Lemma 17.6, sixth-power amplification: its price and its loss map

Hunt #136, lane b78-amplification, 2026-10-21. Lane output, unreviewed by any person. Nothing here is a result until the case log in hunts/README.md says how the hunt ended. Files: price.py (the exact model: closed forms, an exact LP, an exact decision procedure on the whole bin rectangle, the enumerations), test_price.py (pins every fraction and decimal on this page and on guo_h11.md), guo_h11.md (Guo's Hypothesis 11.1).

Grades as in MISSION.md: cited (a line of a source file, quoted, the file's SHA-256 checked by price.verify_citations), derived (exact arithmetic here, pinned by a test), inferred (a reading or standard step no computation here checks), measured (float). A composite claim takes the grade of its weakest step. Nothing here bears on real part 1/2. No improvement of the half-plane is claimed. Every abscissa below is the best target of a stated comparison system with a stated, hypothetically changed input; the lemmas that supply the inputs are not checked.

Sources: the 7/8 preprint paper.tex (16,677 lines, SHA-256 42a5ee0f...a6a3); Liu, arXiv:2610.26835v1 number_fields_v2.tex (c55f17c6...626f); Guo's section 11 and Baier and Young as in guo_h11.md; hunt #134's probe.py read from origin/claude/hunt-134-seven-eighths-pin with git show.

Summary

1. What the lemma says and where it enters. For sixth-power-free rows of norm about U and the inverse polynomial M_u of length D = U^r, sum_u |M_u|^2 << U^(e(r) + eps) with e(r) = max{1, (1 + 5r)/6} (lines 12362 to 12389, cited). The Part II row counts use it only through the long count L(t) = 1 - delta + (alpha - delta)(t - 1), alpha = 5/6 (lines 15101, 15208), and only on 1 <= r <= t <= 3/2 (cited, line 15303). Write e(r) = max{1, e0 + alpha r}: the paper's family moves the slope with e0 = 1 - alpha; a mean square valid for H >= D^theta feeds the same proof and gives alpha = 5 theta/6, e0 = 1/6 (derived).
2. At hunt #134's optimum (the paper's row counts, total accounting, geometry free) the best target is (1507 - 2 sqrt 921)/1653 = 0.8749570698, and (derived, exact in Q(sqrt 921)) d(target)/d(alpha) = 7414/303601 - 181690 sqrt(921)/279616521 = 0.0047006135, d(target)/d(eta) = (sqrt 921 - 478)/551 = -0.8124356047 for an improvement eta of the direct exponent l_x/2 + b/12. Per unit, the direct exponent is worth 172.8360798139 times the slope. To reach 0.8748087 the slope must drop to 144263243506939/179612914571532 = 0.8031897030 in hunt #134's system and in the second-step frame P2, and to 0.8028731275 with the paper's side conditions as printed (frame P1, where the intermediate rows bind first); or the direct exponent must improve by eta = 0.0001826235 at the paper's slope.
3. Inside Liu's refined system, reconstructed exactly (his critical polynomial, his ell*, his kappa quartic and his B_flat quartic are reproduced, derived), the fixed point is B_flat = 0.8748866470 and, with the feedback through kappa = 2B - 1, dB/d(alpha) = 0.0073713973, dB/d(eta) = -0.7727666039. To reach 0.8748087 the fixed point needs alpha = 0.8229295767 (exact rational), but there kappa = 0.7496174000, below the range 7497/10000 <= kappa <= 3/4 of Liu's admissibility; inside that range a step at kappa = 7497/10000 needs alpha = 0.8229135719.
4. To reach 13/15. In hunt #134's system the slope must go to alpha = 0: the target exceeds 13/15 for every alpha > 0 and tends to it (derived). In the threshold family theta = 2/3 attains it, because then e(r) = 1 on the whole detector range and the counts are the ideal 1 - delta. With the paper's side conditions as printed no alpha reaches it: the ceiling is 84031/96156 = 0.8739028246 (frame P1) or 368153/422028 = 0.8723425934 (frame P2). Liu's system with its admissibility as stated stops at 437/500 (one step at kappa = 3/4, decided at alpha = 1/2 and alpha = 0), while Liu's endpoint alone, which has no bins below delta = 1/8, reaches 13/15 as soon as alpha <= 1/8 (derived).
5. Loss map. Every step of the lemma's proof either loses no exponent or is tight relative to the inputs it uses (section 3). The slope is (1 - 1/6) theta_0: the 1/6 is forced, because the only multipliers that leave every row character unchanged are sixth powers (exact enumeration), and theta_0 = 1 is the first width condition of Lemma 17.1, tight for every mean value valid for arbitrary coefficients and open for the Moebius coefficients. No slack step was found, so this lane proposes no sharper input.
6. The only door through this lemma is theta_0, and it is a zero-free-region door. A raw moment valid for H >= D^theta gives, through Lemma 17.6 at U = 1 alone, a zero-free half-plane Re s > 1/2 + 5 theta/12 for every fixed L(s, nu) (derived from the statement; Mellin step inferred). Below theta = 0.8898838649 that trivial route already beats the whole Part II assembly (frame P0). Guo's Hypothesis 11.1 is theta = 99/100 (guo_h11.md).
7. A side correction to hunt #134. At the paper's geometry, hunt #134 reports the endpoint margin 7/30672; that is the value at delta = 79/204, where the paper's completed square is centred. The minimum over the rectangle lies between 228147/10^9 and 228148/10^9 (derived, exact decision), a little lower. The paper's printed bound 49/440640 is unaffected.

