Cubic Phi3 companion-matrix factorisation study

A disciplined computational study of the companion matrix of x cubed minus x minus one, its norm-one order law, prospective factor windows and exact finite baseline. This is a verified reconstruction within established cyclotomic factorisation territory, not a new algorithm.

A bounded computational reconstruction of a cubic cyclotomic factorisation mechanism, with exact order diagnostics, prospective witnesses and carefully limited conclusions.

Cubic Phi3 companion-matrix factorisation study.

A disciplined reconstruction, not a new factoring algorithm.

What is this study actually trying to do?

Turn a matrix into a repeating process.

The hidden factors keep different time.

Use the mismatch as the key.

The failed idea stayed in the record, and made the next test stronger.

Prediction first. Extraction second. Limits kept visible.

MatriX / Rootfinder · Frozen bounded study

Can a cubic companion matrix create a genuinely cubic factorisation channel, and does one fixed element make that channel easier to reach?

The cubic mechanism is real and computationally verified within the study. Its cyclotomic foundation is established prior art. The value lies in the transparent matrix specialisation, prospective tests and exact evidence.

Some whole numbers are made by multiplying two prime numbers together. Finding those hidden factors can be difficult when the number is large. This study asks whether a small mathematical machine, a matrix, can make one hidden factor reveal itself before the other.

The matrix repeatedly transforms a set of numbers. When we calculate using a composite number, the same process is effectively running against both of its hidden prime factors at once. Their patterns can complete at different times. If one has completed while the other has not, ordinary greatest-common-divisor arithmetic can expose the factor during that brief window.

Think of the matrix as a compact set of instructions. Applying it again and again produces a cycle whose length depends on the prime number underneath it.

  • Cubic Phi3 Companion-Matrix Factorisation Study
  • False lead retained
  • Higher dimension did not mean higher-degree behaviour.
  • A₃ₘ = Vₘ(−t, 1)
  • Cubic construction
  • A genuine degree-three algebra was isolated.
  • C³ = C + I · det(C) = 1
  • Prospective witnesses
  • The factor windows were predicted before extraction.
  • Bmin(rₚ) < Bmin(rq)