Probe lane: what a comparison system other than the sextic one is worth

Hunt #136, lane probe, 2026-10-21, revised after two verifier passes the same day. Status: pricing, not proving; unreviewed by any person. Files produced, not landed: probe_table.py, test_probe_table.py, this page.

Grades. cited: a line of the pinned paper.tex (sha256 42a5ee0f...c6a3), quoted in probe_table.CITATIONS and checked by verify_citations (all 91 on their stated line), or a line of a pinned outside source (OUTSIDE_CITATIONS, all 22 checked against Liu's cached sources and their sha256). derived: algebra done here, checked in exact rational arithmetic (fractions, and sympy for symbolic k) by test_probe_table.py. inferred: a reading no computation here checks. measured: a float optimisation, labelled as such. A composite claim takes the grade of its weakest step. Nothing here concerns real part 1/2, nothing here checks a lemma of the paper, and no improvement of the half-plane is claimed: every number below is the price of an input nobody here has proved.

.venv/bin/python hunts/below_seven_eighths/probe_table.py --paper   # tables + citations
.venv/bin/python -m pytest -q -n0 hunts/below_seven_eighths/test_probe_table.py

Summary

1. The target is a function of one structural integer. Write k for the order of the row characters: the Poisson frequencies are u a^k, the signal's pole is that of zeta_F(kz) at z = 1/k, and the Gram bound's exceptional frequencies are k-th powers. Holding every other estimate of the paper at its stated form, the direct-bound target is sigma_0 = 1 - 1/k + (M - 2 ell)/(2k). The algebra is derived, symbolic in k; that it describes a system whose row characters have order k rests on the k-reading of item 2, so as a statement about systems it is inferred. The offset b cancels exactly because the Gram coefficient 1/(2k) is half the pole: both count k-th powers. At k = 6 this is hunt #134's 5/6 + (M - 2 ell)/12.
2. Seven constants of the 7/8 assembly carry a 6 that this lane reads as k; the sextic large sieve is not one of them. The pole 1/6, the slot weight q_p^(-5/6), the Gram term P_a^(1/6) with its two consequences b/12 and 11b/6, the amplification slope 5/6 and the 6 in the plain capacity 6 kappa (section 1). Their values are cited; that each 6 is the row-character order is inferred. The large-sieve exponent (KD)^(2/3) is the same for every order n >= 3 (Blomer, Goldmakher and Louvel, Theorem 1.3, read here from the arXiv copy; the paper follows their Section 3 recursion, line 4728) and enters the 7/8 only through the 11/12 bootstrap, which does not bind.
3. Walls per k (derived, exact LPs with dual multipliers valid for every k >= 2): balanced probe 1 - 1/(2k), per-summand corner 1 - 3/(4k), and the perfect-row-count floor, which depends on the floor bin. The paper's floor bin is a = 51/100, delta_0 = 1/50 (line 15843), the left edge of the held Euler region x_r >= 51/100 that carries the nonprincipal contours (lines 4054 and 4069; that the region fixes the bin is inferred); there the floor is 1 - 25/(32k). Hunt #134 took the limit delta_0 -> 0 ("as the floor approaches 1/2"), which also needs that region extended; there it is 1 - 4/(5k). At k = 6: 11/12, 7/8, and 167/192 at the paper's bin, 13/15 in the limit. So the 13/15 ceiling needs a floor bin nearer 1/2 as well as perfect row counts.
4. With the paper's row counts carried over, every endpoint optimum is measured and bracketed exactly on both sides (section 4): an exact single-point lower bound, and an exact upper bound from a point that meets every constraint of the template, the endpoint by Bernstein subdivision in exact arithmetic. At k = 6 with the plain capacity tied to the target the template returns 0.8749570194, the optimum Liu states for this comparison system in an unrefereed arXiv preprint (2610.12234v2, 0.874957019420098946, cached source line 131) to 1e-12, together with his binding point xi in (0.38858, 0.38859) and his ell and b. With the capacity held at 4/9 it returns Liu's earlier closed form (1507 - 2 sqrt 921)/1653 (2610.12234v1, cached source line 104) to 1e-12. The template's count R_* = 2/3 at x = 1/2 is also Liu's printed count equation, identically in kappa (v2 source line 298). These check the k = 6 transcription where it binds, the endpoint at x = 1/2 near delta = 0.3886 and the tie. The floor rows, the Gram cap, the losslessness and the slot supply are slack there, and nothing in the k != 6 structure is exercised.
5. Levers at k = 6, compared like for like (variant A at the paper's floor bin, measured). One step of the row order, 6 to 5, is worth 0.0255. Each other lever pushed as far as the template allows: removing the Gram exceptional term outright 0.0110, removing the row-count excess outright (alpha -> 0) 0.0052, the plain capacity at its best (c near 1.144, a local best on a grid of step 0.001, where the slot supply binds) 0.0011. Single steps are smaller: alpha from 5/6 to 3/4 buys 0.0004 and to 1/2 buys 0.0027, c from 4/9 to 1 buys 0.0008. Part of the pole's lead is built into the template: at b = 0 the variant-A endpoint, floor and Gram constraints do not involve k, so 1 - sigma_0 = (1 - (M - 2 ell)/2)/k at a k-free optimum geometry (derived). A step in k rescales the whole distance to 1, which no single row-count input can do.
6. Every k <= 5 puts the corner, both floors and the endpoint below 13/15, 0.8748867 and 0.8748087; k = 8 and 12 are worse than 7/8 at every wall. The balanced probe is below 0.8748867 only for k <= 3 and equals 7/8 at k = 4.
7. No system known to the literature realises k <= 5 (section 6, inferred from cited lines plus a literature check). The paper twists by prod_p chi_p^(j_p), with j_p chosen for each prime (lines 1700 and 1703); in Part II the twist is the row character, whose exponent at p is the valuation j_p of the row u (line 8969). This lane reads a system as one exponent j at every prime, which is an inference, not a case the paper treats. On that reading, inside the cubic-theta family the twist is pinned at the sextic one by the Gauss pairing that cancels the theta's angular factor; the cubic twist (pole 1/3) leaves that factor and lands on Groessencharacter L-functions; the classical theta (k = 4 over Q(i)) has no squarefree coefficients; the biquadratic and quintic thetas have coefficients the literature does not determine. Liu's extension to fields containing mu_12 keeps zeta_F^S(6z): a bigger field has not moved the pole.
8. The tied variant has no fixed point at k = 2 and k = 3, and that is a property of the equality, not of the template (section 4). Lemma plain needs only beta_* <= (1 + kappa)/2 (line 12565) and nothing at kappa = 1, so a relaxed tie has admissible points: 0.6093750 at k = 2 (the floor wall 39/64) and 0.7435200 at k = 3 with kappa inside the lemma's stated range, both at the paper's floor bin (measured, bracketed exactly).

