Assembly lane: an exact model of OpenAI's Part II final assembly

Hunt #136, lane "assembly", 2026-10-21. Status: lane output, revised after verifier round 1, unreviewed by any person. Nothing here is a result until the case log in hunts/README.md says how the hunt ended. Files: assembly.py (the model and its solvers) and test_assembly.py. The tests pin the numbers on this page: every decimal and fraction in the text is checked, sign included, against a whitelist in test_assembly.py whose entries are asserted by its tests (or are cited constants of the paper, input values and external reference lines, each marked as such where it appears), and the three tables of sections 4 to 6 are checked cell by cell against the computed values. For prose numbers the check is a whitelist, so it catches an unpinned or mistyped number, not two pinned numbers swapped in a sentence.

Grades as in MISSION.md: cited (a line of the pinned paper.tex, sha256 42a5ee0f...c6a3, checked by assembly.verify_citations), derived (exact arithmetic here, pinned by a test), inferred (a reading no computation checks), measured (float output). A composite claim takes the grade of its weakest step. Nothing on this page is about real part 1/2 or RH, and no improvement of the half-plane is claimed: every abscissa below is a statement about this one assembly with stated inputs, and the lemmas that supply those inputs are not checked here.

Two readings carry every abscissa below 7/8 on this page. Both are inferred, and a sub-7/8 number is graded "conditional on R1 and R2" wherever it appears.

• R1, total accounting (hunt #134): only the unrescaled summand of the low estimate must be lossless (Lemma probe-row-norm, lines 8306 and 8325). Under the paper's own per-summand bookkeeping the best abscissa is exactly 7/8 (section 2).
• R2, the shared lemmas below their range. Every high row of the model is the exponent of Lemma bin-contour-accounting (line 6090), and the floor, small and large rows come from Lemma fixed-bin-contour, the principal-signal lemma and Lemma outer-row-tails. All four are stated only for sigma_0 in [7/8, 1) with beta_* > sigma_0 (lines 5811, 6024, 6178, 6345), and the paper invokes them at sigma_0 = 7/8 (lines 15532, 15823). Using them at a target near 0.87496, or lower, is the bootstrap clause of MISSION dead route 1. The Euler-region inputs that plug in 7/8 are in the same position; hunt #134 (RESULTS section 7) read them as harmless for targets above about 2/3, also inferred. This lane does not check that any of these proofs survive a lower target.

