Results: which step of the quasi-Riemann argument fixes the number 7/8

Hunt #139 (opened as #134), 2026-10-16. Status: answered within its stated scope; unreviewed by any person. Grades: cited (a line of the paper, quoted and checked against the pinned file), derived (algebra done here from cited formulas and checked in exact rational arithmetic by test_seven_eighths_pin.py), inferred (a reading of a lemma's proof that no computation here checks). A composite claim takes the grade of its weakest step. Nothing here is a statement about RH, and nothing here checks the paper's lemmas: they are taken as cited inputs, at the exponents the paper proves them with.

Source: github.com/openai/math at adc7f1241b42e322a6451854ab7e4b4c146bf78a, preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/build/paper.tex, 16,677 lines, sha256 42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3. Line numbers below refer to that file. probe.py --paper PATH checks that each of the 39 cited snippets sits on its stated line (all 39 do).

Landing note, 2026-10-21. Opened as hunt #134 and renumbered #139 at landing, because #134 on main is gauss_positivity and #136 to #138 are taken by other open hunts and landings. The prior-art search this page had not run was run at landing and extended the same day to the three works its first source names and to the follow-ups' own manuscripts (section 10 lists what was read, with commits). Much of this page was public before the hunt opened on 2026-10-16, so it was derived here independently and is not novel:

• a model computation by tomoto0 (7 October 2026) has the target formula of Summary

item 1 with its ζ_F(6z) pole, the floor-row ceiling 13/15 of Summary item 5 at the same corner, a geometry optimum near 0.87496 with the zero-detected rows binding (the two ends of section 6), and on a grid the worst point and margin of Summary item 4 at the paper's geometry;

• J. Guo and B. Liu (8 October 2026) give the exact optimum of the re-optimized geometry

of Summary item 3 and section 4, σ* = (1507 − 2√921)/1653 ≈ 0.8749570698, and the further gain of section 7's tied capacity; Guo derives the binding bin δ* = (49 − √921)/48 at x = 1/2 from the vanishing coefficient of the offset;

• the total accounting of section 3, marked inferred below, is a proof step on the line

M + ℓ = 1 in Z. Gao's draft (7 October) and in Guo's and Argonaut Math's papers (8 October).

Section 10 also records two consequences checked at landing against this hunt's own formulas (the exact bound 0.874942 is valid but loose, and the scan optimum 0.874958 is a grid reproduction of σ*), what was not found in the sources read, and one disagreement with tomoto0 over the slope α. The rest of this page is as the hunt wrote it.

Summary

1. The target is set by the direct ("low") bound, and both stages lie on one line. The paper continues 1/L past a hypothetical zero by comparing a smoothed sum with a Mellin integral Z^{C(s)}. The target is the point where the direct bound on that sum equals C(σ₀). Written in the free geometry (ℓ the total prime-slot length, M the two averaging lengths together, b their offset), that point is σ₀ = 1 − h/6 + b/12 = 5/6 + (M − 2ℓ)/12 (derived). Part I (no slots, M = 1) gives 11/12; Part II (ℓ = 1/6, M = 5/6) gives 7/8. On the paper's line M + ℓ = 1 this is σ₀ = 11/12 − ℓ/4: every unit of slot length lowers the target by a quarter. The 1/6 in h/6 is the pole at z = 1/6 of the sextic factor ζ_F(6z), not the slot length (cited, lines 5586, 8641).
2. The paper's geometry is the corner of two direct-bound conditions, and one of them alone gives exactly 7/8. The reflected dual length must not exceed the row length, M + ℓ ≤ 1 (line 8246, used there as an equality). The row-norm lemma has a second-branch loss (1 + 3ℓ' − 2M')/4 (line 8306); asking for no loss in every rescaled summand means M − 2ℓ ≥ 1/2, hence σ₀ ≥ 5/6 + 1/24 = 7/8 for every choice of ℓ, M and b (derived). The two lines meet at ℓ = 1/6, M = 5/6, which is the paper's geometry.
3. That second condition is stricter than the argument needs, but the high side then binds at the same place. Each rescaled summand also gains Z^{−d} (line 8631), which absorbs the loss; only the unrescaled summand must be lossless, M ≥ (1+3ℓ)/2 (inferred). Under that accounting the direct bound alone would allow targets down to 13/15. The high side does not: the paper's endpoint inequality (Lemma "Compensated endpoint certificate", lines 15994 to 16072) fails below the following, with the paper's row counts held fixed.
4. Exact bound, every geometry with M + ℓ ≤ 1 and b ≥ 0: σ₀ ≥ 5655673/6464057 ≈ 0.874942 (derived; one witness point δ = 223/576, x = 1/2).
5. With the paper's Gram condition b ≤ 3(M − 2ℓ)/8 added: σ₀ ≥ 44117/50422 ≈ 0.874955 (derived; witness δ = 7/18, x = 1/2).
6. Float optimum of the scan: 0.874958 at b ≈ 0.12, ℓ ≈ 0.166836 (measured). The paper's (ℓ, b) = (1/6, 1/8) is within 1.7 × 10⁻⁴ in ℓ and 4.2 × 10⁻⁵ in the target of the best its own high side allows.

