Gravity Seven Gaps Freudenthal Torus Class Mass Tendsto Labeled Summand Zero

The labeled summand limit

In the Recognition Science framework, a torus (a doughnut-shaped geometric object) of growing size is assigned a numerical mass. The mass is built from a sum of smaller pieces, each associated with a particular labeling of the torus's structure. The theorem tendsto_labeledSummand_zero proves that as the torus size parameter N increases, the contribution from any single labeled representative, multiplied by any unit-modulus complex number, tends to zero. The bound is sharp: the mass of a single labeled representative is at most 1/N³, and the sequence of these masses, when multiplied by any sequence of unit-modulus weights, converges to zero.

The proof is a direct squeeze: the mass is nonnegative, it is bounded above by 1/(N+1)³, and that bound itself tends to zero. The theorem is part of a larger consistency gate that separates two distinct objects. The first object is the labeled representative, a single, specific member of an equivalence class of torus labelings. The second object is the class mass, which is the mass of the entire equivalence class, obtained by summing over all labelings in the class. The class mass equals the number of labelings in the class (the fiber cardinality) times the mass of a single representative.

The theorem applies only to the labeled representative. The class mass is not bounded by 1/N³, because the fiber cardinality grows with N. A claim that the class mass is absolutely suppressed as N⁻³ is explicitly marked as false in the framework's own status ledger. The framework's library, a machine-checked collection of formal theorems, records this distinction with a theorem stating that the class mass is at most the fiber cardinality divided by N³, without bounding the fiber cardinality itself.

This distinction matters because it prevents a subtle error: one might be tempted to conclude that the entire mass of the torus class vanishes, which would be a stronger statement with different physical implications. The framework's honest gate keeps the two objects separate in its type system, proving the limit for the labeled piece while explicitly leaving the class mass, and any claims about a continuum limit or a derived substrate measure, as open targets.

/-- **THEOREM (labeled tendsto, summand form; the honest T4 gate).**  The
labeled-representative summand sequence `μ(T_{N+1}) · z_N`, for ANY
sequence of unit-modulus values (e.g. `unitaryWeight` values of any
action), tends to `0`.  LABELED / CLASS DISTINCTION: this is the labeled
representative sequence; NO claim is made about the pushforward CLASS
mass sequence `|fiber|·μ`, whose fiber factor grows. -/
theorem tendsto_labeledSummand_zero (z : ℕ → ℂ) (hz : ∀ n, ‖z n‖ ≤ 1) :
    Filter.Tendsto
      (fun n : ℕ => (mu (freudenthalBoundedComplex (n + 1)) : ℂ) * z n)
      Filter.atTop (nhds 0) := by
  rw [tendsto_zero_iff_norm_tendsto_zero]
  refine squeeze_zero (fun n => norm_nonneg _) (fun n => ?_)
    tendsto_mu_freudenthal_zero
  rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos (mu_pos _)]
  exact mul_le_of_le_one_right (le_of_lt (mu_pos _)) (hz n)

/-- **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 (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 torusClassMassStatus_flags :
    torusClassMassStatus.labeled_member_mass_bounded = true ∧
    torusClassMassStatus.labeled_summand_bounded = true ∧
    torusClassMassStatus.labeled_summand_tendsto_zero = true ∧
    torusClassMassStatus.classMass_identity_proved = true ∧
    torusClassMassStatus.classMass_fiberCard_bound_proved = true ∧
    torusClassMassStatus.pushforward_classMass_absolutely_suppressed = false ∧
    torusClassMassStatus.Z_RS_continuum_limit = false ∧
    torusClassMassStatus.substrate_measure_derived = false ∧
    torusClassMassStatus.gap1_bridge_derived = false :=
  ⟨rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl, rfl⟩

What this page does not claim

The pushforward class mass of the torus class tends to zero. The fiber cardinality is bounded independently of N. The theorem establishes any convergence or dominance claim about 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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