probe_tower: probes of growing degree in the 7/8 assembly

Hunt #150 (rh_moonshot), mechanism 6. 2026-10-21. Status: constructed and pushed to a located obstruction; not challenged yet; unreviewed by any person. Nothing here is a result until the case log in hunts/README.md says how this hunt ended. Files produced on branch hunt/rh-moonshot: tower_template.py (the arithmetic), test_probe_tower.py (the pins), results_probe_tower.json (one run of the report), RUNS.md, this page.

Grades, as in the hunt's MISSION.md: cited (a line of a source, quoted and checked against the pinned file by verify_citations, or a primary source read at the stated place), derived (algebra done here, checked in exact rational arithmetic, with sympy for statements symbolic in k; every derivation here is self-reviewed only), inferred (a reading that no computation here checks), measured (one floating route, mpmath). Nothing here is hardened or kernel-checked. A composite claim takes the grade of its weakest step. The paper is OpenAI, The Quasi-Riemann Hypothesis, 2026-09-30, github.com/openai/math at adc7f124, read by line number from <cache>/openai-math-adc7f124/paper.tex, SHA-256 42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3 (49 cited lines, all found on their stated lines).

0. The answer

1. The tower goes the wrong way. In the paper the comparison pole sits at z = 1/6 because the row characters are sextic: Poisson frequencies are u a^6 (line 3758), the sum over a is zeta_F(6z) (line 4045), and its pole at 1/6 is the principal residue (line 4071). Replacing the order 6 by k where sixth powers are counted, and holding every other estimate at the paper's form, the abscissa is sigma_0(k) = 1 - 1/k + (M - 2 ell)/(2k) (derived; the k-reading is inferred). For every admissible geometry it is strictly increasing in k, with sigma_0(k + 1) - sigma_0(k) = (2 - (M - 2 ell))/(2k(k + 1)) > 0, and it tends to 1. A tower of probes of growing degree drives the abscissa toward 1, not toward 1/2.
2. No order reaches 1/2, and the best order is the smallest. The dual-length condition alone gives sigma_0(k) >= 1 - 2/k (above 1/2 for k >= 5); with the floor rows counted in full, which is what perfect row counts give, sigma_0(k) >= 1 - 4/(5k) in the limit delta_0 -> 0 and 1 - 25/(32k) at the paper's floor bin (derived, exact LPs with dual multipliers, symbolic in k). The infimum over all orders k >= 2 is 3/5, at k = 2, for which no system exists.
3. The formal order k = 1 is RH itself. At k = 1 (no twist: every frequency is principal) the balanced geometry gives sigma_0 = 1/2 exactly, and the direct bound it asks for is square-root cancellation in the Moebius sum, Littlewood's RH-equivalent condition. The template interpolates between that restatement of RH (k = 1) and the trivial bound (k -> infinity): the twist buys a provable direct bound and pays for it with the density 1/k of k-th powers among the frequencies.
4. The large-k estimates. None needs to fail for the tower to fail. Of the two named candidates: the large sieve of degree k does not degrade (Blomer, Goldmakher and Louvel's Theorem 1.3 has the same M + N + (MN)^(2/3) for every n >= 3, read here from the arXiv source); the theta coefficients at squarefree indices are undetermined over number fields for n >= 4 and, over F_13[T] at n = 4, exist but are not multiplicative (hunt #148, uncommitted). That second failure closes the small-k doors (k = 4, 5), the only direction that helps, not the tower.
5. The step that reads the Euler product is the same at every k: the principal row is linear in the target's coefficients and carries their Moebius-weighted squarefree sum (line 3967 supplies mu(c)), which the criterion needs as H/L with sup |H - 1| <= 1/2 (line 413). DH fails it at its zero 0.808517 + 85.699348i (|H - 1| = 1.0000, measured) and, for every finite deletion set, on every half-plane Re s > sigma_0 with sigma_0 < 1 (Saias and Weingartner, cited); Epstein (1,1,6) fails it at 0.953260 + 16.290216i (measured); W_a passes it at its own zero 3/4 + 14.1347i (|H_W - 1| = 0.236 with nothing deleted, measured) and is held off by unitarity in the direct bound, which moves its template target up by exactly 1/4.

Disposition: restricted class obstructed (section 9), with the mission's tower premise refuted inside that class.