So 7/8 is the corner of the direct bound, and the high side's optimum sits 4.2 × 10⁻⁵ below it. The number is not slack left on the table: re-choosing the geometry inside this assembly can gain at most 5.8 × 10⁻⁵.

1. The binding rows are those whose twisted L-functions have a zero near real part 0.69 (δ = 2a − 1 ≈ 0.39) and whose prime-slot polynomials are as large as that zero allows (x = q/δ = 1/2). At the paper's point the margin there is exactly −E_* = 7/30672 ≈ 2.3 × 10⁻⁴, about twice the paper's printed lower bound 49/440640 (derived). The mechanism: in the high exponent the slot amplitude costs +qℓ and the constant C₀ = −1/4 + 5ℓ/4 + b/6 grows with ℓ, while the moment lemmas' row-count gain is capped by their capacities and does not grow with ℓ (cited formulas, lines 15192 and 15729; the reading that the counts are ℓ-free is cited at 15192 and inferred for the lemmas behind it).
2. Even perfect row counts stop at 13/15. Rows with no zero beyond the floor are counted in full (exponent R = 1), which no lemma can improve, and they need C₀ = 1/2 + ℓ/2 − 3M/4 + b/6 ≤ 0. Together with M + ℓ ≤ 1 the smallest target is 13/15 at ℓ = 1/5, M = 4/5, b = 0, where the unrescaled-summand condition is tight too (derived). Scaling the row-count excess R_* − (1 − δ) by λ moves the best target almost linearly from 0.874958 (λ = 1) to 13/15 (λ = 0) (measured, section 6). This floor holds for this assembly's three forms (direct bound, Mellin signal, high-exponent shape), not for the method family.
3. What does not fix 7/8 (section 7): the Euler regions and the contraction of H_η (7/8 is plugged in; they hold down to about 2/3); the contour Re z = 17/50 (it cancels at d = h); the principal margins (contour choices); the 11/12 input (it only caps δ ≤ 5/6; the α = 5/6 in the row counts is the sextic amplification slope (1 + 5r)/6, not the bootstrap); Lemma plain's κ ≥ 3/4; the slot supply (a lower bound only).

1. What was read

The final section, "The seven-eighths bound" (lines 15495 to 16466), in full, and the Part II setup (6807 to 6918), the compensated low estimate (8559 to 8651), the row-norm lemma's second branch (8196 to 8334), the definition of the stage data (5586) and the no-slot inverse count (15276 to 15310) directly. Four separate readings, each writing verbatim formulas with line numbers to a report and checking its own arithmetic, covered the row counts (14985 to 15495 and the statements of Lemmas marked, plain and the detector propositions), the shared analytic lemmas (5460 to 6700, including the Part I conclusion), the probe geometry (3313 to 3723, 6807 to 6918, 8061 to 8651) and the Euler regions and moment hypotheses (3723 to 4201, 8651 to 9209, 9209 to 14985 for hypotheses only). Those reports are model readings; every number this page uses from them is either re-derived in probe.py or marked inferred. The proofs of the moment lemmas (Sections 9 to 12 of Part II, about 5,800 lines) were not checked.

