Field-Morphism Matrix (FMM) investigation

A legacy matrix factorisation experiment is identified exactly as multiplication by a fixed unit in a finite quotient algebra, generalised to k = 2 to 6, verified across 375 local profiles and 2,469 extraction pairs, and compared against an exact unit-group smoothness baseline. The mechanism is valid; no privileged smooth-order advantage was measured.

A bounded computational classification of the Field-Morphism Matrix: exact algebraic identification, closed-form characteristic polynomial, 2,469 verified extractions and an exact smoothness baseline.

The Field-Morphism Matrix investigation.

The mechanism is valid. The advantage was not there.

What is this study actually trying to do?

What is the matrix?

Does the factor appear?

Is it unusually easy?

From a mystery matrix to a bounded classification.

Every number here was computed exactly, then checked against a control.

MatriX / FMM · Frozen bounded study

An old experiment factored numbers with mysterious matrices. What are those matrices really doing, and is the trick genuinely clever, or just correct?

The matrices really do extract factors, and we now know exactly why. But the distinguished element they use behaves like an ordinary one, so no claim of a new or superior factoring method is made.

Some whole numbers are made by multiplying two prime numbers together, and finding those hidden primes is hard. We inherited a piece of old code that claimed to find them using grids of numbers called matrices, with no explanation of how or why it worked.

So we asked three separate questions, and deliberately kept them separate: what are these matrices really? Do their factor discoveries genuinely work? And is the particular element they use unusually lucky, easier to reach than an ordinary one? A method can pass the first two tests and still fail the third, and confusing "it works" with "it is better" is how false claims are born.

It is a compact way of writing "multiply by one fixed number" inside a special number system. Nothing more exotic than that, and now provably so.

  • Field-Morphism Matrix (FMM) Investigation
  • Recovered construction
  • A legacy experiment used matrices nobody had explained.
  • M[i, j] = 1 if i ≥ j, otherwise d
  • Exact identification
  • The matrix is simply multiplication by one fixed number.
  • Executable campaign
  • The factor extraction works, and we measured it exactly.
  • G(B) = gcd(N, entries of M^L(B) − I)
  • Exact smoothness control