1. Where each constant comes from

Origins: sextic (a 6 this lane reads as k = 6), cubic-theta (set by the cubic theta, its pairing with the sextic twist, or the Euler regions of its local identity), generic (form independent of k), choice. The table is rendered from probe_table.PROVENANCE (the tests compare every row), and every line it names is checked.

Readings behind the table:

• The pole and the slot weight are one constant. The principal residue sits at w = 1, z = 1/6 (line 4071); the weight X^(1/2-z) Z^(x+z-1) (line 3825) there gives X^(1/3) Z^(s-5/6), and each slot prime is summed with q_p^(-5/6) = q_p^(z-1) (line 15563), adding Z^(ell/6) after the normaliser A_T (line 15576). Together: C(s) = s + l_x/2 - 1 + h z_* with h = 1 - l_x + ell, the paper's line 5588 at z_* = 1/6 (derived for every z_*, sympy).
• The Gram exponent is half the pole for the same reason. The exceptional frequencies of the Gram bound are sixth powers times a bounded part (line 8535), counted as K^(1/6) (line 8541); Cauchy-Schwarz halves it to P_a^(1/12) (line 8619). Since P_a = Z^b, the low exponent is l_x/2 + b z_*/2, and b cancels from the target. Without that equality b would have a price of its own (section 5).
• The bootstrap ceiling equals the amplification slope for every k. The Part I analogue gives beta_* <= 1 - 1/(2k), so delta <= kappa <= 1 - 1/k = alpha(k) (derived). Hunt #134 noted that the paper's alpha = 5/6 equals 2(11/12) - 1 "by coincidence"; in the template it is an identity, so the bin ceiling delta <= alpha of line 15107 holds automatically.
• The dual length. With M' = M - 2d and ell' = ell - d the subtraction of line 8246 leaves an extra term M + ell - 1 for general parameters (derived, sympy), so M + ell <= 1 keeps the paper's bound; that nothing else in the dual-length argument depends on the geometry is hunt #134's reading, not checked here.
• The large sieve. Line 4717 states the sextic bound with the term (KD)^(2/3), and the proof follows the recursion of Blomer, Goldmakher and Louvel (line 4728). Their Theorem 1.3 has the same term for n-th order characters, "valid for all n >= 3", over a number field containing the n-th roots of unity (section 6, L5). That is why the row is generic rather than sextic.
• Where the held contour and regions stop fitting (held_inputs_fit, derived from lines 4045, 4054, 4056 and 15849). The pole 1/k sits in every row, so the nonprincipal contour Re z = 17/50 must lie right of it: true exactly for k >= 3. For k = 2 it must move into 1/2 < Re z < 3/4, where the frequency slope R + delta/2 - Re z of every row priced here (R >= 1 - delta) stays positive for delta <= 1/2; E(h) does not involve the contour, so the k = 2 prices survive the move, but the intermediate rows that pin the contour near 17/50 in hunt #134's reading are not re-priced. The principal residue z = 1/k lies in region two (z_r >= 33/200) exactly for k <= 6, so every k >= 7 (k = 8 and 12 in the tables here) needs that region re-derived. Region two also asks x_r >= 7/8, so every target below 7/8 needs it extended; Liu v1 states that extension for the sextic system ("Their Euler correction is shown to be holomorphic below 7/8", cached source lines 244 and 245), and for k != 6 it is part of L3.