2. The target on the direct-bound side

Cited:

• C_II(s) = l_x/2 + s − 1 + h/6 = s − 11/16 (line 8641), with h = 1 − l_x + ℓ by definition (line 5586). Part I: C_I(s) = s − 2/3 (line 190).
• Direct bound |I_mod| ≪ Z^{l_x/2 + b/12 + ε} = Z^{3/16 + ε} (line 8570), and C_II(7/8) = 3/16 = l_x/2 + b/12 (line 8642). Part I: Z^{1/4} (line 3717).
• Geometry b = 1/8, h = 13/16, ℓ = 1/6, l_x = 17/48, l_y = 23/48, M = 5/6, M + ℓ = 1 (lines 6860 to 6864).

Derived (exact, test_target_closed_form): with l_x = (M − b)/2, l_y = (M + b)/2,

σ₀ = 1 − h/6 + b/12 = 5/6 + (M − 2ℓ)/12,

so b cancels, σ₀ = 11/12 − ℓ/4 on M + ℓ = 1, and the two stages are the points (ℓ, M) = (0, 1) and (1/6, 5/6) on the same formula.

3. The paper's corner

Cited: line 8246, "Since M'+ℓ'−1=−3d", shows the reflected dual length T_d stays below the row length; with general parameters the same subtraction leaves an extra M + ℓ − 1, so M + ℓ ≤ 1 suffices (derived from the cited display). Line 8306 gives the second-branch energy M' + (1 + 3ℓ' − 2M' − 2Δ_H + θ_N)/4, with M' = M − 2d and ℓ' = ℓ − d for a summand rescaled by Z^d (the paper's own substitution at M = 5/6, ℓ = 1/6 gives its (d − 1/6)/4). The general loss is therefore ((1 + 3ℓ − 2M + d)/4)₊ for 0 ≤ d ≤ ℓ.

• Per summand: the summand with every slot rescaled (d = ℓ, so ℓ' = 0) falls in the lossless first branch (line 8326), but the summands just below it, d = ℓ − ℓ/K with the fine slot mesh the paper takes (K large, line 16246), carry (1 + 4ℓ − 2M − ℓ/K)/4. Zero loss for all of them, as K grows, needs 1 + 4ℓ − 2M ≤ 0, that is M − 2ℓ ≥ 1/2, so σ₀ ≥ 7/8 whatever ℓ, M, b (derived, test_per_summand_reading_forces_seven_eighths). The paper's (1/6, 5/6) satisfies it with equality.
• In total: summand d carries −d + loss(d)/2 in its exponent (f(d) = −d, line 8631), so only d = 0 must be lossless: M ≥ (1 + 3ℓ)/2 (inferred that the low estimate may be assembled this way; the paper never needs to, since its loss is zero; arithmetic in test_total_accounting_needs_only_the_d0_summand).

4. The high side and its endpoint inequality

Cited: the high exponent E(d) = a − 7/8 + h(17/50 − 1/6) − a l_y − (1−a)ℓ − (δ/2 − q)ℓ + d(R + δ/2 − 17/50) with C₀ = −1/48 (lines 15729 to 15734); the row counts D_x = 3 − 17x/9, P_x = (2 − 8x/9)(1 − x) (line 15192), R_* = 1 − δ + (α − δ)δP_x/(2𝒥) with 𝒥 = (α − δ)D_x + δP_x (line 16019); the certificate −E_* ≥ 49/440640 (line 16000) and its completed-square identity (line 16048); and the requirement E_actual(h) − Δ ≤ −49/440640 − (51/64)Δ (line 16101).

The requirement is relative to the hypothetical zero, Δ = β* − 7/8 > 0, and the capacity allowance hΔ/4 is smaller than Δ, so what the argument needs is −E_* ≥ 0 at every (δ, x); the printed 49/440640 is room on top. As Δ → 0 the condition is necessary for this assembly's bound.

Derived, with M free:

−E_* = (−1/2 − ℓ/2 + 3M/4 − b/6) + δ(ℓ + b)/2 − ℓxδ − (h/2)·K(δ, x), K = (α − δ)δP_x/𝒥, h = 1 + ℓ − M/2 + b/2.

