Artificial Wasteland

Nothing there, certificate 02

The Weight That Will Not Circulate

No 110 by 110 circulant weighing matrix of weight 81 exists.

The bound is exact and narrow: order 110, weight 81, circulant matrices only. A theorem compresses the possible first rows to computing orbit assignments. This page enumerates every one.

0matrices found
after live sweep
Each mark is one residue in Z/110Z. Colour identifies its orbit under multiplication by 3.

The certificate

computing the domain
The claim
No CW(110,81) exists: no 110 by 110 circulant matrix over {0,+1,-1} has product with its transpose equal to 81 times the identity.
Domain swept
Generating the multiplier-orbit domain.
Method
Generate the orbits, assign each orbit 0, +1, or -1, prune wrong sign counts, then compute every nonzero periodic autocorrelation.
Positive control
CW(63,16), ready to run with the same orbit searcher.
Row-checker harness probe
One valid cyclic shift can be added to a live order-63 calibration tray.
Result
running the census
Bound
Only CW(110,81). The sweep says nothing about noncirculant W(110,81), or any different order or weight. Removing the multiplier theorem makes this reduced census underdetermined.

I. What is being weighed

A row trying to cancel every shifted copy of itself.

Take a ring of 110 cells. Put +1 in 45 cells, -1 in 36, and 0 in 29. Now turn the ring by one place and multiply overlapping cells. The sum must be zero. Turn it again. Zero again. It must cancel at all 109 nonzero shifts.

The unturned row has dot product 81 with itself, its weight. Its 109 shifted dot products have to vanish. Those 110 dot products are exactly the first row of WWT for the circulant matrix made from the ring.

The sign counts are forced. The row sum squared equals the weight, so its sum is +9 or -9. Reversing every sign changes neither the matrix property nor existence, so this search chooses +9 without loss of generality: 45 plus signs and 36 minus signs.

Operable base

Sweep the claimed empty domain

The browser constructs all orbits and all label assignments. No result table is downloaded. The counter below is produced by this run.

The complete census is queued.
Positive control

The same engine can succeed

Switch only the parameters to order 63, weight 16, multiplier 2. One found row is printed as P and N, then checked against the pinned repository witness.

Not yet run.
Harness probe

Put one success into an empty tray

The tray contains 63 correct-count near misses at order 63. Planting adds a cyclic shift of the valid control row. The same row checker must change from exactly 0 to exactly 1, then back to 0 when removed.

Unplanted tray ready.

II. Why twelve orbits are enough

The theorem is the bridge from a small search to a global claim.

Searching only symmetric rows would normally prove almost nothing. Here the multiplier theorem says every possible CW(110,81) has a cyclic translate fixed by multiplication by 3.

The hypotheses are visible: 81 is the prime power 34, and gcd(110,81)=1. Multiplication by 3 partitions the 110 positions into 12 orbits. A fixed translate must be constant on each orbit, so every possible matrix has a representative among three labels raised to those 12 orbits.

Boundary: orbit inspector

Move the domain before trusting the reduction

Try another order and multiplier. This inspector constructs the partition only. It refuses values outside its stated browser bound and refuses noninvertible multipliers.

Ready.

III. The second lock

Fold the ring. The contradiction gets smaller.

A separate argument in Proposition 3.1 folds the 110 coefficients onto 11 positions. The resulting intersection numbers must obey two equations. A theorem about 3 modulo 10 additionally forces the first number to be -1.

This is not the orbit census in disguise. It is a second route to the same empty answer, and it shows which assumption carries the contradiction. Every arithmetic signature below is only a necessary condition, never a matrix.

y1 + 5(y2 + y3) = 9
y12 + 5(y22 + y32) = 81
Computing integer signatures.
CONSISTENT

IV. A success, for scale

At order 63, the empty center fills.

The parameter switch is a boundary, not decoration. With multiplier 2 and weight 16, the same orbit assignment engine returns eight fixed representatives before equivalence reduction.

The strip below is the first live hit. It matches a witness in Daniel M. Gordon's La Jolla Circulant Weighing Matrices Repository. Green is +1, copper is -1, charcoal is zero.

+1-10
Run the positive control to expose its row.

The check

Recomputed here

  • Orbit count pending.
  • Census pending.
  • Positive control pending.
  • Row-checker harness probe pending.
  • Folding enumeration pending.

Uncertainty and free choices

  • Choosing row sum +9 is a sign convention. Global sign reversal covers -9.
  • Completeness depends on the multiplier theorem. Without it, the reduced search is UNDERDETERMINED.
  • The reversible harness probe calibrates the unchanged row checker on a live order-63 sub-domain. It is not, and cannot be, a witness of the searched-for CW(110,81).
  • The folding signatures are necessary arithmetic conditions only. They are not matrices.
  • No claim is made about noncirculant W(110,81), or about parameters other than the ones explicitly run.
  • The source repository calls weight 81 “CW(110,9)” because its second key is the square root of the weight.

Sources

A closed question, reopened as a check.

In Ming Ming Tan's 2018 update of the Strassler table, the weight-81 entry at order 110 was still marked open. Arasu, Gordon, and Zhang later proved nonexistence in Proposition 3.1. This page does not claim a discovery. It independently re-runs the finite orbit census and the paper's smaller folding contradiction.

  1. K. T. Arasu, Daniel M. Gordon, and Yiran Zhang, “New Nonexistence Results on Circulant Weighing Matrices”, Proposition 3.1 and the CW(63,16) worked example.
  2. Ming Ming Tan, “Group Invariant Weighing Matrices”, the 2018 table in which the case remained open.
  3. La Jolla Circulant Weighing Matrices Repository, pinned witness and nonexistence record, CC BY 4.0.
  4. R. M. Adin, L. Epstein, and Y. Strassler, classification of odd-order CW matrices of weight 16.