$n=20$$n=20$

Bounds

Results in the register

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

  s(n)≥min(⌈n​⌉,n−2⌊n​⌋+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-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.

  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-021 S3 V3 C3 Levy after Burns, Massaccesi · 2026-09-05 · n = 20, 21

  $s(n)\ge 97/20=4.85$$s(n)\ge 97/20=4.85$ for $n=20,21$$n=20,21$

  Claim and records

  Claim

  $s(20)\ge 97/20$$s(20)\ge 97/20$ and $s(21)\ge 97/20$$s(21)\ge 97/20$, from this project's weighted fractional unavoidable-set certificate at container side 97/20 = 4.85. Both sizes held 24/5 = 4.80 from T-020 earlier the same week in this register's verified fields; the movement is +0.05 at each. The certificate does not reach $n=19$$n=19$: its total mass is 19848723/1000000 = 19.848723, above nineteen, so Condition 2 gives it $n=20$$n=20$ and upward and T-020's 24/5 rung, retained beside it, still carries $n=19$$n=19$.
  
  The two sizes take no monotonicity step, for the reason T-020 gives: only Condition 2 mentions n among the five conditions, so an atom set of mass M certifies its side for every integer strictly above M. From $n=22$$n=22$ on the register already holds 5, so the certificate is true there and weaker.
  
  Where the rung came from is worth the record. A pre-registered bisection of [24/5, 9977/2000] ran four rungs unattended. The uniform grid walled at this side -- its restricted optimum crossed twenty at round 31 of 32 and stood at 20.000439 with placements still violated -- while a second site set, the 24/5 certificate's own 2260 atoms scaled by 97/96 and unioned with the grids, converged below twenty at the same side. The certificate is the second construction's; the first construction's crossing is why H-062 asks for two.

  Significance

  Scored at S3 by the calibration note T-020 wrote for exactly this case: "Further sizes from the same generator, absent a new technique or a case that changes character, belong at S3." This is the same generator at the same two sizes, one rung higher, and neither size changes character at 4.85. What it adds beyond a rung is a measurement rather than a theorem, and the measurement is registered as such: the m = 5 covering wall is bracketed to [97/20, 39/8] -- a certificate at 4.85, crossings above twenty on both constructions at 4.875 -- which is the first bracket this project has put around a covering wall, and it sits far below the method's ceiling of 4.9885 rather than at it. That bears on planning, not on the strength of this claim.

  Novelty

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

  Records

  n = 20n = 21registerevidence 1evidence 2

  n = 20n = 21registerevidence 1evidence 2

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

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

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

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

• T-058 S2 V3 C3 Hosono 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 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases

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

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

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

• T-077 S3 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-03 · n = 20, 42, 70

  Rectangle-density lower bounds replayed at $n=20$$n=20$, 42 and 70

  Claim and records

  Claim

  Three rectangle-density certificates in wand125/square-packing-bounds, raised on 2 October 2026, prove $s(20)\ge 49/10$$s(20)\ge 49/10$ = 4.9, $s(42)\ge 2731/400$$s(42)\ge 2731/400$ = 6.8275 and $s(70)\ge 3451/400$$s(70)\ge 3451/400$ = 8.6275. Each raises the certificate T-068 reports at its count, which T-074 replayed, by 0.0025 to 0.0125.
  
  Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The certificates were built with Tokoharu's solver, their weights scaled by one exact factor each to that mass. The same revision raised four more rungs, at $n=59$$n=59$, 77, 91 and 93, which are below what this record holds there: the exact values $s(59)=8$$s(59)=8$ and $s(77)=9$$s(77)=9$, and the same revision's mixed certificates at $n=91$$n=91$ and 92, the latter carried to $n=93$$n=93$ (T-075).
  
  Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, in two cloud batches on 3 October 2026 whose receipts were merged into the packet on 6 October, and each replay reproduced the upstream accepting run's node count, leaf count and lower bound at every direction: confirmed, reproduced with the producer's code. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises.
  
  At $n=20$$n=20$ and 42 the values raised the verified lower bound, from T-074's 1959/400 and 1363/200, and later on 6 October T-104's check2 certificate raised it again at $n=20$$n=20$ and T-090's mixed certificate at $n=42$$n=42$; at $n=70$$n=70$ the mixed certificate of T-091 holds more.
  
  Each was also decided by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions at Tokoharu's threshold 10001/10000 in the census of 3 October, and two mutants of each scaled below that threshold were refused on 6 October: confirmed, independently re-implemented as well. It shares no code with the source's checker and runs the same method, a second implementation and not a second method.
  
  Hosono after Tokoharu and Levy, square-packing-bounds. No issue requests these certificates; the import of 2 October took them in from the source's own commits. The source says parts of the work were produced with AI assistance under human direction.

  Significance

  Raised the verified lower bound at $n=20$$n=20$ and 42, by 0.0025 and 0.0125 over T-074's replayed certificates, and the strongest rectangle-density lower bounds on record at all three counts. Raised rungs from one generator, as T-068's were, at S3; the 2 October review of the afternoon certificates proposed S3. Confirmed on 11 October beside T-068 and T-074.

  Novelty

  previously-published Present in an identified source

  Records

  n = 20n = 42n = 70registerevidence 1evidence 2evidence 3evidence 4evidence 5source 1source 2review 1review 2

  n = 20n = 42n = 70registerevidence 1evidence 2evidence 3evidence 4evidence 5source 1source 2review 1review 2

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

  s(n)≥1/2+n−⌊n​⌋+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; Vlasenko · 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-104 S3 V3 C3 Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-06 · n = 20

  Mixed rectangle-measure lower bound verified at $n=20$$n=20$, on a declared net

  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(20)\ge 981/200$$s(20)\ge 981/200$ = 4.905.
  
  The certificate is a density of 327 uniform rectangles in D4 orbits, and no point mass, of total mass 1999999/100000, on the net T-099's certificate declares: core side 1999/2000 and 832 half-angle tangents of step 1/2006. Since 1999/2000 (1 + 1/2006) = 4011993/4012000 < 1, and the last tangent 831/2006 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 832 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 49/10 = 4.9 the record reported and verified at $n=20$$n=20$, the source's rectangle certificate rect_n20_L49 (T-077), by 0.005.
  
  The certificate was decided here on 6 October 2026 by sqverify-fast, this repository's clean-room measure verifier, at all 832 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 679 and 831, the least-bound node of the source's run among them, and was not run in full: samples that decide those nodes only.
  
  Hosono 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.

  Significance

  The strongest verified lower bound on record at $n=20$$n=20$, by 0.005 above T-077, 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 = 20registerevidence 1evidence 2sourcereview

  n = 20registerevidence 1evidence 2sourcereview

• T-124 S2 V0 C0 Daniel after Couzo · 2026-10-07 · 178 cases

  Reported non-strict local minima for 178 source configurations

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