Summary

1. The model. The final section (lines 15495 to 16463) and the lemmas it cites are rewritten with seven exponents as symbols: the marked capacity slope c_M (paper 1/2), the plain capacity slope c_P (2/9), the amplification slope alpha (5/6), the floor-bin count R_fl (1), the Gram exponent g behind P_a^{1/12} (1/6), the sextic pole z_p (1/6), and, though it is not a row count, the pointwise reflected-numerator cost n of U^{n delta} (1/2, line 15801), the factor this hunt's FOLLOWUPS.md reports the public follow-ups changed. With the geometry (ell, M, b) free, every high exponent is affine in the geometry once the bin data (delta, x) are fixed, so the best abscissa of the assembly is a semi-infinite linear program in three variables. 72 reproduction checks (printed constants and printed identities, rebuilt from the general formulas) agree, and 123 citations sit on their stated lines of the pinned file (derived, cited).
2. The controls reproduce exactly. 7/8 is the target at the paper's geometry and is exactly the best abscissa under the paper's own row-norm bookkeeping (a dual witness gives 7/8 from below, the paper's geometry gives it from above). Also 49/440640 with its completed square, the worst point 7/30672, #134's one-witness bound 5655673/6464057 (0.8749417) and its Gram-limited 44117/50422, #134's scan 0.874958 (the optimum with b held at 3/25 is 0.8749576643), and the perfect-count ceiling 13/15 (derived; the sub-7/8 ones conditional on R1 and R2).
3. The re-optimized assembly has a closed-form optimum. With the paper's inputs the best abscissa is sigma_* = (1507 - 2 sqrt 921)/1653 = 0.8749570698, at delta_* = (49 - sqrt 921)/48, ell = 0.1668383875, b = 0.1234054370, M = 1 - ell (derived, conditional on R1 and R2). The KKT multipliers form an exact dual witness in Q(sqrt 921) whose bound is sigma_* itself; a rational geometry checked exactly on the whole (delta, x) rectangle lies less than 2 parts in 10^14 above it; an LP proposed from a float cut set and decided exactly brackets it within 2 parts in 10^8. #134's 0.8749417 is a lower bound with slack, and its scan value sits at b = 3/25.
4. Guo's number matches this re-optimization with the plain capacity tied to the target. Setting c_P = 1/(6 (2 sigma - 1)) (the dynamic kappa of line 15100 carried below 3/4) and solving the fixed point gives 0.87495701942 (exact bracket of width below 10^-18), which agrees with Guo's 0.8749570194 in all ten printed digits. Measured agreement; that Guo's improvement is exactly this is inferred, since his paper was not read here. Beyond R1 and R2 it uses Lemma plain below its stated range kappa >= 3/4 (line 12532).
5. Shadow prices (derived, exact in Q(sqrt 921); conditional on R1 and R2). Per unit of each input at the optimum: z_p -0.8119602997, n +0.0591792953, g +0.0501294854, c_M -0.0095710837, alpha +0.0047006135, c_P -0.0019776984, R_fl 0 (the floor is slack). Ranked by size: z_p, n, g, c_M, alpha, c_P, R_fl, and the elasticities rank the same. Of the three row-count inputs the marked capacity leads, but the numerator cost is worth about six times more per unit (ratio 6.18). Finite differences of the closed form agree.
6. To beat Xu and Sang (0.8748867) one input alone must move by z_p +0.0000866553, n -0.0011898023, g -0.0013738421, c_M +0.0073016702, alpha -0.0146429748 or c_P +0.0342796610 (derived, conditional on R1 and R2; each crossing lies in regime A, the paper's neighbourhood). To reach 0.87: z_p to 0.1727198769, n to 0.3708105098, g to between 0.0890072152 and 0.0890074160, or alpha to 0.2946252341; c_M and c_P cannot, alone or together, with alpha at 5/6. To reach 13/15: z_p to 0.1767414708 or g to 181/4290; not n alone (every n >= 0 stays at or above 659/759), and not the row-count inputs, alone or together (every value stays at or above 167/192). Each is a candidate about a hypothetically stronger input, not a claim that such an input exists.
7. The reach of the row-count inputs as written is 167/192 = 0.8697916667 (derived, a ceiling of the frozen frame, conditional on R1 and R2). The low side, the floor bin (line 15847), the dual length and b >= 0 carry an exact dual witness that involves none of c_M, c_P, alpha. The bound is attained exactly at every alpha up to 27/100 (checked there; the endpoint rows only weaken as alpha falls), and the regime-B closed form reaches it at alpha = 0.2757692192. The capacities alone (alpha at 5/6) stop at 0.8740524834, set by rows with no prime amplitude (x = 0). #134's 13/15 returns only when the floor bin also goes to delta_0 -> 0 (relaxation inferred). These are ceilings of the frame this lane froze: the bin ceiling is held at 5/6 with no margin, while the paper's bins satisfy delta <= kappa = 2 beta_* - 1 (lines 6825, 15100) and evaluate the intermediate bound at 5/6 only as a convenience (line 16165).
8. The constraint that becomes active next (derived in the model, exact brackets or closed forms; the abscissae there are below 7/8, so conditional on R1 and R2): along z_p and g, the floor bin (z_p = 0.1751510556, g = 0.1567188215); along c_M, the supply hypothesis of Proposition detector-counts (c_M = 0.5361872194); along n, the intermediate range (n = 0.4908887159); along alpha, the offset b reaching 0 (alpha = 0.6098049914), then the floor bin (alpha = 0.2757692192); along c_P, the binding amplitude ratio leaves x = 1/2 (c_P = 0.5261221985).

Corrections after verifier round 1

The first version of this page (the lane run of 2026-10-21 that started at 14:36Z) had three errors that change claims and several overstatements. Each is corrected in place above and below; this is the list.