The offset is bounded below by a hypothesis of the Gram bound, P_a = Y'²/Q ≥ 1 (Prop. probe-gram, line 8345), and P_a = Z^b up to fixed constants (line 8591), so b ≥ 0 (cited). This matters: at the witness below, smaller b helps, so b < 0 would weaken the bound; it lies outside the lemma as stated and would change the direct bound's form (P_a^{1/12} no longer below the other Gram terms), which is not explored here. At fixed (δ, x), both −E_* and σ₀ are affine in (ℓ, M, b). Along a line of constant σ₀ and b, d(−E_*)/dℓ = 1 + δ(1/2 − x) > 0 (test_constant_target_slope), so the best geometry has M + ℓ = 1. There −E_* decreases in ℓ, and each fixed (δ, x) gives an exact lower bound on σ₀ by a two-variable linear program:

The paper's own point has min −E_* = 7/30672 at δ = 79/204, x = 1/2 (derived; the completed square (4vδ − 79)² at v = 51 places it there).

5. The floor-row ceiling 13/15

Cited: the floor bin a = 51/100 uses no zero witness, count R = 1, and E(h) ≤ C₀ + (3/4)δ₀ = −7/1200 (line 15847). Derived: in general the floor rows give E(h) = C₀ + (1 + ℓ − M/2)δ₀ with C₀ = 1/2 + ℓ/2 − 3M/4 + b/6, and as the floor approaches 1/2 the condition becomes C₀ ≤ 0. Rows counted in full cannot be counted better, so this survives any improvement of the moment lemmas. Minimizing σ₀ = 5/6 + (M − 2ℓ)/12 subject to C₀ ≤ 0, M + ℓ ≤ 1, b ≥ 0 gives the infimum 13/15 at ℓ = 1/5, M = 4/5, b = 0, where C₀, the dual-length slack and the unrescaled-summand slack are all zero (test_floor_corner_is_thirteen_fifteenths, test_no_geometry_beats_the_floor_corner).

6. How much better row counts would buy

Scaling the excess R_* − (1 − δ) by λ and re-optimizing (ℓ, b) on M + ℓ = 1 (still optimal for every λ, since the constant-target slope does not involve K), float scan:

The curve is close to linear, about 0.0021 per quarter of the excess removed (measured).

7. What does not fix 7/8

• Euler regions and the contraction of H_η. sup_{Re s > 7/8} |H_η − 1| ≤ 1/2, the second Euler region x_r ≥ 7/8 (line 4056) and B_p = −1 + O(Q^{−7/8}) (line 9074) all plug in the target. The local sextic term Q^{3 − 6s} at z = 1/6 keeps the products under control for targets above about 2/3 (inferred, from the fourth reading). The four shared lemmas headed "σ₀ ∈ [7/8, 1)" (lines 5811, 6024, 6178, 6345) use only those regions (inferred).
• The contour Re z = 17/50. Its coefficient in E(d) is h − d, so it cancels at d = h where the endpoint inequality lives (derived).
• The principal margins m_w = l_y/20, m_z = h/600 (lines 15674, 15675) are contour placements at the edge of the second Euler region and never bind.
• The 11/12 input. It caps bins at δ ≤ 5/6 and gives Δ ≤ 1/24 (line 6817). The α = 5/6 inside R_* is defined at line 15101 and is the sextic amplification slope e(r) = (1 + 5r)/6 = 1 − α + αr (line 15300), equal to 2·(11/12) − 1 by coincidence. Feeding the 7/8 output back would only cap δ ≤ 3/4; the binding point is δ ≈ 0.39. An earlier scratch scan in this session that also lowered α was wrong for that reason and is not used.
• Lemma plain's κ ∈ [3/4, 1] (line 12532). Its baseline capacity 1/(6κ₀) with κ₀ = 2σ₀ − 1 is where every ninth in D_x, P_x comes from. Tying it to the target (c = 1/(3(2σ₀ − 1))) changes the optimum by less than 10⁻⁶ (measured). The one numerical use of κ ≥ 3/4 in its proof survives down to targets of 3/4 (inferred).
• The slot supply ℓ/d ≥ 8/39 > 7/37 (line 16116) is a lower bound on ℓ.

