GATE: two-sided bounds for Lambda_DH

Adjudicated 2026-08-16 (verdict NOT YET, five closure items). Re-adjudicated 2026-08-16 after the repairs, by a session that did not make them. Every field below is at post-repair state. Nothing was accepted on report: the two load-bearing scripts were re-run here and diffed against the pre-re-run files, the prior-art source the first pass called unread was retrieved and text-checked in this session, and the frame arithmetic was recomputed. What the first pass found and what each repair did to it is in the Closure log at the end, together with four places where the repairs fell short of their own standard.

Hardening pass, 2026-08-18, after this gate closed. Three tasks were run against the closed record and every field they touch now carries an update block with the superseded text kept verbatim beside it. (1) The upper bound sharpened by the full factor 2.082, and in-tree rather than by citation: STRIP2.md and strip2.py derive and decide the abscissa sigma_0' = 1.12036249819 from a phase obstruction in the two Euler products, on both backends. (2) M2 is no longer prose: M2-LEMMA.md proves it with decided constants and m2_lemma.py exercises it by four routes and two attacks. (3) The three open prior-art items are closed or reduced, and the novelty sentence survives unchanged. Nothing broke, no decided lower-bound number moved, and the verdict below is unchanged at YES. What did change is the referee's attack ordering, which is re-ranked honestly at the end of this file.

Vocabulary per MISSION.md: measured is one float route, decided is an enclosure whose exact endpoints settle a sign or an integer, cited is somebody else's theorem. A composite takes its weakest grade.

Exact object / normalization

Phi_DH(u) = 4 e^{3u/2} sum_{n>=1} n a_n exp(-pi n^2 e^{2u}/5) with a_n the period-5 coefficients (1, kappa, -kappa, -1, 0), H_t(z) = int_0^inf e^{t u^2} Phi_DH(u) cos(zu) du, and Lambda_DH := inf{t : H_t has only real zeros}; this is the frame in which H_0(z) = Xi_DH(z) = F(1/2 + iz) exactly, checked in the first pass against zeta.epstein.completed_dh at five points (relative 3.9e-32 to 5.8e-32).

This is the narrow frame, s = 1/2 + iz. It is Stopple's published one (arXiv:1301.3158, his Phi(u, chi) and Xi_t(x, chi) at D = 5, quoted verbatim in FRAME.md section 1a), and it is not the frame of Newman, de Bruijn as usually quoted, Rodgers-Tao, Polymath 15 or Dobner, all of which sit at s = (1+iz)/2. The conversion is now derived rather than asserted:

Phi_F(u) = Phi_DH(2u), xi_t^F((1+iz)/2) = H_{t/4}(z/2), Lambda(wide) = 4 Lambda(narrow), Delta(wide) = 2 Delta(narrow), Lambda / Delta^2 invariant.

I checked the derivation algebraically this session and it closes: Dobner's (5) inverts to Phi_DH(2u) because int_R Phi_DH e^{izu} du = 2 Xi_DH(z), and u = v/2 in his (6) turns e^{tu^2} into e^{(t/4)v^2} and cos(zu) into cos((z/2)v). The hunt's own numerical check of the same identity is 15 significant digits against 2 significant digits for the reading the old text licensed (FRAME.md section 5, THEOREM13.md section 6). Every number below travels with its frame.

Closest prior work

Stopple (arXiv:1301.3158) is nearest, is the published source of this hunt's normalization, and publishes -1.13e-7 < Lambda_Kr for quadratic Dirichlet L-functions, so any phrasing of the form "first quantitative bound on a non-zeta de Bruijn-Newman constant" is false and no longer appears. Dobner (arXiv:2005.05142) supplies existence, finiteness, nonnegativity and the closed half-line for all of S# with no numbers; de Bruijn 1950 Theorem 13 (typeset restatements: Dobner Theorem 3, Newman-Wu 2020 Theorem 7) supplies the Delta^2/2 engine.

Bombieri and Ghosh, Around the Davenport-Heilbronn function, Russian Math. Surveys 66:2 (2011), 221-270 is no longer unread. The first pass called it "unread and paywalled" and named it the likeliest place in print for sigma_0. It was retrieved and read between the two passes, and I retrieved and text-checked my own copy in this session rather than take that on report: https://www.mathnet.ru/eng/rm9410 carries a free "English version PDF (1572 kB)" link to getFT.phtml?jrnid=rm&paperid=9410&what=fullteng&option_lang=eng, which served 1,610,005 bytes, 50 pages, no login. On my own extraction:

• sigma(tau_+, 1) = 1.120362 is present, verbatim, in their section 6, as is sigma(tau_-, 1) = 2.3822861089...;
• the strings 1.395, 1.39513, 0.895136 and 2.4779 are absent;
• Bruijn, Newman, heat, Polya, Turan and deformation occur zero times in all 50 pages, case-insensitive.

So the verdict is mixed and both halves are now carried in the artifacts. On Lambda_DH the risk closes in the hunt's favour: the paper has no de Bruijn-Newman or heat-flow content of any kind, and the bracket is not anticipated. On sigma_0 it closes against the hunt: their tau_+ is this hunt's kappa (I confirmed -phi + sqrt(1+phi^2) against zeta.epstein.kappa to relative 1.2e-33), and their Theorem 7 determines the exact least upper bound of the real parts of the zeros of this function, which sigma_0 only bounds from above. sigma_0 is therefore not a new number and is no longer described as one anywhere.

Update 2026-08-18. The two halves above both stand, and the second one has changed character rather than direction. Bombieri and Ghosh's Theorem 7 is no longer the source of a sharper number this hunt cannot use: the necessary half of that theorem, which is all an upper bound needs, is now derived in-tree from the Euler products of the two Dirichlet L-functions plus one Moebius image, with no Bohr theory and no Kronecker theorem (STRIP2.md section 3), and the abscissa is decided on both backends. The equation is theirs and is not presented as new. What is new is the grade: the constant is decided here rather than adopted at six published decimals. Their converse, which turns the bound into an exact supremum, is not used and not claimed. Two of their published numbers are now reproduced as controls that share no machinery with their Theorem 7: the finite claim 6323 / 420 of their section 9, and sigma(tau_-, 1) = 2.3822861089 to all ten digits they print. sigma_0 is still not a new number, and neither is sigma_0'.

