MISSION: is Lambda_DH exactly Delta^2/2, or is the gap real?

Opened 2026-08-18. Hunt #119 (renumbered from #52 on 2026-09-12). Nothing in this directory is a result until the case log in hunts/README.md says how it ended. Probe discipline as in hunts/lambda_dh_bounds/MISSION.md: the strongest words used here are measured (one float route), observed, decided (an enclosure whose exact endpoints settle a sign or an integer) and cited (somebody else's theorem). A composite claim takes the grade of its weakest step. The reserved enclosure word belongs to zeta/rigor.py and appears nowhere in this directory.

This is the theory phase. Nothing below has been measured on the Davenport-Heilbronn function at a new height. Sections 1 to 5 derive; the numerics quoted are polynomial-flow verifications of the derivations plus a calibration against landing times already measured in hunts/flow_repair/ and hunts/lambda_dh_bounds/census_results.json. Section 6 pre-registers what the later phases must find, with numbers, before any of them runs.

id: lambda_dh_exact
question: Does the crowding shave vanish along a sequence of Davenport-Heilbronn zeros whose depth approaches Delta, so that sup t* = Delta^2/2 and the bracket 0.0576 < Lambda_DH <= 0.19242481458 collapses to a point, or does the shave stay bounded away from zero so the gap is real?
frontier: narrow frame, decided both sides: 36/625 = 0.0576 < Lambda_DH <= 0.19242481458026887663805 = Delta^2/2 with Delta = 0.62036249819 sharp (Bombieri-Ghosh 2011 Theorem 7; necessary half re-derived and decided in STRIP2.md); ratio 3.341; best measured landing 0.05765184035 at gamma = 240.4046, y0 = 0.3695261
proposed_attack: derive the landing law dy/dt = -1/y - 2 pi rho from the N-body identity dQ/dt = 2 - 4Q sum 1/((x-a)^2 - Q), calibrate it on ten already-measured Davenport-Heilbronn landings, and decide whether sup over pairs of t* can reach Delta^2/2 given that depth near Delta forces great height and great height forces large rho
dead_routes:
  - raising the floor by surveying above height 600 without first raising the depth: the census of hunts/lambda_dh_bounds found six quadruples in (412, 600) and none beat 0.0576518, and the derived law says depth, not height, is the only variable that helps
  - reading the isolated-pair law y0^2/2 as a ceiling for an individual pair: six of three hundred two-pair configurations land later than y0^2/2 under exact polynomial flow, so only Delta^2/2 is a ceiling
  - the census's linear-in-log-gamma shave fit tstar = (y0^2/2)(1 - (A + B log gamma) y0^2): it is a local linearization of a law that is quadratic in log gamma, and it under-predicts the shave when extrapolated above the heights it was fitted on
  - float64 exact polynomial flow above degree about 30 with closely spaced roots: it reports a wrong landing time that looks fine, and the t = 0 admissibility gate is what catches it
required_oracles:
  - exact polynomial heat flow, coefficients evolved in closed form, checked against the closed-form quartic root
  - the argument principle applied with directed-rounding interval arithmetic on both ball backends
  - de Bruijn 1950 Theorem 13 and Dobner arXiv:2005.05142 Theorem 1, as published
  - Bombieri and Ghosh, Russian Math. Surveys 66:2 (2011) 221-270, Theorem 7 and section 9, as published
  - the landing times already measured in hunts/flow_repair and hunts/lambda_dh_bounds, used as a holdout rather than as training data where they were not fitted
kill_conditions:
  - a single Davenport-Heilbronn pair is found whose decided landing time exceeds the calibrated model prediction by more than a factor 1.25, which retires the model
  - the depth-versus-height law is falsified: a zero with y0 > 0.55 is found below height 10^4, or no zero with y0 > 0.40 is found below height 10^4
  - the no-creation step of section 2 fails, that is, a configuration is exhibited in which forward flow drives two real zeros off the axis, which breaks Lambda_DH = sup t* and reduces every landing time to a lower bound only
  - the sup of landing times is observed to increase with height rather than peak, over at least two decades of height, which reverses the verdict
  - the model's own consistency fails: its sup falls below the decided floor 36/625
agents_may:
  - search
  - derive
  - code
  - attack
  - formalize
  - measure landing times and depths at new heights
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - state a value for Lambda_DH; the deliverable is a bracket and a mechanism
  - edit anything outside hunts/lambda_dh_exact, figures/, and one new docs/NN file

0. What is already fixed, and what this hunt is allowed to move

Everything in this section is inherited, with its grade, from hunts/lambda_dh_bounds/. None of it is re-litigated here.

