RESULTS: hunt #119, `lambda_dh_exact`

Status: OPEN. There is no result here, and this file exists to say so precisely rather than to leave the question to a reader's charity.

The hunt asked whether hunt #61's bracket

0.0576 < Lambda_DH <= 0.19242481458026887663805 (narrow frame)

collapses to its upper endpoint Delta^2/2. It derived a mechanism that predicts no, registered nine numbered predictions in MISSION.md before evaluating any of them, ran one screen at height 10^6, and then stopped: the three evaluation agents and the adjudicator all terminated on API 529 errors on 2026-08-18. Compiled 2026-09-12 from what reached disk.

1. What the hunt is entitled to say

Nothing about Lambda_DH. Hunt #61's bracket is unchanged by anything here, in either direction and in every digit. The pre-registered verdict is an expectation that was never scored, and MISSION.md labels it as one.

2. What was measured, and at what grade

Every number below is measured: one float route, no enclosure carried, no independent implementation, and no adversary. The two reserved regimes of this repository are absent from this directory by construction.

3. The one control that did run, and what it shows

The screen's inner abscissa was varied over a fixed height range, which the argument principle constrains: widening the contour can only add flagged windows, never remove one.

The sets nest strictly, 0.85's inside 0.75's inside 0.55's, and the one window the shallowest screen flags below height 600 is [228, 248]: the window holding the pair at gamma = 240.4046 that hunts/flow_repair measured independently and earlier. That is a real agreement and it was not arranged. It is also the only control this hunt ran on its instrument.

Reproduce the table from the committed artifacts, without re-screening:

.venv/bin/python - <<'PY'
import json, sys
sys.path.insert(0, "hunts/lambda_dh_exact")
import deep_zeros as dz
main = json.load(open("hunts/lambda_dh_exact/deep_zeros.json"))["screen"]
ctl = json.load(open("hunts/lambda_dh_exact/deep_zeros_control.json"))
w85 = [w for w in dz.flagged_windows(main) if w["t_hi"] <= 600.0]
for name, fw in [("0.85", w85),
                 ("0.75", dz.flagged_windows(ctl["screen_c75"])),
                 ("0.55", dz.flagged_windows(ctl["screen_c55"]))]:
    print(name, len(fw), sum(w["count"] for w in fw))
PY

4. The directional datum, stated as weakly as it deserves

The mechanism says depth should fall as height rises, because crowding shaves the landing time. The two deepest depths this laboratory has located are:

Shallower at four thousand times the height, which is the direction the theory predicts. Two points are not a trend, and the second point is censored by its own instrument. Three separate reasons not to lean on that row, the third of which was found while writing the doors section and is the worst of them:

1. The second screen covers 1200 units of height at one place on the line, and it is one zero rather than a supremum over a decade. Prediction 4 in MISSION.md asks for a fitted exponent that nobody fitted.
2. The zero is float grade, not an enclosure.
3. The screen's inner abscissa is Re s = 0.85, so it cannot see any zero of depth y0 <= 0.35 at all, and the zero it found has y0 = 0.35831185907344154. That is 0.0083 inside the wall. So the measurement is left-censored at almost exactly the value it returned: it cannot distinguish "the deepest zero near height 10^6 has depth 0.3583" from "every other zero there is shallower than the instrument's floor and this is the only one visible". A depth below 0.35 is what the theory predicts at that height, and it is precisely the outcome this screen was incapable of reporting. Re-screening below 0.85 is the only way to turn the row into evidence, and the information class below puts it in the class that requires new data.

Read this row as consistency, not as evidence.

5. Limitations, in the order a referee would raise them

1. Nothing was adjudicated and nothing was attacked. No adversary read any number in this directory. Hunt #61's numbers went through four adversaries before they were quoted; none of these did.
2. The zero is float grade. Winding count 1 with defect 2.2e-16 is a convincing float computation and not an enclosure. The instrument that could decide it exists in hunts/lambda_dh_bounds and was not pointed at this zero.
3. The screens are incomplete as a survey. screen_hi over [10^4, 10^5] stopped mid-sweep with complete: false. Neither it nor the low screen ever ran stage_locate, so their 21 and 88 flagged windows are candidate locations and nothing more: no zero in them is located and no window in them is excluded. Any sentence of the form "no deeper zero exists below height X" is unsupported by this directory.
4. The model's reach exceeds its training set. The calibration is fitted at y0 <= 0.37 and the question is about y0 approaching Delta = 0.62036249819. The ceiling 0.147 is an extrapolation across nearly a factor of two in the variable that matters, from a one-parameter fit. MISSION.md's own dead routes already record one earlier fit of this family failing exactly this way when extrapolated.
5. The no-creation step is sampled, not proved. 400 trials with 0 failures is support. The step it supports, Lambda_DH = sup t*, is what converts every landing time from a lower bound into the constant itself; without a proof, every landing time remains a lower bound only. Kill condition 3 is written against precisely this.
6. Lesion evidence is published, not resolved. lesion_degree52 records a planted fault that the detector's own health metric does not flag correctly, inherited from the same blind spot documented in hunts/lambda_dh_bounds/M2-LEMMA.md.

