1 - eta^2 X^2)(1 - eta X)^{-1} H, with no pure X or X^2 left in H - 1: the target sits in the numerator, L(x, eta)/L(2x, eta^2), not 1/L. The size exponents of H - 1 that the tests bound are floats measured at those two primes; no convergence of the correction product over all primes is shown.
• Scope of the construction. That factor shape is specific to this lane's S1, which keeps H = u a^6 with V = Q^{-6z} and twists the s-side character to chi^{4k}. The adversarial review reports that the probe lane's PROBE.md (uncommitted, not opened by this lane) defines S1 as "cubic, j = 4", the twist used here, and prices it with pole 1/3 (as reported, not checked here). D1 is then not about the twist; what remains is normalization and decomposition. The sign in D1 depends only on the (t, k, m) = (1, 0, 0) coefficient, which reads neither the a-variable nor the s-side twist, so it is unaffected.

The shape a zero detector needs, 1/L(x, eta), is absent in both normalizations, but not for the stated reason (disagreement D1 in disagreements.json).

• S2, the quadratic twist (k = 2), at norm 7. gamma_2 gamma_3 has alpha-exponent 1/3; with the paper's row normalization the coefficient has exponent -2/3, and with the conjugate theta -1/3. Since gamma_2^3 = -alpha and alpha has infinite order, no finite-ray phase or integer power of alpha makes it scalar. The attempt to exhibit a scalar identity failed. Consistent with the kill.
• S3, classical theta over Q(i) with the quartic twist, at the prime -1 + 2i of norm 5 (every scalar by direct summation through t = 4) and at 3 + 2i (norm 13). The theta coefficients at p^t are 2, 0, 2, 0, 2 for t = 0..4, so there is no X^1 term in any row. The non-scalar twist values (quartic Gauss sums, alpha-exponent 1/2 and -1/2) occur only at odd t, exactly where the theta vanishes; every surviving coefficient is scalar (the quadratic Gauss sum at t = 2, Ramanujan sums at t = 4), and the u = 1 row starts at X^4. A scalar local factor exists, in Q^{-2x} and Q^{-4x}; the factor 1/L(x, eta) is absent. Consistent with the kill.

4. Planted faults

Both planted faults required by the task turn checks red (tests test_planted_fault_wrong_gauss_sum_sign and test_planted_fault_wrong_local_factor):

• Wrong Gauss-sum sign, G_1 -> -G_1. Six of Lemma 4.2's 21 identities fail, the raw-coefficient route fails in every row, and the C_p table comparison fails where the inner sums are summed directly. The closed-form identity alone catches it only in rows 3 and 4: elsewhere G_1 enters only through the ratio C_p/gamma_1, which is invariant. A check of the closed form by itself is blind to this sign in rows 0, 1, 2 and 5 (section 6, note N1).
• Wrong local factor, R built from a_p instead of a_p^2. The closed-form identity fails in every row, first at X^6 V and X^6 B V.

Further variants also fire: J_2 with an extra rho (row 2 only), (1 + Q^{-1}) for (1 - Q^{-1}) (every row), and an unconjugated Euler factor D = eta chi_p(u) Q^{-x} (caught by the -D cancellation exactly when rho is not real, and by the region-one bookkeeping, whose maximum becomes -51/100).

5. Comparison with the first-wave lane files

Not performed. euler_step.py and probe_table.py (and their tests and notes) were untracked on hunt/below-seven-eighths at every check (HEAD 9c1f21c52 in wave 1, baa41597e at the repair), and the task forbids treating uncommitted files as final. This lane has not opened them. The kill statements compared in section 3 are the task's descriptions of them, not the lane's text; the one report about PROBE.md in section 3 is the adversarial review's, attributed as such. disagreements.json lists the one disagreement with those descriptions and records the comparison as outstanding.