1. The intermediate row was imposed outside its range (verifier issue 1). The paper takes t = 1 with no slots in d <= 1/2 only "for delta <= alpha" (line 16146), and Proposition detector-counts is stated for delta <= alpha (line 15188). Bins with delta >= alpha carry R = 1 - delta at every d <= h (lines 15458 to 15463, line 16168; inferred for delta > alpha), which the endpoint row already holds. The model imposed the intermediate row at delta = 5/6 for every alpha. It now imposes it at delta = min(alpha, 5/6). At the paper's alpha = 5/6 nothing changes. Along the alpha axis five statements were false and are withdrawn: that the row-count inputs together are capped at 84031/96156; that c_M, c_P and alpha cannot reach 0.87 alone or together; that the alpha path meets 84031/96156 and stays; that the intermediate range binds first along alpha; and that 84031/96156 is attained at alpha = 1/2. In the corrected model the abscissa at alpha = 1/2 is 0.8722266776. What survives of 84031/96156 is the four-row bound for alpha >= 5/6 (section 7); Frozen(intermediate="at_delta_max") reproduces the old frame.
2. Sub-7/8 abscissae rested on a second unstated reading (issue 2): R2 above.
3. The exact rectangle check had a sampling hole (issue 3). The sampler skipped a gap between two roots whenever sympy's isolating intervals shared an endpoint, so it could pass a polynomial negative between two close roots. It now samples every interval endpoint and checks each closed cell between samples with a Sturm count, bisecting any cell that holds two roots. Every stored upper-end geometry in the tests was re-checked with the repaired check and passes (the verifier's own dense grid, in an independent model, also found them feasible), so no number moved. The upper ends are derived on the repaired check.
4. The markdown gate ignored negative numbers (issue 4) and was described as stronger than it was. It now reads signs and checks three tables cell by cell; the header says what it cannot catch.
5. Two planted code faults stayed green (issue 5). In the repaired sampler soundness rests on the Sturm guard alone; the interval endpoints and midpoints only make it fast. So the verifier's first fault, gap midpoints removed, is now harmless, and in scratch-copy runs of this repair (not tests of the suite) removing the midpoints, the shared endpoints, or both, left every test green, as it should. A reconstruction of the 14:36Z sampler (from the verifier's description) turned three of the four sampler tests red, and removing the guard turned test_sturm_guard_alone_decides red. The second fault, skipping the bins with delta >= alpha, goes red by the test hook and by deleting the branch. A further gap found in the repair: reading the bins delta >= alpha at d = h instead of d = d_min inside the check of the rows at d < h stayed green, and now goes red (test_planted_fault_inner_large_delta_rows_go_red).
6. The capacity ceiling was called attained at c_M = 9/10 (issue 6) without saying that Lemma marked's second width condition (line 9254) fails there. With z = c_M (1 - r) over the requested r, whose least value sits at x = 1/2 (or at x = 0, where r = 2/3, once c_P > c_M), that condition 2r + 8z <= 3m - c_2 holds only for c_M <= (5 - 4 c_P)/(9 - 8 c_P) and c_M <= 5/8, which is 37/65 at c_P = 2/9 (derived). Every regime-A statement along c_M sits below 37/65; the attainment needs a hypothetical marked lemma without that condition.
7. The reduction of every row to d = h assumed n large (issue 7). The d-slope of each row is at least (1 - z_0) - (1 - n) delta, nonnegative for every bin once n >= 26/125. Below that the rows at d < h are now checked exactly (inner_rows_ok), and the n-axis formula is checked with them.
8. The ceilings belong to the frozen frame (issue 7, second part): item 7 of the summary.

1. The model

Everything is in assembly.py; every function is exact rational arithmetic and accepts Fractions, sympy expressions or the quadratic-field numbers QF.

The general forms (derived from the cited displays, checked by reproduction at the paper's values):