8. Controls

• Reproduction: 38 constants the paper prints are rebuilt from the general formulas at the paper's values, all exactly (test_every_printed_constant_is_reproduced), among them 11/16, 3/16, 7/8, 11/12, 2/3, 1/4, −1/48, −7/1200, −529/2400, 25/96, −49/14400, −79/800, 23/960, 13/9600, 53/32, 51/64, 8/39, 23/1443, 1/48, 2/185, and the certificate's completed-square identity on 101 points.
• Two forms of every central formula (transcribed and closed) agree on exact grids.
• Planted faults: changing the contour 17/50, the slope α or the capacity coefficient 4/9 breaks at least one reproduction (test_planted_fault_is_caught).
• Citations: 39 of 39 snippets on their stated line of the pinned file.
• Cross-reading: the four readings and this page agree on every shared number (C₀, the worst point 79/204, 7/30672, the role of α); the one disagreement found (whether M + ℓ = 1 is required) was resolved in favour of ≤ by reading line 8246.

9. Limitations and kill conditions

• The lemmas are inputs. If Lemma marked, Lemma plain or the detector propositions are wrong, everything here is about a different argument. Nobody here has checked them, and no person has checked the paper (docs/38).
• The general-parameter forms are readings of proofs written at fixed numbers. The ones that only substitute (the target, C₀, −E_*) are checked by reproduction; the ones that change which inequality is used (M + ℓ ≤ 1 suffices, only d = 0 must be lossless, the Euler regions hold below 7/8) are inferred.
• The exact bound 0.874942 is a statement about the endpoint inequality with the paper's row counts. It does not say no other assembly of the same lemmas does better; it says re-choosing (ℓ, M, b) in this one cannot.
• Kill conditions (from MISSION.md): a printed constant not reproduced (none); a parameter treated as free being fixed by an unread lemma that moves the binding constraint (open for the moment-lemma proofs); the target relation failing at either stage (it holds at both).

10. Prior art

As written on 2026-10-16: Not searched. This is a reading of a two-week-old preprint's internal constants; the paper itself does not say which inequality fixes 7/8, and docs/38 records that nobody here had read past its section 3. Original to this laboratory as a reading; whether anyone else has published the same observation is unknown.

Searched 2026-10-21, at landing. Searched: the follow-up that main already cites in hunts/qrh_nonresidue/RESULTS.md, J. Guo's github.com/aster2024/openai-math-followups, directory zero-free-beyond-7-8 (its README as of commit 5750cc81e489, 9 October 2026, unchanged at the repository's head on 2026-10-21); the arXiv abstract of B. Liu, arXiv:2610.12234 (v1 8 October, v2 12 October 2026); and one web search for the Xu and Sang figure that hunt #136 records. No paper's proofs were read. Found:

