The Branch Factor of the Phenomenological-Velocity Radical

The phenomenological-velocity (PV) radical equation\[α lsinβ=√L_+κ\,√L_-/κ,κ=√1-v^2/c^2,\]is standardly analyzed by squaring, which yields the v-free compatibilityconstraint $D=0$ and the conclusion that the equation does not determine v.That conclusion is correct as far as it goes and it discards the equation'sv-dependent content. We first document how v enters and how it acquires a value. The equationis obtained from a v-free height identity Psinβ=√L₊L₋ bysubstituting a factored unity into the radicand and distributing the root. Twounities are involved: the formal 1=κκ⁻¹, which places κ inthe radicand and is algebraically inert; and an asserted √cD/√ D=1,inferred from the joint vanishing of numerator and denominator on the solutionlocus. We show that this inference does not hold: joint vanishing determines adifference and not a quotient, the quotient in question equals √ c ratherthan $1$ wherever it is defined, and the two steps of the inference havedisjoint domains, one requiring $D=0$ and the other D≠0. In natural units$c=1$ the inserted factor is unity and the construction proceeds as intended.Solving for v returns v=±√c²D/√ D=± c, which is the insertedfactor read back, and substituting it into κ gives κ=0,annihilating the factor whose insertion produced the equation. None of thistouches the branch analysis, which depends on how the principal rootdistributes and not on the value of the factor inserted. All of the equation's v-contentresides in the distribution step, which is a suppressed premise of thederivation. That premise is exactlyε=+1 below, and it fails on an identifiable locus: the decoratedequation creates solutions the base geometry lacks in one sign cell anddestroys solutions it has in another. We show that the product of principal square roots satisfies\[√L_+κ\,√L_-/κ=ε\,√L_+L_-,ε∈\{+1,-1\},\]where ε is a locally constant function of v that is notidentically $+1$ and that jumps discontinuously across $|v|=c$. Squaring sendsε↦ε²=1 and erases it. Restricted to the solution locus $D=0$, the mixed sign sectors of(L₊,L₋) become unreachable and ε reduces to\[εsub=(lα),εₛᵤₚ=+1 .\]The unsquared equation then imposes a selection rule on the velocity regime:subluminal solutions exist if and only if sinβ>0, and superluminalsolutions exist if and only if (lα)=(sinβ). One of the foursign cells admits no solution in either regime. Under the natural conventionβ∈[0,π] implied by the height relation h=lsinβ, this collapsesto the statement that subluminal solutions always exist while superluminalsolutions exist if and only if lα>0. The branch discontinuity occurs precisely at $|v|=c$, which is also the valuereturned by computer-algebra evaluation of the associated ratio off thesolution locus, and the point at which L₋/κ becomes undefined. Thesethree facts identify the same locus by three independent routes. Restricting to the locus, the branch factor becomes a function of twoindependently specifiable arguments: the sign of lα and the velocityregime. That table has odd parity, and therefore admits no factorizationε(s,g)=f(s)h(g). This is the same algebraic obstruction thatforbids a locally factorized table from exhibiting a maximallyCHSH-violating sign pattern. We locate the correspondence precisely rather thanby resemblance: ε satisfies the $2$-cocycle identity and representsthe nontrivial class of H¹( C^×; Z/2), the obstruction toa global branch of the square root, while the Bell obstruction is thecohomological obstruction to a global section of an empirical model, evaluatedin the GHZ--Mermin case by a parity computation over deterministic signs. Bothare Z/2 obstructions to a global section, differing in what thesections are sections of. We also correct three disanalogies that do notsurvive examination, including the common claim that Bell's obstructionconcerns expectation values rather than deterministic signs. We then connect the branch structure to the setting-dependent selectionformalism. Since ε is a definedness condition, and since reportingonly defined trials constitutes an acceptance map, we exhibit an assignment ofthe geometric parameters under which the branch-admissibility indicatorcoincides pointwise with the acceptance rule of the sign--cosine construction.The coincidence is exact, and on the admissible subensemble the correlator is-cos(a-b) with unbiased marginals and admissible fractionZ(Δ) having minimum 0.8786. This is a representation rather thana derivation, and we say so: the geometry is fitted to the acceptance rule. Whatit establishes is that the selection map has the form of a branch-definednesscondition, and that the branch factor and the acceptance map are the samenon-outer-product 2×2 table in different coordinates. The composite construction retains Bell-local responses, ameasurement-independent prior, definite bounded outcomes, and unbiased acceptedmarginals --- the last protected by an antipodal symmetry of the base model,the same mechanism by which the pairing symmetry protects the balanced sector,and not extending automatically beyond it. What it relinquishes is the singleensemble. The premise that fails is not locality but the requirement that allfour contexts be evaluated on one measure. We then show that the branch factor is not an artifact of square roots butof asymmetric decoration, by transporting the same move into anoperator-algebraic setting containing no radicals. Conjugating the two wings ofa singlet by $U(v)$ and U(v)⁻¹ rather than by $U(v)$ twice givesE=-cos(a-b+2ϑ(v)), microcausal and non-signalling, withS(ϑ)=2√2\,|cos2ϑ| on the standard quartet and the Bellbound attained exactly at ϑ=π/8, that is v/c=cos(π/8).Symmetric decoration cancels in both settings; asymmetric decoration leaves aresidue, discrete where the decorated object has Z/2 monodromy andcontinuous where it is a rotation angle. We also record that the sixselection-geometry parameters defined elsewhere are independent definitionssharing the Lorentz template rather than consequences of the present equation:the radical's parameter is the light-cone value while theirs are strictlyinterior. Finally, the symbolic solution v=±√c²D/√ D=± c is not adegenerate output. The light cone is simultaneously the branch point ofκ, the locus of the jump in ε, and the point at whichL₋/κ is undefined. The solver returns the boundary between the branchregimes, which is the only v-dependence the relation possesses.