Surviving novelty

A strictly positive two-sided bracket, whose lower side is an enclosure-carrying zero count, for the de Bruijn-Newman constant of a Dirichlet series with a Riemann-type functional equation whose Riemann hypothesis is false. It is not the first non-zeta constant of this type (Stopple), not the first strictly positive one (Newman-Wu compute ln 2 exactly for a three-atom measure), and the upper side is a cited theorem applied to a constant computed here rather than a new mechanism. The adopted sentence is in NOVELTY.md section "The sentence", with the superseded one kept verbatim beside it:

So far as the literature search recorded in NOVELTY.md reaches, these are the first quantitative bounds, from either side, on the de Bruijn-Newman constant of a Dirichlet series with a Riemann-type functional equation whose Riemann hypothesis is false. They are stated in the normalization of Stopple (arXiv:1301.3158) ... and are four times smaller than the same constant in the normalization of Newman, Rodgers-Tao, Polymath 15 and Dobner.

The separation (named surviving item, added 2026-08-16, after this gate closed). In the common wide frame the decided floor beats the best published zeta upper bound: Lambda_zeta <= 0.22 (Polymath 15 Theorem 1.1, pinned at source in POLYMATH-PIN.md) while Lambda_DH > 0.2304 = 144/625 (this hunt), so Lambda_DH > Lambda_zeta unconditionally, with the exact rational core 144/625 > 11/50 (7200 > 6875, cross-multiplied) and the same inequality in the narrow frame (0.055 < 0.0576, the identical cross-multiplication). A dedicated novelty adversary killed the unqualified phrasing "first strict inequality between the de Bruijn-Newman constants of two Dirichlet series" against the function-field literature (Andrade-Chang-Miller arXiv:1310.3477; CMMRSY arXiv:1411.2071) and sanctioned the narrower claim: the first, so far as the searches reach, in which both constants are nonnegative. Full statement, chain, caveat list verbatim and grade: SEPARATION.md. Composite grade cited plus decided, weakest step cited.

The claim to sigma_0 is withdrawn and the claim to the bracket is what is left. The first pass named sigma_0 and the decided off-line zero of H_t at t = 36/625 as "the genuinely new numbers"; half of that is now known to be wrong and the artifacts say so. What survives on the upper side is the derivation, not the quantity: an enclosure-carrying elementary route to a weaker bound on a constant Bombieri and Ghosh determined in 2011.

Update 2026-08-18. The paragraph above is kept as written and one word in it is now wrong: weaker. The elementary route no longer stops short of their abscissa. STRIP2.md reaches it, decided on both backends, at sigma_0' = 1.12036249819, and the deep flint point at P = 10^7 decides 1.1203624981833251, which sits about 4e-17 above the decided lower end of the root enclosure. The withdrawal of the originality claim is unaffected: the quantity was determined in 2011 and this hunt reaches the same one. The novelty sentence and its footnote are unchanged; only the footnote's characterisation of the upper side as "a weaker bound on that same quantity" needed the same correction, and NOVELTY.md carries it.

Lower bound

0.0576 < Lambda_DH narrow, with 0.0576 = 36/625 exactly; 0.2304 < Lambda_DH wide, 0.2304 = 144/625. It rests on a decided argument-principle count N = 1 for H_{36/625} over a rectangle whose interior has Im z >= 3/1024 > 0 in exact rational arithmetic, then monotonicity for Lambda_DH >= t and Dobner's closed half-line for the strict inequality, transported through t -> t/4, which is an increasing bijection and so preserves a closed half-line.

Lower-bound status (rigorous / numerical / failed)

Rigorous modulo one prose lemma, which is named and lesioned. The two analytic steps that failed "proved in-tree, or a correctly-applied cited theorem whose hypotheses were verified" as written in the first pass are both repaired at the level of justification, with no computed number moving:

• the evenness of G, which cancels the vertical legs and is what makes M2 a bound at all, is now derived from F(s) = F(1-s) in three parts (Hecke's theta transformation for the odd primitive character mod 5 and the single real condition on kappa it forces; the Mellin transport showing the transformation and the functional equation are one fact, written in the sound direction Hecke -> functional equation and saying why the converse would need an inversion in a strip the two integrals do not share; and the identity theorem on |Im u| < pi/4, which is what the vertical legs at u = +/- is actually need and the old justification never reached). I checked the algebra: Phi_DH(u) = 4 x^{3/4} omega(x) at x = e^{2u}, so evenness is exactly omega(1/x) = x^{3/2} omega(x), which is what the file derives.
• the citation supplying strictness is now routed through the corrected conversion rather than through a false claim of frame identity.

M2 itself remains prose. It is the single load-bearing analytic step no cross-route exercises, and its failure mode is the one this routine promises never to produce. That is now a recorded blind spot with a measured onset rather than an unexamined assumption, and a necessary-not-sufficient guard sits in front of it with a refusal path. Read as a whole: enclosure-carrying count, cited theorem, one prose lemma inside.

Update 2026-08-18, after this gate closed. The paragraph above is superseded and is kept for the record. M2 is no longer prose and is no longer unexercised: M2-LEMMA.md proves it, with every constant a reported ball and every hypothesis a decided predicate, and four routes plus two falsification attacks stand behind it (m2_lemma.py). The correct reading of the lower bound is now: enclosure-carrying count, cited theorem, and one proved lemma whose only non-elementary input is the cited functional equation already carried as assumption 5. Nothing computed moved; see known assumption 6 for the full replacement text and for what is left.

Upper bound

Sharpened 2026-08-18, in-tree, by the full factor 2.082. The superseded statement is kept verbatim below it.

Lambda_DH <= 0.19242481458026887663805 narrow, <= 0.7696992583210755065522 wide. Both decimals are exact, not outward roundings: the abscissa is decided at the exact rational sigma_0' = 112036249819/100000000000 = 1.12036249819, so Delta = sigma_0' - 1/2 = 0.62036249819 is exact and Delta^2/2 = 3848496291605377532761/20000000000000000000000 terminates. The wide value is that one times exactly four. (STRIP2.md section 5.2 prints the narrow value rounded outward to 22 decimals, 0.1924248145802688766381; it is the same bound, one ulp of display above the exact value.) The ratio upper/lower is 3.3407085864630015 and is frame-free.

What decides the abscissa is a phase obstruction rather than coefficient domination: f = A L(s, chi) + conj(A) L(s, conj chi) with A = (1 - i kappa)/2, so a zero with Re s > 1 forces arg L(s,chi)/L(s,conj chi) = pi + 2 arctan kappa mod 2 pi, while each prime p = 2, 3 mod 5 can supply at most 2 arctan(p^{-sigma}) radians (a Moebius image of a disc, with the maximising point of modulus exactly 1, so the |R| = 1 constraint is free and no modulus-phase trade sharpens it further). The criterion is Theta(sigma) = sum_{p = 2,3 (5)} 2 arctan(p^{-sigma}) < pi - 2 arctan kappa, and Theta is decreasing, so one decided sigma closes the half-plane. The tail is closed by the Euler products of zeta and L(., chi5) themselves, so no prime-counting estimate enters.

