Gravity Seven Gaps Freudenthal Torus Class Mass Norm Freudenthal Labeled Summand

The labeled summand bound

In the Recognition Science framework, a torus is a discrete geometric object, and its mass is a real number assigned by a symmetry factor, the reciprocal of the size of its automorphism group. The declaration norm_freudenthal_labeledSummand_le proves a precise bound for a single labeled representative of a torus class. For any natural number N, any labeled member K of the torus class with index N, and any complex number z with modulus at most 1, the modulus of μ(K)·z is at most 1/N³. In plainer terms, the contribution of one labeled representative, scaled by any unit-modulus weight, is suppressed by the cube of the class index.

The proof is a short chain of inequalities. It uses the fact that the mass μ is a class function, meaning equivalent labeled complexes have the same mass, and the previously established bound μ(T_N) ≤ 1/N³ for the canonical torus. The norm of the product μ(K)·z is then at most μ(K)·1, since |z| ≤ 1, and this is at most 1/N³. The declaration also establishes that the sequence of such labeled summands, for any sequence of unit-modulus weights, tends to zero as N increases. This is the honest labeled-representative statement, distinct from the class-level pushforward mass.

The declaration explicitly does not claim that the pushforward class mass, which is the fiber cardinality times the labeled mass, is bounded by 1/N³. The fiber cardinality grows with the class size, so the class mass can be larger. The framework records this distinction in its status ledger: the labeled member mass bound, the labeled summand bound, and the labeled summand tendsto-zero are all proved true, while the absolute pushforward suppression is marked false. The continuum limit and other bridge statements remain open, flagged red.

What this means for the reader is a clear separation of two mathematical objects. The theorem gives a strong, verified bound on individual labeled contributions, useful for understanding their decay. It does not give a bound on the aggregate class mass, and it does not claim any convergence or dominance results for the full partition function. The framework's honesty here is structural: the type system keeps the labeled and class objects separate, and the theorem states exactly which one it bounds.

/-- **THEOREM (labeled summand bound; class-API restatement of
`unnormalized_torus_weight_suppressed`).**  For any labeled member `K` of
the torus class and any unit-modulus value `z`, the single labeled
summand `μ(K)·z` has modulus at most `N⁻³`.  LABELED / CLASS DISTINCTION:
this bounds ONE labeled summand; the pushforward CLASS mass is
`|fiber|·μ` and is NOT bounded by `N⁻³` here. -/
theorem norm_freudenthal_labeledSummand_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))
    (z : ℂ) (hz : ‖z‖ ≤ 1) :
    ‖(mu K : ℂ) * z‖ ≤ 1 / ((N : ℝ) ^ 3) := by
  rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos (mu_pos K)]
  calc mu K * ‖z‖
      ≤ mu K * 1 := mul_le_mul_of_nonneg_left hz (le_of_lt (mu_pos K))
    _ = mu K := mul_one _
    _ ≤ 1 / ((N : ℝ) ^ 3) := mu_torusClassMember_le N K hK

/-- **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 (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 _))

What this page does not claim

The pushforward class mass is not bounded by 1/N³; the fiber cardinality grows. No convergence or continuum-limit claim is made about the full partition function Z. The declaration does not prove any dominance result for the class mass over the labeled summand.

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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