• target: sigma_0 = 1 - z_p h + (g/2) b, with h = 1 + ell - M/2 + b/2; the row-norm loss adds ((1 + 3 ell - 2M)/8)_+ under total accounting (R1), while the paper's own bookkeeping requires M - 2 ell >= 1/2.
• counts: D_x = 3 - 2x(c_M + 2 c_P), P_x = (2 - 4 c_P x)(1 - 2 c_M x); the crossing of the marked and plain counts, R_short(t) = 1 - delta + (delta P_x/D_x)(3/2 - t) (independent of r, as at line 15417) and R_* = 1 - delta + K/2 with K = (alpha - delta) delta P_x / ((alpha - delta) D_x + delta P_x) (line 16019) for delta < alpha; R = 1 - delta for delta >= alpha (cited at delta = alpha, line 16129; for delta > alpha an inferred reading of the remark at lines 15448 to 15467, which the paper never needs because its alpha equals its bin ceiling).
• high side, measured against the matched target: the endpoint -E_* = -C_0 + delta(ell + b)/2 - ell x delta - (h/2) K - h delta (n - 1/2) with C_0 = 1/2 + ell/2 - 3M/4 + b(1/4 - g/2); the floor E(h) at delta_0 = 1/50 with count R_fl; the intermediate range at d = 1/2 with the no-slot count 76/75 - 2 delta/3, for delta up to min(alpha, 5/6) only (line 16146); small rows h(z_0 - z_p) - l_y/2 + 2 d_min; the supply hypothesis that ell/h covers the largest requested capacity (7/37 at the paper's values); the Gram condition b(3 - 2g) <= M - 2 ell; M + ell <= 1; b >= 0.
• rows at d < h: each row is affine in d, with slope R + n delta - z_0 at least (1 - z_0) - (1 - n) delta, so for n >= 26/125 every row is bounded by its d = h end (derived; the paper states the slope at n = 1/2, line 16133). Below 26/125 inner_rows_ok checks the other ends: the slot rows at d = 1/2, the bins delta >= alpha, the intermediate rows and the floor at d = d_min.
• the comparison used at line 16101: E_actual(h) - Delta <= E_* - (1 - h/4) Delta, so the argument closes for every small Delta > 0 exactly when E_* <= 0 on every bin.

z_p cancels from every high exponent measured against the target (the signal and the high side both carry h z_p); the small-row bound, measured against C(beta_*) and slack, is the only place it stays. g enters only the low side and the Gram condition.

Four solvers. regime_A is the closed form when the only binding high constraint is the endpoint at x = 1/2 with b > 0: one tangency K(delta_*) = k_0 + k_1 delta_* (a quadratic in delta_*), then two linear equations for (ell, b). regime_B is the closed form once b has reached 0 (section 5). sandwich brackets the optimum at any rational inputs: a float LP over a refined cut set proposes a basis, the lower end is an exact rational dual witness and the upper end is a rational geometry whose feasibility on the whole rectangle delta in [1/50, 5/6], x in [0, 1/2] is decided exactly (times J the endpoint is quadratic in delta, so the check is root isolation in x, with every root-free piece sampled and every cell guarded by a Sturm count). lower_bound_from builds a dual witness from named rows only, which is how the ceilings below are shown to hold for every value of an input.

2. Controls

Planted faults (all go red, test_planted_*): a wrong value of any one input (g = 1/12, z_p = 1/5, alpha = 4/5, c_M = 1/3, c_P = 1/4, R_fl = 99/100, n = 1/3) breaks at least one reproduction; a wrong exponent changes the optimum; a geometry 10^-6 past the optimum and a target 10^-10 below sigma_* both fail the exact rectangle check; a geometry with M + ell > 1 or b < 0 is refused; a perturbed multiplier leaves a nonzero residual; the stored crossing dual witnesses fail when used at the wrong end; the sampler catches a cubic negative only between two roots 0.0028 apart, whose isolating intervals share an endpoint, and still catches it with the interval proposal emptied (the Sturm guard alone), while the guard switched off lets it through; the rectangle check fails a target violated only in the bins delta >= alpha, and passes it with that branch switched off; the check of the rows at d < h fails a target violated among them only by a slot row at d = 1/2 and passes one that only the endpoint violates (so pointing it at d = h instead goes red); and it fails a target violated among them only by the bins delta >= alpha at d = d_min (alpha = 2/3, n = 0), so reading those at d = h goes red too. In scratch-copy runs of this repair (recorded runs, not tests of the suite): undoing the delta <= alpha restriction of the intermediate row turned 14 tests red (the alpha axis, regime B, the four-row bound and both inner-row tests), a wrong K_B in regime B turned 10 red, and dropping the end x = 0 from the second width condition of Lemma marked turned the marked-width test red.

