Foundation Recognition Operator Sector Project Eq Id On Quarter Turn Core

The fixed points of projection

In linear algebra, a projection is a map that sends every vector to a subspace and leaves the vectors already in that subspace unchanged. The declaration sectorProject (a projection onto a chosen sector of an eight-dimensional signal space) has a theorem attached to it: on a specific subspace called the quarter-turn core, the projection acts as the identity. In plain terms, if a signal lives entirely in that core, projecting it does nothing; it passes through untouched.

The quarter-turn core is built from four of the eight discrete Fourier modes, specifically the odd-indexed ones (modes 1, 3, 5, and 7). These are the modes that acquire a sign flip under a shift by four positions, which is the discrete analogue of a half-turn. The theorem sectorProject_eq_id_on_quarterTurnCore states this invariance for every structured sector: no matter which modes the sector retains or drops, any signal composed only of odd modes is fixed by the projection.

The proof is a direct consequence of how the projection is defined. The projection acts on each Fourier mode independently: it retains a mode if the sector says to retain it, and zeroes it otherwise. For an odd mode that the sector retains, the mode is already in the target subspace, so the projection returns it unchanged. For an odd mode the sector drops, the projection would zero it, but such a mode cannot appear in a signal that lives entirely in the quarter-turn core. The invariance is therefore structural, not a special property of any particular sector.

This theorem matters because it pins down a stable region for the recognition update, the map that shifts a signal and then projects it. On the quarter-turn core, the shift by four positions sends a signal to its negation, and the projection leaves it alone, so the update composes to a clean negation. That gives a predictable four-step cycle for signals in the core, a fact the framework uses to build its eight-tick recognition cycle.

What the theorem does not claim is just as important. It does not say that every signal is fixed by the projection; signals with even modes can be altered or zeroed. It does not say the quarter-turn core is the only invariant subspace; other subspaces may exist for specific sectors. And it says nothing about the physical meaning of the modes or the signals, only about the algebraic structure of the projection.

/-- Every structured-sector projector fixes the quarter-turn core pointwise. -/
theorem sectorProject_eq_id_on_quarterTurnCore (S : StructuredSector) :
    ∀ {f : Signal8}, f ∈ quarterTurnCore → sectorProject S f = f := by
  intro f hf
  refine Submodule.span_induction ?_ ?_ ?_ ?_ hf
  · intro m hm
    rcases hm with ⟨k, hkodd, rfl⟩
    have hk : k ∈ S.keepModes := S.odd_modes_included hkodd
    simpa [sectorProject_mode, hk] using sectorProject_mode S k
  · exact map_zero (sectorProject S)
  · intro x y hx hy hpx hpy
    calc
      sectorProject S (x + y) = sectorProject S x + sectorProject S y := by
        exact map_add (sectorProject S) x y
      _ = x + y := by rw [hpx, hpy]
  · intro a x hx hpx
    calc
      sectorProject S (a • x) = a • sectorProject S x := by
        exact map_smul (sectorProject S) a x
      _ = a • x := by rw [hpx]

/-- The quarter-turn core is the span of the odd DFT modes. -/
def quarterTurnCore : Submodule ℂ Signal8 :=
  Submodule.span ℂ {m | ∃ k : Fin 8, Odd k.val ∧ m = dft8_mode k}

/-- Projecting a basis mode either keeps it or kills it. -/
lemma sectorProject_mode (S : StructuredSector) (k : Fin 8) :
    sectorProject S (dft8_mode k) = if k ∈ S.keepModes then dft8_mode k else 0 := by
  ext t
  by_cases hk : k ∈ S.keepModes
  · rw [if_pos hk, sectorProject_apply]
    rw [Finset.sum_eq_single k]
    · simp [dft_coefficients_mode, dft8_mode, hk]
    · intro j hj hne
      simp [dft_coefficients_mode, hne, dft8_mode]
    · intro hnot
      exact (hnot hk).elim
  · rw [if_neg hk, sectorProject_apply]
    apply Finset.sum_eq_zero
    intro j hj
    have hj_ne : j ≠ k := by
      intro hj_eq
      apply hk
      simpa [hj_eq] using hj
    simp [dft_coefficients_mode, hj_ne, dft8_mode]

/-- Four shifts act as `-I` on the quarter-turn core. This is the concrete
`P^4 = -I` statement used in the paper. -/
theorem shift_four_eq_neg_on_quarterTurnCore {f : Signal8} (hf : f ∈ quarterTurnCore) :
    cyclicShiftIter 4 f = -f := by
  refine Submodule.span_induction ?_ ?_ ?_ ?_ hf
  · intro m hm
    rcases hm with ⟨k, hkodd, rfl⟩
    rw [cyclicShiftIter_mode, odd_mode_fourth_eigenvalue k hkodd]
    ext t
    simp
  · ext t
    simp [cyclicShiftIter, cyclic_shift]
  · intro x y hx hy hpx hpy
    calc
      cyclicShiftIter 4 (x + y) = cyclicShiftIter 4 x + cyclicShiftIter 4 y := cyclicShiftIter_add 4 x y
      _ = -x + -y := by rw [hpx, hpy]
      _ = (-1 : ℂ) • x + (-1 : ℂ) • y := by simp
      _ = (-1 : ℂ) • (x + y) := by rw [smul_add]
      _ = -(x + y) := by
            ext t
            simp [smul_eq_mul]
  · intro a x hx hpx
    calc
      cyclicShiftIter 4 (a • x) = a • cyclicShiftIter 4 x := cyclicShiftIter_smul 4 a x
      _ = a • (-x) := by rw [hpx]
      _ = -(a • x) := by simp

What this page does not claim

The theorem does not claim that all signals are fixed by the projection. The theorem does not claim the quarter-turn core is the unique invariant subspace. The theorem does not assign physical meaning to the modes or signals.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/RecognitionOperator.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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