1. Target and its relation to RH

Target. A sequence of probes P_k, k = k_1 < k_2 < ..., each verifying the hypotheses of the paper's continuation criterion (Proposition 2.1, lines 400-432) unconditionally at a boundary sigma_0(k), for every primitive finite-order Hecke character of its field, with sigma_0(k) -> 1/2. The Dirichlet transfer (the quadratic transfer proposition, line 6705) then gives zeta(s) != 0 for Re s > sigma_0(k) for every k, so zeta has no zero with Re s > 1/2, and the functional equation places every nontrivial zero on Re s = 1/2: the target implies RH (indeed GRH for Dirichlet L-functions). It is not an RH equivalent with a new estimate attached; the all-k step is the whole content, as the mission says. The question this mechanism had to answer is whether any natural tower makes the all-k step plausible.

2. The construction: the order-k template

2.1 Where the 6 enters (cited) and the reading of it as k (inferred)

That each of these sixes is the order k of the row characters is a reading (inferred), the same one hunt #136's probe lane makes (PROBE.md there, uncommitted, read). Everything else is held at the paper's form: the reflected energy, the dual length M + ell <= 1 (line 8246; that M + ell <= 1 suffices for a general geometry is hunt #134's reading, inferred), the floor-row exponent (lines 15729-15734, 15843-15847), the contour Re z = 17/50 (line 15742), the Euler regions (lines 4054-4056).

2.2 The formulas (derived)

With l_x = (M - b)/2, l_y = (M + b)/2, h = 1 - l_x + ell (line 5586), and z* = 1/k:

• Signal. The weight X^(1/2 - z) Z^(x + z - 1) (line 3825) at z = z*, with X = Z^l_x, times the slot product Z^(ell z*), is Z^(x + c) with c = l_x(1/2 - z*) + (z* - 1) + ell z* = l_x/2 - 1 + h z*. At k = 6 this is line 5588, C(s) = s + l_x/2 - 1 + h/6. Two routes (weight and line 5588) agree identically.
• Direct bound. l_x/2 + b z*/2 (line 8570 at k = 6: l_x/2 + b/12).
• Target. C(sigma_0) = low exponent gives sigma_0 = 1 - h z* + b z*/2 = 1 - 1/k + (M - 2 ell)/(2k). The offset b cancels because the Gram term counts k-th powers exactly as the pole does.
• Floor rows. Line 15729 with 7/8 replaced by sigma_0(k) and 1/6 by z*, at d = h, R = 1, q = delta_0/2, a = (1 + delta_0)/2: E(h) = C_0(k) + (1 + ell - M/2) delta_0, C_0(k) = 1/2 + ell/2 - 3M/4 + b(1/4 - 1/(2k)). The contour 17/50 cancels at d = h. At k = 6 the transcription of line 15729 equals line 15732 identically in (delta, q, R, d).

2.3 Reproduction at k = 6 (cited values rebuilt exactly)

Part I (ell, M, b) = (0, 1, 0): c = -2/3, low exponent 1/4, sigma_0 = 11/12 (lines 190, 509-510, 3717). Part II (1/6, 5/6, 1/8): (l_x, l_y, h) = (17/48, 23/48, 13/16), c = -11/16, low exponent 3/16, sigma_0 = 7/8 (lines 191, 514-515, 6860-6864, 8641-8642), C_0 = -1/48 and the floor value -7/1200 (lines 15734, 15847). For random geometries at k = 6 the target equals hunt #134's 5/6 + (M - 2 ell)/12.

What the reproduction does not pin. A pole read as k/36 also equals 1/6 at k = 6 and reproduces every constant above, yet makes sigma_0 decrease in k (test_planted_fault_a_wrong_k_reading_passes_k6_but_reverses_the_tower). The direction of the tower therefore rests on the structural reading "pole = 1/(character order)", which comes from lines 3758 and 4045 (the sum over the k-th power part a^k of the frequency is zeta_F(kz)), not from any reproduced number. Grade of the direction: inferred reading, derived algebra.

3. The tower question

3.1 Direction (derived)

sigma_0(k) = 1 - (2 - m)/(2k) with m = M - 2 ell. Every admissible geometry has m <= M <= 1, so 2 - m >= 1 > 0 and

sigma_0(k + 1) - sigma_0(k) = (2 - m) / (2 k (k + 1)) > 0, 1 - sigma_0(k) <= 2/k.

At the paper's geometry the increment from k = 6 to 7 is 1/56, and sigma_0(k) = 1 - 3/(4k): 7/8, 25/28, 29/32, 15/16, 31/32 at k = 6, 7, 8, 12, 24. The mechanism behind it: the principal frequencies of an order-k twist are the k-th powers, whose number up to height T is about T^(1/k); the signal is proportional to that density, so it weakens as k grows.

3.2 Walls per order (derived; exact LPs over (ell, M, b), dual multipliers checked

symbolically in k and delta_0)

Constraints: ell, b >= 0, l_x >= 0, M + ell <= 1, and as stated per column. Every value below is computed by tower_template.wall and equals its closed form.