3. The optimum at the paper's inputs

Total accounting (R1), the shared lemmas at the re-optimized target (R2), the paper's inputs, every frozen choice as printed:

• sigma_* = (1507 - 2 sqrt 921)/1653 = 0.8749570698, delta_* = (49 - sqrt 921)/48; ell = 0.1668383875, b = 0.1234054370, h = 0.8119602997, M - 2 ell = 0.4994848376, which is below 1/2: the paper's own bookkeeping forbids this geometry, so every number in sections 3 to 7 rests on R1, and every one below 7/8 on R2 as well.
• multipliers: endpoint mu = 0.1875643953, low side 1 - mu, dual length nu = 0.0625214651. The dual witness with these weights on the low row, the endpoint row at delta_* and M + ell <= 1 has zero residual and bound exactly sigma_*.
• slacks of the other conditions at the optimum: the boundary x = 1/2 (as a d/dx sign) 0.0091161574, floor 0.0058792912, intermediate 0.0026581589, supply 0.0132242767.
• with the plain capacity tied to the target (fixed point, measured agreement with Guo): 0.87495701942.

4. Shadow prices and ranking

The envelope theorem at the regime-A optimum: for the row inputs the price is mu (h/2) dK, for g it is (1 - mu) b/2, for z_p it is -h exactly and for the numerator cost it is mu h delta_* (derived, exact in Q(sqrt 921), conditional on R1 and R2; a finite difference of the closed form at plus and minus 10^-9 agrees to 10^-8).

z_p and g are low-side constants of the sextic structure, not row counts: z_p = 1/6 is the pole of zeta_F(6z) and g = 1/6 comes from the sixth-power form of the exceptional Gram frequencies (K_R^{1/6}, line 8541). Neither is a free parameter of the method; their prices say how much the abscissa leans on them, not that a lemma could move them (inferred). The numerator cost n = 1/2 is the pointwise size of the reflected L-value at Re w = 1 - a; a scalar n below 1/2 stands in for a bound averaged over the bin, which is a change of structure, so its price is the marginal value of such a change, not of a sharper exponent (inferred).

5. The abscissa as a piecewise function of each input

Every abscissa in this section is below 7/8 and conditional on R1 and R2.

Regime A (the paper's neighbourhood), closed form. With g = z_p = 1/6 and only the row inputs moving, put rho = (3 - c_M - 2 c_P)/((2 - 2 c_P)(1 - c_M)) (37/14 at the paper). Then delta_* is the root in (0, alpha) of

(3 - 2 rho) d^2 + (2 rho alpha - alpha - 2/3 + 2 rho/3) d - 2 rho alpha/3 = 0, sigma = 11/12 - (5 - 6 delta_*) / (36 (1 + 2 delta_*)),

so c_M and c_P act only through rho (derived). For general g, z_p the tangency is K(delta) = k_0 + k_1 delta with k_1 = 2 z_p/g, k_0 = (2 z_p - 5g + 6 g z_p)/(3g) (a numerator cost n other than 1/2 subtracts 2(n - 1/2) from k_1), and (ell, b) solve two linear equations (assembly.regime_A). Regime A holds from the paper's value up to the first break:

Past the break:

