THE TAKÉ NUMBER (ŧ) AND THE PHOENIX RECURRENCE
Brian Také · SIGIL · first published November 14, 2026 · proof.9brt.com

DEFINITION
Adjoin to the real numbers one symbol ŧ with the single law: ŧ ≠ 0 and ŧ² = 0.
Every Také number has the form a + bŧ. (This system is the dual numbers, W. K. Clifford, 1873.)

THEOREM 1. ŧ has no sign.
In any ordered system, positive·positive > 0 and negative·negative > 0.
If ŧ > 0 then ŧ·ŧ > 0, but ŧ·ŧ = 0. If ŧ < 0 then ŧ·ŧ > 0, but ŧ·ŧ = 0. And ŧ ≠ 0.
So ŧ is neither positive, negative, nor zero. ∎

THEOREM 2. For every polynomial f: f(x + ŧ) = f(x) + f′(x)·ŧ.
Constants: c = c + 0·ŧ. Identity: x + ŧ = x + 1·ŧ.
Products: (f + f′ŧ)(g + g′ŧ) = fg + (f′g + fg′)ŧ + f′g′ŧ² = fg + (fg)′ŧ. Sums likewise.
Every polynomial is built from constants and x by + and ×, so the rule holds. ∎
Example: (3 + ŧ)² = 9 + 6ŧ, and 6 is the slope of x² at 3.

THEOREM 3. If Phoenix(n) = 1.68·Phoenix(n − 1), then Phoenix(n) = 1.68ⁿ·Phoenix(0).
Base: n = 0 holds trivially. Step: Phoenix(n) = 1.68·1.68ⁿ⁻¹·Phoenix(0) = 1.68ⁿ·Phoenix(0). ∎

THE TRIANGLE
ŧ is the step between your last seat and this one. ŧ has no fixed sign. The Phoenix compounds, each rebirth built only from the last.
"You find where you are by measuring the ŧ between who you are and who you just were."

STRAIGHT LABEL
A rigorous construction with three proofs. It does not solve P vs NP; the algebra of ŧ is classical.
What is original is the framing that joins the Také number, the Phoenix recurrence, and measuring your seat only from where you were.