• The dual-length wall is attained at the degenerate vertex ell = 1, M = b = 0 (no averaging at all); it needs no row count and no reading of the direct bound's bookkeeping, which is why it is the robust statement: whatever the row counts and the loss accounting, an order-k probe of this template cannot pass 1 - 2/k.
• The floor wall: for every delta_0 >= 0 the minimum of M - 2 ell is m* = (2 + 12 delta_0)/(5 + 6 delta_0) at b = 0, ell = (1 - 2 delta_0)/(5 + 6 delta_0), M + ell = 1, so the wall is 1 - 4/((5 + 6 delta_0) k). It is proved for every geometry by the identity (M - 2 ell) - m* = y1 (-floor) + y2 (1 - M - ell) + y3 b with y1 = 12/(5 + 6 delta_0), y2 = 4/(5 + 6 delta_0), y3 = y1 (1/4 - 1/(2k)), all >= 0 exactly when k >= 2. In the limit the corner is (ell, M, b) = (1/5, 4/5, 0) for every k, where the unrescaled-summand condition M >= (1 + 3 ell)/2 is tight as well (so that reading of the direct bound gives the same wall).
• These walls agree with hunt #136's probe lane (its section 3, recomputed here by separate code from the paper's lines) and, at k = 6, with hunt #134 (11/12, 7/8, 13/15). Below 7/8 at k = 6 the row counts matter (hunt #134's endpoint optimum 0.874958); for the tower only lower bounds matter, and the walls are lower bounds for any row counts.
• Best order. 1 - 4/(5k) increases with k, so the infimum over k >= 2 is 3/5 at k = 2. Every wall is above 1/2 for every k >= 2 (the dual-length wall is exactly 1/2 at k = 4, at its degenerate vertex).

3.3 The order k = 1 (derived arithmetic; the reading is inferred)

At k = 1, z* = 1 and the balanced geometry gives c = -1/4, low exponent 1/4, sigma_0 = 1/2. There is no twist, so every frequency is principal and the probe is the signal itself: Z^(-1/4) times a smoothed sum of mu(n) eta(n) of length Z. The direct bound Z^(1/4) it asks for is square-root cancellation in that Moebius sum, which for eta = 1 is Littlewood's condition M(x) << x^(1/2 + epsilon), equivalent to RH (Littlewood 1912; Titchmarsh, The Theory of the Riemann Zeta-Function, section 14.25; classical, quoted, not rechecked here). The floor-wall multipliers fail at k = 1 (y3 < 0), so k = 1 is outside the walls' proof, as it must be. The reading is that the template is a one-parameter family whose k = 1 member is a restatement of RH and whose k = 6 member is the paper; moving k toward 1 is moving toward the restatement, which supplies no estimate.

3.4 The price of 1/2 (derived, exact LPs)

Three generic constraints meet at the floor corner (ell, M) = (1/5, 4/5): the dual length M + ell <= 1 (D), the floor rows at delta_0 -> 0 (F, high side), and the direct bound's loss condition (L, low side), which hunt #134 reads either as "only the unrescaled summand lossless", M >= (1 + 3 ell)/2 (total accounting, tight at the same corner), or as "every summand lossless", M - 2 ell >= 1/2 (per-summand, the paper's own corner, which gives 1 - 3/(4k) and is higher still). How far they would have to be relaxed, by a common eps in powers of Z, for some geometry to reach sigma_0 = 1/2, i.e. M - 2 ell = 2 - k:

Closed forms (each checked against its LP for k = 2..12): max((5k - 8)/16, (k - 4)/2); (5k - 8)/12 for k <= 4; 5k/4 - 2; (5k - 8)/7; k - 3/2. Every column increases with k. The two columns without L are relaxations of the template, not readings of it: with L imposed in either reading, a gain on the floor rows alone moves nothing (the wall stays 1 - 4/(5k) under total accounting and 1 - 3/(4k) per summand), because L is tight at the same corner. Improving F and L together by Z^g gives 1 - (4 + 3g)/(5k) up to g = 2 and the dual-length wall 1 - 2/k after it; a dual length longer by Z^e with F exact gives 1 - (4 + 2e)/(5k) (derived, checked against the LPs). In words: at the paper's order 6, reaching 1/2 needs all three generic constraints relaxed by 22/7 (total accounting) or 9/2 (per summand) in powers of Z, more than the whole range of the geometry, and no gain on the floor rows and the direct bound together takes the template below the dual-length wall 2/3. At k = 2, for which no system exists, the joint price is 2/7 or 1/2.