The one inherited fact this hunt is entirely built on, and it is cited, not derived here. Bombieri and Ghosh determine sigma(tau_+, 1) = 1.120362 as the least upper bound of the real parts of the zeros of F. STRIP2.md re-derives and decides only the necessary half of their Theorem 7, that is, no zero lies to the right of sigma_0'. Their converse, which is what makes Delta an attained-in-the-limit supremum rather than merely an upper bound, is neither used nor claimed anywhere in lambda_dh_bounds. This hunt's whole question presupposes that converse: without it there may simply be no zeros with y0 near Delta, and the bracket would be open for the dull reason that its upper end was never sharp. So the premise is cited and not decided, and every conditional below carries it.

1. Why every landing time is a lower bound for Lambda_DH

Phi_DH is real and even (measured to 4.2e-51 for |u| <= 0.5 at dps 50, flow_repair section 0), so H_t(conj z) = conj H_t(z) and the nonreal zeros of H_t occur in conjugate pairs. Write a pair as x(t) +- i y(t) with y(t) > 0.

Dobner (arXiv:2005.05142, Theorem 1) gives, for this class, that {t : H_t has only real zeros} is a closed half-line [Lambda_DH, inf). So if H_t has a nonreal zero then t < Lambda_DH.

Fix a pair P present at t = 0 and let

t*(P) := inf{ t >= 0 : the pair P is no longer nonreal } .

t*(P) is finite: Newman-Wu Theorem 7 (de Bruijn 1950 Theorem 13 restated) gives y_max(t) <= sqrt(max(Delta^2 - 2t, 0)), so every pair has reached the axis by t = Delta^2/2. For t < t*(P) the pair is nonreal, so H_t has a nonreal zero, so t < Lambda_DH. Letting t rise to t*(P),

Lambda_DH >= t*(P) for every pair P, hence Lambda_DH >= sup_P t*(P).

That is the whole content of item (a), and it needs nothing beyond Dobner's half-line. It is why flow_repair's nine measured landings are floors and why lambda_dh_bounds could turn one of them into a decided one.

2. Does Lambda_DH equal that sup? The no-creation step

Write T*_0 := sup_P t*(P) over the pairs present at t = 0, and T* := sup{t : H_t has a nonreal zero}. Dobner's half-line says Lambda_DH = T*. Section 1 says T* >= T*_0. Equality needs exactly one thing:

(NC) No creation. For t' > t >= 0, no nonreal zero of H_{t'} fails to be the continuation of a nonreal zero of H_t. Equivalently: forward flow never drives two real zeros off the axis.

Given (NC), for t > T*_0 every pair of H_0 has landed and nothing new is off the axis, so H_t has only real zeros, so Lambda_DH <= T*_0; with section 1, Lambda_DH = T*_0.

Why (NC) should hold, derived rather than asserted. In this repository's sign convention H_t = exp(-t d^2/dz^2) H_0, because d^2/dz^2 cos(zu) = -u^2 cos(zu) turns the kernel factor e^{t u^2} into exp(-t d^2/dz^2). Restricted to the real axis, u(x, t) := H_t(x) therefore solves

du/dt = - d^2u/dx^2 ,

which in the reversed time s = -t is the ordinary forward heat equation du/ds = d^2u/dx^2. Sturm's classical theorem on the heat equation (Sturm 1836; in the modern form Matano 1982, Angenent, J. reine angew. Math. 390 (1988) 79-96) says the number of sign changes of a solution on an interval is non-increasing in s, with interior zeros never created. Reversing time: under our flow the number of real sign changes is non-decreasing in t, and two real zeros can never merge and leave the axis in the interior. A landing does the opposite, converting one nonreal pair into two real zeros, which the theorem permits. So (NC) is the time-reversed lap-number statement.