2. The template and its formulas (derived)

For an order-k system, with l_x = (M - b)/2, l_y = (M + b)/2, h = 1 - l_x + ell, z_* = 1/k and a = (1 + delta)/2:

• signal C(s) = s + l_x/2 - 1 + h/k; low exponent l_x/2 + b/(2k) + f(d), f(d) = -d;
• target sigma_0 = 1 - h/k + b/(2k) = 1 - 1/k + (M - 2 ell)/(2k);
• high exponent (line 15729 with sigma_0 and z_* free) E(d) = a - sigma_0 + h(17/50 - z_*) - a l_y - ell/2 + q ell + d(R + delta/2 - 17/50);
• floor rows C_0(k) = 1/2 + ell/2 - 3M/4 + b(1/4 - 1/(2k)), so the floor condition is C_0 + (1 + ell - M/2) delta_0 <= 0;
• endpoint -E_* = -C_0(k) + delta(ell + b)/2 - ell x delta - (h/2) K, with K = (alpha - delta)_+ delta P_x / J, J = (alpha - delta) D_x + delta P_x, D_x = 3 - (1 + 2c) x, P_x = (2 - 2cx)(1 - x), and c the plain-capacity coefficient (4/9 in the paper). The endpoint covers delta_0 < delta <= 1 - 1/k, x in [0, 1/2] (line 15949 at the paper's bin).

test_closed_forms_symbolically_for_every_k re-derives all of these in sympy with k, ell, M, b, delta, x, alpha, c and the contour as symbols, and checks that the balancing t of line 15964 makes R_short = L = R_*; test_two_routes_agree_exactly... checks the transcribed and closed forms against each other on random rational points for k = 2 to

1. Two structural facts fall out: the contour cancels at d = h, and the pole enters

E(h) only through the b-coefficient of C_0, because the signal and the row bound move together under a change of pole. The constant-target slope 1 + delta(1/2 - x) is free of k, so M + ell = 1 is optimal for every system, as hunt #134 found for k = 6.

3. Walls per system order (derived)

The floor walls are exact LPs over (ell, M, b) with the dual-length, Gram-cap, d = 0 losslessness (inferred, as in hunt #134) and floor constraints. For every floor bin 0 <= delta_0 <= 1/2 the minimum of M - 2 ell is m = (2 + 12 delta_0)/(5 + 6 delta_0), so the wall is 1 - 1/k + m/(2k), attained at b = 0 and ell = (1/4 - delta_0/2)/(5/4 + 3 delta_0/2), and proved optimal by the dual identity

M - 2 ell - m = y1 (-(C_0 + (1 + ell - M/2) delta_0)) + y2 (1 - M - ell) + y3 b,

with y1 = 12/(5 + 6 delta_0), y2 = 4/(5 + 6 delta_0) and y3 = y1 (1/4 - 1/(2k)), nonnegative exactly when k >= 2 (checked in sympy, symbolic in k and delta_0, and exactly for k = 2 to 39). At the paper's bin m = 7/16 at ell = 3/16, M = 13/16 (multipliers 75/32 and 25/32); in the limit m = 2/5 at ell = 1/5, M = 4/5 (multipliers 12/5 and 4/5). At delta_0 = 1/2 the vertex is ell = 0, M = 1 and the floor wall is the balanced probe 1 - 1/(2k). Above 1/2 the floor rows cannot be met at all: they are smallest at ell = 0, M = 1, b = 0, where they read -1/4 + delta_0/2 <= 0 (derived; the LP is infeasible at delta_0 = 51/100). Under the stricter per-summand reading the floor wall coincides with the corner for both bins.

Read against the mission's thresholds: the corner and both floors are below 13/15 and 0.8748867 for every k <= 5; at k = 6 both floors (167/192 and 13/15) are below 0.8748867; the balanced probe is below 0.8748867 only for k <= 3 and equals 7/8 at k = 4. This hunt's follow-ups lane (FOLLOWUPS.md) reports a newer record of 0.8748087 for Dirichlet L-functions; every k <= 5 corner, floor and endpoint value here is below that too. That page measures the room left for row counts at k = 6 against 13/15 (0.0081420); with the floor bin held at 51/100 the same room is 0.0050170, the distance from 0.8748087 to 167/192 (derived here, a thread for that lane).

