Algebra F2 Power Axis1 Weight

A single true bit

The declaration axis1_weight is a small, machine-checked proof about a specific three-bit string. The string is written ![true, false, false], meaning it has a true value in its first position and false values in the other two. The theorem states that the Hamming weight of this string is 1. The Hamming weight of a bit string is simply the number of positions that hold a true value, so the statement is that exactly one of the three positions is true. The proof is immediate: the definition of the string and the definition of Hamming weight are unfolded, and the result is decided by computation.

This theorem lives inside a larger structure called F2Power, which is a discrete record of three binary coordinates where addition is pointwise XOR. In this structure, the string ![true, false, false] is named axis1, and it represents one of the three coordinate axes in a three-dimensional space over the two-element field. The theorem axis1_weight is one of three similar results, one for each axis, and together they establish the 1+3+3+1 weight decomposition of the seven nonzero elements of this space. The count of elements with weight one is three, matching the three axes, and the count with weight two is also three, with one element of weight three.

The practical use of this theorem is to support a count in narrative theory. The framework's library proves that the seven nonzero elements of this three-dimensional binary space correspond to seven basic plot families. Each nonzero element generates a one-dimensional subgroup containing just itself and the zero element, and because there are seven nonzero elements, there are exactly seven such subgroups. The theorem axis1_weight is a small piece of this proof: it confirms that the axis1 element has weight one, which is part of the weight decomposition that organizes the space. Without this and the companion weight theorems, the count of seven would rest on an unproved assertion rather than on a verified computation.

What this declaration does not claim is any physical or geometric content. It does not assert that the first coordinate of ordinary three-dimensional space is special, nor that any narrative plot family is preferred over another. It only states a combinatorial fact about a particular bit string within an abstract algebraic structure. The theorem does not say that the string ![true, false, false] is an axis in any physical sense; the name axis1 is a label chosen for convenience within the module. The proof establishes a fact about a finite set of bits, nothing more.

theorem axis1_weight : hammingWeight axis1 = 1 := by
  unfold hammingWeight axis1
  decide

/-- The Hamming weight of `v`: the number of coordinates equal to
    `true`. -/
def hammingWeight (v : F2Power D) : ℕ :=
  (Finset.univ.filter (fun i => v i = true)).card

/-- 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 declaration does not assign physical meaning to the first coordinate of ordinary space. The theorem does not imply that any narrative plot family is more fundamental than another. The name axis1 is a label within the module, not a claim about a physical axis.

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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