Bounds

Results in the register

• T-007 S3 V0 C1 Nagamochi · 2026-08-31 · 321 cases

  \(s(n) \ge \min(\lceil\sqrt{n}\rceil, \sqrt{n - 2\lfloor\sqrt{n}\rfloor + 1} + 1)\)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for \(4 \le n \le 324\)$4\le n\le 324$

• T-016 S3 V3 C3 Massaccesi after Burns · 2026-09-03 · n = 18, 19

  \(s(n) \ge 22529/5000\)$s(n)\ge 22529/5000$ for \(n = 18, 19\)$n=18,19$, by monotonicity from T-015

  Claim and records

  Claim

  \(s(18) \ge 22529/5000\)$s(18)\ge 22529/5000$ and \(s(19) \ge 22529/5000\)$s(19)\ge 22529/5000$, by monotonicity from T-015 (a packing of n >= 17 unit squares contains a packing of 17). At \(n = 19\)$n=19$ this replaces Nagamochi's 1 + sqrt(12) = 4.4641... since (22529/5000 - 1)^2 = 307265841/25000000 > 12; at \(n = 20\)$n=20$ the certificate does not improve 1 + sqrt(13) = 4.6055... since that square is below 13, and nothing changes.

  Composition

  Derived: the minimum over T-015 and the monotonicity step, which is one recorded line carried in the evidence limitations and the n-018 and n-019 case bodies; T-015 sits at V3/C3, so the derived claim keeps the rungs.

  Next rung

  Rises with T-015; a bespoke \(n = 18\)$n=18$ or \(n = 19\)$n=19$ certificate stronger than the inherited bound would be a new result, not a rung change. Superseded as the verified lower bound on 2026-09-04 by T-019, which carries 459/100 to \(n = 18\)$n=18$ and \(n = 19\)$n=19$ by the same monotonicity step this result uses, from a larger base. This result stays true as stated and keeps its rungs.

  Significance

  The same movement carried to \(n = 18\)$n=18$ (+0.079587 over T-002) and \(n = 19\)$n=19$ (+0.0416984 over Nagamochi's 1 + sqrt(12), the first movement of that case's verified lower bound since 2005); the derivation adds one line beyond T-015 and inherits its source and credit.

  Novelty

  previously-published Present in an identified source

  Records

  n = 18n = 19registerevidence 1evidence 2

  n = 18n = 19registerevidence 1evidence 2

• T-019 S4 V3 C3 Levy after Burns, Massaccesi · 2026-09-04 · n = 17–19

  \(s(n) \ge 459/100 = 4.59\)$s(n)\ge 459/100=4.59$ for \(n = 17, 18, 19\)$n=17,18,19$

  Claim and records

  Claim

  \(s(17) \ge 459/100\)$s(17)\ge 459/100$, and \(s(18) \ge 459/100\)$s(18)\ge 459/100$ and \(s(19) \ge 459/100\)$s(19)\ge 459/100$, from this project's weighted fractional unavoidable-set certificate at container side 459/100 = 4.59.
  
  This displaces Massaccesi's 22529/5000 = 4.5058 (T-015) and the two cases T-016 carried from it, all three of which had held that value until 2026-09-04. The movement past the published value is +0.0842 at each of the three sizes.
  
  A stronger public candidate, anabologyco-maker's side 4.5705 (nine thousand one hundred forty-one two-thousandths) of 16 August 2026 (GitHub; not replayed here), was outside the record's search corpus when this was registered and was found on 2026-09-07; it lies 0.0195 below this result. Two rungs are retained below it, 451/100 and 229/50.
  
  The September 2026 DS7 audit identifies a stronger reported n19 bound from Theorem 10 at k=4, approximately 4.6172815, with an unresolved theorem/table discrepancy and missing proof. T-019 does not improve that report; its public source displacement is supported at n17 and n18, while all three certified inequalities and their movements from the prior register remain valid.
  
  The three sizes do not need a monotonicity step. Only Condition 2 mentions n among the five conditions, so an atom set of mass 423327/25000 certifies the side for every integer above it, which is 17 and upward; monotonicity would give the same thing and is not what is used.
  