3.5 The first test that could fail

test_tower_hypothesis_fails_sigma0_increases_to_one encodes the mission's hypothesis in its negation: for 200 random rational admissible geometries and k = 1..64 the increment is the closed form above and positive, 1 - sigma_0(k) <= 2/k, sigma_0 > 1/2 from k = 5 on, and sigma_0(10^6) > 0.999997. It would have failed had the pole entered the template in any way that made the signal stronger with k (the planted k/36 reading does exactly that and is caught by the comparison at k = 7). It passed.

4. Which estimate fails at large k

None has to: the signal's density 1/k ends the tower before any estimate is reached. For the record, the candidates the mission named and the others the template holds:

• Theta coefficients at squarefree indices, n >= 4 (L1). Over number fields the Fourier coefficients of the n-fold theta at squarefree indices are not determined for n >= 4 (David, Dunn, Hamieh and Lin, arXiv 2306.11875, as quoted in hunt #136's PROBE.md section 6, uncommitted, read; the obstruction traced there to Deligne's non-uniqueness of Whittaker models and Kazhdan and Patterson). Over F_13[T] at n = 4 they exist and are not multiplicative up to phases: on three linear primes their modulus is 0 for 234 triples and nonzero for 52, so no scalar local identity of Lemma 7.1's type exists (hunt #148 ff_probe, S4.md, uncommitted on branch hunt/ff-probe, exact computation, unreviewed, read here). Grade here: cited from uncommitted sources. For a tower k = 2n -> infinity, L1 is missing at every level from k = 8 on; but it is the small-k doors (k = 4, 5) that it closes, which matters more (section 5).
• Large sieve of degree k. Does not fail. Blomer, Goldmakher and Louvel, L-functions with n-th order twists, arXiv 1112.1650, Theorem 1.3 (label thm:thm3, read in the arXiv TeX source nth-order-twist.tex, SHA-256 bd846eb6f570c40cc2b571595260cdd250c633f337c88c33e261e95f7979a6cb, fetched into scratch): for every n >= 3 and a number field containing the n-th roots of unity, sum* over N a <= M of |sum* over N b <= N lambda_b chi_a(b)|^2 << (MN)^epsilon (M + N + (MN)^(2/3)) sum* |lambda_b|^2, over squarefree ideals prime to S. The exponents do not depend on n (the constants may). Cited.
• Amplification slope and plain capacity. The template's alpha = 1 - 1/k and the capacity k kappa (hunt #136's readings, inferred) both get worse as k grows: they only raise the endpoint above the walls.
• Euler region two. The principal residue z = 1/k lies in the region z_r >= 33/200 (line 4056) exactly for k <= 6 (derived): from k = 7 on the paper's correction product is not shown to converge at the residue point, so the scalar Euler identity's region would have to be re-derived for an order-k identity that does not exist (L1).
• Field degree. An order-k twist needs the k-th roots of unity in the base field, of degree phi(k) over Q for k outside {1, 2, 3, 4, 6}; the template does not price the field degree (inferred). Liu's extension to fields containing mu_12 keeps zeta_F^S(6z) (as hunt #136's probe lane records Liu, arXiv 2610.26835): a bigger field has not moved the pole.

5. The small orders, the only useful direction

So the direction that lowers the abscissa has no working system below k = 6, and even a working k = 2 system with perfect row counts would stop at 3/5.

6. The function-field analogue

Hunt #148 (ff_probe, branch hunt/ff-probe, files uncommitted at the time of reading) built the F_7[T] transplant of the cubic probe and checked Lemma 7.1 exactly there, and found the biquadratic case unusable (section 4). What that settles for this mechanism:

• The walls are the method's, not the zeros'. In F_q[T] RH is a theorem (Weil) for every L-function concerned, while the template's algebra (Mellin weight, Poisson frequencies u a^k, pole of zeta(kz) at 1/k) is the same; so the template would price sigma_0(k) -> 1 there too, against the truth 1/2 at every k. The tower's failure is a property of the probe's signal, independent of whether RH holds (inferred from the shared algebra; hunt #148's ledger records the k = 6 excess, 7/8 priced against the truth 1/2).
• The small-k door at n = 4 is closed in F_13[T] (squarefree coefficients not multiplicative). That is evidence, not proof, about Q(i).
• What it cannot settle: nothing about number-field L1 for n >= 4 beyond evidence, and nothing new about large k, which the algebra already settles negatively.

