Gravity Seven Gaps Quotient First Z Labeled Z Eq Sum Fiber Card Mul Mu

The labeled path sum

In the Recognition Science framework, a ledger is a discrete record of events, and a path sum is a way of adding up contributions from all possible paths in a system. The declaration labeledZ_eq_sum_fiberCard_mul_mu is a theorem in the framework's machine-checked library of formal theorems. It proves that a particular labeled path sum, Z, which counts paths while tracking individual labels, is exactly equal to a sum over equivalence classes of paths, where each class is weighted by the number of distinct labeled paths it contains, multiplied by a symmetry factor.

This is a precise accounting identity. It says that if you group paths into classes that are the same up to relabeling, then the total labeled sum is not simply the sum over classes with a naive per-class weight. Instead, each class contributes its size, the number of labeled paths in it, times a measure that is independent of which representative of the class you pick. The theorem states this equality holds for any class-constant weight function, and it is a proved statement in the framework's library, with no unproved assumptions beyond the standard axioms of the ambient type theory.

The theorem is the honest replacement for a stronger claim that was once considered but is not true in general. The framework's own status record marks that the unconditional equality between the labeled sum and a per-class quotient sum with a 1/|Aut| factor is not derived. The new theorem shows the labeled sum equals the quotient sum only when an explicit excess term, the sum over classes of (|fiber q| - 1) * mu(out q) * wq q, is zero. That excess is not assumed to vanish, and in fact the library proves that for at least one edge class, the fiber size is greater than one, so the excess is not generally zero.

What this means in plain terms: the framework has a precise, machine-checked rule for how to move between a labeled count and a class-based count, and it does not pretend the two are the same without a correction term. This is a contribution to the framework's internal consistency, and it is a model of how to state a bridge between different counting conventions without overclaiming.

/-- **Bridge to the labeled sum, with the mandatory fiber factor.**  For a
class weight `wq`, the standing labeled path sum with pulled-back weight is
the quotient sum weighted by the pushforward class mass
`|fiber q| · μ(out q)`. -/
theorem labeledZ_eq_sum_fiberCard_mul_mu (B : ℕ)
    (wq : TriangulationClass B → ℂ) :
    Z B (fun K => wq (Quotient.mk (relabelSetoid B) K)) =
      ∑ q : TriangulationClass B,
        (((fiberCard (relabelSetoid B) q : ℂ) * (mu (Quotient.out q) : ℂ))
          * wq q) := by
  classical
  have hw : ∀ K K' : BoundedComplex B, Equivalent K K' →
      wq (Quotient.mk (relabelSetoid B) K) =
        wq (Quotient.mk (relabelSetoid B) K') := by
    intro K K' h
    exact congrArg wq (Quotient.sound h)
  calc
    Z B (fun K => wq (Quotient.mk (relabelSetoid B) K))
        = ∑ q : TriangulationClass B,
            (PathSum.classMass q : ℂ) *
              wq (Quotient.mk (relabelSetoid B) (Quotient.out q)) := by
          simpa using PathSum.Z_eq_classPushforward B
            (fun K => wq (Quotient.mk (relabelSetoid B) K)) hw
    _ = ∑ q : TriangulationClass B,
        (((fiberCard (relabelSetoid B) q : ℂ) * (mu (Quotient.out q) : ℂ))
          * wq q) := by
          refine Finset.sum_congr rfl fun q _ => ?_
          rw [PathSum.classMass_eq_fiberCard_mul_mu]
          rw [show wq (Quotient.mk (relabelSetoid B) (Quotient.out q)) = wq q
            from congrArg wq (Quotient.out_eq q)]
          simp only [Complex.ofReal_mul]
          norm_num

/-- **Exact relation.**  The labeled class-constant path sum is the
quotient-first path sum plus the labeled-fiber excess.  This is the honest
replacement for the killed unconditional claim `Z = Σ_q wq/|Aut q|`. -/
theorem labeledZ_eq_Zq_plus_fiberExcess (B : ℕ)
    (wq : TriangulationClass B → ℂ) :
    Z B (fun K => wq (Quotient.mk (relabelSetoid B) K)) =
      Zq B wq + fiberExcess B wq := by
  classical
  rw [labeledZ_eq_sum_fiberCard_mul_mu, Zq, fiberExcess, ← Finset.sum_add_distrib]
  refine Finset.sum_congr rfl fun q _ => ?_
  let f : ℂ := fiberCard (relabelSetoid B) q
  let m : ℂ := mu (Quotient.out q)
  let z : ℂ := wq q
  calc
    (((fiberCard (relabelSetoid B) q : ℂ) * (mu (Quotient.out q) : ℂ)) * wq q)
        = (f * m) * z := rfl
    _ = m * z + ((f - 1) * m) * z := by ring
    _ = (mu (Quotient.out q) : ℂ) * wq q +
        ((((fiberCard (relabelSetoid B) q : ℂ) - 1) *
          (mu (Quotient.out q) : ℂ)) * wq q) := rfl

/-- **IFF form of the exact relation.**  The quotient-first object equals
the standing labeled sum for a pulled-back class weight exactly when the
explicit fiber excess vanishes. -/
theorem Zq_eq_labeledZ_iff_fiberExcess_vanishes (B : ℕ)
    (wq : TriangulationClass B → ℂ) :
    Zq B wq = Z B (fun K => wq (Quotient.mk (relabelSetoid B) K)) ↔
      fiberExcess B wq = 0 := by
  rw [labeledZ_eq_Zq_plus_fiberExcess]
  constructor
  · intro h
    have h' : Zq B wq + fiberExcess B wq = Zq B wq + 0 := by
      simpa using h.symm
    exact add_left_cancel h'
  · intro h
    rw [h, add_zero]

What this page does not claim

The theorem does not claim that the labeled sum equals the quotient sum without a fiber factor. It does not claim that the fiber excess is generally zero, and in fact the library proves it is not for at least one edge class. It does not claim that an orbit-stabilizer theorem for the full bounded triangulation class setoid is derived here.

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/Gravity/SevenGaps/QuotientFirstZ.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.

Derived articles

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