Decided on both backends at P = 10^5, 4814 class primes: python-flint (Arb) 192 bits gives the root in [1.1203624981833869487276, 1.1203624981833869487332] (65 sign decisions) and mpmath.iv at dps 40 gives [1.1203624981833854, 1.1203624981841131] (38 sign decisions); the intervals overlap. At the headline rational, Theta = [2.5880182946402392454052004, 2.5880182946415650528147533] (flint) against the target [2.5880182946927479869541106, 2.5880182946927479869541107], margin 5.12e-11. A flint-only deep point (320 bits, P = 10^7, 332442 class primes) decides sigma = 1.1203624981833251, giving 0.1924248145761280189989039 narrow / 0.7696992583045120759956154 wide.

Superseded, kept verbatim (the state through 2026-08-17): "Lambda_DH <= 0.4006343708899557 narrow, <= 1.6025374835598228 wide. It is Delta^2/2 with Delta = sigma_0 - 1/2, rounded outward: the decided flint interval is [0.4006343708899556944469547527, 0.4006343708899556944469548120] and the headline decimal sits above its upper endpoint, so the inequality is safe; the wide interval is that one times four and its headline likewise sits above the endpoint. The ratio upper/lower is 6.9554578 and is frame-free."

Upper-bound status (rigorous / numerical / failed)

Rigorous, with the frame stated. The looseness the first pass recorded is removed, and what replaces it is a smaller and differently located looseness. Every numerical step is enclosure-carrying (kappa at 500 bits, sigma_0' bisected on both backends with exact rational sign decisions and the headline decided at an exact rational rather than at a bisection endpoint), the phase argument in STRIP2.md is elementary and complete, and Theorem 13's hypotheses hold for Phi_DH exactly as before. Sections 3(d) and 3(e) of STRIP.md, the gamma factor and the trivial zeros, are reused unchanged, so the two derivations share their reflection step and differ only in how the zero-free half-plane is reached.

Three things about the new route that a reader is entitled to before trusting it:

• The equation is Bombieri and Ghosh's, term for term (their Theorem 7 at q = 1, xi = kappa), and STRIP2.md section 7 says so. Only the necessary half is used and it is derived here; their converse, which makes the abscissa exact rather than an upper bound, is not used and not claimed. So the number is theirs and the grade is this hunt's.
• The instrument reproduces two of their published numbers by routes that share nothing with their Theorem 7: their section 9 finite claim (threshold prime 6323, cardinality 420) and the sibling abscissa sigma(tau_-, 1) = 2.38228610898712387152... against their published ten digits. Nothing in the instrument was built around the second constant.
• It forces one correction against an in-tree artifact, not against the paper. At P = 10^7 and 320 bits it decides that BOMBIERI-GHOSH.md check B's two 29-digit re-solves each sit on the wrong side of their own root, by about 1.2e-17 and 6e-18. Bombieri and Ghosh print six and ten decimals and this instrument reproduces both exactly; what moves is FRAME.md's 18-digit derived row, which agrees with the decided replacement to 17 digits.

The natural alternative sharpening was tried, decided, and rejected on the mathematics rather than on taste. Regrouping the series into period-5 blocks and applying the mean value theorem gives |B_k| <= |s|[3(5k+1)^{-sigma-1} + kappa(5k+2)^{-sigma-1}], correct and O(n^{-sigma-1}) per block, but the |s| is not an artifact (both pairs share the midpoint 5k+5/2 and their first-order terms add), so it yields a height-restricted strip that climbs back to the old sigma_0 = 1.3951361582... as T grows: 1.19585459 at T = 10, 1.39224106 at T = 10^5. de Bruijn's theorem consumes a half-plane statement, which this cannot give. It is recorded in STRIP2.md section 2 as decided and useless.

Superseded, kept verbatim (the state through 2026-08-17): "Rigorous, with the frame stated, and now known to be loose by a factor of at least 2.08. Every numerical step is enclosure-carrying (kappa at 500 bits, sigma_0 bisected on both backends with exact rational sign decisions, outward rounding checked by exact rational comparison), the strip argument in STRIP.md is elementary and complete including the boundary equality case and the trivial zeros, and Theorem 13's hypotheses hold for Phi_DH. What changed since the first pass is not the bound but what is known about it. Bombieri-Ghosh's sigma(tau_+, 1) = 1.120362 is the exact abscissa sigma_0 bounds, and feeding it through the same engine gives 0.192424814576128011 narrow / 0.769699258304512045 wide, a factor 2.08203. It is correctly not adopted, and the artifacts say why: their value is six published decimals of Mathematica output resting on Bohr-Kronecker machinery unverified in-tree, so the improvement is cited plus measured, not decided, and swapping it into a decided headline would launder a grade. I checked both halves of that judgement here. The arithmetic is right, but it is right against BOMBIERI-GHOSH.md's own 29-digit re-solve 1.12036249818332508773010350311, not against the six decimals the FRAME.md row cites as its input (those alone give 0.192424505522). And their root is reproducible: my own truncated sum over primes p = 2, 3 mod 5 with an integral tail correction converges to 1.1203623 at P = 10^7, matching all six published digits."

What did not change. The grade is unchanged at cited plus decided, weakest step cited, because the weakest step was and remains de Bruijn 1950 Theorem 13. STRIP.md remains correct and is kept: it is the weaker of two valid derivations and a reader should be able to see both.

Independent cross-check

Four, and the adjective is now measured rather than asserted. MISSION.md's "two independent winding routes" has been replaced everywhere by the radius that harness/independence.py reports, and INDEPENDENCE.md carries the layer lists so a reader can check them.

The first pass reported 8 of 11; independence_decl.py re-runs that coarser merge and reproduces 8 shared of 11, radius 8 exactly, so the two counts are one declaration at two granularities and not a disagreement. The invariant neither depends on: the two winding routes duplicate none of the evaluator.

The independence the claim actually rests on is routes 3 and 4, both of which were scratchpad code at the first pass and are now landed as reproducible scripts with results files. I re-ran both here. Route 3 (mpmath dps 130, no shared layer, 5814 quadrature nodes, 64 samples per edge) returns N = 1 on both published boxes, t = 23/400 and t = 36/625, reproducing with zero non-timing differences; that is the first and only second witness the headline value 0.0576 has. Its own margin is worth quoting because it is the tightest thing in the cross-check: the largest consecutive argument step is 0.3946 radians against the routine's 0.40 threshold, and the minimum |H_t| on the boundary is 8.64e-85, which is the scale that makes the count need balls rather than floats. Route 4 (Gauss-sum kappa, Hurwitz-zeta H_0, acb_series Taylor in t with an explicit remainder) overlaps instrument.H_ball at all eight boundary points of the t1 box; I re-ran it here, 8 of 8, agreement 40.3 to 42.7 decimal digits, which corrects the first pass's "about 22 significant digits" (that was the printed width, not a measurement).