• z_p (raised): the optimum is the floor vertex, which involves no z_p, so the abscissa is affine: sigma(z_p) = C - z_p H with H = (23762198 - 8190 sqrt(22829)) / 27128539 and C = 83492716/81385617 - 2240 sqrt(22829) / 27128539, at delta = -740/5573 + 35 sqrt(22829) / 11146, until z_p = 0.2735042735 where the endpoint multiplier reaches zero (derived). 0.87 is reached inside regime A at z_p = 0.1727198769; 13/15 at z_p = 28260493/155678520 - 329 sqrt(22829) / 10378568 = 0.1767414708, bracketed by dual witnesses at 0.1767413646 and 0.1767415654.
• n (lowered): (measured, float LP) the endpoint and the intermediate range together, then the floor and the intermediate range, where all rows are affine and sigma(n) = (2250 n^2 + 76293 n + 1137725) / (72 (600 n + 18397)) (84031/96156 at n = 1/2); b reaches 0 at n = 0.3484739370, and then sigma(n) = (18 n + 659) / (3 (6 n + 253)) (derived formulas). 0.87 is reached at n = -4301/500 + sqrt(181150489) / 1500 = 0.3708105098, bracketed by dual witnesses at 0.3708106102 and 0.3708104095. The last formula is the exact optimum at n = 0, 1/10, 1/5, 13/50, 3/10 and 1/3: at each the four-row dual witness gives it from below and its vertex geometry passes the rectangle check and the rows at d < h with no slack (derived at those six points; in between it is the same vertex, inferred). The floor rows weaken as n falls, so the n = 0 dual witness bounds every n >= 0 by 659/759 = 0.8682476943: n alone cannot reach 13/15.
• g (lowered): the floor vertex (now g-dependent) down to 0.0857605178, then the floor and the Gram condition: sigma(g) = (418 g - 1601) / (6 (106 g - 309)) down to g = 4/53, where the supply hypothesis binds: sigma(g) = (12922 g - 11513) / (6 (2650 g - 2221)) (derived). 0.87 falls in the floor-vertex regime, between g = 0.0890072152 and 0.0890074160; 13/15 at g = 181/4290 = 0.0421911422 in the supply regime. Dropping the uniform supply hypothesis removes a constraint, so 13/15 would then need a g at least as close to 1/6 (not computed).
• c_M (raised): the supply hypothesis of Proposition detector-counts binds: the largest capacity it must cover grows with c_M while ell/h cannot. A stronger capacity can always be used as a weaker one, so the abscissa is the running minimum of the fixed-capacity optimum; on a grid its minimum sits at c_M = 3023/5625 (0.537422, measured), where a checked geometry gives 0.8745869525 (derived upper end). With the supply hypothesis dropped entirely, regime A runs to 0.5468059250, the binding x then leaves 1/2 and the path flattens toward the capacity ceiling 0.8740524834 (path measured, ceiling derived). A per-bin hypothesis, which would only need the capacity at t_* (inferred), lies between the two frames and was not computed. The regime-A values along c_M and the envelope point lie below 37/65, where Lemma marked's second width condition stops holding; the flattening past 0.5468059250 was scanned only in float and may cross 37/65, and the ceiling's attainment at c_M = 9/10 lies well past it (section 7).
• alpha (lowered): regime B, closed form. Once b = 0 the endpoint at x = 1/2 needs ell <= (1 - K)/(5 + 3K) in every bin, so the binding bin maximises K: at delta_B = alpha (37 - sqrt 518)/23, K_B = alpha (714 - 28 sqrt 518)/529, and sigma = 11/12 - (1 - K_B)/(4 (5 + 3 K_B)) (derived, exact in Q(sqrt 518)). Regime B starts at alpha = 247/573 + 3 sqrt(518)/382 = 0.6098049914 (abscissa 18589/20919 - 14 sqrt(518)/20919), crosses 0.87 at alpha = 85/546 + 5 sqrt(518)/819 = 0.2946252341, and meets the floor at alpha = 51/350 + sqrt(518)/175 = 0.2757692192, where the abscissa is 167/192 and stays for every smaller alpha. At alpha = 3/5, 1/2, 2/5, 3/10 and 7/25 the regime-B dual witness gives the closed form exactly from below and a rational geometry checked on the whole rectangle lies within 2 parts in 10^13 above it; the generic vertex solver agrees. The intermediate row, now at delta = alpha, does not bind on this axis: its slack, 0.0026581589 at the paper's alpha, is larger at every other point checked exactly (three in regime A, five in regime B, three in the floor-limited stretch). An independent LP model (the verifier's, built from paper.tex without this code) gave values within 10^-8 below these at alpha = 4/5, the verifier's 0.6076005515, 1/2 and 3/10, as a grid relaxation should (measured agreement).
• c_P (raised): the binding amplitude ratio moves inside (0, 1/2) (derived); beyond, the capacity ceiling 0.8740524834 bounds the path (derived), which the float LP shows it reaching (measured). Lemma marked's second width condition holds along this whole axis: it needs c_M <= (5 - 4 c_P)/(9 - 8 c_P) from the amplitude end x = 1/2 and c_M <= 5/8 from x = 0, and c_M = 1/2 meets both for every c_P < 1 (derived).

6. What each input must reach, alone

Every entry is exact: regime-A crossings to 10^-12 from the closed form, the alpha crossing of 0.87 in closed form, the others from a dual witness above the level at one end and a checked geometry below it at the other. All are conditional on R1 and R2. The improvements in the first column are candidates about this assembly with a hypothetically stronger input, not claims that such an input exists.