1. The lemma and its use, read

• eq:raw-moment (lines 12345 to 12358, cited): Lemma 17.1 at Z = H, m = 1, no slots, gives sum_{q_u <= H} |M_u(D; W)|^2 << H (HD)^eps for H >= D^(1+c). Lemma 17.1 needs r + 2z <= m - c_1 and 2r + 8z <= 3m - c_2 (line 9254); with z = 0, m = 1 the first is the binding one, r = log_H D < 1.
• The proof (lines 12391 to 12469): scale supremum by a Sobolev inequality; at least c_S P amplifiers a with q_a <= P; the exact column identity M_u(D) = sum_{d | rad a} mu(d) psi_u(d) q_d^(-1/2) M_{u a^6}(D/q_d) (line 12428), from psi_{u a^6}(m) = psi_u(m) 1_{(m,a)=1} (line 12420); coefficient mass d(a) << P^delta (line 12430); injectivity of (u, a) -> u a^6 without coprimality (line 12443); q_{u a^6} << U P^6 << H; then sum_u |M_u|^2 << (H/P)(UD)^eps = U^(1/6) H^(5/6) (UD)^eps (line 12449) with H = max(2U, D^(1+c)), P = (H/U)^(1/6) (line 12369).
• The use (Proposition 19.1, lines 15249 to 15310; endpoint, lines 15928 to 15992): for r >= 1, Markov's inequality on the amplified moment gives the count 1 - alpha + (alpha - delta) r (line 15289); with r <= t this is L(t); balancing L(t) against the short count R_short(t) (line 15206) gives R_* = 1 - delta + (alpha - delta) delta P_x/(2 J) (line 16019). For a general law e(r) = max{1, e0 + alpha r} the same minimisation gives the excess R_* - (1 - delta) = delta P_x (delta_e - delta)_+ / (2 J) with delta_e = 3 alpha + 2 e0 - 2 and J = (alpha - delta) D_x + delta P_x (derived; price.excess, checked against a direct minimisation over t at 380 bin points for four laws).
• In the paper alpha = 5/6 coincides with 2(11/12) - 1 but plays a different role (hunt #134): the bins satisfy delta <= kappa (line 15503); alpha is only the amplification slope.