Two things the radius does not carry, both recorded in INDEPENDENCE.md: route 2 ran at one t and its own box, so t = 36/625 has no route-2 witness at all; and route 3's kappa is a different implementation of the same equation, so only route 4 makes the kappa equation independent.

Known assumptions

1. de Bruijn 1950, Duke Math. J. 17, Theorem 13, all-zeros form, transcribed in THEOREM13.md by a visual read of an image-only scan. Independently re-read from the same scan by one adversary, and corroborated by two typeset restatements: Dobner's Theorem 3 and Newman-Wu 2020 Theorem 7.
2. Its hypotheses for Phi_DH: integrability; hermitian symmetry, here evenness, now derived in-tree from F(s) = F(1-s) including the complex-u case the vertical legs need, and measured at dps 260 to relative 1e-261 near the origin; and O(e^{-|u|^b}) with b > 2, discharged by a 201-point grid standing in for an unbounded-range claim. The fact itself is immediate from the artifact's own domination |Phi_DH(u)| <= 4 e^{3u/2} e^{-(pi/5)e^{2u}} S(0), so this stays a discharge gap and not a doubt.
3. Dobner 2020 Theorem 1, the closed half-line, used only for the strictness of the lower bound and now applied through the corrected conversion t -> t/4. S# membership of DH is verified condition by condition in-tree and by an adversary against Dobner's verbatim text. Without this citation only Lambda_DH >= 0.0576 survives, from in-tree monotonicity (Theorem 13 at Delta = 0) or Polya.
4. Dobner Theorem 2 (Lambda_F >= 0): quoted, load-bearing for neither side.
5. Classical facts used without in-tree proof: Gamma has no zeros and only the simple poles at s = -1, -3, ...; F is entire with F(s) = F(1-s). kappa is decided by an Arb linear solve and agrees to 40 digits with the Gauss-sum route, and to 33 digits with Bombieri-Ghosh's closed form tau_+ = -phi + sqrt(1+phi^2), which I checked here.

Added 2026-08-18, with the sharpened upper bound. The new route uses one further classical fact that the old one did not: the Euler products of L(s, chi) and L(s, conj chi) for the odd primitive character mod 5, absolutely convergent and nonvanishing on Re s > 1. That is textbook and is the only addition; the decomposition a_n = A chi(n) + conj(A) conj(chi)(n) it rides on is decided in-tree as five acb residual balls containing 0 with radius below 1e-40, with chi taken from flint's own Dirichlet character table rather than from a remembered list of values, and it is separately pinned by the suite through zeta/epstein.py's dh_f docstring. f itself has no Euler product, and none is used.

1. The analytic bound M2 in winding.py. Rewritten 2026-08-18: the blind spot recorded here is closed in the form it was recorded, and what replaces it is narrower and named. M2-LEMMA.md states the bound as Lemma M2 and proves it: differentiation under the integral sign with an explicit dominating function, Cauchy's theorem on the shifted contour with the far side bounded rather than asserted to vanish, the vertical-leg cancellation, separate proofs of the two majorants for the theta-like sum, and the panel-plus-tail split. Every constant of the proof is a reported Arb ball and every hypothesis is a decided predicate, so no step of it rests on an unverified numerical claim. It is exercised by four routes rather than none (m2_lemma.py, m2_lemma_results.json): an independent re-implementation of the bound, an unshifted majorant that reaches a valid bound without Cauchy's theorem and without the evenness, a pointwise Arb enclosure of H_t'' that makes the cushion decided, and the pre-existing float quadrature guard. Two attacks are wired in beside them: the identity H_t'' = -int e^{tu^2} Phi_DH(u) u^2 cos(zu) du is checked against second central differences of H_ball (worst relative gap 7.66e-07 against h^2 = 9.54e-07), and both majorants for the theta-like sum are attacked against a sharp truncated enclosure at 24 probe points with no refutation.

The independent re-derivation returns 1.1886319406e-78 at t1 against winding.py's 1.1886642645e-78, a relative -2.7e-05 traced to exactly one panel of 800, the crossover between the two majorants, where winding.py keeps the larger of the two valid bounds because its ball comparison does not decide. winding.py's value is therefore the more conservative one and no decided winding number moves.

The cushion stands and is now decided rather than measured: 55.65 at t1, 53.79 at t2, from a decided sup |H_t''| >= 2.1358e-80 (433-point grid of Arb enclosures) against M2 = 1.1887e-78. The old measured value 2.1357367685579024e-80 is reproduced to all 17 digits by an enclosure at the same point. Across a 40-unit span of Re z, over which |H_t''| falls by 14 orders of magnitude, the cushion stays between 33 and 204, because M2 depends on x_lo only through e^{-x_lo v} with v = pi/4 - 1/256 against the strip half-width pi/4. The cushion is a structural constant of the bound, not a property of these two rectangles.

What is left, stated at its true size. (i) One cited classical input is load-bearing for M2 and for nothing else in the route: the evenness Phi_DH(-u) = Phi_DH(u), which is Hecke's theta transformation plus F(s) = F(1-s) transported through the Mellin transform, that is, assumption 5 above. If it failed the vertical legs would not cancel and M2 would be false, and M2-LEMMA.md section 3 step 2 shows there is no numerical substitute: the quantity that must vanish is identically zero, so enclosing it to 1e-78 would take of order 1e78 subdivisions. (ii) The proof is written prose plus decided arithmetic, at the hardened rung. It is not kernel-checked and it has been read by no human. (iii) Control 5's lesion table is retained and its meaning has changed: with M2 proved it measures the guard's sensitivity to an implementation fault, not the exposure of an unproven assumption. The detector still cannot see a corrupted M2 by itself, which is why the guard and its refusal path stay.

Superseded text, kept verbatim (the state through 2026-08-17): "an in-tree shifted-contour derivation whose numerical ingredients are ball-computed and whose derivation is prose. It is exercised by no cross-route. Its stated reason for the evenness of G is repaired, and its docstring's cushion is corrected from 'three digits of slack' to the measured factor 55.7 (measured sup |H_t''| = 2.1357e-80 against M2 = 1.1887e-78; 53.9 at t2). It is now covered by two things that did not exist at the first pass: control 5, which lesions it, and winding.measured_h2_guard, a necessary-not-sufficient float sample of |H_t''| by quadrature of the defining integral, wired into winding.main() with a refusal that voids the floor if it fails. Standing blind spot, recorded rather than repaired; the lower bound carries it."