  It does not improve \(n = 20\)$n=20$, where Nagamochi's 1 + sqrt(13) = 4.6055... is larger -- the certificate does prove \(s(20) \ge 459/100\)$s(20)\ge 459/100$, but that is the weaker statement.

  Composition

  Primary at all three sizes: one certificate, one accepted verdict, the bound being the container side directly. Its mass 423327/25000 is below seventeen, and Condition 2 is the only condition that mentions n. The same certificate therefore applies directly at n17, n18 and n19; monotonicity would also prove the latter two inequalities but is unnecessary for this certificate.

  Next rung

  A second machine method decides it, the interval-certified decision, the first here that is not exact-algebraic. V4 and C4 need two adversarial AI reviews by distinct reviewers and a human oversight record, and none is retained.
  
  The bound itself: the retained certificate's total is 423327/25000 = 16.933080, leaving 0.066920 below seventeen, and that margin is what a further rung has to be found inside.
  
  One figure from the ladder is worth carrying, because it contradicts the obvious reading. Margin is not monotone in the side. The 451/100 rung two below this one has total 16.593620 and margin 0.406380 -- an order of magnitude more headroom at a side 0.08 lower -- and the 229/50 rung between them has margin 0.034265, half of what 459/100 has at a higher side. So the covering value at a given side is set by the site set at least as much as by the side itself, and a better site set at a higher side can open the margin back up rather than close it. The \(n = 12\)$n=12$ ladder shows the same shape in the same direction: margin 0.007175 at 197/50, then 0.029410 at the higher 79/20.
  
  Massaccesi's own atoms are the reference point and not a limit: they give a covering value of 203/12 at 4.5058, tight by construction, which bounds nothing about what a different site set reaches at a larger side.
  
  A further rung is therefore a search question, and the quantity to read is the restricted optimum a converged run reaches, never an extrapolated rate.
  
  One such question was asked and answered on 2026-09-04, at \(n = 18\)$n=18$ and side 117/25 = 4.68, and it is recorded here as measurement rather than as a claim about tau*. Three site sets in succession -- 538, 578 and 618 orbits -- returned a restricted optimum of exactly 18.000000, the third of them after 157 row-generation rounds that grew the row set from 15888 to 27516 and cost 7056 s. Adding sites can only lower a restricted optimum, and across those three it did not move at all.
  
  Two readings are open and the run did not separate them. Either the covering value at that side is at or above eighteen, in which case no certificate exists there; or the site sets are still short of it and the optimum is sitting on a degenerate vertex, which the collapse of pricing from 90 s to 1--3 s suggests -- a dual that sparse is what a degenerate basis looks like. Exactly round values are the known artefact signature in this pipeline, 18.0 having been one for \(n = 17\)$n=17$'s grid-31 optima, so the second reading cannot be dismissed.
  
  What is not in doubt is the cost of settling it: two hours for the round that did not move. The run was stopped for that reason rather than for its answer, and the side is available to a later block with a cheaper decision path.
  
  The method's own ceiling is not close here. A certificate for n cannot exist above ceil(sqrt(n)) * B, a wider container holding ceil(sqrt(n))^2 pairwise disjoint axis-parallel B-squares whose masses Condition 5 forces past n; for \(n = 17\)$n=17$ that is 5B = 4.9885, so 459/100 still has 0.3985 of runway. Unlike \(n = 12\)$n=12$, where the ceiling sits below the conjectured value and forecloses the case, nothing structural stops this one.
  
  Where the next rung is worth spending on, though, is not \(n = 17\)$n=17$. Of the five conditions only Condition 2 mentions n, and the covering program behind the search does not contain n at all -- minimising total mass subject to every admissible B-square carrying mass at least 1 is a question about L, B and the net. So one atom set proves \(s(n)\)$s(n)$ >= L for every integer n above its mass, this certificate's 16.933080 reaching 17, 18 and 19 directly, and a larger n is strictly easier at the same side.
  
