Refinement Consistency Obstructs Nonlinear Power Lifts of Finite History Measures

Abstract

A finite branching system is a finite set of local states equipped with an admissibility relation and a row-stochastic transition kernel. From an initial law, such a system induces a canonical positive valuation on admissible histories. We prove that, whenever at least one branching state is reachable by a history of positive $\mu$-valuation, this valuation is the unique member of its own power-lift family, $\mu^\alpha$ for $\alpha>0$, that is Kolmogorov-consistent: summing the lifted weight of every admissible one-step extension of a history recovers the lifted weight of the history itself, so that $\{\mu_n^\alpha\}_n$ forms a projective family of measures. The proof is an elementary comparison of $x^\alpha$ against $x$ on $[0,1]$, strict at every state with at least two positive-probability successors. Consequently, no unnormalised pointwise power lift of a branching measure whose positive-valuation support reaches a branching state remains a projective family of measures; in particular the square-root modulus does not inherit classical cylinder consistency from the underlying measure whenever it branches on that support. This does not obstruct the use of such a lift as an amplitude modulus unless that modulus is additionally required to obey classical additive marginalisation: a row-renormalised (``escort'') power lift restores Kolmogorov consistency trivially, at the cost of generally no longer coinciding with the pointwise lift once path-dependent normalisation factors accumulate (Section sec:escort), a fact already known in the escort-distribution literature of nonextensive statistical mechanics (Abe 2000; Tsallis, Plastino \& Alvarez-Estrada 2009) and situated here, not claimed as new in isolation. We instantiate the unnormalised obstruction on the non-backtracking Heisenberg-word automaton of the Cosmochrony trajectory-branching programme, under an explicitly named, non-derived maximal-entropy/ Parry postulate (Parry 1964) we call H2, giving an exact real-algebraic proof that certifies every radicand in ${{\mathbb{Q}}(\sqrt{2})}$. Independently of H2, we show, with explicit witnesses on words of length up to $4$, that the endpoint quotient and the shadow quotient of this automaton are mutually incomparable, and that relabelling $X\leftrightarrow X^{-1}$ transports the endpoint by an exact group automorphism, forcing every uniform-probe phase sum weighted by a real shadow-only valuation, over histories sharing a final $b_n$, to be real. Under H2, we report a bounded exact search, on words of length up to $8$, finding no destructive cancellation among such histories. No result here establishes a Born rule, selects a complex carrier, or constructs a coherent sum; each remains exactly as open as it was in the companion states-and-effects-tomography note this paper succeeds. Interpretive outlook. The following reading is structural, not a further result: within the Cosmochrony programme's search for an amplitude modulus, this obstruction means the intuitive move of silently taking a square root of a branching probability is not free. Any future construction must therefore either replace the pointwise power lift by a genuinely different construction — escort renormalisation being one example — or decline to require classical cylinder-measure consistency of the lifted modulus, with that choice justified independently.