RESULTS: the rightmost zeros of the prime zeta function

Run of 2026-08-16. Instruments: decide.py and theorem_inputs.py (decided constants), controls.py (WP5), witness.py (WP6); raw numbers in decided.json, theorem_inputs.json, controls_results.json, witness_results.json; the mathematics in THEOREM.md; the pinned sources in SOURCE.md. Vocabulary contract throughout (MISSION.md): measured is one float route, decided is an interval or ball enclosure whose exact endpoints settle a sign, stated with backend and precision, heuristic is an order-of-magnitude model with measured inputs. Nothing in this directory is a repo-level result until the case log in hunts/README.md says how the hunt ended.

The headline, rewritten 2026-08-16 after an adjudication of prior art. The mathematics in this directory is prior art. The wall at sigma_c = 1.779544653546994116445898786965... , its triangle-inequality proof, the existence of zeros filling every window below it, and the value of the constant were all in print before this hunt opened. Belovas, Cepaityte and Sabaliauskas, "On the zero-free region and the distribution of zeros of the prime zeta function", An. St. Univ. Ovidius Constanta Ser. Mat. 33(2) (2025), 27-44, state the identical constant by the identical argument as their Theorem 1; and Sepulcre and Vidal, "On the real projections of zeros of analytic almost periodic functions", Carpathian J. Math. 38 (2022), no. 2, 489-501 (preprint arXiv:1805.02041, 2018), own the general theorem that implies all of it and more. MISSION.md's kill condition 2 has therefore fired, the hunt's core is reclassified as rediscovery, and the OEIS correction now cites those sources for the constant rather than this work (section 8).

The refutation of OEIS A107311 stands, and so does every theorem and every proof below: none of the mathematics is wrong, and none of it is new. What survives as this hunt's own is narrow, and it is exactly five items (section 7.1):

1. the line-by-line refutation of A107311's two conjectures, which no source in the literature engages (new as a connection, zero new mathematics);
2. Theorem C2 and Corollary C3, that the tail subsets {p >= p_k} have thresholds >= log2(3 p_k / (5 log p_k)) growing without bound, so no constant bounds the zeros across all prime subsets (new, small, elementary corollary of a published framework plus Rosser-Schoenfeld);
3. the constant sigma_3 = 1.82522595607384576238787271088... for {p >= 3}, which out-walls the full series by more than 0.045 (new instance of prior-art theory);
4. two-backend enclosures of sigma_c, sigma_3 and x* with exact endpoint sign logic (rediscovery in sharper form: Belovas et al. print 15 digits and note that any precision is available);
5. one observation about the literature rather than about mathematics: Belovas et al.'s Conjecture 1, left open in their paper, is a corollary of Sepulcre and Vidal's Theorem 4.3, whose other direction already contains their Theorem 1 (section 7.2). This is the most publishable item here.

The substance that is now known to be prior art, restated once so the record is readable on its own: the supremum of the real parts of the zeros of P(s) = sum_p p^(-s) is not x* = 1.7286... (the root of zeta(x) = 2) but sigma_c (decided here on both backends), the root of the balance P(sigma) = 2^(1-sigma), which exceeds x* by more than 1/20 (decided); the supremum is approached by infinitely many zeros and attained by none (Theorems A and B, THEOREM.md). The subset conjecture fails without bound: {p >= 3} already has zeros beyond sigma_c, and the tail subsets {p >= p_k} have thresholds tending to infinity (Theorems C1, C2). The refuting zeros live in Re s > 1, where every reading of "the prime zeta function" agrees with the series, so the refutation is reading-independent. No explicit zero is exhibited; existence is by a Bohr-Kronecker-Rouche argument with every numeric inequality decided on both backends.

Ownership item by item is in PRIOR-ART.md section 8, which also verifies that the general theorem really does specialize to P and that the bridge in item 5 has no gap.

