Algebra F2 Power One Dim Subspace Closed

The two-element subgroup

The elementary abelian 2-group of rank D, written F2Power D, is the set of all binary strings of length D, where the group operation is bitwise XOR. This is a standard object in algebra: each string is a vector over the two-element field, and adding two strings flips the bits where they differ. The framework's library of formal theorems defines this group and proves its basic properties, including that it has exactly 2^D elements.

For any nonzero vector v in this group, the set {0, v} forms a one-dimensional subspace, a subgroup of size two. The theorem oneDimSubspace_closed proves that this set is closed under the group operation: if you take any two elements from {0, v}, their sum is also in {0, v}. Since v + v = 0 in this group, the only possible sums are 0, v, and 0 again, so closure holds trivially. The theorem is proved in the machine-checked library with no axioms beyond the standard logical ones.

This closure property matters because it makes the count of such subgroups exact. In rank 3, the group F2Power 3 has 2^3 - 1 = 7 nonzero vectors, and each generates a distinct two-element subgroup. The library proves this count directly, which the framework's companion paper uses to derive the number 7 for basic plot families in narrative geometry. The theorem itself, however, is purely algebraic: it says nothing about stories, plots, or aesthetics. It only establishes the group-theoretic fact that each nonzero vector spans a closed two-element subgroup.

What the theorem does not claim is that these subgroups are the only subspaces, or that the count 7 has any meaning outside the algebraic setting. It does not assert that the framework's narrative applications are correct; that is a separate modeling choice. The theorem is a building block, not a conclusion about the world.

/-- The 1-dimensional subspace is closed under addition. -/
theorem oneDimSubspace_closed (v : F2Power D) (a b : F2Power D)
    (ha : a ∈ oneDimSubspace v) (hb : b ∈ oneDimSubspace v) :
    a + b ∈ oneDimSubspace v := by
  unfold oneDimSubspace at ha hb ⊢
  simp [Finset.mem_insert, Finset.mem_singleton] at ha hb ⊢
  rcases ha with ha | ha <;> rcases hb with hb | hb <;>
    subst_vars <;> simp [add_self]

/-- `F2Power D` has `2 ^ D` elements. -/
theorem card_eq : Fintype.card (F2Power D) = 2 ^ D := by
  unfold F2Power
  simp [Fintype.card_bool, Fintype.card_fin]

/-- At `D = 3`, the non-zero count is `7`. The seven Booker plot
    families bijection in `Aesthetics.NarrativeGeodesic` chains off
    this corollary. -/
theorem nonzero_card_three :
    (Finset.univ.filter (fun v : F2Power 3 => v ≠ 0)).card = 7 := by
  have h := @nonzero_card 3
  -- h : … = 2 ^ 3 - 1
  have h2 : (2 : ℕ) ^ 3 - 1 = 7 := by norm_num
  rw [h2] at h
  exact h

What this page does not claim

The theorem does not claim that the count 7 has any meaning outside the algebraic setting. It does not assert that the framework's narrative applications are correct. It does not claim that {0, v} is the only type of subspace in the group.

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/Algebra/F2Power.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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