What that derivation does not cover, stated plainly. Sturm's theorem in the Angenent form is proved on a bounded space-time rectangle for a solution that does not vanish on the lateral boundary. H_t is entire, not compactly supported, and lives on the whole line; the standard route for entire functions is Polya-Wiman theory rather than parabolic comparison. The derivation above is therefore a mechanism, not a proof for this object, and the honest statement is:

• Lambda_DH >= sup_P t*(P) is unconditional (section 1), and this is what every floor in this tree actually uses.
• Lambda_DH = sup_P t*(P) holds under (NC), for which the lap-number argument is the reason to expect it and a proof for entire functions of this class is not in hand here.

The numerical check, run. Exact polynomial heat flow (coefficients evolved in closed form by p_t = sum_k (-t)^k/k! p^{(2k)}, roots by numpy.roots, a t = 0 admissibility gate rejecting any configuration whose starting roots are not recovered):

• 400 of 400 admissible random configurations of 2 to 6 real zeros plus one conjugate pair at depth 0.5 to 3.0: the count of real zeros never decreased at any of 121 sampled times in [0, 3]. An earlier ungated sweep of 200 configurations gave the same answer.
• The four-root case (z^2-a^2)(z^2+Y^2) is exactly solvable and was used as a control. Its landing time is the positive root of 12t^2 - 2(Y^2-a^2)t - a^2Y^2 = 0, that is t_+ = [(Y^2-a^2) + sqrt((Y^2-a^2)^2 + 12 a^2 Y^2)]/12; the flow reproduces it to nine digits at (a, Y) = (0.1, 1.0), (0.05, 1.2), (0.3, 0.8), and the real pair survives in all three (four real zeros immediately after landing).

Verdict on (b). Lambda_DH = sup_P t*(P) under (NC), and (NC) is what the time-reversed Sturm zero-number theorem says and what 600 sampled polynomial configurations show. So the hunt's question is exactly a question about that sup. Without (NC) the sup is still a floor, so no phase of this hunt is wasted if (NC) turns out to be false: only the word "equals" is.

3. The isolated-pair law, derived and verified

Let H_0 carry a single conjugate pair at x +- i y0 and nothing else, so p_0(z) = (z-x)^2 + y0^2. Then p_0'' = 2 and p_0^{(4)} = 0, so the flow is a single term:

p_t(z) = exp(-t d^2/dz^2) p_0 = p_0 - t p_0'' = (z-x)^2 + y0^2 - 2t .

Roots x +- sqrt(2t - y0^2): nonreal for t < y0^2/2, a real double root at t = y0^2/2, two real roots after. Hence

t*(isolated) = y0^2 / 2, exactly.

Feeding y0 = Delta gives Delta^2/2, which is the same number de Bruijn's Theorem 13 produces from the strip. That coincidence is the reason the collapse question is worth asking at all: the upper bound is the isolated-pair landing time of a pair sitting at the very edge of the strip.

Verified, exact polynomial flow, bisection to 1e-13 on max |Im root| > 0:

(theory.py stage 1; the residual is the bisection tolerance, not a defect in the law.)

Cross-route, second implementation: zeta.heatflow.polynomial_heat_flow run on the real pair +-a flowed backward, which is the same law read in the other direction. Its collision_t came back at -a^2/2 to within exactly one grid step (2.5e-3) for a = 0.3, 0.7, 1.0. Two implementations, opposite directions, same constant.

4. The crowding correction, derived

4.1 The pair identity

For a pair z_1, z_2 the contour moments give q_1 = z_1 + z_2, q_2 = z_1^2 + z_2^2, and

Delta_pair := 2 q_2 - q_1^2 = (z_1 - z_2)^2 , Q := Delta_pair / 4 .

Off the axis z_{1,2} = x +- i y gives Q = -y^2; on the axis Q > 0; the landing is Q = 0, and Q is analytic through it, which is why flow_repair tracks Q and not y.

