Chapter CoherentDynamics

import Definitions.Def_ChapterCoherentOverlapComplex
import Definitions.Def_ChapterCoherentFidelity
import Definitions.Def_ChapterA4
import Mathlib


/-!
# Chapter "The Coherent State of Attention" — the symmetry group of attention

The Born weights of the coherent-state head are built from the Bargmann kernel
`⟨q|k⟩`, and the kernel only sees norms and inner products.  Everything that
preserves those leaves attention invariant.  This module identifies three such
symmetries and records what each means physically.

Deliverables (all `sorry`-free, `axiom`-free):

* `coherentOverlapC_isometry`, `bornWeightC_isometry` — **unitary invariance**:
  a common unitary change of frame on the query and all the keys changes no
  attention weight;
* `phaseRotate` — the free harmonic evolution of a coherent state,
  `α ↦ e^{iθ}α` (with `θ = -ωt`), together with `phaseRotate_isometry`
  identifying it as a unitary;
* `coherentOverlapC_phaseRotate`, `bornWeightC_phaseRotate` and
  `bornWeightC_evolution_const` — **attention is a constant of the motion**: the
  weights of a head whose query and keys evolve freely under the same
  Hamiltonian do not depend on the time;
* `fidelityC_phaseRotate` — the same statement for the fidelity readout;
* `bornWeightC_translation_invariant` — **displacement covariance**: displacing
  the query and every key by a common vector (a Weyl/Heisenberg translation of
  phase space) leaves the weights unchanged, so only *relative* positions in
  phase space are physical.

Everything here is `sorry`-free and `axiom`-free (only `propext`,
`Classical.choice`, `Quot.sound`).
-/

open scoped BigOperators

noncomputable section

namespace BookProof.ChapterCoherentDynamics

open BookProof.ChapterCoherentOverlapComplex BookProof.ChapterCoherentFidelity

variable {n m : ℕ}

/-! ## Unitary invariance -/







/-! ## Free evolution: the phase rotation -/

/-- The **free harmonic evolution** of a coherent state on the Bargmann
parameters: `α ↦ e^{iθ}α`, with `θ = -ωt` for a mode of frequency `ω`. -/
def phaseRotate (theta : ℝ) (q : EuclideanSpace ℂ (Fin n)) : EuclideanSpace ℂ (Fin n) :=
  Complex.exp (theta * Complex.I) • q















/-! ## Displacement covariance -/



end BookProof.ChapterCoherentDynamics

end