0. The source

OEIS A107311, "Decimal expansion of the solution to zeta(x) = 2." Fetched 2026-08-15 as JSON and refetched live 2026-08-16; the two fetches are byte-identical (empty diff). Entry revision 55, last modified 2024-12-29T23:50:41-05:00; the conjecture comment is dated Dec 21 2024 inside the entry. The full JSON body is pinned byte-for-byte in SOURCE.md section 1. The comment field, word for word as served (spelling as served: "partials sums", "Riemman", "the anyone subset" are in the source):

From Artur Jasinski, Dec 21 2024: (Start) Borwein et al. (2007) proved (Theorem 3.1) that the real parts of the zeros of the partials sums of the Riemman zeta functions are not greater than this constant. Conjecture 1: the real parts of the zeros of the prime zeta function are not greater than this constant. Conjecture 2: the real parts of the zeros of the anyone subset of the prime zeta function are not greater than this constant. (End)

"This constant" is x*, the real root of zeta(x) = 2, whose correct role is the partial-sums-of-zeta threshold (Borwein, Fee, Ferguson, van der Waall 2007, Theorem 3.1, pinned through two independent secondary quotes in SOURCE.md section 3: Platt-Trudgian 2016 and Gonek-Ledoan 2010).

The prior literature on zeros of P itself, as it stands after the adjudication sweep (SOURCE.md section 4, corrected): Belovas, Cepaityte and Sabaliauskas (2025) prove the sigma_c threshold and conjecture that it is sharp; Sepulcre and Vidal (2022, preprint 2018) prove the general theorem that gives both; Moreno (1973) supplies the polygon step both rest on; and Froberg (1968) is the earliest source touching zeros of P at all (four numerically observed roots, "very little is known"). The hunt's own first search sweep found only Froberg and concluded from that emptiness that no threshold was in the literature. That conclusion was wrong; SOURCE.md sections 4.5 and 4.6 keep the failed queries and record why they failed.

1. The decided constants

Backends, deliberately independent code paths: flint = python-flint 0.9.0 (arb balls) at 350 bits, arb zeta direct for x*, Moebius series K = 120 with a proved tail bound for P; iv = mpmath.iv 1.3.0 at dps 40, zeta by finite Dirichlet sum plus Euler-Maclaurin tail with a decided remainder (iv.zeta raises on call in mpmath 1.3.0), Moebius K = 40. Bracket bookkeeping is exact Fractions on both legs. Derivations of every tail bound are in instrument.py.

Cross-checks, all decided: each flint interval lies inside its iv interval; the OEIS entry's 102-digit value interval lies inside both x* enclosures. Each constant is the unique root of a strictly decreasing function on its bracket (zeta termwise-exactly on (1, 2); h and u by decided derivative guards on [1.7, 1.9] and [1.80, 1.85], both backends), and each bisection first decided its bracket endpoint signs.

The decided comparisons feeding THEOREM.md, every one decided on both backends:

• separation: sigma_c - x* > 1/20, exact rational compare of enclosure endpoints; lower bound of the difference 0.050897414548810498 (flint), 0.05089741454877 (iv).
• margin: h(x*) = P(x*) - 2^(1-x*) > 1/60, with h evaluated over the whole x* enclosure; flint enclosure [0.016907377213856298105802854273997, 0.016907377213856298105802854274113].
• D1: h(7/4) in [0.009465637293734518587514821979513, 0.009465637293734518587514821979514] (flint, width 1.275e-65; iv confirms); > 1/128.
• D2: u(9/5) in [0.004312491729413063275865581947004, 0.004312491729413063275865581947005] (flint, width 1.897e-67; iv confirms); > 1/256.
• D3: log2(69/(5 log 23)) in [2.137903503656002856060611381367, 2.137903503656002856060611381368] (flint width 1.423e-104; iv width 1.837e-40, agreeing to all displayed digits); > 17/8.
• W1-W5 (window compares, exact Fractions against outward-rounded endpoints): hi(x*) < 173/100; lo(sigma_c) > 7/4; lo(sigma_c) > 177/100; hi(sigma_c) < 89/50; lo(sigma_3) > 91/50.