7. The distinguishing step and the rival gate

7.1 The step

The continuation criterion (Proposition 2.1) is a shared lemma: hunt #120 showed it goes through for any function whose reciprocal is bounded on Re s >= 2. The Euler product is consumed where its hypotheses are manufactured. In the probe, the theta coefficient at a squarefree index is gamma_2(c) = mu(c) alpha(c) G(c)/gamma_1(c) (line 3967), so the target enters the principal row through mu(c) eta(c) and, by the scalar Euler identity (lines 4043-4047), as 1/L^S(x, eta) times a correction H that the criterion needs within 1/2 of 1 (line 413). The target enters the row linearly through eta(A) at the completed index (hunt #136 referee, section 2), at every order k. So the step, in the form that can be tested on any coefficient sequence f, is:

(E_k) The principal row's dependence on the target is R_f(s) = sum over squarefree n of mu(n) f(n) n^(-s), and the criterion needs R_f = H_f / L_f with sup over Re s > sigma_0 of |H_f - 1| <= 1/2, for some finite deletion set.

It does not depend on k. For a linear combination f = c1 eta1 + c2 eta2 of two Euler products, linearity gives R_f = c1 H1/L1 + c2 H2/L2, whose poles are the constituents' zeros, never the combination's. Below, the constituents' corrections H_i are set to 1 (over Q, for characters, exactly) or computed (degree 2). For unitary characters the paper's full correction at the principal point converges absolutely from Re x > 2/3 by the closed form's monomials (hunt #148 derived this for F_7[T]; the number-field monomials are the same, inferred; principal_correction_abscissa pins the arithmetic), which covers the DH points used here. For the degree-2 constituents of Epstein (1,1,6) and for W_a the paper writes no identity; section 7.5 says how unitarity enters there.

7.2 Control: zeta and Dirichlet L-functions

f completely multiplicative: R_f = 1/L_f exactly, H_f = 1. Measured control: the squarefree Moebius sum to n = 20000 at s = 2 equals 1/zeta(2) to 6.9e-8, inside its tail bound 5e-5.

7.3 Davenport-Heilbronn (fails, at every k)

f = c1 chi + c2 chibar with chi mod 5, chi(2) = i, c1 = (1 - i kappa)/2, from zeta.epstein.

• Linear structure, second route (measured). At s = 2 the squarefree Moebius sum of the DH coefficients to n = 20000 is 0.9755806880; c1/L(2, chi) + c2/L(2, chibar) is 0.9755806815 (gap 6.5e-9, inside the tail bound 5e-5); 1/DH(2) is 0.9999316669 (gap 0.0244). The probe sees the combination of reciprocals, not the reciprocal of DH.
• At the zero rho = 0.80851718 + 85.69934849i (measured, dps 30). |DH(rho)| = 6.5e-40 by dh_f and 6.5e-40 as c1 L(chi) + c2 L(chibar); |L(rho, chi)| = |L(rho, chibar)| = 0.70220 (equal, as they must be at a zero of the combination); the probe-visible reciprocal is finite, |R| = 0.77832; so H = DH R has |H| = 5.1e-40 and |H - 1| = 1.0000. The closed form H - 1 = c1 c2 (L1 - L2)^2/(L1 L2) gives the same. The contraction fails on every half-plane Re s > sigma_0 with sigma_0 < 0.8085.
• On the 7/8 line, nothing deleted (measured). On Re s = 7/8, t in [1, 300], step 1/4 (1197 points): max |H - 1| = 3.68 (at t = 256) and |H - 1| > 1/2 at 572 points. The paper's own normalization hypothesis cannot be met for DH even at 7/8.
• With primes deleted (measured, then cited). With S = {2, 3} the same scan gives max |H^S - 1| = 0.276 (t = 178) and no point above 1/2; with S = {p <= 100}, max |H^S - 1| = 0.018 (t = 100.5) and again no point above 1/2. Deleting primes makes the two constituents' ratio close to 1 at every height a computation reaches, so no finite scan shows the failure once small primes are deleted. The failure is an all-height fact, at the placement over Q used here: the coefficients of DH^S (DH with n restricted to n prime to S) are periodic, and DH^S is not a Dirichlet polynomial times one L-function (its coefficients at large primes take the values 1, kappa, -kappa, -1, of two different moduli, while a Dirichlet polynomial times L(s, psi) has coefficients b_1 psi(p) of one modulus at every large prime; derived), so by Saias and Weingartner (Acta Arith. 140 (2009) 335-344, arXiv 0807.0783, abstract read here; proof not checked) there is eta(S) > 0 such that every strip sigma_1 < Re s < sigma_2 with 1/2 < sigma_1 < sigma_2 < 1 + eta(S) holds >> T zeros of DH^S up to height T. At those with Re s > 1 the constituents have no zeros, so R^S is finite and H^S = 0. Hence for every finite S and every sigma_0 < 1 + eta(S) the contraction fails (cited theorem plus derived steps). The placement over F = Q(sqrt(-3)) through the norm, which hunt #136's referee used, processes c1 L(chi) L(chi chi_-3) + c2 L(chibar) L(chibar chi_-3) instead; its coefficients are not periodic, and nothing is cited here for its zeros.