  The headroom said the rest, when this was written. \(n = 17\)$n=17$ had 0.0855 to its best known packing at 4.675530; \(n = 18\)$n=18$ had 0.2329 to 4.822876 and \(n = 19\)$n=19$ had 0.2956 to 4.885618, both still carrying this certificate's side in the verified register; and \(n = 20\)$n=20$ had 0.3944 to the trivial 5, holding Nagamochi's closed-form 1 + sqrt(13) = 4.605551. A run at a side whose covering value lands between 17 and 18 would raise \(n = 18\)$n=18$ and leave \(n = 17\)$n=17$ where it is; one landing between 19 and 20 would displace a published closed form.
  
  The resumed search ran the same day and is registered as T-020, at side 24/5 with retained certificate mass 18.922620, between eighteen and nineteen. It supersedes this result at \(n = 19\)$n=19$ and moves \(n = 20\)$n=20$ and \(n = 21\)$n=21$ as well. What this result still holds alone is \(n = 17\)$n=17$ and \(n = 18\)$n=18$: T-020's atoms are too heavy for either, which is the reach rule above read in the other direction. This result stays true as stated at all three of its sizes and keeps its rungs.
  
  For \(n = 17\)$n=17$ the reading is unchanged and the runway is 0.3985 to the ceiling and 0.0855 to the packing. For \(n = 18\)$n=18$, T-030 now certifies 4679/1000; T-029's 1871/400 remains as the previous rung. 117/25 remains the open probe this next_rung named.

  Significance

  Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value". One certificate moves three registered cases, and it replaces a published value rather than an inherited or closed-form one -- the only bound this project has displaced that someone else had put in print.
  
  A stronger public candidate found afterwards, anabologyco-maker's 9141/2000 = 4.5705 of 16 August 2026 (GitHub; not replayed here), was outside the search corpus at scoring time, so the displacement at n17 and n18 is 0.0195 over that public report and 0.0842 over the one this record had adopted. At n19, the DS7 audit now retains a stronger source-reported bound than T-019; its 0.0842 movement refers to the prior register. The family-level score is unchanged.
  
  Held below S5 because none of \(n = 17\)$n=17$, 18, 19 is a central open case, the movement is 0.0842 rather than a qualitative change, weighted resource counting predates this project, the recent pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, and the case does not change character at the new value.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 17n = 18n = 19registerevidence 1evidence 2

  n = 17n = 18n = 19registerevidence 1evidence 2

• T-020 S4 V3 C3 Levy after Burns, Massaccesi · 2026-09-04 · n = 19–21

  \(s(n) \ge 24/5 = 4.80\)$s(n)\ge 24/5=4.80$ for \(n = 19, 20, 21\)$n=19,20,21$

  Claim and records

  Claim

  \(s(19) \ge 24/5\)$s(19)\ge 24/5$, \(s(20) \ge 24/5\)$s(20)\ge 24/5$ and \(s(21) \ge 24/5\)$s(21)\ge 24/5$, from this project's weighted fractional unavoidable-set certificate at container side 24/5 = 4.80.
  
  Two of the three cases previously held Nagamochi's 2005 general closed form in this register's independently verified lower fields -- min(ceil(sqrt(N)), sqrt(N - 2*floor(sqrt(N)) + 1) + 1), which gives 1 + sqrt(13) = 4.6055... at \(n = 20\)$n=20$ and 1 + sqrt(14) = 4.7416... at \(n = 21\)$n=21$ -- and the third held the 459/100 = 4.59 this project certified earlier the same day as T-019. The movement is +0.21 at \(n = 19\)$n=19$, +0.194449 at \(n = 20\)$n=20$ and +0.058343 at \(n = 21\)$n=21$, relative to those verified fields.
  
  The September 2026 DS7 audit records stronger external reports below 4.80 separately, with missing source proofs and table discrepancies explicit; no claim of historical priority follows from the earlier source omissions.
  
  The three sizes take no monotonicity step. Only Condition 2 mentions n among the five conditions, so an atom set of mass 946131/50000 certifies the side for every integer strictly above that mass, which is 19 and upward. From \(n = 22\)$n=22$ on the register already holds 5, so the certificate is true there and weaker; these three are where it moves anything.
  
  One rung is retained rather than a ladder. The side was reached in a single resumed column-generation run, not climbed to, so there is no weaker certificate at this size to compare it against.