The zeros obey dz_k/dt = 2 sum_{j != k} 1/(z_k - z_j) (the sign pinned by zeta/heatflow.py's four-way check: exp(-t D^2) pairs with +2, roots repel). Write d = z_1 - z_2. The neighbour terms telescope,

1/(z_1-a) - 1/(z_2-a) = -d / ((z_1-a)(z_2-a)) ,

so dd/dt = 4/d - 2 d sum_a 1/((z_1-a)(z_2-a)), and with (z_1-a)(z_2-a) = (x-a)^2 - Q,

dQ/dt = 2 - 4 Q sum_a 1/((x-a)^2 - Q) .

That is the identity flow_repair section 3 used, re-derived here from the N-body law rather than recalled.

4.2 The shave, in one integral

Off the axis put Q = -y^2 and S(y) := sum_a 1/((x-a)^2 + y^2). Then dQ/dt = -2y dy/dt = 2 + 4 y^2 S(y), so

dy/dt = -(1 + 2 y^2 S(y)) / y ,

and separating,

t* = int_0^{y0} y dy / (1 + 2 y^2 S(y)) . (*)

With S = 0 this is y0^2/2, recovering section 3. When every neighbour is real, S > 0 and t* < y0^2/2: the shave is a theorem about the sign, not an empirical trend, and it explains flow_repair's P1 holding 9 of 9.

4.3 The leading term, and the answer to "nearest neighbour at distance d"

Expanding (*) for y0 small against the nearest-neighbour distance, with S_0 := S(0) = sum_a 1/(x-a)^2:

t* = y0^2/2 - S_0 y0^4/2 + O(y0^6) ,

relative shave = 1 - t*/(y0^2/2) = S_0 y0^2 + O(y0^4) = y0^2 sum_a 1/(x-a)^2 ,

and for a pair whose only close neighbours are two zeros at distance d on either side, S_0 = 2/d^2 and the leading term is 2 (y0/d)^2. The shave is the square of depth over neighbour distance. This is the derivation behind flow_repair's empirical sentence "the shave tracks y0^2 times local zero density", which was observed there and is derived here.

4.4 The deep regime, which is the one that matters

The leading term is useless for a deep pair, because S_0 y0^2 exceeds 1 long before y0 reaches Delta. Take instead y >> h, h the local mean gap. The sum becomes an integral over a sea of density rho = 1/h,

S(y) -> rho int da/((x-a)^2 + y^2) = pi rho / y ,

so 2 y^2 S = 2 pi rho y and (*) collapses to a law with no free parameter but the density:

dy/dt = -1/y - 2 pi rho , t* = [V - log(1 + V)] / (2 pi rho)^2 , V := 2 pi rho y0 = y0 * L ,

where L := 2 pi / h. Two limits, both worth stating:

• V << 1: t* = y0^2/2 - (2 pi rho) y0^3/3 + ..., the shallow regime.
• V >> 1: t* -> y0 / L. A deep pair lands in time proportional to depth and inversely proportional to the log-density, not to depth squared.

The mechanism is the constant -2 pi rho in dy/dt: a dense real sea pulls a pair toward the axis at a rate that does not care how deep it is.

4.5 The density is not a free parameter for this function

Deriving h rather than recalling it. F(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2) f(s) and N(T) ~ (1/pi) Im log[(pi/5)^{-(s+1)/2} Gamma((s+1)/2)] at s = 1/2 + iT. The first factor contributes (T/2) log(5/pi); Stirling on Gamma(3/4 + iT/2) contributes (T/2) log(T/2) - T/2. So

N(T) ~ (T / 2pi) log(5T / (2 pi e)) , h(T) = 2 pi / log(5T / (2 pi)) .

Checked twice against counts this tree already made, and the right comparison is with the total strip count, since S sums over all zeros:

zeta/epstein.py's _dh_mean_spacing carries the same formula, which is a third witness and not an independent one.

4.6 Calibration, and what it says about where a Davenport-Heilbronn pair sits

The pure lattice at phase 1/2 (a pair sitting midway between two line zeros) over-predicts the shave badly: it reproduces the nine measured landings only to a factor 1.03 to 1.42, always low. Giving the model one parameter, a local gap of half-width d with lattice spacing h beyond it, and fitting d on each of the nine:

d / h = 1.4284 mean, sd 0.0749, range [1.3350, 1.5682] over nine pairs.