Preregistration P1 and P3 check boxes recorded in decided.json: all true.

2. The replacement theorem

Full statements and proofs in THEOREM.md; the shape. Every item carries its ownership tag, per the map in PRIOR-ART.md section 8. The proofs are unchanged and remain correct; the tags say who got there first.

• Theorem A (the wall, proved there). For prime q and sigma_c(q) the unique root of the balance q^(-sigma) = sum_{p>q} p^(-sigma): P_q has no zero with Re s >= sigma_c(q). Triangle inequality for Re s > sigma_c(q); the equality case on the line Re s = sigma_c(q) is excluded by multiplicative independence of the primes (Lemma 3, from unique factorization). Exact mathematics, no numeric input; the numerics only locate sigma_c and sigma_3 and compare them with x*. Ownership: pure rediscovery. Part (a) is Belovas et al. (2025) Theorem 1 for q = 2, same constant and same proof; earlier still, it is the "only if" half of Sepulcre and Vidal Theorem 4.3 (2022, preprint 2018). Part (b) is pure rediscovery by adjudication (Sepulcre and Vidal, JMAA 437 (2016), Prop. 5 and Cor. 6), with one precision note recorded rather than argued: those results are stated for finite exponential polynomials and the finite-to-infinite step is not automatic, so the honest phrasing is "the same argument gives the infinite-series case" rather than a verbatim citation.
• Theorem B (existence up to the wall, proved there). For every window (sigma_1 - eps, sigma_1 + eps) inside (1, sigma_c(q)), P_q has infinitely many zeros with Re s within eps/2 of sigma_1, imaginary parts unbounded. Proof: phase the whole series to vanish at sigma_1 (polygon Lemma 2 with the tail aggregated as one side), then steer actual vertical translates onto the phased model by Kronecker-Weyl (Lemma 4, proved via Weyl's method with positive lower density) and transfer the zero by Rouche. Corollary B1: sup of the real parts equals sigma_c(q), not attained. Ownership: pure rediscovery, and the prior art is stronger. Sepulcre and Vidal Theorem 4.3 gives that the closure of the real projections, intersected with (1, infinity), is exactly (1, sigma_c]; that statement implies Theorem B's "infinitely many per window" clause, which in turn implies its existence clause, and neither converse holds (PRIOR-ART.md section 6(vi)). Corollary B1 is likewise pure rediscovery (Sepulcre and Vidal JMAA (2016), Prop. 5 and Cor. 6, as the finite ancestor). The only clause not delivered by Theorem 4.3 is the positive lower density in t of admissible heights (Lemma 4), and that is not claimed as new either: it sits in classical almost-periodic territory (Jessen and Tornehave) which has not been searched. Recorded as an unverified lead, not a credit.
• Corollary B2 (A107311 Conjecture 1 is false). Ownership: new as a connection, zero new mathematics. No source in the literature engages the OEIS entry. Belovas et al. do not refute it: their bound 1.7795 is weaker than the conjectured 1.72864, and their numerics stop at sigma_M = 1.6826..., below x*, so their paper is consistent with the entry as far as it goes.
• Theorem C1 (subsets, proved there). The same pair of theorems for q = 3: zeros of sum_{p>=3} p^(-s) fill windows below sigma_3, in particular (1.78, 1.82), which is beyond sigma_c: the decided enclosures give sigma_3 - sigma_c > 0.0456813 (exact rational compare on the flint legs). Ownership: new instance of prior-art theory. The theory is Sepulcre and Vidal's; the constant sigma_3 and the observation that a subset out-walls the full series are this hunt's.
• Theorem C2 (unbounded walls, proved there). For p_k >= 23, sigma_c(p_k) >= log2(3 p_k / (5 log p_k)) -> infinity, from Rosser-Schoenfeld Corollary 3, inequality (3.8). Decided instance: the {p >= 23} wall exceeds 17/8 (D3). Corollary C3: no constant bounds the zeros over all subsets. Ownership: new, small, elementary corollary of a published framework plus Rosser-Schoenfeld. No prior art was located by four independent searches, which after the episode recorded in section 8 is a statement about the searches and not about the world.
• Lemmas. Lemma 1 (U has a unique root) is Belovas et al.'s Lemma 1: pure rediscovery. Lemma 2 (the polygon lemma) is Moreno's Geometric Principle, Compos. Math. 26 (1973), p. 71, used verbatim as the aggregated-tail device in the proof of Sepulcre and Vidal Theorem 4.3: pure rediscovery, and the aggregation trick is theirs too. Lemma 3 ({log p} is Q-linearly independent) is unique factorization and was never a claim. Lemma 4 (phase steering with positive lower density) is classical Kronecker-Weyl: pure rediscovery, with the density residue as noted above.