1. Residual float-grade steps, none load-bearing: the dps-112 locating pass that places the boxes; the WP3 census over heights 412 to 600; calibration.json's re-derivation of the Delta^2/2 dictionary. All three are labelled measured in the artifacts. Added 2026-08-18: three of STRIP2.md's eight controls are measured rather than decided (the series against the two L-functions, the phase lemma's circle sample, the Euler-product tail identity) and the artifact labels them so; none of the five decided ones, and none of the headline, depends on them.
2. Tooling: python-flint 0.9.0 (Arb), mpmath's iv context, and the exact-rational plumbing. Both first-pass complaints here are repaired. The mpmath.iv cross-leg now evaluates instrument._truncated_integrand, the callable the integrator actually receives, instead of phi_ball, which no decision path calls; the hoist moved no number (H_ball midpoint 4.3658433958660405e-83 and radius 1.4894837452822593e-124 are unchanged in validation.json, and both winding boxes reproduce every decided field). The two hand-derived tail bounds, which previously sat 36 to 41 orders below the delivered ball radius so that nothing could see them, now have a standing domination check against high-precision true remainders: 18 of 18 rows dominate, and the blindness factor is 8.02, so a tail bound deflated by less than about eight passes it. Both repairs are measured and neither can upgrade the thing it checks. There is still no second rigorous integrator; route 4 answers that as a route, not as a backend.
3. Prior art. Rewritten 2026-08-18: all three items this entry carried as open are now closed or reduced, and none of them touched a computed number. Bombieri-Ghosh 2011 was already read. The three that remained:

• academia.edu preprint 166936409 is read in full. Every earlier route failed again (direct fetch HTTP 403 under two user agents, no Wayback snapshot), but r.jina.ai against the full slug URL returned the record, which names the author, Mesut Ismail, and a DOI, 10.5281/zenodo.21679490. The paper is open access on Zenodo (Ismail_rh_pf_v18.4.pdf, 758,872 bytes, 2026-07-29) and was downloaded and read. Its subject is the classical wide-frame Lambda_zeta; Davenport-Heilbronn appears only as instrument and negative control. It contains no Lambda_DH, no bound on one from either side, and no claim about one. Its two DH off-line lifetimes, tau1 = 0.1819 and tau2 = 0.0449 in the wide frame, are labelled upper bounds on those lifetimes inside a Numerical Observation, which is the wrong direction to give a lower bound on Lambda_DH; taken at face value anyway, 0.1819 wide is 0.045475 narrow, below this hunt's decided floor of 0.2304 wide / 0.0576 narrow by a factor 1.267. Its witnesses are the same zeros zeta/epstein.py pins, and its frame was confirmed here by recomputing 2 h1^2 = 0.190365478578 (mpmath, mp.dps = 30, measured) against its tabulated 0.190. A side benefit: its Lemma 3.2 states this hunt's factor-4 frame dictionary independently, with the same warning that a clock off by 4 would "prove" Lambda <= 1/8. Risk closed, in the hunt's favour. Details in NOVELTY.md section 2.
• Bombieri-Mueller 2008 is identified and read at abstract and reference-list level. It is E. Bombieri and J. Mueller, On the zeros of certain Epstein zeta functions, Forum Math. 20:2 (2008), 359-385, DOI 10.1515/FORUM.2008.018, Zbl 1217.11040, MSC 11E45 and 11M41. Per the zbMATH summary it bounds the rate of approach of zeros to the boundary of the zero-free half-plane for Epstein zeta functions of class number 2, by Bohr's method for the lower side and a diophantine-type result for the upper. Its five deposited references name neither de Bruijn nor Newman. So: no de Bruijn-Newman or heat-flow content, and no rival value for the quantity sigma_0 bounds either, since it is a different family and a different quantity. Its full text is still unread (De Gruyter answers HTTP 202 behind a human-verification wall, r.jina.ai gets HTTP 405, no mirror found), so this is reduced to a small named residual, not fully closed.
• The Dobner forward-citation sweep is no longer single-source. It was re-run across three independent indexes plus a web sweep: Semantic Scholar (5 records, queried by DOI and arXiv id separately and agreeing), OpenCitations/COCI (1 record), Google Scholar (cluster 5851792251239807498, 2 distinct records before it rate-limited), and a web-search sweep that surfaced one preprint none of the three indexes carried. OpenAlex still returned HTTP 429 with $0 daily budget and retryAfter: 72359, and was not worked around. The union is 7 distinct citing works, none of which attaches a quantitative Lambda to any non-zeta object; the per-work verdicts are tabulated in NOVELTY.md. One citing item is unread rather than unfound: Voronov, A Crowding-Normalized Reformulation of Neighboring-Gap Dynamics for the de Bruijn-Newman Flow (ResearchGate, 2026), whose abstract places it on the zeta-side real spectrum and whose full text is behind Cloudflare.

New negative evidence, stronger than any single query. The sweep turned up Tao, Trudgian and Yang's ANTEDB (teorth.github.io/expdb), chapter 18 of which is The de Bruijn-Newman constant, works in this hunt's wide frame (H_0(z) = (1/8) xi(1/2 + iz/2)), and tabulates the complete known bound history from Newman 1976 to Rodgers-Tao/Dobner >= 0 on the lower side and Platt-Trudgian 2021 <= 0.2 on the upper. Every entry is zeta's. A maintained community database of de Bruijn-Newman results with no non-zeta constant in it is the best available evidence that none is in print.

Two consequences outside this item, recorded and not acted on here. ANTEDB's Lambda_zeta <= 0.2 (Platt-Trudgian 2021) is sharper than the <= 0.22 (Polymath 15) that NOVELTY.md and SEPARATION.md quote, so the separation corollary Lambda_DH > Lambda_zeta gains headroom rather than losing any (0.2304 > 0.2); SEPARATION.md should cite Platt-Trudgian. And FRAME.md now has an independent published source for its factor-4 dictionary.

The novelty sentence survives unchanged. Only its footnote and the one-line short form moved.

Reproduction command/artifacts

Backend first, from the repo root: .venv/bin/python -c "from zeta import rigor; print(rigor.BACKEND, rigor.available_backends())" must print python-flint. Then .venv/bin/python hunts/lambda_dh_bounds/{strip,strip2,validate,winding,winding_quad,controls,m2_lemma,census,calibrate_theorem13,independence_decl,crosscheck_quadfree,crosscheck_dhflow_winding}.py, each of which rewrites its own *_results.json. The claim's two decided numbers are strip2_results.json (headline.upper_bound_delta_sq_over_2_narrow, about one minute) and winding_results.json (decided_floor_t = 36/625, t1_run/t2_stretch_run both status: decided, N: 1). strip_results.json still holds the superseded and still-correct coefficient-domination constant, and m2_lemma.py (about 105 seconds) writes m2_lemma_results.json and touches no winding number.