2. The price

2.1 The comparison systems, so that they can be rebuilt

System P (price.SystemP), variables (ell, M, b), objective the target 5/6 + (M - 2 ell)/12 - eta (hunt #134's closed form of 1 - h/6 + b/12, lowered by eta). An improvement eta of the direct exponent lowers the target by eta at fixed geometry and raises every high exponent by eta, since they are measured against Z^C(sigma_0) (lines 15694 to 15696). Constraints, every one affine in (ell, M, b) at a fixed bin:

• endpoint -E_* = -C_0 + delta(ell + b)/2 - ell x delta - h X >= 0 on the bin rectangle, with C_0 = 1/2 + ell/2 - 3M/4 + b/6 + eta, h = 1 + ell - M/2 + b/2, X the excess above, c = 4/9 (hunt #134);
• M + ell <= 1; 2M >= 1 + 3 ell (hunt #134's total accounting, inferred there); b >= 0; 8b <= 3(M - 2 ell) (Gram); l_x > ell.

Frame P0 is hunt #134's: bins 0 <= delta <= 5/6, 0 <= x <= 1/2. Frame P1 adds the paper's side conditions as printed: bins from delta = 1/50, the floor bin at delta_0 = 1/50 with count 1 (line 15847), the intermediate rows d <= 1/2 with R = 76/75 - 2 delta/3 up to delta = 5/6 (lines 16146 to 16165), the supply ell/h > 7/37 and 5 ell > h, the small rows. Frame P2 is P1 with every bin ceiling at delta <= 3/4, the bins of a step justified by beta_* <= 7/8 (as Guo and Liu use them).

System L (price.SystemL), Liu's refined system: variables (ell, b) with M = 1 - ell, target 11/12 - ell/4 - eta, endpoint (lines 8987 to 8994) E = C_0 + delta/2 + delta x {ell + c_kappa (h/4 - l_y/2)} - (3h/4)(1 - R_kappa) + eta <= 0 on 1/8 <= delta <= 3/4, C_0 = -1/4 + b/6 + 5 ell/4, c_kappa = 1/(3 kappa), with R_kappa built from the same law. Frame L0 has only b >= 0 and ell <= 1/5 besides; frame L1 adds delta <= 5/6, the small-delta endpoint (line 8999) and admissibility (1), (2), (4) (lines 9010 to 9023). The moment parameter is either fixed or tied at the fixed point kappa = 2B - 1 of Liu's chain.

Exact LP and decision. price.sandwich brackets the best target: an exact LP on a finite set of bins (a float LP proposes the basis; optimality is decided in rationals by primal feasibility of every row and nonnegative dual weights), cuts added at the worst bin until no float violation is left (lower end), and a nearby rational geometry whose constraints are decided on the whole rectangle by exact interval subdivision in x with the delta-quadratic minimised exactly (upper end). Every sandwich below has width under 10^-12.

2.2 At hunt #134's optimum

Only the endpoint at x = 1/2 binds, at the bin where the endpoint does not depend on b: X(delta, 1/2) = delta - 1/3, i.e. 288 delta^2 - 588 delta + 185 = 0 at the paper's law (derived). Closed forms (derived, exact in Q(sqrt 921)):

The target is B_alg of Guo and Liu (FOLLOWUPS.md), here as the optimum of hunt #134's system. By the envelope theorem d target/d alpha = mu h dX/d(alpha) with dX/d(alpha) = (delta P_x)^2/(2 J^2) at the critical bin, and d target/d eta = mu - 1; finite differences of the closed form agree.

In regime A the target is 11/12 - (5 - 6 delta)/(36(1 + 2 delta)), delta the root of 48 delta^2 - (90 alpha + 23) delta + 37 alpha = 0, so a rational target gives a rational delta and then a rational alpha: the alpha for 0.8748087 is 144263243506939/179612914571532, and for Xu and Sang's 0.8748867 it is 0.8186903585 (derived). The latter, and the ceiling 84031/96156 below, agree with this hunt's assembly lane (ASSEMBLY.md, an independent implementation, read uncommitted on 2026-10-21).

