Black and Gold VIII: Form, McKay Placement, and the Withdrawals under the Common Integrity Gate

Black and Gold VIII: Form, McKay Placement, and the Withdrawals under the Common Integrity Gate C. J. Tully Independent Researcher, Engineer Tamworth, NSW, Australia ORCID: 0009-0007-5661-7332 10th October 2026 Abstract This paper is the eighth of the black-and-gold series. It places the geometry of the Neolithic carved stone balls and megalithic circles, the McKay reading of their symmetries, and the explicit withdrawals under the common Quasi-Equivalence Index gate. A six-knob ball is an octahedron. A four-knob ball with equal gaps is a tetrahedron. A stone circle is a ring of discrete reflectors. The McKay correspondence places the binary tetrahedral and octahedral groups on affine E_6 and E_7; the binary icosahedral group sits on affine E_8, which remains a hypothesis for the corpus. The same E_8 lattice appears as one factor of the heterotic gauge lattice; that is a citation, not a derivation from the stone. Acoustic levitation, the norm-4 shell partition, the characteristic-zero uniqueness of the five-fold bracket, and the smooth golden-ratio spectrum remain withdrawn. Form is admitted as geometry. Use remains a hypothesis. No claim is made that the corpus encodes string compactification, nor that the gate certifies archaeological intention. 1. Introduction Papers I–VII established the common gate and placed navigation, guidance, spectral fingerprint, swarm coherence, golden orbits, telemetry, and external ranging under it. The present paper records the geometric form of the carved stone balls and megalithic circles, the McKay placement of those symmetries, and the list of claims that stay withdrawn. Form is the result. Use is the experiment. The gate certifies algebraic integrity; it does not certify historical intention.