The written record is RESULTS.md with results.json (each claim carrying value / frame / grade / backend / precision / source file). Sources and hypothesis checks: FRAME.md, STRIP.md, STRIP2.md, M2-LEMMA.md, THEOREM13.md, NOVELTY.md, BOMBIERI-GHOSH.md, INDEPENDENCE.md, SEPARATION.md, POLYMATH-PIN.md, KAPPA-CLOSED-FORM.md, MISSION.md. Adversary write-ups: attack_adversary1_normalization.md, attack_adversary3_upperbound.md, attack_adversary4_instrument.md, attack_adversary5_priorart.md.

What would a skeptical referee attack first

Re-ranked 2026-08-18. The 2026-08-16 ordering, kept below in full, put M2 first and the looseness of the upper side second. Both of those moved, in opposite directions from the reader's point of view: M2 became a proved lemma and the upper bound sharpened by 2.082 in-tree, so the two attacks the first ranking named as the best ones available are the two that were spent. What is left is dominated by things this directory cannot fix by computing harder, and the honest ranking says so. In order:

1. The citations, which are now unambiguously the weakest steps. The composite grade was always cited plus decided and the cited half is no longer sharing the stage with a prose lemma or a loose constant. Four citations carry the whole structure and none is verified in-tree: de Bruijn 1950 Theorem 13 in its all-zeros form, transcribed from an image-only scan (corroborated by an independent adversary re-read of the same scan and by two typeset restatements, Dobner Theorem 3 and Newman-Wu Theorem 7, but not by an authoritative typeset original); Dobner 2020 Theorem 1, which supplies the strictness of the lower bound and nothing else; the evenness Phi_DH(-u) = Phi_DH(u), which is Hecke's theta transformation plus F(s) = F(1-s) transported, on which the vertical-leg cancellation and hence the whole size of M2 depends, and for which M2-LEMMA.md section 3 step 2 shows there is no numerical substitute; and, for the separation only, Polymath 15 Theorem 1.1. An error in any one of them moves or erases a headline while moving nothing else in this directory.
2. Two new proofs, both written by agent sessions, both attacked only by scripts written in the same sessions, both read by no human. The phase obstruction of STRIP2.md section 3 and Lemma M2 of M2-LEMMA.md are the two pieces of mathematics this hunt now claims for itself, and each is prose plus decided arithmetic at the hardened rung, not kernel-checked. The specific places to press: in STRIP2.md, the Moebius-image lemma and the claim that arg of the absolutely convergent product is the sum of the factors' principal arguments modulo 2 pi; in M2-LEMMA.md, the contour-shift step and the two Omega majorants. Both files hand a referee their own falsification runs rather than making them build them, which is the right posture and is not the same thing as an outside reading.
3. The detector still cannot see a corrupted M2 by itself. This is item 1 of the old ranking, reduced to what survives: the lemma is proved but the implementation could still be wrong, control 5's lesion table still shows three wrong and silent integers from deflation 75, and the health metric still moves the wrong way across the lesion (chord margin 0.02 to 1.11). The two implementations agree exactly on 799 panels of 800 and differ on the crossover panel in the conservative direction, with winding.py holding the larger value, so no decided winding number moves; but a reader should know that the agreement is between two implementations and not between two mathematical routes. The guard and its refusal path stay.
4. What is left of the looseness, and where it now sits. The bracket ratio is 3.341, down from 6.955, and the strip constant is no longer where the slack is: the sieve-limit sweep shows the abscissa converging like P^{1-3 sigma}, so accuracy there is essentially free. Two honest gaps remain. The engine: with Delta = 0.62036249819 the de Bruijn theorem returns 0.19242481458 narrow while the deepest measured DH zeros reach |Im z| = 0.347, which would give 0.0602 against a decided floor of 0.0576, so most of the remaining factor is the engine plus the sparsity of the extreme zeros. And the converse: this argument bounds the supremum of the real parts of the zeros and does not show it is attained, which is Bombieri and Ghosh's converse and is neither used nor claimed. If their converse holds, Delta cannot be improved at all.
5. Whether the new derivation is independent of the paper it reproduces. STRIP2.md's criterion is Bombieri and Ghosh's Theorem 7 equation, term for term, at q = 1 and xi = kappa. The artifact states this plainly and claims only the grade rather than the equation, and it reproduces two of their published numbers by machinery their Theorem 7 does not share (the 6323 / 420 finite claim and sigma(tau_-, 1) to ten digits). A referee can still ask whether the necessary half was genuinely re-derived or reconstructed from a known answer. The answer available is the derivation itself, which uses only the Euler product, absolute convergence and one Moebius image, and the fact that the deliberately-tried alternative (the five-term block regrouping) failed and is recorded as failing.
6. The remaining prior art, largely spent. See item 3 of the old ranking below, whose 2026-08-18 update stands: what is left is the full text of Bombieri-Mueller 2008 (publisher wall) and of one citing preprint (Voronov 2026, Cloudflare), both unread, neither able to touch a computed number.

Superseded ranking, kept verbatim (2026-08-16, with its own 2026-08-18 in-place updates):

1. M2. It is prose, it is load-bearing, no cross-route touches it, and the artifacts now hand the referee the exact lesion table rather than making them build it. The honest reading of that table is the uncomfortable one: the guard trips at deflation 55.7 and the first wrong integer appears at 75, so the guard happens to fire first on this box, by luck rather than by structure, and a derivation wrong by a factor under 55 would pass it and could still be wrong. winding.py says exactly that.

Update 2026-08-18. The first two clauses are no longer true and the paragraph is kept for the record. M2 is proved in M2-LEMMA.md and is exercised by four routes. What a referee should attack instead, in order: the cited evenness Phi_DH(-u) = Phi_DH(u) on which the vertical-leg cancellation and hence the whole size of M2 depends; the fact that the proof is written prose rather than kernel-checked, and has been read by no human; and the implementation agreement between winding.py and m2_lemma.py, which is exact on 799 panels of 800 and differs on the crossover panel in the conservative direction. The lesion table survives as a measurement of the detector's sensitivity to a corrupted M2, which is still real: the winding routine cannot see one by itself.

1. Looseness of the upper side. 0.4006 is a factor 2.082 above a constant published in 2011, and further above the truth: the deepest measured DH zeros reach |Im z| = 0.347 against Delta = 0.895, and a phase-minimum refinement in the hunt's own materials already gives 0.3871 narrow. The calibration that shows how loose the method is, rather than this instance of it, is running the same coefficient domination on zeta: it returns Delta^2/2 = 3.0191480758 in the wide frame where the truth is 1/2, a factor 6.04. A referee will ask why the sharper published constant is not used, and the answer, that it is cited-plus-measured while the headline is decided, is a good one that has to be given rather than assumed.
2. The remaining prior art. Update 2026-08-18: this item is largely spent. academia.edu 166936409 is read in full and contains no Lambda_DH and no bound on one; Bombieri-Mueller 2008 is identified and carries no de Bruijn-Newman content; the Dobner sweep now runs on three independent indexes. What a referee can still press on is small and named: the full text of Bombieri-Mueller 2008 (publisher wall) and the full text of one citing preprint (Voronov 2026, ResearchGate, Cloudflare), both unread. Neither can touch a computed number, and the superseded wording of this item is kept below.

