Metamorphosis Part I — The Closure Law

The Metamorphosis Trilogy concludes a research trajectory initiated in February 2022. Over nearly four years, the Sc-Rubs programme advanced through iterative theoretical development, numerical engines, computational prototypes, and analytical consolidation. These three papers formalise and archive the complete body of work produced during this period, ensuring that every stage of the programme now enters the public scientific record.

This paper establishes a geometric closure law connecting classical Euclidean ratios, scalar-field emergence behaviour, and differential growth identities. By analysing the relationships between squares, circles, cubes, and spheres, the study demonstrates that three independent derivations—integral geometry, scalar-field persistence, and d/dr growth rates—converge on the same constants: 4/π, 6/π, and the composite field value 2.915. This identifies the closure constant as an invariant underlying both classical and emergent geometry.

This work forms Part I of the Sc-Rubs Metamorphosis Trilogy.