The C() Spectral Family: Higher-Spin LQC Bounce Corrections and Their CMB Scale Predictions (Paper 4 of the LQG–LQC Intertwiner Series)

Introduction Paper I of this series Hillard 2026a derived the intertwiner-mediated window function W () =³ (²−2) / C sinh (/2) from Loop Quantum Gravity first principles. The derivation assigns thedominant LQC bounce to a = 1 Pöschl–Teller potential, corresponding to the spin-½ four-valentvertex with dominant intertwiner channel k = 1. Paper 2 then eliminated the last free parameter, deriving the exact median ||² = 1/13 from the SU (2) -invariant Haar measure on the Livine–Spezialeintertwiner space. LQC holonomy corrections beyond leading order are not negligible in principle. The LQC effectiveHamiltonian in the improved -scheme contains holonomy operators whose expansion in powers ofthe connection generates contributions from higher spin representations j = 1, 2, 3, …. Each suchcontribution produces a Mukhanov–Sasaki correction with Pöschl–Teller coupling = j, and thus awindow function W () with its own normalization constant C (). This paper pursues three questions. (1) What is the amplitude F () for the -th correction? Previousdiscussions Singh 2009; Diener et al. 2014 treated this as requiring external input from spin-jholonomy matrix elements. We show this is unnecessary: F () is determined entirely internally1from the SU (2) Casimir eigenvalue structure and the fixed -scheme area gap. (2) Do higher-spinspectral features overlap with lower-spin features, or can they be observationally separated? Weprove strict non-overlap. (3) Does the sum over all converge, and under what conditions? The central new result is the Triangular Amplitude Theorem: F () = T · F (1), T = (+1) /2 (1. 1) where T are the triangular numbers: 1, 3, 6, 10, 15, 21, …. Every amplitude in the C () family isthus a rational multiple of F (1) = 0. 2398, itself derived from, , and the area gap (Paper I, §6). No external computation is needed at any spin level. Notation. Throughout this paper, denotes the PT coupling index (equal to the edge spin jof the holonomy representation). The symbol Λ (uppercase) denotes the ̄-scheme area gap Λ² =4√3P². The symbol T = (+1) /2 denotes triangular numbers (§3 onwards). The transmissionamplitude is written _ () (§2 only). These notations do not conflict but are typographicallysimilar; readers should attend to context. Thepaper is organized as follows. Section 2 derives the = j PT coupling from the LQC Hamiltonian. Section 3 states the C () spectral family and proves the Triangular Amplitude Theorem. Section4 derives CMB scale predictions. Section 5 proves spectral monotonicity and non-overlap. Section6 writes the complete power spectrum. Section 7 examines the amplitude structure of higher-spincorrections, including the conditional Casimir bound and the actual amplitude growth from theHaar measure. Section 8 addresses convergence. Section 9 presents new falsifiability conditions. Section 10 summarizes.