An exact laboratory for a rank upper bound

The Forty-Eighth Product

Multiply two 4 by 4 matrices through a published 48-product decomposition, then inspect the exact tensor certificate that makes every output entry correct. A second bench moves the same identity from complex coefficients to a rational representative and odd-prime arithmetic, keeping the field restrictions and the unresolved optimality question in view.

The spectacle is one missing multiplication. The substance is the certificate: every possible input is covered by a finite identity that this page reconstructs in full.

Layer one: put numbers through itThree roads to one matrix

Choose the published 48-product table

Loading exact coefficients.

A
B

Preparing the tables.

Layer two: examples are not evidenceCheck the whole tensor

A random match can be luck. A bilinear identity is settled by its coefficients. Select a valid lane and rebuild every coordinate of the tensor. Odd-prime mode clears the rational denominators in a field where two is invertible.

Field and certificate bench

Complex exact arithmetic uses Gaussian integer pairs after clearing eighths.

The display is one 256-cell projection. The adjacent live text reports the complete 16 by 16 by 16 check.

Universal certificate

waiting

No coordinates checked yet.

Repeat the block rule

The product count compounds. This is not a runtime benchmark.

The check

The numbers below are decoded and recomputed by this page from pinned coefficient tables. They are not result labels copied into the markup.

Conventions, choices, uncertainties, and what remains open
  • Indexing convention. Inputs use row-major indices. The AlphaEvolve output factor uses the published transpose convention. The rational LRP output is row-major. The target entry is one exactly when the shared inner index joins an input row to an output column.
  • Exactness convention. The complex table is stored as integer real and imaginary pairs after multiplication by two. Its triple products are compared after clearing a denominator of eight. The rational table clears denominators two, one, and eight for L, R, and P.
  • Free choices. The editable starting matrices and random range are interface choices. The odd primes offered are examples, not privileged primes. Recursion depth is a product-count illustration only.
  • Cost model. The counts include only products between input-dependent quantities. Additions, multiplications by fixed coefficients, basis changes, allocation, memory traffic, and hardware costs are excluded. A complex product is not one real hardware multiplication. No measured speedup is claimed.
  • Evidence boundary. Entered and random matrices demonstrate the rule. The full coefficient identity is the universal certificate. The two characteristic-zero 48-product papers are preprints. The AlphaTensor and Strassen comparison papers are peer reviewed.
  • Field boundary. The peer-reviewed 47-product AlphaTensor result is modulo two. It is not a characteristic-zero counterexample. This page does not run that table because this build did not pin and independently audit its official coefficients.
  • Still open. These decompositions prove a characteristic-zero tensor-rank upper bound of 48. They do not prove minimality. Whether the rank is 48 or lower remains open here. Practical crossover and numerical stability also remain unsettled by rank.
Primary sources and review status