Gravity Seven Gaps Freudenthal Torus Class Mass Tendsto Mu Freudenthal Zero

The labeled summand's limit

In the Recognition Science framework, a ledger, a discrete record of events, assigns each recognized object a symmetry factor: a number that shrinks as the object's internal symmetry grows. The declaration tendsto_mu_freudenthal_zero proves that this symmetry factor for a particular family of tori, written T_N, tends to zero as N grows without bound. Concretely, the theorem states that the sequence of values mu(T_{N+1}) approaches 0, and it does so by squeezing the factor between 0 and 1/(N+1)³, a bound that itself shrinks to zero.

The proof is a standard squeeze argument, and the result is a theorem in the framework's machine-checked library of formal theorems. The key inequality is mu(T_N) ≤ 1/N³, which holds for every positive integer N. Because the cube in the denominator grows faster than N itself, the bound 1/N³ is eventually smaller than 1/N, and the squeeze theorem forces the symmetry factors to converge to zero. This is a statement about each labeled representative of the torus family, not about a quotient object that lumps many representatives together.

The distinction matters. The framework also defines a class mass for the torus, which multiplies the symmetry factor by the number of labeled representatives in the class, a quantity called the fiber cardinality. That fiber cardinality grows with N, so the class mass is bounded only by |fiber|/N³, which does not necessarily shrink to zero. The theorem tendsto_mu_freudenthal_zero therefore does not establish any suppression of the pushforward class mass, and it makes no claim about convergence of a continuum limit or about dominance of any partition function.

In plain terms, the theorem says: take a single torus from the family, compute its symmetry weight, and let the family index grow. That weight vanishes. But if you collect all tori that share the same label, their combined weight may not vanish, because there are more of them as N grows. The framework's own status record marks the stronger claim as false, and it leaves three related targets open: a continuum limit for the Z function, a derived substrate measure, and a bridge from gap one. The theorem's value is precision: it pins down exactly which object vanishes and which one does not, so that later work does not build on a false suppression.

/-- **THEOREM (labeled, per-`1/|Aut|`).**  Every labeled member of the
torus class carries symmetry-factor mass at most `N⁻³`: μ is a class
function (`mu_congr`), and the translation embedding gives
`μ(T_N) ≤ 1/N³`.  This is a statement about EACH labeled representative,
not about the pushforward class mass. -/
theorem mu_torusClassMember_le (N : ℕ) [NeZero N]
    (K : BoundedComplex (7 * N ^ 3))
    (hK : Quotient.mk (relabelSetoid (7 * N ^ 3)) K =
      Quotient.mk (relabelSetoid (7 * N ^ 3)) (freudenthalBoundedComplex N)) :
    mu K ≤ 1 / ((N : ℝ) ^ 3) := by
  rw [mu_congr (equivalent_of_mk_eq hK)]
  exact mu_freudenthal_le_inv_cube N

/-- **THEOREM (labeled tendsto, measure form).**  The symmetry-factor mass
of the canonical torus LABELED representative tends to zero:
`μ(T_{N+1}) → 0`.  (Squeeze between `0` and `1/(N+1)³ ≤ 1/(N+1)`.) -/
theorem tendsto_mu_freudenthal_zero :
    Filter.Tendsto (fun n : ℕ => mu (freudenthalBoundedComplex (n + 1)))
      Filter.atTop (nhds 0) := by
  refine squeeze_zero (fun n => le_of_lt (mu_pos _)) (fun n => ?_)
    tendsto_one_div_add_atTop_nhds_zero_nat
  calc mu (freudenthalBoundedComplex (n + 1))
      ≤ 1 / (((n + 1 : ℕ) : ℝ) ^ 3) := mu_freudenthal_le_inv_cube (n + 1)
    _ = 1 / (((n : ℝ) + 1) ^ 3) := by rw [Nat.cast_add, Nat.cast_one]
    _ ≤ 1 / ((n : ℝ) + 1) := one_div_cube_le_one_div n

/-- **THEOREM (class mass, identity form).**  The pushforward class mass
of the torus class is the labeled fiber cardinality times the symmetry
factor of the canonical torus:
`classMass(⟦T_N⟧) = |fiber(⟦T_N⟧)| · μ(T_N)`. -/
theorem torus_classMass_eq_fiberCard_mul_mu (N : ℕ) [NeZero N] :
    classMass (Quotient.mk (relabelSetoid (7 * N ^ 3))
        (freudenthalBoundedComplex N)) =
      (fiberCard (relabelSetoid (7 * N ^ 3))
          (Quotient.mk (relabelSetoid (7 * N ^ 3))
            (freudenthalBoundedComplex N)) : ℝ) *
        mu (freudenthalBoundedComplex N) := by
  rw [classMass_eq_fiberCard_mul_mu]
  congr 1
  exact mu_congr (equivalent_of_mk_eq (Quotient.out_eq _))

/-- **THEOREM (the honest class-mass bound).**
`classMass(⟦T_N⟧) ≤ |fiber(⟦T_N⟧)| / N³`.  The fiber cardinality is NOT
bounded here; in particular this does NOT yield the (killed) absolute
`N⁻³` suppression of the pushforward mass. -/
theorem torus_classMass_le_fiberCard_div_cube (N : ℕ) [NeZero N] :
    classMass (Quotient.mk (relabelSetoid (7 * N ^ 3))
        (freudenthalBoundedComplex N)) ≤
      (fiberCard (relabelSetoid (7 * N ^ 3))
          (Quotient.mk (relabelSetoid (7 * N ^ 3))
            (freudenthalBoundedComplex N)) : ℝ) / ((N : ℝ) ^ 3) := by
  rw [torus_classMass_eq_fiberCard_mul_mu, div_eq_mul_one_div]
  exact mul_le_mul_of_nonneg_left (mu_freudenthal_le_inv_cube N)
    (Nat.cast_nonneg _)

What this page does not claim

The pushforward class mass of the torus is suppressed as N⁻³. The labeled summand sequence vanishing implies any convergence of the class mass sequence. The theorem establishes a continuum limit or dominance for the partition function Z.

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

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