Proof status of each step (THEOREM.md section 6): decided on both backends: every numeric input (section 1 above; no claim rests on a measured-only number). Proved in THEOREM.md with complete proofs: Lemmas 1, 1a (Euler), 2 (polygon), 3 (independence), 4 (Kronecker via Weyl), 5 (triangle equality); Theorems A, B, C1, C2; Corollaries B1, B2, C3. Cited without reproof: the fundamental theorem of arithmetic, Rouche, Stone-Weierstrass on the torus, Rosser-Schoenfeld (3.8), and standard analysis facts; hypotheses checked explicitly where used.

Gaps and honesty notes, as declared in THEOREM.md section 8, restated here at full prominence:

• Citation numbering from memory. The theorem/section numbers for Hardy-Wright, Rouche's textbook placements and Rudin (items 2, 4, 5 of THEOREM.md section 7) were not re-checked against copies this session. The load is bibliographic, not mathematical: every argument using them is proved in full in THEOREM.md or has its hypotheses checked against a stated standard form. The one externally load-bearing count, Rosser-Schoenfeld (3.8), was re-checked against the paper's scanned text this session.
• Not kernel-checked. No Lean formalization of any step exists; the ladder's top rung is untouched. A natural later rung: Lemma 2 plus Theorem A.
• No explicit witness zero (section 5 below; per MISSION.md its absence changes nothing).
• Instrument defect, found and fixed. The first D3 run produced disjoint enclosures on the two backends: MISSION.md kill condition 3 fired and correctly marked the instrument. iv_endpoints in instrument.py had converted interval endpoints through the global mp context (dps 15), collapsing iv results to one 53-bit float. Sign decisions were never at risk (nearest rounding at relative precision cannot move a value across zero); value enclosures beyond 15 digits were misreported. Fixed to read the raw _mpi_ endpoints; decide.py re-run reproduced decided.json identically except timings.
• The Rouche step closes (kill condition 1 did not fire): the model series carries its own tail (aggregated as one polygon side), so the remaining errors are the steering term and the beyond-Y remainder, both cut after the isolating minimum m is fixed. No assumption left undischarged.

3. The refutation ledger

What is not refuted or claimed: nothing about zeros with Re s <= 1 or about the continuation beyond the series' half-plane; nothing about finite subsets (the conjecture quantifies over all subsets, so the infinite witnesses settle it); nothing about RH (the zeros produced are zeros of P and of subset series, not of zeta); and x* keeps its correct role as the partial-sums threshold of Borwein-Fee-Ferguson-van der Waall.

4. Calibration control and lesions (WP5)

All in controls_results.json; the solver under test is decide.bisect_decreasing, the same function object decide.main() calls (asserted identical to instrument.bisect_decreasing at import), and the baseline sigma_c re-solve reproduced the recorded interval strings exactly.