Superseded, kept verbatim: "The remaining prior art. academia.edu 166936409, Bombieri-Mueller 2008, and a single-source forward-citation sweep. Each is named in NOVELTY.md with what it might contain. None of them can touch a computed number"."

Note added 2026-08-16, after this gate closed: the separation Lambda_DH > Lambda_zeta (SEPARATION.md) offers the same referee nothing new to attack but inherits both exposures above through its two kinds of links. Through its decided link (the winding floor 36/625) it inherits the M2 blind spot of item 1 in full; through its cited link it stands on Polymath 15's Theorem 1.1 as published, unverified in-tree, so an error in that paper's 0.22 would erase the separation while moving nothing else in this directory.

2026-08-18: the separation is untouched by the hardening. It rests on the lower bound and on the cited zeta upper bound, and neither moved. The sharpened upper bound changes only the last, unused link of its chain, from <= 1.6025374835598228 to <= 0.7696992583210755065522 in the wide frame, which is printed there so the claim travels with the whole bracket. Its two inherited exposures are now item 1 (the citations, including Polymath 15) and item 3 (the detector's blindness to a corrupted M2) of the ranking above.

Closure log

The first pass listed five closure items. The re-adjudication brief split them slightly differently, and both letterings are recorded here so neither reader is lost: the first pass's (b) bundled the winding repair and the missing control, and carried Bombieri-Ghosh as its own (d); the brief splits those into (b) winding repair and (c) missing control, and merges the novelty restatement and the prior-art item into (d). The brief's lettering is used below.

Everything in this section was verified by re-running or re-reading in this session. Not one item was accepted on the repairing session's report.

(a) Frame. CLOSED.

FRAME.md is new and carries the two conventions, the scaling law derived from scratch (Phitilde(u) = (c/a)Phi(u/a) gives Htilde_t(z) = c H_{a^2 t}(az), hence Lambda -> Lambda/a^2 and Delta -> Delta/a), a four-row conversion table, and a numerical check of every row. THEOREM13.md section 6's false sentence, "so the two share zeros and share Lambda exactly", is preserved verbatim inside a correction box with the derivation that replaces it and a record of what was routed through it, which is the repo's correction style rather than a deletion. Frame banners are in MISSION.md, STRIP.md and NOVELTY.md, each saying explicitly that the preregistration below it is unaltered. NOVELTY.md now prints the zeta record in both frames, so the comparison that flattered by a factor of 4 cannot recur, and it prints the consequence the old text concealed: in the common wide frame, Lambda_zeta <= 0.22 < 0.2304 < Lambda_DH.

Verified here: the derivation closes algebraically; 4 * 36/625 = 144/625; 4 * the narrow Delta^2/2 interval is the wide one endpoint for endpoint; the ratio 6.9554578 is identical in both frames. A side finding the repair volunteered and that runs against tidiness: Newman-Wu's own kernel is narrow, so the 1/2 on their page 9 and Stopple's page 4 are wide-frame numbers carried across a frame change unconverted by two refereed papers. That is the best argument for FRAME.md existing and it was not asked for.

(b) Winding repair. CLOSED.

The false parenthesis, "n a_n e^{-pi n^2 e^{2u}/5} is even in u termwise", is superseded by a three-part functional-equation derivation and is kept verbatim in a CORRECTION block. The overstated cushion is corrected to 55.7 with the measurement described (41 x 9 grid plus a 65-point left-edge sweep, mpmath dps 140, direct quadrature sharing no code with the shifted-contour bound), and the file states plainly that the correction costs about three extra halvings of h and no decision.

Re-run here. winding.py reproduces with zero non-timing differences against the pre-re-run file: t1 decided N = 1, 71 segments, chord margin 0.02, ball margin 40.32, M2 = 1.1886642645115153e-78, guard PASS at ratio 55.66; t2 decided N = 1, 79 segments, chord 0.02, ball 39.87, M2 = 1.137082903400534e-78, guard PASS at 53.88; decided_floor_t = 36/625. The repair moved no number, as claimed.

(c) Missing control. CLOSED, and the blind spot is recorded as blind.

Control 5 exists in controls.py and controls_results.json, holds the t1 box, t, precision, instrument and subdivision rule fixed and divides only M2. Re-run here, zero non-timing differences:

Three rows are wrong and silent; the wrong-answer onset is deflation 75; and both health metrics move the wrong way across the lesion, chord 0.02 to 1.11 and ball 40.32 to 42.61. (2026-08-18: the table is unchanged and the history above stands as written. What it measures changed when M2 became a proved lemma; see known assumption 6.) The record does not soften any of that: the verdict field is BLIND SPOT, not a pass, all_pass is now false with controls_1_to_4_pass preserving the older true statement unchanged, and the blind_spot record names the perverse metric, the countermeasure, and three things still blind including that the guard-before-failure ordering is luck. Controls 1 to 4 all still pass. This is the item where the repair had the clearest opportunity to flatter the hunt and did not take it.

(d) Novelty restatement. CLOSED on substance; residual named and bounded.

"L-function" and "first quantitative" are gone from the deliverable, the gate's recommended sentence is adopted with Stopple's kernel spelled out, the superseded sentence is kept verbatim with all four of its faults itemised, and NOVELTY.md ends with three standing instructions ("Do not use 'L-function'. Do not drop 'so far as the search reaches'. Do not drop the frame.").

The prior-art half went further than the item asked. It asked for a human to read Bombieri-Ghosh or for the sigma_0 originality claim to be dropped; both happened. The sigma_0 claim is withdrawn in NOVELTY.md, FRAME.md, RESULTS.md and results.json, and a misattribution the reading surfaced (Righetti's 2.3822861089 is originally Bombieri-Ghosh's) is corrected. Two caveats stated plainly rather than buried: the reading was done by an agent session and verified by a second agent session, not by a human, so if the gate's word "human" was load-bearing it is still outstanding; and what makes that tolerable is that the decisive checks are mechanical absence counts over 50 pages, which I ran on my own independently retrieved copy and which agree.