Held fixed at the mean, that one-parameter model reproduces all nine to

pred/meas in [0.9942, 1.0094], rms deviation 0.54% ,

and on a holdout it was not fitted to, the census pair at gamma = 531.27972689652, y0 = 0.34695380309204904, whose landing 0.05033975468118168 was measured by the N-body null control:

calibrated model 0.049944269, pred/meas 0.9921 (0.79% low) , census's own linear-in-log-gamma fit 0.048403, pred/meas 0.9615 .

Two things follow, and the second is a correction.

1. An off-line pair of F sits in a locally sparse patch. The nearest line zero is about 1.43 mean gaps away, against 0.5 for a generic point. Natural, since the quadruple takes zeros off the line locally, and checkable directly at a new height.
2. The census's shave model is a local linearization. Its form tstar = (y0^2/2)(1 - (A + B log gamma) y0^2) is the section 4.3 leading term with S_0 linear in log gamma; but S_0 ~ 1.74/h^2 is quadratic in log gamma. The two agree across the heights it was fitted on (S_0 of 1.22 versus 1.27 at gamma = 245, 1.62 versus 1.64 at gamma = 545) and part company above them (3.55 versus 2.99 at gamma = 10^4). Extrapolating the linear fit under-predicts the shave.

4.7 The bias of the frozen-neighbour approximation, measured

() freezes the neighbours. They move. Exact polynomial flow against () with the same neighbour list, degree kept at or below 30 and every configuration gated at t = 0:

The frozen model always under-predicts t*, by 1% at light crowding and saturating near 17% at heavy crowding, because the neighbours repel away from the pair while it descends. The DH calibration of 4.6 absorbs this into the fitted d/h, which is why d/h is bigger than the 0.5 a static lattice would give and why the calibrated model must not be read as a claim about actual neighbour positions until someone measures them.

Lesion, recorded because it fired silently. At degree 52 with spacing 0.5, float64 numpy.roots on the heat-evolved coefficient vector reported 14 nonreal roots at t = 0 where there are 2, and max |Im| of 0.967 where it is 0.600; the resulting landing time was wrong by a factor 3.3 and looked perfectly ordinary. The t = 0 admissibility gate catches it and every table above is gated. Any later phase using exact polynomial flow must gate.

5. What closes the bracket, and what keeps it open

Combining sections 2 and 4: under (NC),

Lambda_DH = sup over pairs of t*(y0, gamma), t* given by (*) with rho = rho(gamma).

t* is increasing in y0 at fixed gamma, and decreasing in gamma at fixed y0 (checked: at y0 = 0.35 the calibrated model runs 0.055394, 0.054569, 0.052795, 0.050357, 0.046041, 0.042932, 0.037521 across gamma = 86, 120, 240, 600, 3e3, 1e4, 1e5; at y0 = Delta, 0.147092 down to 0.074538 over the same heights). So the sup is a competition between depth, which helps, and height, which hurts.

5.1 The collapse criterion

The bracket collapses if there is a sequence of pairs with y0 -> Delta whose shave tends to 0, and (given 5.3's delay mechanism) that is the only route to collapse that does not require a pair to land later than its own y0^2/2, which needs a second pair sitting nearly above it. By 4.4 the shave tends to 0 only if V = y0 L(gamma) tends to 0, and with y0 -> Delta > 0 that forces L(gamma) -> 0, that is gamma bounded, in fact gamma -> 2 pi / 5 = 1.2566, far below the height of any zero.

But below any fixed height F has finitely many zeros, so a sequence with y0 -> Delta must have gamma -> infinity, hence L -> infinity, hence V -> infinity, hence by 4.4 t* -> y0/L -> 0.

The deepest pairs land the fastest. sup t* is not approached along a depth-maximizing sequence; it is attained at finite height and intermediate depth.

So the collapse scenario requires exactly what Bombieri and Ghosh's section 9 denies: deep zeros at low height. Their measurement is that for xi = 0 no zero with Re s > 1 occurs at all below height 10^4, and that reaching one needs the arguments of p^{it} aligned near 0 mod pi for hundreds of primes at once. The Davenport-Heilbronn function proper sits at the hard end of their difficulty scale: prime-sum targets are 0.2767872 for tau_- (easy, and its deepest zero below 10^4 is already at 99.60% of its Delta), 1.2940091 for tau_+, and 1.5707963 for xi = 0.