• The re-optimized geometry was public before this hunt opened, so it is not novel here. Guo's README states that optimising the skew parameter (the offset b here) gives the endpoint σ* = (1507 − 2√921)/1653 ≈ 0.8749570698, and that widening Lemma plain's fourth-moment range from [3/4, 1] to [37/50, 1] gives σ† ≈ 0.8749570194, the root of 7884s³ − 18819s² + 14643s − 3686 in (0.8749570194, 0.8749570195). It credits B. Liu with σ* independently (8 October) and Z. Gao with 20999/24000 at ℓ = 1001/6000, b = 1/8 (a draft of 7 October), whose fixed-offset envelope it gives as about 0.8749572006. Liu's v2 abstract varies b, ℓ and κ, gives σ† in closed form as B_new = (18ξ + 7)/(18ξ + 9) with ξ the root of 1188ξ³ − 2160ξ² + 171ξ + 190 in (0.38858, 0.38859), and states that its optimality is proved "within the explicitly specified exponent comparison system". Gao's 20999/24000 is this page's target line 11/12 − ℓ/4 at ℓ = 1001/6000, exactly.
• Checked at landing against this hunt's own formulas (the last three tests of test_seven_eighths_pin.py, which run the exact code at 45-digit rational approximations of the closed forms):
• The exact bound of Summary item 3 is valid but loose. The one-witness program holds at every witness where its b-coefficient is at most zero. At x = 1/2 that coefficient vanishes at δ* = (49 − √921)/48 ≈ 0.3885837, and the recorded witness 223/576 ≈ 0.3871528 sits below it. At δ* the same program returns σ* itself: with the paper's row counts, no geometry with M + ℓ ≤ 1 and b ≥ 0 passes the endpoint inequality below σ* (derived, at the grade of the recorded bound; the closed forms come from an exact symbolic computation at landing and are pinned to 35 digits; they are Guo's, as the extension below found). The inequality is tight at δ* for ℓ* = 11/551 + 8√921/1653 ≈ 0.1668384 and b* = −4/29 + 230√921/26709 ≈ 0.1234054. So re-choosing the geometry gains at most 7/8 − σ* ≈ 4.29 × 10⁻⁵, inside the "at most 5.8 × 10⁻⁵" of Summary item 3, and Guo and Liu state that this gain is attained (cited). That the inequality holds at every other (δ, x) at that geometry is shown in Guo's section 8 by an exact decomposition of the endpoint numerator (read in the extension, not re-run or checked here); a refinement of this page's float scan at landing agrees (measured, RUNS.md).
• The scan optimum 0.874958 at b = 0.12 is a grid reproduction. The offset grid of 0.01 leaves it 5.9 × 10⁻⁷ above σ* (measured).
• Section 7's tied capacity is Liu's κ. Liu's κ_new = (18ξ + 5)/(18ξ + 9) = 2B_new − 1 is the tie c = 1/(3(2σ₀ − 1)). With it, this page's endpoint formula at Liu's ξ, ℓ_new and b_new is zero at δ = ξ and x = 1/2, a minimum in δ and flat in b, with B_new on the target line. So Liu's ξ is this page's binding δ, and the "less than 10⁻⁶" measured in section 7 is the step of 5.0 × 10⁻⁸ from σ* to σ†.

Extended 2026-10-21, the same day. The first search stopped at Guo's README, whose related-work paragraphs name three more works, and it read none of the follow-ups' own manuscripts. Read in the extension, as data, with the pinned paper.tex (sha256 as above) at hand for the cited lines:

• tomoto0, github.com/tomoto0/quasi-riemann-hypothesis-7-8-verification at b83a73c6cb62 (the repository's only commit, 7 October 2026): the README, and reports/barrier_analysis.md and reports/improvement_exploration.md in full (both in Japanese); its other reports, scripts/numerical_checks.py and data/numerical_checks_results.json searched, with the grid check C18 read;
• Argonaut Math, github.com/Argonaut-Math/argonaut-math-quasi-riemann-boundary at 5971383edcb0 (8 October 2026): the README with its announcement; in preprints/Beyond_the_Seven_Eighths_Barrier.tex the introduction, the outline and section 3 (the low side) read, the rest searched;
• LW56e2, github.com/LW56e2/quasi-riemann-improved at b83dbbd914c5 (8 October 2026): the README (the repository holds Lean files and no manuscript);
• Guo's paper source, in the directory and commit above: section 8 (the quadratic endpoint) in full, the end of section 3's low-side proof (the row-energy bound and the sum over rescaled tuples) and the first subsection of section 11 (the limit of his envelope) read; the rest of sections 3 and 11, and sections 1 and 6, searched;
• Z. Gao, github.com/Gaozhongpai/seven-eighths-refinement-certificates at 309a757ea539 (the version Guo cites, committed 8 October 2026, draft dated 7 October): the low-side lemma of manuscript section 2 and its use in section 3, the opening of the README and the closing section of REPORT.md read; the rest of those files and of the manuscript searched.

