The Quotient Dictionary: Wall 5 as a Representation Kernel

Research Note 43 in the "Geometry of the Critical Line" programme. TITANIUM (T4). The SCT Hecke algebra satisfies the Bost–Connes axioms at the abstract algebraic level, but cannot be faithfully represented on the critical sector Hilbert space (Kills #60–64). This note reframes Wall 5 as a quotient problem. The SCT geometry determines a specific algebraic ideal I_α — the kernel of the natural representation — generated by the fractional-momentum operators that Kill #64 annihilates. The quotient algebra is the largest algebraic Bost–Connes-type quotient that acts faithfully on the SCT Hilbert space. The central result: the range projections E₏⋒ = μₚʳμₚ^*r (projecting onto winding sectors divisible by pʳ) survive the quotient, and the KMS weights p^−r/2 live on these surviving projections. The kernel generation theorem identifies the obstruction generators, and the KMS descent theorem proves that the unique KMS state at β = 1/2 vanishes on the entire ideal and descends to the quotient with weights p^−r/2 intact. Wall 5 is crossed algebraically.