 109    The Dedekind eta function η(τ) = q^{1/24} ∏ₙ (1 − qⁿ) counts
 110    the microstates of a single mode; raising to the 24th power counts
 111    all 24 directed flux modes on the voxel. -/
 112structure ModularDiscriminantBridge where
 113  /-- The exponent in η²⁴ -/
 114  eta_exponent : ℕ := 24
 115  /-- It matches the directed flux count -/
 116  matches_flux : eta_exponent = directed_edge_count D := by rfl
 117
 118/-- The bridge certificate: 24 directed fluxes = η²⁴ exponent. -/
 119def modularDiscriminantBridge : ModularDiscriminantBridge := {}
 120
 121/-! ## §3. The Leech Lattice Connection -/
 122
 123/-- The Leech lattice has dimension 24, matching Q₃ directed flux.
 124
 125    The Leech lattice Λ₂₄ is the unique even unimodular lattice in
 126    dimension 24 with no vectors of norm 2. Its uniqueness properties
 127    mirror the uniqueness of the Q₃ double-entry structure. -/
 128def leech_lattice_dimension : ℕ := 24
 129
 130theorem leech_dimension_eq_directed_flux :
 131    leech_lattice_dimension = directed_edge_count D := by
 132  simp [leech_lattice_dimension, directed_edges_Q3]
 133
 134/-! ## §4. Why String Theory's Interpretation Is Wrong -/
 135
 136/-- String theory claims D_crit = 26 = 24 + 2 (transverse + longitudinal).
 137    RS claims D = 3 and 24 = directed flux on Q₃.
 138
 139    The mathematical content (the number 24 and its role in partition
 140    functions) is identical. The physical interpretation differs:
 141    - String theory: 24 transverse spatial dimensions
 142    - RS: 24 directed flux modes on the Q₃ double-entry ledger

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