7. Ceilings

All rows of this table are ceilings of the frozen frame (bin ceiling 5/6 with no Delta margin, the paper's floor bin, contour and supply hypothesis), conditional on R1 and R2.

The first is attained exactly: the geometry (3/16, 13/16, 0) passes every row at the abscissa 167/192 with no slack at alpha = 1/4, 1/5 and 27/100, and fails at alpha = 7/25, just past the floor point. The second is attained in the float LP at c_M = 9/10 with the supply hypothesis dropped, and a geometry checked on the whole rectangle confirms it there; but at c_M = 9/10 the second width condition of Lemma marked fails (2r + 8z - 3 = 215/149, which must be negative), so that attainment is a statement about a marked lemma without that condition, not about Lemma marked.

8. Limitations

• The inputs are the lemmas' conclusions at their printed exponents; nothing here checks Lemma marked, Lemma plain, the amplification lemma, the Gram bound or the detector.
• Every abscissa below 7/8 rests on R1 and R2, both inferred. Under the paper's own bookkeeping the abscissa is exactly 7/8 and every row-input price is zero.
• The general-parameter forms of the supply hypothesis, the intermediate count and the floor are readings of displays written at fixed numbers; they reproduce the printed values.
• The model carries only the first width condition of Lemma marked; the second binds above c_M = 37/65 (at c_P = 2/9) and is checked only as a separate test.
• The bin ceiling is frozen at 5/6 with no margin. A dynamic treatment, in which a bin with delta above 2 sigma_0 - 1 exists only when Delta is large enough and so carries a margin of order (1 - h/4) Delta, would relax every large-delta row; it is not modelled here.
• The c_M envelope minimum, the c_P path past its break and the n-axis stretch between regime A and the floor-and-intermediate regime are float scans, labelled measured.
• Guo, Liu, and Xu and Sang are reference lines from the campaign's external map; their papers were not read in this lane. This hunt's FOLLOWUPS.md records a smaller public value, 0.8748087 (not checked here); the "beat 0.8748867" column uses MISSION's reference line and was not recomputed against it.
• For delta > alpha the count 1 - delta is an inferred reading of the remark at lines 15448 to 15467; it matters only on the alpha axis below 5/6.

The doors

Candidate status: the paper's inputs give the exact optimum 0.8749570698 over all geometries with the frozen choices as printed, conditional on R1 and R2 (both inferred); proof status: exact arithmetic on transcribed formulas, unreviewed.

Active constraints at the optimum

1. The endpoint inequality at (delta_*, x = 1/2), weight mu = 0.1875643953. Its inputs' prices are the table of section 4: z_p, n and g dominate, then c_M, alpha, c_P.
2. The dual length M + ell <= 1 (line 8246), price nu = 0.0625214651 per unit of relaxation.
3. Slack, in the order they bind as inputs improve: the floor bin (0.0058792912; first along z_p and g, second along alpha), the intermediate range (0.0026581589; first along n), the supply hypothesis (0.0132242767; first along c_M), the boundary x = 1/2 (0.0091161574; first along c_P). Along alpha the first to bind is b >= 0, which is not a slack of the optimum but the sign of b.

The frozen-constant inventory

The information class

• Inside the current data: the geometry and the frozen choices above. The geometry is exhausted at 0.8749570698. The intermediate range, the bin ceiling and the supply hypothesis are bookkeeping of the final section whose relaxation is a reading, not a new lemma; they are the cheapest doors, and the next binding constraints on the row-count path.
• Inside the current family, better inputs: c_M, c_P and alpha. Their combined reach as written is 167/192, attained through alpha alone; with the floor at delta_0 -> 0 as well, 13/15. The capacities alone stop at 0.8740524834, because rows with no prime amplitude get no help from them. A smaller alpha is a stronger amplification lemma, which this lane does not propose.
• Reading more: z_p and g are among the largest prices but belong to the sextic structure of the probe (the pole of zeta_F(6z), the sixth-power exceptional frequencies), which also fixes alpha; moving them is a different probe, not a stronger lemma (inferred). The numerator cost n carries the largest price that is not sextic structure; lowering it means bounding the reflected L-values on average over a bin instead of pointwise, new analytic input of the kind this hunt's FOLLOWUPS.md records in the public follow-ups. No candidate for such an input is proposed here.