The delay route is not excluded by this argument and is priced separately in 5.3: two pairs both at depth Delta, 3.0 apart in the real coordinate, reach 93.55% of Delta^2/2, so a delayed configuration can come close. It needs two deep quadruples in near-coincidence, and quadruples arrive at about 0.032 per unit height, so it raises the sup by a few percent rather than closing the gap.

Therefore the pre-registered theory verdict is: the bracket does not collapse, Lambda_DH < Delta^2/2 strictly, and by a large factor.

5.2 The numerical criterion a later phase can test

Three tests, in increasing cost.

C1 (cheap, decides the shape). For the calibrated model, the height above which no pair, at any depth up to Delta, can land later than a given floor:

If the model is right, the entire search for a better floor lives below height about 3e6, and a floor above 0.12 can only come from below height about 600, where the census is already complete. C1 is falsified by any measured landing that violates a row.

C2 (the model ceiling). Even the most favourable configuration the strip allows, a pair at exactly y0 = Delta sitting at the lowest height at which F has any off-line zero (gamma = 85.6993), gives

t*_model(Delta, 85.6993) = 0.14709209 = 76.44% of Delta^2/2 .

So the calibrated model says crowding alone costs at least 23.6% of the upper bound, for any pair, anywhere. A measured landing above 0.15 would break the model outright; a measured landing above 0.1924 would break de Bruijn's theorem and would be a defect in the instrument, not a discovery.

C3 (the sup). Model the deepest available depth as y_max(gamma) = Delta (1 - c_p / L(gamma)^p), anchored at y_max(600) = 0.3695261, and maximize t*(y_max(gamma), gamma) over gamma:

Two things to notice. First, the answer is remarkably stable across a factor 4 in p: the sup lands in [0.056, 0.071] narrow, always between heights 3e3 and 1e4, always at depth 0.40 to 0.50. Second, p = 0.5 is already refuted by this tree's own decided floor: its sup 0.055750 is below 36/625 = 0.0576, and sup t* >= Lambda_DH >= 36/625 is decided. So the data forces p above about 0.6, and the surviving band is

sup t* in [0.058, 0.071] narrow = [0.232, 0.282] wide.

That is 30% to 37% of Delta^2/2. The bracket does not collapse; it closes from above, toward a value only a little over the current floor.

5.3 What would keep it open, or reverse the verdict

• (NC) false. Then sup_P t*(P) is a floor and not the constant, and a created pair could land later than anything present at t = 0.
• A deep pair at low height. One zero with y0 > 0.55 below height 10^4 refutes the depth-versus-height law and puts the sup back in play; the model would then give t* = 0.0789 at (0.55, 10^4), still well under Delta^2/2, but the shape of the argument would be gone.
• Delay by a deeper neighbour, which is real and is a mechanism this hunt found. If a neighbour is itself off-axis at u +- i v, its contribution to S is 2 Re[1/((x-u-iv)^2 + y^2)], whose sign is the sign of (x-u)^2 - v^2 + y^2: negative when v^2 > (x-u)^2 + y^2, that is when a deeper pair sits nearly above the tracked one. Then crowding delays the landing and t* can exceed y0^2/2. Measured, exact polynomial flow: 6 of 300 admissible two-pair configurations landed later than y0^2/2. Two conjugate pairs both at depth Delta, separated by 3.0 in the real coordinate, land last at 0.18000977, which is 93.55% of Delta^2/2. So the de Bruijn bound is nearly saturated by pairs of deep pairs, not by one. With DH quadruples arriving at about 0.032 per unit height near gamma 500, two of them within the required proximity is a few-percent event, so this is a correction to the sup rather than a reversal, but it is the mechanism most likely to make the model low.
• The cited converse fails. If Bombieri and Ghosh's supremum is not approached by actual zeros of F, the upper end Delta^2/2 was never sharp and the gap is uninteresting rather than real.

6. Pre-registered predictions

Registered 2026-08-18, before any Davenport-Heilbronn evaluation at a height above 600. Every number is from the calibrated model of section 4.6 or from the laws of sections 3 and 4, both fixed before this list was written.

