IndisputableMonolith.Quantum.RecognitionFirst.EightTickWeyl
Formalizes the finite Heisenberg–Weyl algebra on the eight-tick recognition cycle Z/8Z. Introduces a primitive eighth root of unity together with clock and shift operators, and proves their Weyl commutation relation. Cited by anyone deriving quantum kinematics from the recognition substrate: occupation and cost-rate fail to commute by that phase. Argument is elementary circle arithmetic plus modular evaluation.
claimOn the recognition cycle $\mathbb{Z}/8\mathbb{Z}$, let $\omega=e^{2\pi i/8}$ be a primitive eighth root of unity, and let shift and clock be the standard finite Heisenberg–Weyl generators. Then $\mathrm{clock}\circ\mathrm{shift}=\omega\cdot(\mathrm{shift}\circ\mathrm{clock})$, with $\omega^8=1$ and $\omega\neq 1$. The cost-rate advances by this phase once per recognition tick.
background
Recognition Science forces an eight-tick octave (forcing chain T7: period $2^3$). Per tick the cost-rate advances by a fixed complex phase; that phase is a primitive eighth root of unity on the unit circle.
This module works entirely in the discrete setting $\mathbb{Z}/8\mathbb{Z}$. Shift advances the tick index; clock multiplies by the phase power of the current residue. Their composite realizes the finite Heisenberg–Weyl pair whose commutator phase is exactly $\omega$.
The Quantum facade re-exports the construction as the bridge from the recognition substrate to standard quantum structure: occupation and cost-rate no longer commute once the eight-tick phase is present.
proof idea
Definitions introduce $\omega$ via the complex circle, then the shift and clock maps on $\mathbb{Z}/8\mathbb{Z}$. Two short lemmas record $\omega^8=1$ and $\omega\neq 1$. The main identity is the operator equation clock after shift equals $\omega$ times shift after clock; a companion statement packages the same fact as canonical noncommutativity. All proofs evaluate both sides on residues, using Mathlib facts about the circle group and modular arithmetic. No continuum analysis is required.
why it matters in Recognition Science
Supplies the Quantum facade with the finite Heisenberg–Weyl relation on the 8-tick cycle, the first concrete quantum structure derived from the recognition substrate. Also imported by RecogPhysicsStaging, the verified bridge that turns prose derivation steps into checked Lean. Lands forcing-chain T7 in operator language: the same period $2^3$ that forces the octave becomes the order of the Weyl phase. Without this discrete noncommutativity, occupation and cost-rate would commute and the recognition-first route to quantum kinematics would have no algebraic starting point.
scope and limits
- Does not construct continuous-time Schrödinger dynamics or the Weyl CCR on L²(ℝ).
- Does not derive ħ or any continuum limit from the eight-tick algebra alone.
- Does not prove representation uniqueness beyond the explicit Z/8Z model.
- Does not treat spin, statistics, or multi-particle Fock structure.