2.3 Inside Liu's refined system

The reconstruction (derived): from Liu's endpoint with the law left free, dE/db = 0 at x = 1/2 is X = delta (1 + c_kappa/6) - 4/9, whose numerator is -2 cr(delta, kappa) with Liu's cr (line 9694); there E = -5/12 + delta/2 + ell (3/4 + delta (1 + c_kappa)/2), so ell* = (5 - 6 delta)/(9 + 6 delta + 6 c_kappa delta), which equals Liu's ell*(delta, kappa) modulo cr; eliminating delta from cr = 0 and 11/12 - ell*/4 = (1 + kappa)/2 gives the factor 16686 kappa^4 - 27405 kappa^3 + 9936 kappa^2 + 1290 kappa - 275 (line 9721), and the root brackets Liu's B_flat quartic (line 8863). With that check the price, with the law generalised:

dF/d kappa = 0.0015104302 at the fixed point (Liu bounds it by 0.0065/4, line 9968), so the feedback factor 1/(1 - 2 dF/d kappa) changes the one-step price by under one percent. The slope is worth 1.57 times as much here as in system P. The counts enter Liu's endpoint with weight 3h/4 instead of h, but its ell-slope lambda = 1.0890028233 is smaller than system P's 3/4 + 3 delta_c/2 = (73 - sqrt 921)/32 = 1.3328755684 (both derived; which effect dominates is a reading of the formulas, inferred).

What alpha must reach for 0.8748087 (derived; at the fixed point kappa = 2 L - 1 is rational, the critical bin is rational and the critical equation is linear in alpha):

and the direct exponent alone at the paper's slope: eta = 0.0001008670 (fixed point).

2.4 The price curve

Best target, exact sandwiches (10 decimals; both ends agree to 10^-12), paper's law e0 = 1 - alpha:

At alpha = 3/4 the P0 value is 223/255 (the critical quadratic has a rational root there, delta_c = 37/96; derived). Active sets: P0 is still in regime A at alpha = 2/3 and sits on b = 0 by alpha = 1/2; P1 leaves regime A when the intermediate rows at delta = 5/6 bind (between 41/50 and 4/5), and from alpha = 1/2 down sits at the vertex of the intermediate rows, the floor bin and M + ell <= 1, which involves no row count; P2 does the same with the ceiling delta <= 3/4, from alpha = 1/3 down.

2.5 What alpha must reach

P1's entry is the exact bracket [336749397/419430400, 2155196141/2684354560] from bisection on sandwiches; the others are exact. The target is nondecreasing in alpha (every bin's excess grows with it, dX/d alpha = (delta P_x)^2/(2 J^2) > 0), which is what makes the brackets valid. For 13/15 in P0: the unique minimiser of the target over the floor condition, M + ell <= 1 and b >= 0 is (ell, M, b) = (1/5, 4/5, 0) (hunt #134), and there -E_*(delta, 1/2) = -h X < 0 for every 0 < delta < alpha; so the target exceeds 13/15 whenever alpha > 0 (derived), and the curve shows it tending to 13/15.

3. The loss map

Each step of the proof of Lemma 17.6 (and of the two places its output is used), with what it loses and a verdict. "Tight" means a configuration consistent with every input the step uses attains the loss; "no exponent" means only (UD)^eps, logarithms or constants are lost.

Reading. The exponent of the lemma is alpha = (1 - 1/6) theta_0 with theta_0 = 1: step 4 fixes 1 - 1/6, step 1 fixes theta_0, and step 7 is where the price is paid. Within the information the proof uses (a ball mean value, sixth-power invariance, |chi| <= 1), no step has slack: the max-plus configuration and, at U = 1, a hypothetical zero of a fixed L(s, nu) on Re s = 11/12 attain every bound. A smaller slope needs information the lemma does not read: how the ball's mass is spread over its rows, or a mean value with fewer rows than columns. No slack verdict is given, so no sharper input is proposed and the rival gate has nothing of this lane's to test; it is applied to Guo's hypothesis in guo_h11.md section 4.