7.4 Epstein (1,1,6), discriminant -23 (fails)

Z_Q/2 = (zeta_K + 2 L_psi)/3 with K = Q(sqrt(-23)) and psi the cubic class character, so R_f = (H_1/zeta_K + 2 H_psi/L_psi)/3 with H_1, H_psi the degree-2 corrections (absolutely convergent on Re s > 1/2), computed over p <= 10^4 from exact representation counts. At rho = 0.9532604748 + 16.2902157204i (measured, dps 25): |Z_116(rho)| = 3.1e-29; |zeta_K(rho)| = 1.45600 by the lattice route (Z_116 + 2 Z_213)/2, agreeing with zeta(rho) L(rho, chi_-23) by the Hurwitz route to 5.2e-27; |L_psi(rho)| = 0.72800; |H_1| = 1.4163, |H_psi| = 1.0963; the probe-visible reciprocal is finite, |R| = 0.70349; H_E = (Z_116/2) R has |H_E| = 1.1e-29 and |H_E - 1| = 1.0000. The contraction fails for every sigma_0 < 0.9533, past 7/8: the step refuses this rival exactly where hunt #120 located the refusal for discriminant -15.

7.5 W_a = zeta(s + 1/4) zeta(s - 1/4) (passes the step; held off by unitarity)

The coefficients a(n) = n^(-1/4) sigma_(1/2)(n) are multiplicative, so R_W = H_W / W_a with H_W = prod over p of (1 - p^(-2s)/((1 - p^(1/4 - s))(1 - p^(-1/4 - s)))), absolutely convergent on Re s > 1/2. Second route: at s = 3 the squarefree Moebius sum to n = 20000 equals H_W/W_a to 1.0e-10 (measured). At the zero s_1 = 3/4 + 14.1347251417i, |W_a(s_1)| = 1.7e-21, and over p <= 10^5 (omitted factors within 0.0064 of 1): |H_W - 1| = 0.236 with nothing deleted, 0.0017 with p <= 10 deleted, 0.0024 with p <= 100 deleted (measured). The pass is uniform, not just at the zero: for Re s >= sigma > 1/2 each omitted factor is 1 + e_p with |e_p| <= p^(-2 sigma)/((1 - p^(1/4 - sigma))(1 - p^(-1/4 - sigma))), so deleting the primes up to P makes sup |H_W - 1| at most exp(c(P) sum_{n > P} n^(-2 sigma)) - 1, which is below 1/2 for P large enough at every sigma > 1/2 (for instance below 0.26 on Re s >= 3/4 with P = 100, summing crudely over all integers; derived). W_a passes (E_k) on every half-plane Re s > 1/2 + delta: its probe-visible reciprocal has the pole at its off-line zero that the criterion needs (this tests the bare Moebius identity; the paper writes its full identity only for unitary GL(1) targets, and the cube-completion terms of a degree-2 target are not computed here). What keeps the template away from W_a is unitarity, which the paper uses twice: the direct bound uses |eta| = 1 through the mean square of the theta row, and the local identity's parameters rho, a_p, b_p have absolute value one (line 3870). For the GL(1) factor zeta(s - 1/4), whose character n^(1/4) is not unitary, both shift by exactly 1/4, as a change of variables (derived): the direct bound and the target rise by 1/4, and the region where the principal correction converges absolutely moves from Re x > 2/3 to Re x > 11/12 (its term a_p^2 Q^(4 - 6x - 6z) at z = 1/6 grows by Q^(6/4)). For the degree-2 placement the same rise of 1/4 in the direct bound is inferred (coefficients up to d(n) n^(1/4)). W_a's zeros sit at 1/2 + 1/4, so they are within reach only if sigma_0 < 1/2, which no wall allows (the best is 3/5 + 1/4 = 17/20 > 3/4). W_a is a rival that lacks a hypothesis the argument uses, not a counterexample; what it shows is that the tower needs unitarity at every k, beside the Euler product.

7.6 What the gate says

Read two-sidedly (docs/08 section 4.3): DH and Epstein (1,1,6) violate the conclusion and fail the step that uses the Euler product, at every k, so they are not counterexamples and the step is genuinely distinguishing. W_a satisfies the step and violates the conclusion's analogue, but lacks unitarity, which the direct bound and the local identity use at every k. No rival passes every hypothesis. The gate does not bear on the tower's failure, which is internal to the template.

