Algebra F2 Power Axis123 Weight

The weight of the all-ones vector

The declaration axis123_weight is a theorem in the framework's machine-checked library of formal theorems. It concerns axis123, a specific vector in the elementary abelian 2-group of rank 3, which is the set of all 8 possible triples of booleans (true/false values) with a coordinate-wise XOR operation. The vector axis123 is the triple [true, true, true], the one with every coordinate set to true. The theorem states that its Hamming weight, the number of true coordinates it contains, equals 3. This is proved by direct computation: the definition of the vector and the weight function are unfolded, and the resulting equality is checked by the kernel.

The theorem is part of a larger structure in the framework's algebra module. The module defines the entire group F2Power D for any natural number D, proves that its cardinality is 2^D, and proves that the number of nonzero elements is 2^D - 1. At D = 3, this gives the count 7, a number that the framework's narrative theory associates with Booker's seven basic plot families. The module also proves the 1+3+3+1 weight decomposition: exactly one vector has weight 0, three have weight 1, three have weight 2, and one has weight 3. The theorem axis123_weight is the last piece of this decomposition, identifying the unique vector of weight 3.

In Recognition Science, this algebraic fact is not an isolated curiosity. The framework models the space of narrative possibilities as the nonzero elements of F2Power 3, and the weight of a vector corresponds to a count of active plot elements. The vector axis123, with all three coordinates active, represents the maximal configuration. Its weight being 3 is a formal prerequisite for downstream modules that chain off this count, such as the narrative geodesic and cube bridge constructions. The theorem is tagged THEOREM, meaning it is proved in the library with zero unproved assumptions and no framework-specific axioms.

What the theorem does not claim is important. It does not assert that the number 7 has any narrative meaning; that identification is a separate modeling choice, not a mathematical consequence. It does not claim that the vector axis123 is special in any way beyond having all coordinates true. It does not claim anything about physical space, despite the word "axis" in the name. The theorem is purely combinatorial: a counting statement about a finite set of triples, proved by computation. Its significance in the framework comes from what later modules build on it, not from anything intrinsic to the statement itself.

theorem axis123_weight : hammingWeight axis123 = 3 := by
  unfold hammingWeight axis123
  decide

/-- `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]

/-- The unique weight-0 element: the zero vector. -/
theorem card_weight_zero_three :
    (Finset.univ.filter (fun v : F2Power 3 => hammingWeight v = 0)).card = 1 := by
  have hsubset :
      (Finset.univ.filter (fun v : F2Power 3 => hammingWeight v = 0)) = {0} := by
    ext v
    simp [Finset.mem_filter, Finset.mem_univ, Finset.mem_singleton, weight_zero_iff]
  rw [hsubset]
  rfl

What this page does not claim

The theorem does not assign narrative meaning to the number 7; that is a modeling choice. The theorem does not claim anything about physical space despite the word 'axis'. The theorem does not establish that axis123 is unique in any property other than having weight 3.

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)

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