4. The threshold door, priced

The proof of Lemma 17.6 turns a raw moment valid for H >= D^theta into e(r) = max{1, 1/6 + 5 theta r/6} (derived). Since only r <= 3/2 matters, theta <= 2/3 already gives e(r) = 1 there, the ideal counts 1 - delta and, in P0, exactly 13/15. Best targets (exact sandwiches) against the trivial route 1/2 + 5 theta/12 (Lemma 17.6 at U = 1):

The thresholds that reach 0.8748087 are theta = 0.9958275690 (P0, P2; exact rational 1292195122714234/1297609308016905, bracketed by sandwiches at plus and minus 10^-9) and 0.9982147398 (Liu, fixed point, with the same kappa caveat as for alpha). The trivial route wins below theta in [0.8898838649, 0.8898838701] in P0 and in [0.8936222185, 0.8936222258] in P2 (exact brackets): for a mean value that much stronger than the paper's, the Part II assembly is no longer the best use of it.

Information class. A raw moment with theta < 1 is a statement about the Moebius coefficients that no orthogonality argument can supply (step 1), and by the trivial route it carries a zero-free half-plane for every L(s, nu). So this door is not inside the data the current family reads: it needs zero-density or zero-free information about the row L-functions themselves.

5. Controls

• Citations: every snippet in price.CITATIONS sits on its stated line of its pinned file, and every file's SHA-256 matches (measured).
• Reproductions: B_alg, delta_c, ell*, b* equal Guo's and Liu's closed forms (FOLLOWUPS.md); hunt #134's single-point bounds 5655673/6464057 and 44117/50422 lie below the optimum and the paper's geometry lies above it; Liu's cr, ell*, both quartics; Guo's identity, squares and bound; the paper's R_* (line 16019), Liu's R (line 8438) and Guo's T_99 as cases of price.excess.
• Two exact decision procedures (interval subdivision in x, and exact root isolation of six polynomials) agree at the paper's geometry on the margins 0 and 49/440640 (accepted) and 1/4000 (refused); the subdivision procedure places the minimum there between 228147/10^9 and 228148/10^9.
• Planted faults turn the tests red: a slope of 5/6 + 1/100 breaks the B_alg reproduction; the weight h in place of 3h/4 breaks Liu's cr; a margin above the true minimum is refused by the decision; an unpinned number on either page is caught.

6. Limitations

• The paper's lemmas, Liu's and Guo's inputs are taken at their printed exponents and are not checked. Total accounting of the row norm (needed for every abscissa below 7/8) is hunt #134's reading, inferred.
• The general-law forms of the counts and of Liu's endpoint are readings of displays written at fixed numbers, checked by reproduction at the printed values.
• Frames P1 and P2 inherit the paper's frozen side conditions (the intermediate count, the floor at 51/100); their ceilings are about those choices, not about the method.
• Liu's fixed point below kappa = 7497/10000 uses his formulas outside the range where he verified his tangent family; that is flagged where it occurs.

The doors

Candidate status: the price of the amplification slope and of the direct exponent at two optima; parameter ranges as in section 2.1; proof status: exact arithmetic on transcribed formulas, unreviewed.

Active constraints at the optimum

1. The endpoint at the critical bin, x = 1/2 (delta_c = 0.3885837123 in P, 0.4693448587 in Liu), the only binding row family at both optima. Its multiplier is 0.1875643953 (P) and 0.2295678162 (Liu, 1/(4 lambda)).
2. M + ell <= 1, the dual length, always active.
3. Next to bind as alpha falls: in P1 the intermediate rows at delta = 5/6; in P2 the intermediate rows at delta = 3/4; in P0 b >= 0; in Liu's system as stated, the principal region B > 437/500 and the kappa range.

The frozen-constant inventory

The information class

• Inside the current data: re-choosing geometry, the bin ceiling, Liu's kappa range and the intermediate-row treatment. These move the abscissa only through the endpoint, at the prices above.
• Inside the current family, better inputs: the direct exponent, worth 172.8360798139 times the slope per unit at hunt #134's optimum, and the row-count inputs, which this hunt's other lanes price. The amplification slope itself is worth 0.0047006135 per unit there.
• Reading more: any smaller slope from this lemma needs a mean value with fewer rows than columns for the Moebius coefficients, which is zero-free-region strength by itself.