• Calibration (known answer, same code path): PASS. The zeta-partial-sums balance (leading term 1 against sum_{n>=2} n^(-sigma), root of zeta(sigma) = 2) run through the same solver with sigma_c's bracket recovers x*: flint enclosure [1.7286472389981836181351030102976660, 1.7286472389981836181351030102977450] (350 bits, width 7.889e-32), OEIS value interval inside both backend enclosures; 31 of the entry's digits reproduced on the flint leg, 13 on the iv leg. P2 held.
• Lesion 1 (input sensitivity): PASS. Dropping the p = 3 term from the tail moves the root to [1.4293161356678019330620811031492544, 1.4293161356678019330620811031493235] (flint, 350 bits): a decided shift from sigma_c of [0.3502285178791921833838176838161, 0.3502285178791921833838176838163], decidedly below x* as well. The solver reads its input, not a cache.
• Lesion 2 (the mis-port made mechanical): PASS. Keeping the zeta series but balancing P's leading term 2^(-sigma) against the tail after it lands at [2.4241112509134051299681251496785051, 2.4241112509134051299681251496785939] (flint, 350 bits), decidedly equal to neither x* nor sigma_c: the balance template and the series fed to it both matter.
• Precision response: PASS. The sigma_c code path at 60 / 120 / 200 bits achieves widths 1.421e-15 / 6.163e-34 / 1.020e-57, strictly shrinking, all intervals nested consistently with the 350-bit run.
• Rerun reproducibility: PASS. decide.py re-run in a subprocess reproduced decided.json line for line with only time_s lines differing.

5. The witness hunt (WP6) and prediction P4

Budget-capped bonus hunt for an explicit zero near sigma_1 = 7/4; absence was the pre-registered expectation and changes nothing above.

• Margin analysis (decided, flint 350 bits, iv confirming). The coherence budget at sigma_1 = 7/4 is eps_0 = h(7/4) in [0.009465637293734518587514821979513, 0.009465637293734518587514821979514]: the p >= 3 resultant must shed less than one percent of its fully aligned modulus (R in [0.306767416044414785266889814619632, ...633]) to land on the p = 2 term (r_2 in [0.297301778750680266679374992640118, ...119]). eps_0 > 0 decided on both backends: cancellation is feasible in principle.
• Expected height (heuristic, labeled as such). A von Mises tilted importance-sampling model of the phase-coherence probability (checked against plain Monte Carlo at loose eps: agreement within 0.05 combined standard errors) puts Prob(loss <= eps_0) at about 4.1e-16 on the line budget, giving an expected first-witness height of order 1.6e16 (line-exact) to 1.7e10 (window-wide budget h(x*)); expected zero count below t = 1e8: 6.1e-9 (line) to 0.0059 (window). The margins are parts in a thousand, as MISSION.md anticipated.
• Screen (measured, numpy float64). sigma = 7/4 exactly, t in [0, 1e8], 146,961,923 points at step 0.6804; head primes p <= 47 with the pointwise bound ||P| - |F47|| <= 0.013200; two refinement passes (p <= 1e4, then p <= 1e6 with all-integer tail bound 4.216e-5). Global refined minimum |P| = 0.010021 at t = 56316681.51 (float interval [0.009979, 0.010063], above the eps_0 budget). No point reached the candidate gate |P| < 1e-4: zero candidates.
• Box counter (validated, never fired in earnest). The argument-principle counter (python-flint acb, 192 bits, Moebius K = 25) passed both controls: a null box at a coarse-grid |F| maximum decided winding 0, and a planted-zero box (P minus the exact dyadic midpoint of its ball value) decided winding 1.
• Verdict: no witness; witness_decided_exists: false. P4 confirmed: no explicit witness zero below t = 1e8.

