The Capstone Master Formula for Pure-Gauge Glueballs: Parity-Odd Topology and the Complete Five-Channel Spectrum on L(Z3)

The first two papers of this glueball series established the universal mass formula mdressed N = (2N−1)ΛQCD for parity-even pure-gauge N-body Fock states on the Truncated Cubic Hon- eycomb (TCH) gauge web ( 1) and the Threshold Bound State Theorem establishing the intrinsic non-locality of glueball decay ( 2). This third paper extends the framework to the parity-odd sector and consolidates the complete pure-gauge bound-state spectrum into a single capstone master formula. We prove the Pseudoscalar Exclusion Theorem (the 0−+ glueball cannot exist as a single-cell pure-gauge construction; A1u multiplicity van- ishes on the 12-dimensional directed-edge space of a single cubic unit cell), establish the Edge-Overlap Binding Criterion (a Fock state of closed cycles is bound if and only if the Gram matrix of pairwise edge overlaps has non-zero off-diagonal entries), and state the Parity-Even/Parity-Odd Bifurcation Principle (no parity-odd pure-gauge bound state can be a local single-cell resonance; parity-odd channels are necessarily multi-cell topological knots). We then prove the Macroscopic Grover Coin Uniqueness Theorem (the only real- valued permutation-symmetric unitary scattering operator on the degree-6 vertex of the macroscopic gauge web is the Grover coin, yielding the analytic inter-cell hopping ampli- tude t= 1/3) and identify the Universal Delocalization Constant δ≈0.155ΛQCD per shared edge, empirically channel-independent across the parity-odd manifold to within 1.4%. The capstone master formula mdressed N = (2N−1)ΛQCD + Nshared (t−δ) ΛQCD unifies the complete LQCD lightest-glueball spectrum: 0++ at 1660 MeV (vs 1710 ±50, 3.0%), 1+− at 2988 MeV (vs 2980 ±40, 0.2%), 2++ at 2324 MeV (vs 2390 ±70, 2.8%), 0−+ at 2560 MeV (matches LQCD 2560 ±35 exactly), 1−− at 3830 MeV (matches LQCD 3830 ±40 exactly). Five out of five canonical glueball channels reproduced from substrate topology with four structurally-quantised parameters (N, Nshared, t = 1/3 analytic, δ ≈ 0.155 universal) and one empirical scale (ΛQCD). The substrate derivation of δvia Brillouin- zone integration of the inter-cell hopping operator is the named downstream target, unified with the continuum-limit decay-width calculation of the companion paper. Keywords: Glueball spectrum; pure-gauge bound states; parity bifurcation; Grover coin; Schur’s lemma uniqueness; Edge-Overlap Binding Criterion; Threshold Bound State; Holo- graphic Circlette framework; discrete substrate quantum gravity; lattice gauge theory. v2.0 (2026-06-12): an in-PDF dated status/erratum note has been added reflecting the June 2026 canon audit (DRIFT/ANCHOR ledger); see the paper's status note for the specific corrections, supersessions, or upgrades. 2026-06-20 legacy erratum: This version adds a canon erratum note to the legacy paper. Gauge/spectroscopy claim; check against SMG/TCH continuum status The body is preserved as a historical derivation trail; the erratum note identifies the current ANCHOR/DRIFT status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.