4. The endpoint wall per k, with the paper's row counts carried over

Variants of the row counts: A moves only the pole and the Gram terms (alpha = 5/6, c = 4/9 held); B also moves the amplification slope to alpha = 1 - 1/k; C also ties the plain capacity to the target, c = 2/(k kappa_0) with kappa_0 = 2 sigma_0 - 1 (the inferred reading of 6 kappa), the limit of Liu's iteration ("Iteration reaches the limiting equality", v2 source lines 310 and 311). Constraints: M + ell = 1 (optimal), b >= 0, the Gram cap, the d = 0 losslessness ell <= 1/5, the floor rows, the endpoint on its whole range and the slot supply ell/h above the largest requested capacity (7/37 at c = 4/9, line 16116). Measured values are float optimisations (affine in ell at fixed (delta, x, b); concave in b). Every valid row is also bracketed exactly: best_exact_bound gives a single-point relaxation, a lower bound, and exact_upper_bound steps back from the measured point until point_is_feasible passes every constraint in exact arithmetic (the endpoint by Bernstein subdivision of J (-E_*), a polynomial of degree 2 in delta and 3 in x). Each bracket holds the measured value and is narrower than 2e-8 (test_endpoint_optimum_and_exact_bracket, 38 rows: every valid row at both floor bins, the unchanged k >= 5 limit rows included).

• k = 6 reproduces Liu's optimum of this comparison system (an unrefereed arXiv preprint; it is not the current record, which this hunt has at 0.8748867 and 0.8748087 from other inputs). Variant C at k = 6 gives 0.8749570194, matching Liu v2's 0.874957019420098946 to 1e-12, with binding delta in Liu's interval (0.38858, 0.38859) and his ell and b (v2 source lines 302 to 304) to 1e-6 (test_k6_tied_variant_matches_liu...). Variant A gives 0.8749571, Liu v1's closed form (1507 - 2 sqrt 921)/1653 to 1e-12 (his v1 fixes kappa = 3/4, source line 2485, which is the held c = 4/9), with exact bracket 4995360671043583/5709263753593183 <= sigma_0 <= 0.8749570699, above hunt #134's exact bounds 5655673/6464057 and 44117/50422, which single_point_bound without the floor reproduces exactly at hunt #134's witness points delta = 223/576 (no Gram cap) and delta = 7/18 (Gram cap), x = 1/2. Hunt #134 reported a float optimum of 0.874958 in its RESULTS.md, from a b-grid of step 0.01; that figure is quoted, not re-run here. Liu's count equation (v2 source lines 295 to 298: "For a critical class delta = xi and x = 1/2, the count equation R_kappa = 2/3 becomes" a quadratic in xi) is the template's R_* = 2/3 at x = 1/2 with c = 1/(3 kappa), identically in kappa (sympy), and at the measured binding point the template's R_* is 2/3 to 1e-6. The floor bin moves none of the k >= 5 rows.
• For k <= 6 the endpoint is slack at the corner (its ell exceeds 1/6), so under the per-summand reading the optimum is the corner 1 - 3/(4k) exactly; for k = 8 and 12 the endpoint binds below ell = 1/6 and both readings agree. Both hold for both floor bins.
• The excess over the floor is nearly k-free. The endpoint's largest ell stays between 0.16 and 0.20 in every row of the table, so the target is roughly 1 - 1/(2k) - (3/(2k)) ell with ell near 1/6; removing the row-count excess entirely (alpha = 0) returns the floor wall of section 3 to 1e-12 for k = 4, 6, 12 and both bins (a consistency check of the float optimiser against the exact LP).
• The tied variant has no fixed point at k = 2 and k = 3; a relaxed tie has admissible points. At k = 2 the equality asks for c = 1/(2 sigma_0 - 1), which is at least 2 for every target the template allows there (sigma_0 <= 3/4), so the plain count stops decreasing in m (line 15342 generalised; derived). At k = 3 the capacity it requests exceeds the slot supply (measured). Neither is a property of the template. Lemma plain at kappa needs only beta_* <= (1 + kappa)/2 (line 12565; Liu v2 keeps it, source line
• and no zero-free hypothesis at kappa = 1, and a larger kappa only shrinks its condition n_1 + n_2 + 6 kappa z <= M (line 12561; derived), so c = 2/(k kappa) is admissible for every kappa whose hypothesis is already known. Liu's iteration is of this kind, each step using the bound the step before established. Admissible points (measured, each bracketed exactly on both sides, narrower than 1e-8, the lower end with cutting planes; test_relaxed_tie_points_and_their_exact_brackets):

They are points, not optima: a larger c helps until the slot supply binds, which it already does at k = 2 with kappa = 3/4 (c = 4/3). Each needs the order-k moments (L6).