6. Kill conditions: which fired

None, because none was evaluated. Listing them explicitly so that "no kill condition fired" is not misread as "the hunt survived its kill conditions":

7. What whoever resumes should do first

In this order, because each step makes the next one worth doing:

1. Decide the height-10^6 zero with ball arithmetic on both backends, using the winding instrument from hunts/lambda_dh_bounds. One enclosure turns the only evaluation datum from float grade to decided.
2. Run stage_locate on the completed low screen. Its 21 flagged windows are already paid for and would give the depth distribution below height 10^4 that kill condition 2 needs.
3. Finish or abandon screen_hi explicitly. A screen marked complete: false in a committed artifact is an invitation to misread it.
4. Only then extend the depth-versus-height fit, and only with the exponent prediction 4 registered.

Nothing in this directory is evidence about the Riemann Hypothesis, and nothing in it is evidence about Lambda_DH either.

The doors

Required of every hunt that measures a ceiling (CLAUDE.md, "Door analysis: what every ceiling hunt owes"), and it earns its place here rather than filling a slot: the ceiling this hunt reports is 0.7644 of hunt #61's upper bound, and writing this section out is what showed that one of the five kill conditions already fires.

Active constraints at the optimum

The model's predicted range for sup t*, pre-registered in MISSION.md as [0.058, 0.075] narrow, is not a confidence interval. It is the span swept by one unfitted exponent. theory.py stage 6 assumes a depth-versus-height law

y_max(gamma) = Delta (1 - c_p / L(gamma)^p), L(gamma) = log(5 gamma / 2 pi)

and reports the sup separately for p = 0.5, 1.0, 1.5, 2.0:

Ranked by how much each binds:

1. p, and nothing else comes close. It is the only quantity in the model that moves the sup across the whole registered range, and it is the only one nobody fitted. Its lowest sampled value is already refuted, not by data but by the hunt's own decided floor: at p = 0.5 the model's sup falls below 36/625, which is kill condition 5. So the registered range is really the p = 1 to 2 span, and its lower end is set by an assumption rather than a measurement.
2. The height at which the ceiling is quoted. The headline 0.14709208930872253 is t_star_gap(Delta, gamma = 85.6993), evaluated at the lowest height in flow_repair's census, which is the most favourable height available. It is the value of the landing law at one height, not a supremum over heights. Nothing in the hunt shows that no height gives more, and the crossover table says the opposite of a comfortable margin: the height above which no pair at any depth up to Delta can beat the floor 0.0576518 is 3.05e6, and this hunt's only located zero sits at 1.0e6.
3. The sup is a maximum over an eight-point grid (600, 3e3, 1e4, 1e5, 1e6, 1e8, 1e12, 1e20), and for p = 0.5 and p = 1.0 it is attained at 3e3, an interior grid point with neighbours a factor 5 and 3 away. The true maximum is off-grid and the reported sup is a lower bound on the model's own sup.

The frozen-constant inventory

Every number in the construction that was chosen rather than optimized, with what relaxing it trades against. The first is the only one with real trade shape.

The information class of each door

Whether a door stays inside the data this hunt already holds, or requires reading more.

Inside the current data, recomputation only. The lattice phase, the truncation K, the sup grid resolution, and re-fitting DBAR with its spread rather than its mean. All four are answerable from the committed artifacts and theory.py alone, none needs a new zero, and together they decide whether the reported sup is the model's real sup or an artifact of three convenient choices. Do these first because they are nearly free and they bound how much the expensive door can be worth.

Inside the current data, but underexploited. stage_locate on the completed low screen. Its 21 flagged windows are already paid for and would give the depth distribution below height 10^4, which is the input kill condition 2 asks for and the beginning of a fit for p. This is the highest ratio of value to cost in the whole hunt: no new screening, one stage of an existing script.

Requires reading more. The exponent p itself, past what 21 windows can say. A fit needs depths across at least two decades of height, which means finishing screen_hi over [10^4, 10^5] and screening at least one decade above it, at roughly the 763 s per 1200 units of height that screen_1e6 measured. Also in this class: re-screening below Re s = 0.85, which is the only way to learn whether the height-10^6 zero's depth of 0.3583 is the real depth there or the instrument's wall, and deciding that zero with ball arithmetic, which changes its grade rather than the model.

The door to go through next is stage_locate on the low screen, because it is inside data already held, it is the input p needs, and it is the cheapest thing here that can move a number.