A registered result in its source context

VGEO.SHARPEN.01

Original ledger classification: v_geo capstone [E]/[O] (Direction 6: v_geo the single dimensionful input across matter+gravity, irreducibly primitive by No-Unit; final tally 0 dimensionless dials + 1 unit + pi vs SM ~26)

Original claim text

v_geo IS THE SINGLE DIMENSIONFUL INPUT OF THE WHOLE THEORY (2026-06-23, Direction 6 of next-plan.md), now confirmed across matter AND gravity, and irreducibly primitive. The No-Unit Theorem (v153) already established that the dimensionless compiler cannot output an absolute scale, so v_geo is a forced metrology primitive, with the matter residuals collapsing onto it (U_point~v_geo, 1/G~v_geo^2, m/mu=e^{3/4} a pure ratio). The genuine SHARPENING here, after the parameter-free gravity (v358/v359/v361): the GRAVITY sector adds NO new dimensionful input either. [E] THE COMPILER IS DIMENSIONLESS: c3=1/(8pi) and g_car=5 are mass-dimension 0, so by the No-Unit Theorem an absolute scale CANNOT be an output -- v_geo is a forced INPUT (a chosen unit), not derivable. [E] GRAVITY ADDS NO NEW DIMENSIONFUL INPUT: the Einstein coefficient 1/c3=8pi and the Lambda prefactor (8pi)^2*48c3^4=3/(4pi^2) are dimensionless, and rho_Lambda/Mbar^4=(3/(4pi^2))e^{-2 alpha^-1} is a dimensionless ratio; so 1/G~Mbar^2~v_geo^2 and Lambda~(dimensionless)*v_geo^4 both reduce to v_geo times atoms -- the gravity sector introduces NO scale beyond v_geo. [E] THE FINAL TALLY: every dimensionful quantity in the theory (masses, 1/G, Lambda, U_point, the EW/flavor/Planck scales) = (a dimensionless compiler ratio)*(a power of the ONE unit v_geo); TFPT's complete free content is {v_geo (one mass unit), pi (a fixed math constant)} with ZERO dimensionless dials -- versus the SM's ~26 free parameters, a ~26->1 reduction. [O] v_geo IS IRREDUCIBLY PRIMITIVE: NOT an open gap -- a dimensionless theory provably cannot derive a scale (No-Unit), so v_geo is the one necessary external unit (the choice of ruler, like the metre); 1/G is UV-sensitive (Sakharov induced gravity, v68), a cutoff-dependent metrology readout, not a clean prediction. RESULT: v_geo is the single dimensionful input of the complete theory (matter via v153 + gravity via v358/v359/v361), irreducibly primitive; the theory is 0 dimensionless dials + 1 unit + pi. Direction 6 is clarity (the last dimensionful anchor named and shown to be the only one), not a new derivation.

Original classification detail

[E] the dimensionless-compiler / gravity-adds-no-scale / final-tally facts; [O] the irreducible-primitive status of v_geo (No-Unit consequence) + the Sakharov UV-sensitivity of 1/G. A capstone sharpening of v153 after the parameter-free gravity; v_geo stays the one declared unit (not a gap). Direction 6 closed -- clarity, not a new derivation.

Dependencies as recorded

v153; v358; v359; v361; v60; v68; v78