  Composition

  Primary at all three sizes: one certificate, one accepted verdict, the bound being the container side directly. There is no monotonicity or composition step anywhere between the artifact and the claim -- the three sizes come out of Condition 2, which is the only condition that mentions n. Two evidence entries whose method values differ decide it, the exact event-cell sweep and the interval branch and bound, both run in this repository; that is C3, with the two methods shown beside the rung. Unlike T-017 and T-019 no independent evaluator reviewed the scoring, and no review record is retained, so rung 4 waits on two adversarial AI reviews and a human oversight record.

  Next rung

  This result's 24/5 rung has total 18.922620, leaving 0.077380 below nineteen, and that margin is where a further rung had to be found. It is a wide margin by this register's standards -- \(n = 17\)$n=17$'s top rung has 0.066920 and \(n = 12\)$n=12$'s has 0.001040 -- which says the side was not pushed to where the covering value stops it, only to where the run was stopped. That is the honest reading of how this one was obtained.
  
  The column generation was halted at round 9 with a restricted optimum of 18.916941, because four more rounds would have cost about 3.75 h to buy margin nothing needed. The side above it was never attempted.
  
  Where the ceiling sits differs across the three sizes and it matters. A certificate for n cannot exist above ceil(sqrt(n)) * B, which for all three is 5B = 4.9885, so 0.1885 of structural runway remains. But at \(n = 19\)$n=19$ the best known packing is 4.885618, and no certificate can exceed a side a packing achieves, so the real runway there is 0.085618. At \(n = 20\)$n=20$ and \(n = 21\)$n=21$ the best known packing is the trivial 5, so the ceiling binds first: the method could in principle close either to within 0.0115 of the upper bound and can never reach it. That asymmetry is the targeting instruction.
  
  A run above 4.885618 that succeeded would contradict the retained \(n = 19\)$n=19$ packing and is a refutation to look for rather than a rung to expect; a run between 4.80 and 4.9885 that succeeds moves \(n = 20\)$n=20$ and \(n = 21\)$n=21$ and stops at \(n = 19\)$n=19$. The quantity to read is the restricted optimum a converged run reaches, never an extrapolated rate, and the cost was the exhaustive tier: the 24/5 rung's exact sweep took 5378 s at its own 2260-atom set, against 173 s for the interval route on the same bytes. The same evening the sweep was rewritten to decide in integers on the weights' common scale, in parallel over directions, and returns the same least covered mass in 38.7 s on the same box.
  
  The margin was found and taken the following day: T-021 certifies 97/20 = 4.85 at \(n = 20\)$n=20$ and \(n = 21\)$n=21$ from a site set seeded with this certificate's own atoms, and supersedes this result at both sizes. What this result still holds alone is \(n = 19\)$n=19$, whose bound stays 24/5 because T-021's atoms are too heavy for it; the rung itself is retained at cases/n20_fractional_certificate/certificate-24-5.json and replays by name. On 2026-10-02 wand125's replayed rectangle certificate, \(s(19) \ge 963/200\)$s(19)\ge 963/200$ (T-045), superseded it at \(n = 19\)$n=19$ as well. This result stays true as stated at all three of its sizes.

  Significance

  Scored against the rubric's anchor for S4, "a reusable technique, bound family, or resolved disputed value", and by parity with T-019, which has the same shape: one certificate moving three registered cases and displacing a published value.
  
  This one displaces more. The movement is +0.21 at \(n = 19\)$n=19$ against T-019's +0.0842, the largest single-case movement in the register; and at \(n = 20\)$n=20$ and \(n = 21\)$n=21$ it displaces the peer-reviewed 2005 closed form previously used in the verified register. The September 2026 source audit corrects the earlier claim that no size-specific bounds had been reported; it does not change these measured displacements of the verified fields.
  
  Held below S5 for the reasons T-019 was: none of the three is a central open case, the weighted-resource lineage predates this project, the pure-atomic rational direction-net architecture follows Burns, the LP parameter line follows Massaccesi, and no case changes character at the new value.
  
  A calibration note for whoever scores the next one. This is the fourth result from one instrument in a day, and the S4 anchor is being carried here by the displacement rather than by the technique, which T-017 already banked. Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3.