6. Notes for the owner and the verifier

• N1. Because the closed form is free of Gauss sums, a route that checks only (local-P) = closed form cannot see a sign error in G_1 outside rows 3 and 4. The inputs need their own check (here Lemma 4.2 and the raw coefficient).
• N2. The worktree's branch was not created from origin/main fb647bfc7, as the task's override states (the repair task's override repeats it); its base is d5d1dfc81. No file outside referee/ was touched.
• N3. A process listing taken to find this lane's own background job displayed another session's command lines, which named tests and planted-fault strings of the uncommitted euler_step.py and test_euler_step.py. This lane did not open those files and shares no module or function with them; the disclosure is made because the route is meant to be independent. The adversarial review compared the two files and reports six textually identical lines: the pinned SHA-256 constant, the from __future__ import, three lines of generic Z[omega] arithmetic, and the omega_p exponent line of (local-P), variable name om_exp included (as reported; this lane still has not opened euler_step.py). That exponent line transcribes the exponent tk - e_0 l - l(l-1)/2 of paper line 4002.
• N4. The adversarial review reports reproducing the prime-level checks with its own code, which uses neither PARI nor python-flint, at further primes, and the global coefficient one level deeper than here, row 5 included. That code is outside the repository and this lane has not inspected it, so nothing in this report rests on it and no grade here is raised by it.
• Rerun at another prime. local_identity_pari.py --norm 31 runs the prime-level battery (Lemma 4.2, the lift, the C_p table, the closed form for all rows and phases, the raw route, H_p) and both planted faults; the slow test tier includes norm 31. Default tier: .venv/bin/python -m pytest -q -n0 hunts/below_seven_eighths/referee -m "not slow" (about three minutes on the campaign box, measured once, not pinned).

7. Corrections after the adversarial review (2026-10-21)

An independent adversary reviewed wave 1 and returned "holds with fixes": one major issue and six minor ones. Every experiment of wave 1 is kept; what changed is listed here.

1. Rival verdict (major), withdrawn and replaced. Wave 1 called the single-prime identity a shared lemma for DH and the separation into prime factors the distinguishing step. Its own data contradicted that: DH fails the single-prime closed form at every tested prime where chi(N p) is not real, and fails separation only at pairs of such primes. Section 2 now says that both steps fail, on the same primes, and pins the residue structure (DH) and the split-prime structure (Epstein) by two new tests.
2. D1, three additions (minor). The review's report that the probe lane's S1 uses the same chi^4 twist; the computation that S1 does not separate in the paper's row normalization (pair phase conj(R(p, q)), new test); and the scope of this lane's S1 construction. Section 3 and disagreements.json.
3. Grade wording (minor). The bold S1 factorization sentence is now scoped to norms 7 and 13 and the stated box, the size exponents are labeled as floats measured at those primes, and the argument for the sign at every squarefree c is written out.
4. Citation (minor). gamma_2 gamma_4 = 1 is on paper line 4124, not 4122 (section 1.1 and the check's label in lemma_4_2_checks), pinned by test_gamma_2_gamma_4_is_cited_at_line_4124; lines 3416 and 4002 are now pinned too.
5. Versions (minor). 2.17.2 is the PARI library; the cypari2 package is 2.2.4 (section 0).
6. Push status (minor). The header paragraph now records that the branch was pushed after wave 1, not by this lane.
7. Wording (minor). The rival placement is stated as this lane's extension of the rule on line 6725, and the raw and global routes as distinct derivations sharing one module, not independent implementations (sections 1.4 and 2).

8. Open issues

• The comparison with euler_step.py and probe_table.py is outstanding: neither is committed on the branch.
• D1 is unresolved and goes to the owner. After the review it is narrowed to normalization (whether the probe keeps the conj(alpha) and conj(G) insertions) and decomposition (one Euler product or a finite combination over T-hat).
• The T-hat reading of line 3416 in section 3 is inferred; no finite combination over T-hat is computed here. The bare-row S1 factor shape is checked at norms 7 and 13 in a box, and no convergence of its correction product is shown.
• Lemma 7.1 is checked only at the stated primes and inside finite boxes. The global coefficient is compared with the local summand only for t <= 4 at norm 7 and t <= 1 at norms 13 and 25, and row 5, which needs t = 6, is not computed globally here.
• The analytic part is exponent bookkeeping given the closed form, not a proof of normal convergence; above the directly summed depths the inner Gauss sums use the paper's lift.
• The rivals are placed by this lane's extension of the rule on line 6725; the placed series are not the rivals themselves, and no zero of either is computed. Beyond the computed pairs, the step-by-step verdict of section 2 rests on the coefficient structure together with the reduction of separation to f(pq) = f(p) f(q), which is checked at the computed pairs only.