Filament transverse widths should scale with embedded mass through collapse-expansion balance, yet catalogs and simulations find widths systematically off at fixed mass or nearly scale-invariant across wide mass ranges (Bond 1996; Cautun 2014; Zhu 2024, 2025). A width that barely responds to mass suggests mass never set it.
The model has exactly one width-setting mechanism: gravitational collapse onto the filament spine equilibrating against expansion and velocity dispersion. Width must therefore track mass. When it does not, the only escape routes are environmental physics, baryonic feedback, or blaming the filament finders, because the framework contains no way for a filament to be born with its width.
SCT supplies precisely the missing option: filament width is set at deposition by collision geometry, not afterward by collapse. In a head-on collision, the filament's length traces the combined pocket extent along the collision axis while its width scales with the smaller pocket's self-gravity (P33), a property of the colliding bodies fixed at the moment of thermalization. The embedded halo mass accumulates later, as matter drains along and onto the deposited structure, so width and present-day mass are connected only loosely, through the cascade-stage hierarchy that set both, rather than tightly, through equilibration.
Near-scale-invariance falls out naturally: cascade stages drawing from a narrow band of the virial mass ladder deposit filaments of similar width across a wide range of eventual embedded mass, flattening the width-mass relation in exactly the way the collapse picture cannot. The deposition geometry is laid out in Paper 1, From Chaos to Convergent Foundations, and the cascade structure-seeding mechanics in Paper 4, From Chaos To Collisothermal Cosmogenesis. This is the same deposited-not-grown inversion that carries the filament length function (recid 72) and the connectivity statistics (recid 74); the three are one geometric statement read along three different axes of the web.
Keystone economy: P33 does the work alone, with P34 supplying the population statistics. A structure born with its width does not need feedback, environment, or finder artifacts to explain why mass never tuned it.
Stacked DESI, 4MOST, and Euclid filament profiles carry the kill: if the width-mass relation steepens to full consistency with collapse-equilibration scaling across the whole mass range, with the near-invariant regime dissolving into finder artifacts, the deposition picture is refuted here. SCT requires the flat regime to survive clean methodology; a width that tracks mass the way collapse demands means the filaments were grown after all.