Foundation Universal Forcing Narrative Realization Narrative Cost Symm

The narrative cost symmetry

In ordinary storytelling, the distance between two moments feels directional: the gap from a quiet opening to a shocking twist is not the same as the gap backward. The Recognition Science framework's narrative cost, a measure of how many beats separate two points in a story, deliberately ignores that direction. Its formal definition is stark: the cost is 0 if two beats are the same event, and 1 if they are any two different events. This is a discrete record, a ledger, of narrative distance where every distinct pair costs exactly one unit.

What the declaration narrativeCost_symm establishes is that this cost is symmetric: the cost from beat a to beat b equals the cost from beat b to beat a. In the formal library, this is a verified theorem, not an assumption. The proof checks the two cases: when a and b are identical, both sides are 0; when they differ, both sides are 1. This mirrors the reciprocal symmetry that the framework's central cost function J(x) = (x + 1/x)/2 - 1 obeys, where swapping the two arguments leaves the value unchanged. The narrative realization is a lightweight model showing that the same structural pattern, symmetry under exchange, appears when the carrier is simply the count of beats generated by an inciting event.

In Recognition Science, this is one instance of a broader claim: that narrative order carries the same forced Peano object, the natural numbers, that underlies arithmetic and physical cost. The framework models narrative as a beat count, and the symmetry theorem is a small but necessary step in that modeling. It does not, however, claim that real stories are symmetric in any psychological or dramatic sense. A reader's experience of a plot twist is not reversible; the formal cost only tracks whether two beats are the same or different, nothing more.

What the declaration does not claim is equally precise. It does not assert that narrative cost is a metric, since it lacks the triangle inequality. It does not say that all narratives have the same cost structure, only that this particular realization, where cost is 0 or 1, is symmetric. And it does not derive the golden ratio, the eight-tick cycle, or any of the framework's larger constants from narrative alone. The theorem is a single, verified symmetry property within a much larger forcing chain, and its role is to show that the pattern of reciprocal symmetry, which the framework proves for cost functions, also appears in this minimal narrative model.

theorem narrativeCost_symm (a b : NarrativeBeat) : narrativeCost a b = narrativeCost b a := by
  by_cases h : a = b
  · subst h; simp [narrativeCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [narrativeCost, h, h']

def narrativeCost (a b : NarrativeBeat) : Nat :=
  if a = b then 0 else 1

What this page does not claim

Narrative cost satisfies the triangle inequality or is a metric. Real stories are symmetric in dramatic or psychological experience. The golden ratio or the eight-tick cycle is derived from narrative cost alone.

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

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