Liu's note itself was not read, only its abstract, and no proof was checked. Every translation below between this page's variables and a source's is pinned in exact arithmetic by the block of tests just before the landing checks in test_seven_eighths_pin.py; the sources' own numbers are cited, not re-derived. Found:

• tomoto0 had much of this page nine days before the hunt opened. It calls its figures model computations that use the paper's exponent formulas outside their stated ranges. As far as those files show:
• the target σ₀ = 1 − h/6 + b/12 at both stages, with the 1/6 from the pole of ζ_F(6z), and σ₀ = 2/3 + l_x/2 + b/4 on M + ℓ = 1, the same line as 11/12 − ℓ/4 (barrier_analysis.md section 1): Summary item 1;
• the floor-row ceiling (section 2, its "barrier B"): rows near Re s = 1/2 are counted with R = 1, which no row count improves, and E = 1 − (5/2)l_x − (13/12)b < 0 forces σ₀ > 13/15, the limit at l_x = 2/5, b = 0, that is ℓ = 1/5, M = 4/5. Its E is this page's C₀ on M + ℓ = 1. Scenarios 5 and 6 of improvement_exploration.md give 0.86979 with ideal counts R = 1 − δ and the floor 51/100 binding, and the infimum 13/15 with the floor removed: Summary item 5 and the λ = 0 end of section 6;
• a geometry optimum 0.87496 at (l_x, b, ℓ, h) = (0.3545, 0.1241, 0.1668, 0.8123), 4 × 10⁻⁵ below 7/8, with the adaptive range (the zero-detected rows) binding (barrier_analysis.md section 3, scenario 1): the scan optimum of Summary item 3 and the λ = 1 end of section 6;
• at the paper's geometry, the minimum of −E_* on a 401 × 401 grid, 2.28 × 10⁻⁴ at δ = 0.3875, x = 1/2, about twice 49/440640 (README section 2, check C18), where Summary item 4 has the exact 7/30672 at δ = 79/204.

These were derived here independently (the hunt read only the paper, RUNS.md) and are not novel.

• The low-side loss behind section 3 is handled in three follow-ups. Gao's draft, Guo's section 3 and Argonaut Math's section 3 each bound the row energy of a summand rescaled by Z^d by M' + (5ℓ − 1 + d)₊/4 on the line M + ℓ = 1, which is this page's loss ((1 + 3ℓ − 2M + d)/4)₊ on that line, and show that the rescaling absorbs it, −d + (5ℓ − 1 + d)₊/8 ≤ −7d/8, which holds once ℓ ≤ 1/5. That is the total accounting section 3 marks inferred, used there as a proof step of an unreviewed preprint, so its grade here does not change. In the parts read, none of them says that per-summand losslessness is what places the paper's geometry at 7/8. tomoto0 explains the Part II corner by the paper's sufficient condition l_x − ℓ ≥ 3/16, that is h ≤ 13/16 (barrier_analysis.md section 1). That condition is M − 2ℓ ≥ 3/8 + b: the same line as this page's M − 2ℓ ≥ 1/2 at the paper's b = 1/8, and a different one at any other b.
• Guo names the binding bin. His section 8 takes d_* = (49 − √921)/48, the root of 288δ² − 588δ + 185 where the coefficient of b in the endpoint numerator at x = 1/2 vanishes, with ℓ_* = (33 + 8√921)/1653, the same b_* as above and σ_* = 11/12 − ℓ_*/4. His section 11 shows that at this bin the high exponent is (3/4 + 3d_*/2)(ℓ − ℓ_*) whatever b is, so every admissible geometry has ℓ ≤ ℓ_*: section 4's one-witness bound, placed at its best witness, inside his envelope with M + ℓ = 1.
• Argonaut Math and LW56e2 (both 8 October 2026, with Lean files) state the boundaries 7/8 − 1/4000000 = 3499999/4000000 and 4374785349/5000000000, a rational just above σ*. Argonaut Math moves ℓ up and M down by τ = 10⁻⁶ along M + ℓ = 1 at b = 1/8. LW56e2's README records an exact two-bin obstruction to lowering its boundary by 10⁻¹² with the same two parameters; its Lean files were not read.
• A disagreement with tomoto0, over the slope α. tomoto0's bootstrap scenarios (barrier_analysis.md section 3 and improvement_exploration.md scenarios 2 and 3: 0.87451 at α = 3/4, and a fixed point near 0.8745) set α = 2β* − 1 from the prior bound on the zero. The paper, at lines 15100 to 15104, sets κ = 2β* − 1 = 3/4 + 2Δ and α := 5/6, and says: "The parameter κ remains dynamic; α is the fixed slope from sixth-power amplification." On that reading (section 7), feeding 7/8 back caps the bins at δ ≤ 3/4, far above the binding δ ≈ 0.39, and through κ moves only Lemma plain's capacity, which is section 7's tied capacity (less than 10⁻⁶, and only if that lemma's range extends below κ = 3/4). The follow-ups read agree: Gao and Argonaut Math both take κ = 3/4 from the 7/8 bound and keep α = 5/6 (Argonaut Math's bins stop at δ ≤ 3/4), and Guo's envelope keeps 5/6 while κ = 2β − 1 varies. So, as read here, tomoto0's 0.87451 and 0.8745 do not follow from the paper as written. This page's own early scratch scan made the same substitution, reached the same apparent fixed point near 0.8745 and was discarded for that reason (RUNS.md).
• Not found in the sources read, and so this hunt's own as far as this search goes: the reading that 7/8 is the corner of M + ℓ ≤ 1 (read as an inequality from line 8246; Gao, Guo, Argonaut Math and tomoto0 all fix M + ℓ = 1) and of per-summand losslessness M − 2ℓ ≥ 1/2, which alone forces 7/8 for every (ℓ, M, b), with section 4's constant-target slope putting the optimum on M + ℓ = 1; the exact rational bounds 5655673/6464057 and 44117/50422 and the exact margin 7/30672 at δ = 79/204; the pricing of the row-count excess between the two ends of section 6; and the use of lines 15100 to 15104 against the bootstrap figure. This is a statement about the sources listed, not a novelty claim.
• Open check. Xu and Sang's 0.8748867, as hunt #136 (below_seven_eighths/, open, on main since #359) records it, and the 0.874887 for Hecke L-functions over every number field that the same record attributes to Liu, both lie below σ*, the bound above for every geometry with the paper's row counts. Each must therefore change an input beyond the geometry (row counts, capacities, the comparison system or the probe). Neither source was read here, and the web search did not find Xu and Sang's. If either changed only the geometry, that would refute this page's bound, which is a kill condition.