Residual, and it is bounded because every part of it is named: academia.edu 166936409 unread, Bombieri-Mueller 2008 unconsulted, and the Dobner forward-citation sweep single-source. All three are prior-art risks. A prior-art risk bounds novelty, not correctness: none of them can move an enclosure, and the claim is conditioned on the search by its own opening words. The worst case is that the novelty sentence loses the word "first" and the bracket stands unchanged.

(e) Independence. CLOSED.

INDEPENDENCE.md is new, independence_decl.py declares four routes against harness/independence.py, and the adjective is replaced by the measurement everywhere including MISSION.md's WP1 note and RESULTS.md's P2 row. Re-run here: radius 9 of 12 for routes 1 and 2, radius 0 for the other three pairs, matches_gate_8_of_11: true at the coarser merge, anchors all present. Both validation gaps are repaired and both repairs re-run clean: validate.py reproduces with zero non-timing differences, all_pass = True, six checks; the iv cross-leg now hits the decision-path integrand at 7 of 7 points; the tail-domination check dominates in 18 of 18 rows with minimum ratio 8.0169, which is the reported blindness factor 8.02. Route 4 re-run here, 8 of 8 boundary points, 40.3 to 42.7 digits.

Both repairs come with their blindness radius attached, which is the point: the iv leg misses a planted recurrence fault at u = 5/2 (1 of 7 points, because at large u the sum is its own n = 1 term), and a tail bound deflated by less than about 8 passes check 6.

What the repairs got wrong or left short

Four things, none of which moves a number, all of which are one-line fixes:

1. STRIP.md does not carry the Bombieri-Ghosh finding. It is the file that derives and presents sigma_0, and its "Honest scope" section does not say that a sharper exact value has been in print since 2011. It makes no originality claim, so nothing in it is false, and the finding is in four other artifacts. Still, a reader who opens STRIP.md alone would overrate the constant by omission. One line in section 6.

FIXED 2026-08-18. STRIP.md now opens with a superseded-constant banner naming both facts: Bombieri and Ghosh's published 1.120362, and that the same abscissa is now derived and decided in-tree in STRIP2.md. It says in as many words that the upper bound of record is the one in STRIP2.md, and why this page is kept anyway.

1. MISSION.md's preregistered deliverable still reads "any RH-violating L-function" with no superseding note, while that same file carries notes for items (a) and (e) from this round. The sentence is true as written and MISSION.md is explicitly a preregistration, but the file is inconsistent with itself about which corrections get a note.
2. FRAME.md's Bombieri-Ghosh row prints 18 digits from an input it labels "1.120362 (cited, six decimals)". The digits are right, but they come from BOMBIERI-GHOSH.md's own 29-digit re-solve, not from the citation, and the row does not say so. Six decimals alone give 0.192424505522.

SUPERSEDED 2026-08-18, and the underlying number corrected. The row is no longer a cited row: FRAME.md section 6 now prints the decided in-tree value. The 18 digits it used to carry, 0.192424814576128011, are the tau_+ re-solve's, and that re-solve is now decided to sit about 1.2e-17 above its own root, so the correct replacement from the deep flint point is 0.1924248145761280190, agreeing with the old row to 17 digits. The headline is the P = 10^5 two-backend value, which is a different and slightly more conservative number and is exact.

1. results.json open item 2 is itself stale, in the safe direction: it warns that validation.json may predate the item (e) repairs. It does not; the stored file already carried both, and re-running produced zero non-timing differences. Discharged by this adjudication.

PUBLICATION-CANDIDATE RESULT: YES / NO / NOT YET

YES. Items (a), (b), (c) and (e) are closed, verified here by re-running the two load-bearing scripts to zero non-timing differences and by re-reading every repaired file, and item (d)'s residual is a bounded, named prior-art risk rather than an unknown.

The first pass's verdict was that the mathematics survived every attack while two defects stood in the sentences a reader would quote. Both of those sentences are now repaired, with the false text preserved rather than deleted, and no computed number moved except by the factor of 4 the frame demands. The prior-art source that pass called "the unresolved risk" has been read, and I retrieved and text-checked my own copy rather than trust that: it does not anticipate the bracket, and it does displace sigma_0, which the artifacts now say against their own interest.

What the reader is being handed, stated at its true strength: a bracket whose lower side is an enclosure-carrying integer count made strict by a cited theorem and resting on one prose lemma, and whose upper side is a decided constant fed to a cited theorem and known to be loose by at least 2.08. Grade cited plus decided, taking the weakest step. Two things must travel with it and both are in the artifacts: the frame, because the number quadruples without it, and the M2 blind spot, because the lower bound carries it and the detector's own health metric moves the wrong way when it breaks.

Amended 2026-08-18. The second of those two is now smaller than it was. M2 is a proved lemma (M2-LEMMA.md) whose constants are all decided, so what must travel with the bound is no longer "a prose lemma inside" but the named cited input the proof needs, the evenness Phi_DH(-u) = Phi_DH(u), which is assumption 5 restated. The composite grade of the bracket is unchanged at cited plus decided, taking the weakest step, because the weakest step was and remains a citation.

Amended again 2026-08-18, after the hardening pass, and the verdict is still YES. Nothing in the three tasks broke a claim. What they changed:

• the upper side is no longer "known to be loose by at least 2.08". That factor is taken, in-tree and on both backends, and the bracket is 0.0576 < Lambda_DH <= 0.19242481458026887663805 narrow and 0.2304 < Lambda_DH <= 0.7696992583210755065522 wide, ratio 3.341;
• the lower side is unchanged in every digit. The separation Lambda_DH > Lambda_zeta rests on the floor alone and is untouched;
• one word in an old sentence is now wrong and is corrected in place: the upper side is no longer "a weaker bound on a constant Bombieri and Ghosh determined", it is a decided in-tree bound at the same abscissa;
• the referee's first attack is no longer M2 and no longer the looseness. It is the citations, and after them the fact that this directory's two new proofs are prose plus decided arithmetic that no human has read.

What the reader is being handed, restated. A bracket whose lower side is an enclosure-carrying integer count made strict by a cited theorem and resting on one proved lemma whose own weakest input is a citation, and whose upper side is a decided constant, reached by an elementary in-tree derivation, fed to a cited theorem. Grade cited plus decided, taking the weakest step. Two things must travel with it and both are in the artifacts: the frame, because the number quadruples without it, and the detector's blindness to a corrupted M2, because its own health metric moves the wrong way when the implementation breaks.

What remains is three kinds of residual. The four short items above are cosmetic; two of them are now fixed and the other two still move no number. The M2 blind spot is narrower than it was, real as an implementation exposure, and disclosed at every headline that depends on it. The two remaining unconsulted sources bound only the word "first". None of that is a reason to withhold the work, and all of it is a reason to publish the work with its caveats attached rather than sanded off, which is the state the artifacts are now in.

Pending external verification, which is not ours to award.