• P1 (the headline). The bracket does not collapse: sup t* < 0.75 * Delta^2/2 = 0.14432 narrow. Stronger form, the one being bet on: sup t* lies in [0.058, 0.075] narrow, that is [0.232, 0.300] wide. Refuted by any decided landing above 0.075 narrow.

• P2 (the deepest zero below height 10^4). y_max(10^4) lands in [0.40, 0.52], point estimate 0.45, that is beta_max in [0.90, 1.02] with point estimate 0.95. Below 0.40 or above 0.52 refutes. For scale: y_max(600) = 0.3695261 today, and the strip ceiling is Delta = 0.62036249819.

• P3 (the shave at that depth). The calibrated model at (y0, gamma) = (0.45, 10^4) predicts t* = 0.061084, a shave of 39.7% against the isolated y0^2/2 = 0.10125. Predicted band for the measured landing of whatever the deepest pair below 10^4 turns out to be: model prediction to within a factor [0.85, 1.25]. Outside that band the model is retired (kill condition 1).

• P4 (the floor rises, modestly). A complete census to height 10^4 raises the best measured landing from 0.05765184035 to a value in [0.055, 0.079], point estimate 0.061, which is where the model's maximum over the whole grid below 10^4 sits (0.061084 at (0.45, 10^4), 0.060861 at (0.4214, 3e3)). It does not reach 0.10.

• P5 (where the sup sits). The maximizing pair has height between 10^3 and 10^5 and depth between 0.40 and 0.52. Landing times measured above height 10^6 are all below the current floor 0.0576518; above gamma = 3.05e6 no pair at any depth up to Delta can beat it. This is the falsifiable core: t* versus height must peak and then decline.

• P6 (the local gap is a real structural fact, not a fitting artefact). At a newly located off-line quadruple, the distance from the pair's real coordinate to the nearest line zero, divided by h(gamma), lands in [1.1, 1.8] rather than near 0.5. Measured on the nine fitted pairs it is 1.4284 +- 0.0749; this predicts it out of sample, and it is cheap.

• P7 (no creation). No configuration will be found, in F or in polynomial surrogates, in which forward flow drives two real zeros off the axis. Standing at 0 of 600 sampled polynomial configurations.

• P8 (the delay mechanism appears in F). At least one Davenport-Heilbronn pair below height 10^4 will be found whose measured landing exceeds its own y0^2/2, because a deeper quadruple sits nearly above it. Expected rate a few percent of quadruples. If the rate exceeds 15%, the section 5.2 sup is too low and P1's band must widen upward.

• P9 (what does not happen). No number produced by this hunt is evidence for or against the Riemann hypothesis (docs/08; Littlewood). Lambda_DH is a property of a function where RH is already known to be false, and flow_repair section 3 already measured that the landing clock reads zero geometry and not arithmetic.

6b. Reproduction

.venv/bin/python hunts/lambda_dh_exact/theory.py # about 3 minutes .venv/bin/python hunts/lambda_dh_exact/theory.py --full # wider grids

Seven stages matching sections 2 to 5 above, writing hunts/lambda_dh_exact/theory_results.json. Every polynomial-flow table is gated at t = 0; the degree-52 lesion of section 4.7 is exercised as a negative control inside stage 2, so the gate is checked to fire rather than assumed to.

7. Scope

This hunt may write: hunts/lambda_dh_exact/, figures/, and one new docs/NN-*.md if it closes with something worth a document. Nothing in zeta/, ontology/, harness/, meta/ or lean/ without explicit permission, and nothing in hunts/lambda_dh_bounds/ or hunts/flow_repair/, whose files are read as evidence here and must stay as they were recorded.

Handoff: hunts/lambda_dh_bounds/GATE.md is the adjudication this hunt inherits, STRIP2.md is where Delta comes from, SEPARATION.md is the claim that must not be disturbed (it rests on the floor, which this hunt can only raise), and hunts/flow_repair/NOTES.md is where the nine landings and the N-body null control live.

Grades: everything in sections 3, 4.7 and 5.2 is measured, float route, polynomial surrogates. Everything in section 4.6 is measured plus a fit. The inherited constants in section 0 are as graded there. Section 2's (NC) is derived with a named gap and is the one place where the hunt's central sentence is conditional.