  Novelty

  apparently-novel Not found in the recorded search, subject to its stated gaps

  Records

  n = 19n = 20n = 21registerevidence 1evidence 2

  n = 19n = 20n = 21registerevidence 1evidence 2

• T-045 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 15 cases

  Rectangle-density lower bounds replayed at 15 counts in \(n = 18\ldots78\)$n=18\dots 78$

• T-046 S3 V0 C0 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 48 cases

  Rectangle-density lower bounds reported for 48 counts in \(n = 18\ldots95\)$n=18\dots 95$

• T-058 S2 V3 C3 wand125 after Tokoharu, Daniel · 2026-09-29 · 100 cases

  Rectangle-certificate ceiling α·UB(n) proved for \(n = 1\)$n=1$..100; B·UB(n) on 64 grid rows

• T-068 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases

  Rectangle-density lower bounds verified at 34 counts in \(n = 19\ldots95\)$n=19\dots 95$

• T-074 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 31 cases

  Rectangle-density lower bounds replayed at 31 counts in \(n = 19\ldots95\)$n=19\dots 95$

• T-083 S3 V3 C3 Karakuş · 2026-10-02 · 301 cases

  \(s(n) \ge 1/2 + \sqrt{n - \lfloor\sqrt{n}\rfloor + 1/4}\)$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare \(8 \le n \le 324\)$8\le n\le 324$

• T-085 S3 V3 C3 Karakuş; chelokot · 2026-10-02 · 315 cases

  Nagamochi 2005, Lemma 1 is false for every container with \(a > 3\)$a>3$ and \(b > 2\)$b>2$

• T-100 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-06 · n = 19

  Mixed rectangle-measure lower bound verified at \(n = 19\)$n=19$, past the best 18-square packing

  Claim and records

  Claim

  A rectangle density of wand125/square-packing-bounds, checked at coverage one on a net it declares and published on 6 October 2026, proves \(s(19) \ge 48229/10000\)$s(19)\ge 48229/10000$ = 4.8229.
  
  The certificate is a density of 313 uniform rectangles in D4 orbits, and no point mass, of total mass 1899999/100000, on the net T-096's certificate declares: core side 999/1000 and 416 half-angle tangents of step 1/1001, with 999/1000 (1 + 1/1001) = 500499/500500 < 1. The source reports every core at a net angle capturing mass at least one: at all 415 oblique net angles by code/mixed_rotated_verify.cpp, its research copy of Tokoharu's verify.cpp at coverage threshold one, run by the same driver as T-096's certificate, and at the axis by exact integer tables.
  
  The value is above the 1927/400 = 4.8175 of the source's rectangle certificate rect_n19_L48175 (T-074), the reported and verified lower bound at \(n = 19\)$n=19$ before it, by 0.0054. It is also above (7 + sqrt 7)/2 = 4.8228757..., the side of Hämäläinen's packing of 18 squares and the verified upper bound at \(n = 18\)$n=18$, by about 2.43e-5, so \(s(18)\)$s(18)$ < \(s(19)\)$s(19)$, as the source notes.
  
  The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 416 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, independently re-implemented. The verifier shares no code with the source's checker and runs the same net-and-shrink method, so it is a second implementation and not a second method. The 6 October review of the certificate found no defect. The source's own checker was replayed here at 10 of the 416 directions, each returning the certificate's own record, and not in full.
  
  wand125 after Tokoharu and Levy, square-packing-bounds. Registration was requested in a comment of 6 October 2026 on jlevy/squares#366. The source says parts of the work were produced with AI assistance under human direction.

  Composition

  One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 416 directions of the net the retained candidate declares: one interval-certified method, replayed here by an independent implementation, C3, on the census route, whose per-certificate conditions for a declared net the soundness review of the declared-net change set and this certificate meets. The source's checker was replayed at 10 directions and its axis tables were not run in full here, so no second method and no complete reproduction with the producer's code stands beside it.
  
  The separation \(s(18)\)$s(18)$ < \(s(19)\)$s(19)$ is cited from \(n = 18\)$n=18$'s record, whose verified upper bound is E-lifted-q7-upper, the exact verification of Hämäläinen's packing; this entry adds no evidence for it beyond its own bound.