6. Predictions P1-P4, settled

• P1 (decided enclosures land in the pre-registered windows): held. sigma_c in [1.77954465, 1.77954466] and x* in [1.72864723, 1.72864724] on both backends; separation > 1/20 decided on both.
• P2 (calibration recovers the partial-sum threshold): held. The OEIS value interval lies inside both same-code-path enclosures; 31 digits reproduced on the flint leg (against "within its published digits": the enclosure width, 7.889e-32, is what bounds the digit count, and every reproduced digit matches the entry).
• P3 ({p >= 3} threshold in [1.82522, 1.82523]): held. Both backends.
• P4 (no explicit witness below t = 1e8): held. The screen found no candidate; the pleasant surprise did not occur.

7. Honest scope and ownership

• Which reading of "zeros of the prime zeta function" is covered. All of them. The refuting zeros have Re s > 1, inside the half-plane of absolute convergence, where the defining series, Glaisher's continuation and every branch of it agree; a zero there is a zero under every reading (SOURCE.md section 2). Nothing is claimed about 0 < Re s <= 1, where the continuation has logarithmic branch points and the line Re s = 0 is a natural boundary, and nothing about finite subsets.
• Novelty caveat. SUPERSEDED 2026-08-16. Its text is quoted below, word for word, as the record of what was believed at the time. Two sources were located after this file was written and they own the core: Belovas, Cepaityte and Sabaliauskas (2025) Theorem 1 for the wall and its constant, and Sepulcre and Vidal (2022) Theorem 4.3 for the general statement, with Moreno (1973) supplying the polygon step both of them rest on and Sepulcre and Vidal's JMAA (2016) paper as the finite-sum ancestor. MISSION.md kill condition 2 fired and the finding is reclassified as a rediscovery. The adjudication, the verbatim sources and the statement-by-statement ownership map are in PRIOR-ART.md; the corrected public page is docs/30-prime-zeta-rightmost-zeros.md. The paragraph that follows is false as to novelty. It is quoted rather than deleted, because editing a record to match a later finding is editing evidence; it is quoted rather than left as running text, so that nothing in it can be read as a live claim.

The wall-plus-Bohr-steering mechanism is classical for general Dirichlet series, and a specialist may regard the sigma_c threshold as folklore-derivable; the searches logged in SOURCE.md section 4 (2026-08-16: exact-phrase, zbMATH API, MathWorld, citation hunts) found no source stating any rightmost-zero threshold for P, and only Froberg 1968 touching its zeros at all, but that is a record of queries run, not a completed search of record (no MathSciNet, no zbMATH full text). The deliverable therefore stands as: the OEIS correction, plus the first explicit treatment found for P specifically, with decided constants. Original is claimed; novel is claimed only as "no prior source found by the logged searches".

What is false in it, named so the correction is checkable: "found no source stating any rightmost-zero threshold for P" is true of the searches and false of the world (Belovas et al. state exactly that threshold); "the first explicit treatment found for P specifically" is false outright; and "novel is claimed only as no prior source found" is a claim this file no longer makes at all, since the core is prior art.

• Composite grade. Decided numeric inputs (both backends, exact endpoint sign logic) glued by classical mathematics proved in THEOREM.md or cited with checked hypotheses. By the certainty ladder a composite takes its weakest step: this is a decided-plus-proved composite, not kernel-checked, and the heuristic height model in section 5 is a labeled aside that carries no claim. Nothing here uses the reserved enclosure vocabulary of zeta/rigor.py, and nothing here bears on RH.
• Original is still claimed where it is due, and only there. By the repo's ruling, original is a claim about provenance and novel is a claim about the world. This hunt derived its results itself, and the five items in 7.1 are the ones for which the world had nothing. For the rest the correct word is neither: published mathematics was re-derived without knowing it, which is a rediscovery, and it is named as one.

7.1 What survives as this hunt's own

Five items, no more, each with the grade the adjudication and the verification in PRIOR-ART.md support.

Item (b)'s "no prior art located" rests on four independent searches, and after section 8 that phrase should be read for exactly what it is: a statement about the searches, not about the world.