• Which variants need Lemma plain below its stated range (on the inferred reading c = 2/(k kappa)). Variants A and B hold c = 4/9, which is kappa = 9/(2k): 9/4, 3/2 and 9/8 at k = 2, 3 and 4, above 1 and admissible through kappa = 1, which needs no zero-free hypothesis; 9/10 at k = 5, inside the stated range 3/4 <= kappa <= 1 (line 12532), with hypothesis beta_* <= 19/20 (line 12565), which the order-5 Part I analogue 9/10 supplies; 3/4 at k = 6, with hypothesis beta_* <= 7/8, which the paper's theorem supplies. Liu v1 makes exactly that move for the sextic system ("After the original argument establishes beta_* <= 7/8, the unweighted moment permits kappa = 3/4", source lines 245 and 246) and warns against substituting kappa = 2 beta_* - 1 "below the allowed moment range" (source line 2488). At k = 8 and 12 the held c = 4/9 means kappa = 9/16 = 0.5625 and 3/8 = 0.375, below the stated range; those orders are dead by price already, and the c of kappa = 3/4 (1/3 and 2/9) only raises their A and B values (measured). The tied variant C asks for kappa_0 = 2 sigma_0 - 1 below 3/4: 0.7499 at k = 6, inside Liu v2's extension to 13/18 <= kappa <= 1 (v2 source line 282), and 0.6982 at k = 5 and 0.6194 at k = 4, below it. For k != 6 every variant still needs the order-k moments (L6).

5. Price of each structural constant at k = 6 (measured)

Like-for-like gains at the paper's bin: the step in k 0.0255; the Gram term removed 0.0110; the row-count excess removed 0.0052; c at its best 0.0011 (in the limit the second and third are 0.0116 and 0.0083). Exact shadow prices at the paper's own geometry (derived): d sigma_0 / d z_* = -(h - b/2) = -3/4 with the Gram term tied to the pole, -h = -13/16 with it held; d sigma_0 / d g = b, so removing the Gram term outright at fixed geometry would be worth 1/96 there. With the Gram exponent at 0 the perfect-row-count wall drops to 1601/1854 at the paper's bin and to 31/36 in the limit (exact LPs). These are prices of inputs, not candidates: the Gram exponent's exceptional term is dominated by C of bounded norm (its sum over C_0 converges like C_0^(-1/6), line 8551), where the frequencies k are sixth powers and the residual character is principal, so the complete sums are full lattice counts with no cancellation to find (inferred). A smaller Gram exponent needs a different additive factor, which is a different probe.

6. Systems, the lemmas they would need, and what kills them

The pole is the order of the row character, and in the paper that character is the twist of the completed cubic-theta sum. The paper twists by Psi(n) = Psi_0(n) prod_p chi_p(n)^(j_p), with an exponent j_p in {0, ..., 5} chosen for each prime (lines 1700 and 1703); in Part II the twist is the row character chi_n(u), whose exponent at p is the valuation j_p of the row u (line 8969, where reciprocity writes chi_n(u) through prod chi_p(n)^(j_p)). This lane reads a candidate system as one exponent j at every prime, a twist chi^j; that reading is an inference, not a case the paper treats. On it, two cited facts constrain j: the reflection's local column factor at p is chi_p^(-(j_p+2)) (line 1790; principal Ramanujan factors at j_p = 4, line 1791), and the scalar Euler identity needs the theta coefficient gamma_2(c) alpha-bar(cn^3) (line 3819) times the Poisson Gauss sum of the twist to be an explicit phase. At j = 1 the pairing gamma_1 gamma_2 = -alpha(p) chi_p(4)^- gamma_3 (line 4118) supplies alpha(p), which cancels the angular factor, and the dual chi^(-3) is quadratic, which the quadratic large sieve (line 2648) handles without loss. That is why the paper's k is 6 (inferred from these lines).

The lemmas an S4- or S5-type system would need, beyond the walls it is priced at:

