Submitted:
15 October 2026
Posted:
16 October 2026
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Abstract
For positive running charged-fermion masses in one scheme and at one scale, define \(A_s=\tfrac14\ln(m_{s,1}m_{s,3}/m_{s,2}^2)\) and \(\Delta_\Pi=A_\ell+2A_d-A_u\). A specified four-channel Pati--Salam source map identifies the primitive correction normal to this plane with the unique route-odd coefficient, \(\delta\Delta_\Pi=-a_-/4\). An exact finite matching identity retains noncommuting kinetic determinants and middle singular values. Adjoint Sp(18) supplies conditional mass reconstruction from current and scalar observables, full-rank neutral--link vacua and a quantitative canonical tree gap under explicit metric bounds. In the distinct antisymmetric \(\mathrm{USp}(6)\) construction, a quark-source pencil and current projections reduce the independent normalization inputs. A spectator-active vacuum preserves matter parity and baryon triality; one tensor operator lifts both strong moduli without new preons. Directional mixing leaves one light Higgs pair and a nonsingular coloured \(H_{15}\) block. An exact four-doublet electroweak example has 13 positive physical scalar directions and three gauge zeros. These results establish compatible local sectors in specified tree effective theories. Ultraviolet coefficient selection, strong-sector backreaction, the full quantum vacuum and physical pole matching remain to be determined. No absolute charged mass follows from the present inputs. A detailed comparison states each construction's achievements and next physical tests without excluding unexamined actions or phases.
Keywords:
Pati–Salam symmetry
; composite fermions
; route parity
; canonical normalization
; fermion mass relations
; Sp(18)
; USp(6)
; vacuum stability
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