The doors

The hunt locates a ceiling of one assembly, so this section is owed. Candidate status: the target 7/8 is the corner of the direct bound, and the high side's optimum is 0.874958 (measured) with exact lower bound 0.874942 (derived) over every geometry, with the paper's row counts; parameter range ℓ, M, b ≥ 0 with M + ℓ ≤ 1; proof status: exact arithmetic on transcribed formulas, unreviewed.

Active constraints at the optimum

Ranked by what relaxing them buys:

1. The endpoint inequality at δ ≈ 0.39, x = 1/2. Zero margin at the optimum. Its shadow price: at the robust witness ∂(−E_*)/∂ℓ = −6464057/4850304 ≈ −1.333 on M + ℓ = 1, and ∂σ₀/∂ℓ = −1/4, so each 10⁻⁴ of margin added there buys about 1.9 × 10⁻⁵ of target. Removing a fraction of the row-count excess buys about 0.0083 per unit (section 6), down to 13/15.
2. The direct-bound corner M − 2ℓ = 1/2 (per summand) and M + ℓ = 1. Tight at the paper's geometry. The first is a bookkeeping choice that total accounting removes, but removing it buys nothing while (1) binds.
3. The floor rows, C₀ ≤ 0. Slack 1/48 at the paper's point; binds only once (1) is gone, at 13/15.
4. The slot supply ℓ/h > 7/37. Slack 23/1443; a lower bound, never binding.

The frozen-constant inventory

The information class

• Inside the current data, recomputation only: re-choosing ℓ, M, b, the contours and the floor. Exhausted: at most 5.8 × 10⁻⁵ below 7/8 with the paper's counts, and the corner 7/8 is where the direct bound puts it.
• Inside the current family, better inputs: stronger moment lemmas (larger capacities z_M = (1 − r)/2, z_P = (1 − 2m)/(6κ)), or a better amplification slope than (1 + 5r)/6, shrink the excess in R_*. The whole excess is worth 13/15 − 7/8 ≈ −0.0083, no more.
• Reading or building more: below 13/15 needs a different direct bound, a different Mellin signal or a different high-exponent shape: the h/6 comes from the pole of ζ_F(6z), and the floor rows are counted in full. That is a different probe, not a re-tuning of this one, and no candidate for it is proposed here.