• L1 coefficients and pairing. Fourier coefficients of the degree-n theta at squarefree indices, in closed form, with a Gauss-sum identity making (coefficient) x (Gauss sum of the twist) an explicit phase that cancels the theta's angular factor. Literature (cited from the arXiv HTML of David, Dunn, Hamieh and Lin, 2306.11875v5, fetched on 2026-10-21 into scratch and read here, not in the campaign cache; quoted, with the quotes normalised to ASCII): for the cubic theta "all the Fourier-Whittaker coefficients were computed in closed form by Patterson [Pat77]", while the quartic ones "are in general not all determined!" (section 1.2); after Hecke theory and periodicity there remain "n/2-1 (respectively (n-1)/2-1) undetermined coefficients" for even (odd) n (section 1.2); "Deligne [Del80] ... explained that the inaccessibility of the cases n>=4 on GL_2 was owed to the fact that local representations do not have unique Whittaker models", and "Kazhdan and Patterson [KP84] generalised Deligne's work and showed the periodicity property held in greater generality" (section 4.5); Suzuki's closed form covers the quartic coefficients at squares m^2 of squarefree m (Remark 4.6). For n = 6 Broker and Hoffstein (arXiv 1312.0568, abstract read here) give "partially proven, partially conjectured values". Not known for any n >= 4.
• L2 twisted reflection. The analogue of Proposition completed-reflection with explicit dual local factors, whose dual character for the chosen twist is quadratic, so that the reflected energy is Z^(M') by the quadratic sieve. Not known; depends on L1.
• L3 scalar Euler identity. The analogue of Lemma local-euler with zeta_F(kz) L(w, chi(u))/L(x, eta chi-bar(u)) and Euler regions containing z = 1/k, the contours and the target (section 1). Depends on L1.
• L4 Gram bound of order k. The proof pattern of Proposition probe-gram with k-th power exceptional frequencies; routine given the correlation lemma for the new Gauss sums (inferred).
• L5 large sieve of order k. Known for every n >= 3 over a number field containing the n-th roots of unity: Blomer, Goldmakher and Louvel, arXiv 1112.1650, Theorem 1.3, read here from the arXiv PDF (fetched on 2026-10-21 into scratch, sha256 1ab29e36..., not in the campaign cache; text extracted locally from the decompressed file, the relation signs read through the PDF's font encoding; quoted with ASCII signs): "The following theorem, which is valid for all n >= 3", with the bound (MN)^eps (M + N + (MN)^(2/3)) over squarefree ideals coprime to S. For quartic characters BGL also cite Gao and Zhao (their [GZ]; arXiv 1101.3646, not read here).
• L6 the Part II moments. Analogues of Lemma marked and Lemma plain for order-k rows. Under variant C they would also have to reach kappa_0 below 3/4 (section 4): 0.6982 at k = 5, below the 13/18 to which Liu extends the sextic lemma (2610.12234v2 source line 282; the cyclotomic version, 2610.26835v1 source line 148). Not known; these are the bulk of the paper.

What would kill each candidate. S1 and S2: confirming that no twist of the cubic theta other than j = 1 cancels the angular factor (a local Gauss-sum computation in the style of lines 4101 to 4126, not done here). S3: nothing left to kill. S4 and S5: an L1 that exists but fails the pairing, or a reflection whose dual character for the twist is principal (as at j = 4 for the cubic theta) or of order above 2 (the energy would then pay the (KD)^(2/3) term), or any of these systems running into the Davenport-Heilbronn rival at the step that uses the Euler product (hunt #120). S6 and S7: none needed.

7. Controls

• Reproduction at k = 6 (test_template_reproduces_the_paper_at_k6): 14 values rebuilt from the general-k formulas. Twelve are printed in the paper, and test_reproduction_sources_are_honest finds each printed fragment on its stated line: 11/16, 3/16, 7/8, 2/3, 1/4, 11/12 (line 509), h = 13/16, C_0 = -1/48, the floor -7/1200, the large-delta formula -1/48 - delta/16, the supply 8/39 and the inverse capacity 7/37. Two are values hunt #134 derived and the paper does not print, and the same test finds them absent from paper.tex: the worst point 7/30672 at delta = 79/204, and 13/15. The paper's completed-square identity holds on 105 points.
• Outside values: Liu's two optima and closed forms, (1507 - 2 sqrt 921)/1653 with kappa fixed and B_new with kappa tied, reproduced by variants A and C at k = 6; Liu v2's printed count equation, identical to the template's; hunt #134's exact bounds 5655673/6464057 and 44117/50422, reproduced exactly at its witness points (test_hunt_134_exact_bounds_at_their_witness_points).
• Two routes: transcribed and closed forms agree exactly on random rational points for k = 2 to 12; the polynomial J (-E_*) used by the upper bounds agrees exactly with the closed form; the float optimiser re-implements the endpoint in floats while both bracket ends go through the fraction functions, and every measured value lies in its bracket; the float optimiser meets the exact floor LP when the row-count excess is removed; the slot supply has a closed form and a grid route that agree; and point_is_feasible agrees, constraint by constraint, with an independent route (the floor through the transcribed E(h), the slot supply through an exact capacity grid, the large-delta corners through the closed form) on 400 random rational points on which every constraint is both met and missed.
• Defects of the first pass, fixed in the first revision. The float slot-supply test passed points whose requested capacity was at least 2/3, which no ell can supply; no table row reached it, and test_slot_supply_beyond_two_thirds_is_infeasible pins the fix. In the limit convention the first pass scanned the endpoint only from delta = 1/50, leaving 0 < delta < 1/50 unchecked; the scan now starts at the floor bin, and every value of the first pass's table comes back as the limit column (MEASURED_LIMIT). Starting at 0 needed a guard for J = 0 at delta = alpha = 0, which the alpha = 0 control exercises.
• Defects of the first revision, fixed in this one. The k = 2 and 3 cells of variant C said "outside the derivation", which holds only for the equality tie (section 4). Fractions and line numbers on this page were not tied to the computation: the verifier's four typos passed every test. The floor and slot-supply parts of the exact checker were never fault-tested. "Region two misses z = 1/k for k >= 8" is true from k = 7. The floor-wall formula was stated for every delta_0 >= 0 and holds for 0 <= delta_0 <= 1/2. The summary called Liu's optimum a check of the whole transcription. Variants A and B at k = 8 and 12 were missing from the Lemma plain bullet. BGL's theorem had not been read, and hunt #134's exact bounds were not reproduced.
• Planted faults (test_planted_faults_turn_the_reproductions_red and the tests after it): dropping the Cauchy-Schwarz half of the Gram term, moving the pole to 1/(k+1), shifting the signal by 1/1000, adding 1/100 to P_x, or moving the floor bin to 1/100 each turns at least one reproduction red; dropping K from the upper-bound polynomial makes it accept a point that the true check rejects; source mutants of probe_table.py that make the floor, slot-supply or plain-count check vacuous, or loosen or alter the Gram-cap, loss or large-delta corner check, each disagree with the independent route (test_point_is_feasible_mutants_are_caught); edits of this page are flagged by prose_problems (test_prose_faults_are_caught), among them the verifier's four typos (a fraction of section 5, a cell of the walls table, the S1 wall and a line number of section 6), a line of the section 1 table, a swap of two fractions that are both pinned, a relaxed-tie value and the earlier decimal edits. By hand, a scratch copy of probe_table.py with the Gram coefficient set to 1/k printed four FAIL lines (low exponent 19/96, target 85/96, C_0 = -1/32, worst point 653/61344).
• The page pinned to the computation (test_probe_md_numbers_are_the_pinned_ones): every decimal must sit in a phrase that the tests compute or quote; every fraction must be a computed, cited or quoted value; every line reference must name a line that a citation check reads (paper or outside source); and the tables of sections 1, 3, 4 and 5 and the walls of section 6 are rendered from the computation. It cannot catch a swap of two allowed fractions outside the rendered tables and phrases, or an integer that no phrase carries.
• Lexical rules: a test reads the bytes of this lane's three files (ASCII only, no reserved word, no em dash).

