Cost of Deepening: The Koszul Bridge Between Simplicial Recognition Dynamics and LQC Spectral Geometry (Paper 47 of the LQG–LQC Intertwiner Series)

AbstractTwo independent mathematical frameworks — one built on simplicial topology, the otheron spectral theory of the LQC bounce — produce matching cost ratios for dimensionaldeepening. In the simplicial framework, the Homological Imprint Theorem establishes thatthe first crossing from d − 1 to d increases β1 by exactly d − 2: the ground sacrifices oneH1 class while vertex genesis creates d − 1 new ones. The cumulative cost ratio (classeskilled to classes created) across all crossings from dimension 3 to dimension d is 2/(d + 1).In the spectral framework, the self-energy ratio of the LQC bounce window function isΣ = κ20/M0 = 2/12 = 1/6, where M0 = 12 is the zeroth spectral moment.The cumulative simplicial cost ratio equals Σ = 1/6 exactly when d+1 = 12 = M0. Thisnumerical correspondence is proved as an algebraic identity.The paper then constructs an algebraic bridge between the two frameworks. The simplicialboundary operator ∂ is identified with the CAR (fermionic) annihilation operator c(f)via the classical Koszul complex — both are nilpotent (∂2 = 0, c(f)2 = 0), and the Koszulcontraction ratio d/n at (d, n) = (κ20,M0) recovers Σ = 1/6 through a third independentroute. A K-theory obstruction (K1 = {0} for all von Neumann algebra factors) blocks thena¨ıve H1 ↔ K1 map and motivates the Koszul alternative.The fermionic structure required by the Koszul identification is shown to be alreadypresent: the intertwiner qubit of the LQC bounce satisfies the CAR, and Paper 37’s twostateBoltzmann mechanism is exactly the Fermi–Dirac distribution for a single fermionicmode with energy V4 = 280. The recognition cascade, realised as an inductive limit of CARalgebras in the KMS state, produces the unique hyperfinite Type III1 factor — the samealgebra the Recognitive Consciousness Framework derives from first principles. An honestgap remains: the inductive limit gives the correct algebra type but does not yet encode thestructural dynamics (which vertices connect to which).All claims are epistemically tiered. The cost ratios, the Koszul identification, the CARstructure of the intertwiner, and the Fermi–Dirac form are proved. The interpretation of thematch as non-coincidental is postulated. The full dynamical functor is an open construction.