  Next rung

  V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record. One is retained, the review of 6 October that read the certificate. A complete replay of the source's own checker, the bundle's driver over all 416 nodes by devtools.audit_wand125_declared_net replay (13.4 CPU-hours priced here), would add a reproduction with the producer's code beside the rung; a second machine method would be a method-distinct decision of rotated coverage.

  Significance

  The strongest verified lower bound on record at \(n = 19\)$n=19$, by 0.0054, and the first to pass the side of the best packing of 18 squares, which separates \(s(18)\)$s(18)$ from \(s(19)\)$s(19)$ (Evan Daniel's question on jlevy/squares#281), by about 2.43e-5. The certificate kind and checker of T-069 and later on T-096's declared net; no new technique, at S3 as T-074 is, the separation a corollary of this bound and Hämäläinen's packing rather than a method or a family of bounds. The 6 October review confirmed the draft.

  Novelty

  previously-published Present in an identified source

  Records

  n = 19registerevidence 1evidence 2sourcereview

  n = 19registerevidence 1evidence 2sourcereview

• T-103 S3 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-06 · n = 19

  Mixed rectangle-measure lower bound verified at \(n = 19\)$n=19$, on the finest declared net yet

  Claim and records

  Claim

  A rectangle density of wand125/square-packing-bounds, checked at coverage one on a net it declares and published on 6 October 2026, proves \(s(19) \ge 193/40\)$s(19)\ge 193/40$ = 4.825.
  
  The certificate is a density of 341 uniform rectangles in D4 orbits, and no point mass, of total mass 1899999/100000, on a net it declares itself, finer than any it declared before: core side 4999/5000 and 2073 half-angle tangents of step 1/5002. Since 4999/5000 (1 + 1/5002) = 25009997/25010000 < 1, and the last tangent 1036/2501 is past tan(pi/8), every unit square contains a concentric core at a net angle strictly in its interior.
  
  The source reports every such core capturing mass at least one at all 2073 directions, by sqverify-proof-net, its copy of this repository's sqverify_fast changed to read the declared net, and ships no record of its C++ checker for this certificate.
  
  The value is above the 48229/10000 = 4.8229 of T-100, the source's earlier certificate and the reported and verified lower bound at \(n = 19\)$n=19$ before it, by 0.0021, and supersedes it.
  
  The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 2073 directions of the net the candidate declares, and two mutants scaled below coverage one were refused: confirmed, re-implemented sharing the producer's components. The source's check is a copy of the same crate with one change, so the two share the sqverify_fast crate; their records agree direction by direction, and their agreement is not a second implementation. The source's C++ checker, the producer's own code but sharing none with the crate, verified the candidate here at threshold one at nodes 1021 and 2072, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.
  
  wand125 after Tokoharu and Levy, square-packing-bounds. No registration was requested: an intake pass of the source found it. The source says parts of the work were produced with AI assistance under human direction.

  Composition

  One primary certificate on its reported entry and its replay entry. The replay is sqverify-fast at all 2073 directions of the net the retained candidate declares: one interval-certified method, replayed here by an implementation that shares the producer's components, C3, on the census route, whose conditions for a declared net this certificate meets. The source's own check is a copy of the same crate, and its C++ checker was run here at sampled directions only, so no second implementation and no second method stands beside it.

  Next rung

  V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record; one is retained. A complete run of the source's C++ checker over all 2073 directions (devtools.audit_wand125_declared_net cpp-sample, about 97 CPU-hours at the samples' rate) would add a check that shares no code with sqverify_fast beside the rung, though it is the producer's own code.

  Significance

  The strongest verified lower bound on record at \(n = 19\)$n=19$, by 0.0021 above T-100, and the strongest reported. The certificate kind of T-099 on a declared net, checked at the source by a copy of this repository's verifier rather than its C++ checker; no new technique, at S3 as T-096, T-099 and T-100 are.

  Novelty

  previously-published Present in an identified source

  Records

  n = 19registerevidence 1evidence 2sourcereview

  n = 19registerevidence 1evidence 2sourcereview

In the frontier survey On GitHub