8. Limitations

• The template holds the cubic-theta estimates (reflected energy, row-norm loss, Euler regions, floor bin) at their k = 6 form for every k. That is a pricing assumption, not a claim that any other system satisfies them; section 1 says where the held contour and regions stop fitting, and section 6 why no system is known to satisfy them.
• The mapping of the plain capacity 6 kappa to k kappa is inferred from lines 12594 to 12599 and 14378 (exceptional rows counted as sixth powers); variants A and B use it only in the Lemma plain bullet of section 4.
• The reasons the paper's twist is j = 1 (section 6) are readings of cited lines, on the uniform-exponent reading of the twist; the local computation that would confirm them for j = 2 to 5 was not done.
• The exact brackets are statements about the transcribed template, not about the paper's lemmas: they show where its formulas put the optimum, nothing more. The relaxed-tie rows are admissible points; the relaxed optimum, where the slot supply binds, was not computed.
• Prior art: searched only for the lemmas in section 6 (BGL, Gao and Zhao, DDHL, Broker and Hoffstein) and the public follow-ups cached by the campaign. The order-k pricing itself was not searched for; nothing here is said to be new.

Open issues

• The local Gauss-sum computation for the twists j = 2 to 5 (lines 4101 to 4126 style), which would confirm or kill S1 and S2, was not done.
• The k-reading of the sextic constants, the 6 kappa to k kappa mapping and the uniform-exponent reading of the twist are inferred, and every statement about a system other than S0 inherits that grade.
• The Euler regions were not re-derived for k != 6; the template holds them.
• Liu's B-flat 0.874886647 and the record 0.8748087 that FOLLOWUPS.md reports are quoted, not reproduced.
• The sharpness of the Gram exceptional term is inferred, not proved.
• The relaxed-tie optimum at k = 2 and 3 was not computed.
• The page checks leave the blind spots named in section 7.

The doors

Candidate status: within the order-k template, with the paper's row counts carried over, the endpoint optimum is the table of section 4 (measured, bracketed exactly on both sides); parameter range ell, M, b >= 0 with M + ell <= 1 and the Gram cap; proof status: exact arithmetic on transcribed formulas, unreviewed; information class below.

Active constraints at the k = 6 optimum (ranked)

1. The endpoint inequality at delta = 0.3886, x = 1/2 (binding, with the offset b = 0.1234 interior where -E_* is flat in b). Its pole price is -0.7618 per unit of 1/k; its row-count prices are alpha (0.0004 from 5/6 to 3/4, 0.0052 to 0) and c (0.0008 from 4/9 to 1, 0.0011 at its best).
2. The slot supply ell/h = 0.2055 against 7/37: slack here, binding once the Gram exponent is lowered or c reaches 1.144 (section 5).
3. The Gram cap b <= 3(1 - 3 ell)/8 = 0.1873 against b = 0.1234: slack.
4. The floor rows C_0 = -0.0209, and -0.0059 with the paper's bin: slack; they bind only at perfect counts (167/192 at the paper's bin, 13/15 in the limit).