8. Known obstructions engaged, and what differs from refuted forms

• Hunt #134 (13/15 floor of the k = 6 assembly, 0.874958 endpoint): reproduced at k = 6 as the floor wall; the tower generalizes its floor to 1 - 4/(5k).
• Hunt #136's probe lane (uncommitted, read) priced the order-k template first: its formula, walls and system table agree with this page, which recomputes them from the paper's lines by separate code. Added here: the mission's tower question answered in the negative with the row-count-free bound 1 - 2/k; the price of 1/2; the order k = 1 as Littlewood's restatement; the cubic-twist factorization's non-firing; the large sieve read at the source; and the step tested at the rivals' zeros in its analytic form.
• Hunt #120 placed the Euler-product step at the Moebius identity and Lemma 7.1 and refused the discriminant -15 Epstein zeta there; hunt #136's referee refused DH and the -15 form coefficient by coefficient; hunt #148 refused a DH analogue over F_7[T] at X^2. This page evaluates the same step analytically at each rival's actual zero, adds Epstein (1,1,6) and W_a, and shows that deletion hides DH's failure from any scan.
• docs/09 section 5.1's evasions. Direct estimation: the tower's endpoint k = 1 is exactly a direct estimate of the Moebius sum. Zero-importing: no zero enters any formula. Finite approximants: every statement in sections 3 and 4 is uniform in k by closed form, not a finite scan; the rival scans are labelled measured and finite.
• Refuted forms in the mission's list. EPP/Herglotz (composing archimedean and prime-side positivity) was refuted by W_a; the tower is not a positivity mechanism, and its direct bound uses unitarity, which W_a lacks. Jensen wedges and the heat-flow clocks succeed for DH; the tower's step refuses DH. Neither refuted form is the tower, and the tower fails for its own reason, the signal density 1/k.

9. Disposition

Restricted class obstructed. The class: probes in which the target is twisted by a family of nontrivial power-residue characters of order k >= 2 (so the principal frequencies are the k-th powers and the comparison pole is z* = 1/k <= 1/2), whose signal, direct bound, dual length M + ell <= 1 and floor rows keep the paper's forms. The proof: the LP of section 3.2 with its dual multipliers (derived, symbolic in k, self-reviewed). That its hypotheses apply to such probes is the inferred reading of section 2.1. Within the class the tower premise is refuted (sigma_0 increases to 1) and 1/2 is unreachable (sigma_0 >= 3/5). Outside the class nothing is claimed: a probe whose principal frequency set had density above 1/2, or whose dual length or floor rows were handled differently, is not covered.

10. The furthest true statement reached

In the order-k template, for every k >= 2 and every admissible geometry, sigma_0(k) = 1 - 1/k + (M - 2 ell)/(2k) >= max(1 - 2/k, 1 - 4/((5 + 6 delta_0) k)) >= 3/5, with 3/5 attained only at k = 2 in the limit delta_0 -> 0, at (ell, M, b) = (1/5, 4/5, 0); and no common gain on the floor rows and the direct bound's loss condition takes it below the dual-length wall 1 - 2/k; sigma_0(k) is strictly increasing in k and tends to 1; the formal order k = 1 gives 1/2 exactly, where the direct bound is Littlewood's RH-equivalent Moebius bound.

Grade: derived (exact, symbolic in k, self-reviewed) on cited formulas, with the k-reading and the sufficiency of M + ell <= 1 inferred; composite grade inferred. Beside it, at the rival gate: the step (E_k) is k-independent; DH and Epstein (1,1,6) fail it at their zeros (measured), DH for every finite deletion set by Saias and Weingartner (cited, with derived steps); W_a passes it (measured) and is held off by unitarity (inferred), a shift of exactly 1/4 (derived).

11. Next step

The tower is closed inside its class, and its best member (k = 2) stops at 3/5 even with perfect row counts, so as a route to RH this mechanism has no next construction step that the template contains; continuing it is the owner's call. Section 3.4 prices the generic doors: at the realized k = 6, 1/2 needs the dual length, the floor rows and the direct bound's loss condition all relaxed by at least 22/7 in powers of Z, and no improvement of the floor rows and the direct bound together gets below 2/3 without a longer reflected dual. So the next step is not an RH step. The one decidable step left is a half-plane step, and it belongs with hunt #136, which owns the half-plane: since the floor rows and the loss condition are tight at the same corner, any attempt below 13/15 must improve both at once, for instance a mean value over the row family in place of the floor rows' pointwise treatment (R = 1, each row at its full size; Blomer, Goldmakher and Louvel's Theorem 1.3 applied to the floor rows' Moebius polynomials is the obvious candidate) together with a row-norm lemma whose second-branch loss (1 + 3 ell' - 2M')/4 (line 8306) is smaller. The exact LP of section 3.2 then prices the pair, 1 - (4 + 3g)/(5k) for a common gain Z^g. Recorded here as the hand-off; the mechanism as obstructed for RH.

