$n=21$$n=21$

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-034 S3 V3 C3 Levy after Burns, Massaccesi · 2026-09-23 · n = 21

  $s(21)\ge 122/25=4.88$$s(21)\ge 122/25=4.88$

  Claim and records

  Claim

  $s(21)\ge 122/25$$s(21)\ge 122/25$, from this project's weighted fractional unavoidable-set certificate at container side 122/25 = 4.88. The case held 97/20 = 4.85 from T-021 in this register's verified field, and the movement is +0.03. The certificate does not reach $n=20$$n=20$: its total mass is 5036431/250000 = 20.145724, above twenty, so Condition 2 gives it $n=21$$n=21$ and upward, and T-021's 97/20 rung, retained beside it, still carries $n=20$$n=20$.
  
  The size takes 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.
  
  The site set is the plainest the generator offers: auto grids (34, 46, 56) at inset 1/2 with no seed and no windows, converged below twenty-one in 36 LP rounds. At the same side a T-021-seeded set ran out its deadline unconverged, and a five-window set converged but stalled the interval route on a seam where atom rows sit exactly B apart; neither run decides anything about the side.

  Significance

  Scored at S3 by the calibration note T-020 wrote and T-021 applied: "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 one of the same sizes, one rung higher, and $n=21$$n=21$ does not change character at 4.88. What it adds beyond the rung is a reading about the instrument rather than a theorem: the stock generator with no seed and no windows converged where the seeded and windowed site sets did not decide. The side also sits above 39/8 = 4.875, where both of H-062's constructions crossed twenty at $n=20$$n=20$; a mass above twenty is consistent with that wall and says nothing about $n=20$$n=20$.

  Novelty

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

  Records

  n = 21registerevidence 1evidence 2

  n = 21registerevidence 1evidence 2

• T-050 S3 V3 C3 Daniel after Burns, Massaccesi · 2026-09-29 · n = 21

  $s(21)\ge 5000/1001=4.995004995\dots$$s(21)\ge 5000/1001=4.995004995\dots$

  Claim and records

  Claim

  $s(21)\ge 5000/1001$$s(21)\ge 5000/1001$ = 4.995004995..., by Daniel's weighted point certificate of 23 September 2026. It raised the verified bound by 2878/25025, about 0.115, over T-034.
  
  The certificate is 4,604 D4-invariant rational points in 580 orbits, total weight 260057/12500 = 20.80456 < 21, decided over the rational angle net at $N=6000$$N=6000$ by the method of T-049.
  
  It was replayed here on 27 September 2026 by the source's Rust verifier at $N=6000$$N=6000$, least captured weight 10000083/10^7 at bin 684 as the source records, and by its exact Python re-check on four bins including the binding one; the two are one angle-net method.
  
  Daniel, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as its CREDITS.md says.

  Significance

  Moved $n=21$$n=21$ by 0.115 over T-034 to within 5/1001 of the grid's 5, past this repository's own fixed-shrink ceiling of 4.9885 for the format. A substantive case result, S3; superseded within two days by the same author's exact value. Confirmed on 11 October: supersession does not lower a score, and the step, 0.115 at one count, is a case result as T-021 and T-034 are.

  Novelty

  previously-published Present in an identified source

  Records

  n = 21registerevidence 1evidence 2sourcereview

  n = 21registerevidence 1evidence 2sourcereview

• T-052 S4 V3 C3 Daniel after Burns, Massaccesi · 2026-09-29 · n = 21

  $s(21)=5$$s(21)=5$, by a mixed cover of points and grid-line segments

  Claim and records

  Claim

  $s(21)=5$$s(21)=5$: the lower half by Daniel's mixed cover of 27 September 2026, the upper half by the $5\times 5$$5\times 5$ grid.
  
  The cover is 7,536 weighted points of [0,5]^2 plus mass spread uniformly along 1,872 segments of length 1/50 on the interior grid lines, total 522368729933/(25*10^9) = 20.894749197 < 21, exactly D4-invariant. Every closed unit square in [0,5]^2 captures mass at least one, a boundary point and a segment along an edge counting in full.
  
  The source certifies that at margin zero with two separately written checkers, zm_mixed.py in exact rational arithmetic and the Rust zmx2 with outward-widened binary64 enclosures, and Lean proves minSide 21 = 5 from the one checker statement.
  
  zmx2 was replayed here in full on 28 and 29 September 2026, D4-reduced over 2,500 roots and unreduced over 20,000, every root's census equal to the source's; zm_mixed.py was only sampled.
  
  Daniel, evand/square-packing, building on Burns's and Massaccesi's method, with an AI agent under human direction as its CREDITS.md says.

  Significance

  An exact value for a case that was open, the second of the s(k^2 - 4) family after T-051, and the release that introduced line mass on the grid lines, which is what makes a zero-margin cover below the count possible at an integer side. S4 by the anchor "a reusable technique"; T-053 reuses it at $n=45$$n=45$. Confirmed on 11 October: line mass is the idea later exact values at an integer side reuse (T-053, T-062, T-066, T-067, T-064, T-081), which is what a reusable technique at S4 means here, though not every one does: the point-only routes (T-054, T-055) and the families proved by other measures (T-084, T-086) do without it. T-073's segments at $n=83$$n=83$ and 101 to 105 are an extension within a framework and are S3.

  Novelty

  previously-published Present in an identified source

  Records

  n = 21registerevidence 1evidence 2evidence 3sourcereview

  n = 21registerevidence 1evidence 2evidence 3sourcereview

• T-055 S2 V3 C3 Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · n = 21

  $s(21)=5$$s(21)=5$ by a point-only route

  Claim and records

  Claim

  $s(21)=5$$s(21)=5$ by a second, point-only route: the lower half by Hosono's point-only measure, completed on 28 September 2026 with a Lean 4 reduction, the upper half by the $5\times 5$$5\times 5$ grid.
  
  The lower half is Daniel's s21_lower_4.9950.txt support (T-050) scaled by 1001/1000 and re-weighted, 4,604 D4-invariant entries of total 2624862500021/125000000000, with capture threshold q = 249987/250000 over every closed unit square in [0,5]^2, so that 21q exceeds the total by 999979/125000000000.
  
  The capture statement is decided by the source's own exact rational replay of a retained certificate tree over 5,000 root boxes, and Lean proves minSide 21 = 5 from that one hypothesis.
  
  That replay was run here in full on 29 September 2026, the source's runner unchanged: every stage passed, and its records match the source's M1 run, 45,436 of the 45,446 output files byte for byte and the rest in every certified quantity. The Lean overlay was not built here.
  
  Hosono after Daniel, square-packing-bounds. The source names Daniel's mixed-cover proof (T-052) as first, claims no priority, and says its code was developed with Codex.

  Significance

  A second, point-only route to a value T-052 already holds, with a positive-margin threshold rather than a zero-margin cover and a Lean reduction of its own. A citable detail that changes no theorem. Confirmed on 11 October beside T-054.

  Novelty

  previously-published Present in an identified source

  Records

  n = 21registerevidence 1evidence 2evidence 3sourcereview

  n = 21registerevidence 1evidence 2evidence 3sourcereview

• 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-081 S4 V0 C1 Daniel after Burns, Massaccesi · 2026-10-03 · 14 cases

  s(k2−4)=k$s(k^2-4)=k$ for every integer $k$$k$ from 5 up; $k=5\dots 18$$k=5\dots 18$ are the cases held here

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

  Reported non-strict local minima for 178 source configurations

In the frontier survey On GitHub