Frozen-constant inventory

Information class

• Inside the current data: re-choosing (ell, M, b) is exhausted (hunt #134 and the exact brackets here: the k = 6 template optimum is 0.8749571 with fixed capacity and 0.8749570194 with the tied one).
• Inside the family, better inputs: the row-count lever is worth at most 0.0052 at the paper's bin (0.0083 in the limit), single steps of alpha and c 0.0004 to 0.0027; the Gram exponent would buy up to 0.0110 but its exceptional term looks sharp; the floor bin matters only after the row counts.
• Reading more: only a different row-character order moves the targets by a structural amount, and it needs a theta-type completed sum that the literature does not supply for any order below 6 (section 6). Larger fields (Liu) and larger orders move the other way or not at all. Compared with other mechanisms open to this hunt, this door is the most valuable per unit of success and the least supported; it should not get automatic priority over the row-count lane.

Run manifest

id: below_seven_eighths-2026-10-21-probe
hunt: below_seven_eighths
started: 2026-10-21T14:36Z
finished: 2026-10-21T15:28Z
ran:
  - .venv/bin/python hunts/below_seven_eighths/probe_table.py --paper --quick
  - .venv/bin/python hunts/below_seven_eighths/probe_table.py --paper
  - .venv/bin/python -m pytest -q -n0 hunts/below_seven_eighths/test_probe_table.py
  - planted-fault run of a scratch copy of probe_table.py with the Gram coefficient 1/k
outcome: priced the order-k template; k = 6 reproduces the paper and Liu; every order below 6 is cheaper in principle and none is available
artifacts:
  - hunts/below_seven_eighths/PROBE.md
  - hunts/below_seven_eighths/probe_table.py
  - hunts/below_seven_eighths/test_probe_table.py

id: below_seven_eighths-2026-10-21-probe-repair-r1
hunt: below_seven_eighths
started: 2026-10-21T16:44Z
finished: 2026-10-21T17:41Z
ran:
  - .venv/bin/python hunts/below_seven_eighths/probe_table.py --paper
  - .venv/bin/python -m pytest -q -n0 hunts/below_seven_eighths/test_probe_table.py
  - planted-fault runs of the test file against scratch copies of PROBE.md and probe_table.py
  - curl of the arXiv HTML of 2306.11875v5 and the abstract of 1312.0568 into scratch, read as data
  - .venv/bin/python -m pytest -q -n0 -m "not slow" tests/test_huntspec.py tests/test_hunt_probe_discipline.py tests/test_docs_numbering.py tests/test_doors.py
  - .venv/bin/python scripts/make_context.py --check
outcome: repaired the probe lane after verification; floor-bin convention made explicit with both walls, two-sided exact brackets added, two float-route defects fixed, grades and citations corrected
artifacts:
  - hunts/below_seven_eighths/PROBE.md
  - hunts/below_seven_eighths/probe_table.py
  - hunts/below_seven_eighths/test_probe_table.py

id: below_seven_eighths-2026-10-21-probe-repair-r2
hunt: below_seven_eighths
started: 2026-10-21T22:00Z
finished: 2026-10-21T22:46Z
ran:
  - OMP_NUM_THREADS=1 .venv/bin/python -m pytest -q -p no:cacheprovider -n0 hunts/below_seven_eighths/test_probe_table.py
  - .venv/bin/python hunts/below_seven_eighths/probe_table.py --paper
  - scratch scripts for the relaxed ties, hunt 134's witness points, Liu's count equation and the cutting-plane brackets
  - WebFetch of the arXiv abstract and PDF of 1112.1650, decompressed with qpdf and its text extracted in scratch, read as data
  - planted-fault runs: page typos, a table line, a fraction swap and source mutants of the exact checker, all inside the test file
  - three scratch copies of the lane, each with one planted fault (a page fraction, a relaxed-tie hypothesis bound, the sign of a large-delta corner term), each run against the full test file
  - .venv/bin/python -m pytest -q -n0 -m "not slow" tests/test_huntspec.py tests/test_hunt_probe_discipline.py tests/test_docs_numbering.py tests/test_doors.py
  - .venv/bin/python scripts/make_context.py --check
outcome: repaired the probe lane after the second verification; the k = 2 and 3 tie verdicts corrected with relaxed-tie points, the page's fractions, line numbers and tables tied to the computation, the exact checker fault-tested, wording and grades corrected
artifacts:
  - hunts/below_seven_eighths/PROBE.md
  - hunts/below_seven_eighths/probe_table.py
  - hunts/below_seven_eighths/test_probe_table.py