12. Tests, commands, files

.venv/bin/python hunts/rh_moonshot/probe_tower/tower_template.py --paper --write
.venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/probe_tower/test_probe_tower.py
.venv/bin/python -m pytest -q -n0 -m "not slow" hunts/rh_moonshot/probe_tower/test_probe_tower.py

test_probe_tower.py: 23 fast tests (citations, the k = 6 reproduction, both signal routes for k = 1..40, b-freeness, the line 15729/15732 transcription, the floor closed form, the symbolic identities, the tower's direction, the k = 1 endpoint, the walls and their dual multipliers, the best order, the price of 1/2 in five modes, the relaxed walls of the doors, region two, the S1 remark, the W_a shift and the 2/3 and 11/12 correction abscissae, four planted faults, the committed report and this page's quoted numbers against the code, the lexical rules) and 6 slow ones (DH at its zero, the DH Moebius-sum route, the DH 7/8-line scan near t = 85, Epstein (1,1,6) at its zero, W_a at its zero, the W_a Moebius-sum route); 29 in all, all passing (RUNS.md). Planted faults: a pole read as k/36 (passes k = 6, reverses the tower), the floor rows dropped (the k = 2 wall falls from 3/5 to 0), a Gram term held at 1/12 for every k (sigma_0 starts depending on b), a perturbed dual multiplier (the identity breaks). RUNS.md has the run manifests.

13. Sources, and what was searched

• The paper (pinned above). Hunt #134 RESULTS.md (branch origin/claude/hunt-134-seven-eighths-pin, committed). Hunt #136 REFEREE_W1.md (committed on hunt/below-seven-eighths, b6c4972a8) and PROBE.md (uncommitted in its worktree, read, quoted as uncommitted). Hunt #148 S4.md, RIVAL.md, LOCAL_IDENTITY.md (uncommitted in its worktree, read). Hunt #120 RESULTS.md.
• Blomer, Goldmakher, Louvel, arXiv 1112.1650, Theorem 1.3, read in its TeX source (SHA-256 above). Saias and Weingartner, arXiv 0807.0783, abstract read. Littlewood's criterion, quoted as classical.
• Novelty: not searched. No arXiv, zbMATH or OEIS search was made for the order-k pricing or the tower statement. Original to this laboratory as a reading; overlapping hunt #136's probe lane as stated in section 8.

The doors

The page measures a ceiling of a construction family, so it owes this section. Candidate status: the walls of section 3.2 are lower bounds for the template's abscissa, proved by LP duality; parameter range k >= 2, ell, b >= 0, l_x >= 0, M + ell <= 1, delta_0 >= 0; information class below; proof status derived on an inferred reading, unreviewed.

Active constraints at the optimum

Ranked by what relaxing them buys toward 1/2 (section 3.4):

1. The comparison pole z* = 1/k (the density of k-th powers among the frequencies). It scales the whole distance to 1: 1 - sigma_0 = (2 - m)/(2k). Its shadow price at the paper's geometry is d sigma_0/d z* = -(h - b/2) = -3/4. Lowering k is the largest lever; raising it, the tower, is the wrong sign.
2. The floor rows and the direct bound's loss condition, together. The floor rows C_0(k) + (1 + ell - M/2) delta_0 <= 0 (multiplier y1 = 12/5 in the D-F system) and the unrescaled-summand condition M >= (1 + 3 ell)/2 are tight at the same corner, so neither moves the wall alone; a common gain of Z^g moves it to 1 - (4 + 3g)/(5k), down to the dual-length wall at g = 2. Price of 1/2 with the dual length: 2/7 at k = 2, 22/7 at k = 6 (total accounting); 1/2 and 9/2 per summand.
3. The dual length M + ell <= 1 (multiplier y2 = 4/5 in the D-F system; a longer dual by Z^e moves the wall to 1 - (4 + 2e)/(5k)). Alone it bounds sigma_0 by 1 - 2/k, above 1/2 for k >= 5.

The frozen-constant inventory

The information class

• Inside the data the template reads: the geometry, the floor bin and the order are exhausted by the walls; no choice inside the class reaches 1/2, and the increasing tower moves away from it.
• Reading more: a mean-value treatment of the floor rows across the row family together with a smaller row-norm loss (the next step above, a half-plane step), a reflection with a longer dual length, or a twist family whose principal frequencies have density above 1/2. The last cannot be a power-residue family (density 1/k <= 1/2); at density 1 it is the Moebius sum itself, k = 1, whose direct bound is RH. This door gets no automatic priority over the other mechanisms of the hunt.