7.2 The bridge: two papers that jointly close a stated open problem

The most publishable item here, and it is a claim about the literature, not about mathematics. Stated at the confidence the verification supports and no higher.

Belovas et al. (2025) prove that P has no zeros with Re s > sigma_c (their Theorem 1) and leave open, as their Conjecture 1, whether that bound is sharp: whether sigma_T = max{sigma : P(sigma + it) = 0, |t| < T} tends to sigma_c as T grows. Their own numerics reach only sigma_M = 1.6826... at |t| < 200000, which is 0.097 short, which is why the conjecture looked open.

Sepulcre and Vidal Theorem 4.3 settles that conjecture, and its other direction re-proves their theorem. Specialized to P, the "only if" direction says every sigma_0 > sigma_c fails their condition (4.8) and so is not in the closure of the real projections of zeros, which is precisely Belovas et al.'s Theorem 1. The "if" direction puts sigma_c itself in that closure, so there are honest zeros at finite (uncontrolled, possibly astronomical) heights with real part arbitrarily close to sigma_c; since sigma_T is non-decreasing in T and bounded above by sigma_c, its limit is sigma_c. Their theorem and their open conjecture are the two directions of one published characterization that predates both, and neither paper cites the other: one is filed under number theory, the other under almost periodic functions (MSC 30B50, 30D20).

Grade, stated exactly. The specialization of Theorem 4.3 to P and the derivation above were proved in PRIOR-ART.md (sections 6 and 7) against both sources read verbatim, and no gap was found in either. It is not refereed and not kernel-checked, and it carries pending external verification. This hunt did not prove Theorem 4.3 and did not prove Conjecture 1; it noticed that one implies the other. Say it in those terms and no stronger.

8. The kill condition fired, and what that cost

Recorded rather than fixed quietly, in the register this repository uses for its own withdrawals.

MISSION.md pre-registered kill condition 2: a literature source is found proving the sigma_c threshold for P, in which case the finding is reclassified as a rediscovery and the OEIS correction cites that source instead of this work. It fired on 2026-08-16, hours after the hunt closed, and against two sources rather than one. Both consequences have been applied: the reclassification throughout this file, and the citation change in OEIS-CORRECTION.md sections 1 and 2.

What it cost. Work package 3 and the larger part of work package 4 re-derived published mathematics. THEOREM.md is retained in full and unedited: a proof does not become wrong by being second, and the write-up is the evidence that this hunt actually derived the statements rather than paraphrasing a source it had already found. What was lost is the claim, not the work. Unaffected and retained: the decided instruments, the calibration and lesion controls, the witness screen, and the refutation of the OEIS entry, which no published source performs.

What the pre-registration bought. The kill condition was written before the search, named the exact evidence that would fire it, and named the consequence in advance. When the evidence arrived there was nothing to negotiate, and this file has no room to argue for its own result. That is what writing kill conditions down is for, and it is worth more than the claim it cost.

Why the search failed, so that it is not repeated. The hunt's exact-phrase query was a literal substring of the title of the paper that owns its main theorem, and the engine did not return it; the general theorem was out of reach of every query the hunt could have phrased, because it is filed under almost periodic functions and never names a prime. Countermeasures for both failures are in SOURCE.md section 4.6. The uncomfortable part is the asymmetry: the skepticism reflex was fully present in the numerics, where nine instrument defects were caught during the run and kill condition 3 fired once on a backend disagreement, and it was absent in the literature search, where an empty result was read as an absence rather than as a failed query.

Disposition

Instruments (instrument.py, decide.py, theorem_inputs.py, controls.py, witness.py) retained. OEIS-CORRECTION.md drafted alongside this file; posting anything to OEIS is an operator action, not this hunt's (MISSION.md, agents_may_not). No claim is promoted by this file; the case-log entry in hunts/README.md and any docs/ page are the close-out steps that remain.