Refinement Consistency Obstructs Nonlinear Power Lifts

Successor to Finite Criteria for Simplicial States and Local Tomography

The companion paper characterises the finite, classical, simplicial layer above admissibility completely, and its own conclusion names the amplitude modulus as the natural next step, outside that contract. A recurring, rarely audited move at that next step is to read a positive branching measure directly as an amplitude by taking some power of it — canonically a square root.

This note asks exactly when such a power lift is compatible with the branching structure that produced it, using one elementary, theory-neutral condition: Kolmogorov consistency, the ordinary marginal-consistency identity a family of measures on a growing history space must satisfy.

Status. The central theorem is general and self-contained, independent of the Cosmochrony corpus. Its Heisenberg-automaton instantiation is conditional on an explicitly named, non-derived postulate (H2). No Born rule, no complex-carrier selection, and no coherent-sum construction is established anywhere in this note.

Six results

Refinement-consistency theorem. For any finite branching system, the power-lift family $\mu^\alpha$ of its induced valuation is Kolmogorov-consistent, at a branching state reached with positive $\mu$-valuation, if and only if $\alpha=1$ — an elementary comparison of $x^\alpha$ against $x$ on $[0,1]$, strict wherever a state has at least two positive-probability successors. No phase, complex numbers, or Born rule involved.
Escort lifts, situated relative to the literature. A row-renormalised power lift is trivially Kolmogorov-consistent for every $\alpha$, but generally does not coincide with the pointwise lift once path-dependent normalisation factors accumulate — the same order relation underlying the classical escort-distribution/Rényi literature of nonextensive statistical mechanics, cited and situated here, not claimed as new in isolation.
Exact instantiation under H2. Specialised, in exact $\mathbb{Q}(\sqrt2)$ arithmetic, to the non-backtracking Heisenberg $b$-shadow automaton of the trajectory-branching note, under an explicitly named, non-derived maximal-entropy/Parry postulate. An explicit two-parameter counterfamily of primitive Markov kernels on the same support — recovering H2's own stationary law at one point — shows that support, primitivity, and the Pell recurrence alone do not select a unique kernel.
Endpoint/shadow quotient incomparability. The endpoint quotient and the $b$-shadow quotient of the automaton are mutually incomparable, with explicit witnesses on words of length up to 4.
Exact relabelling automorphism and reality corollary. Relabelling $X \leftrightarrow X^{-1}$ transports the Heisenberg endpoint by the exact group automorphism $(a,b,\gamma) \mapsto (-a,b,-\gamma)$, from which every uniform-probe phase sum weighted by a real shadow-only valuation, over histories sharing a final $b_n$, follows to be exactly real.
Bounded exact cancellation search. Among all 13120 non-backtracking words of length up to 8 ($q=17$), 64 classes share a final $b_n$, 62 carry genuine nontrivial relative phase, and none exhibits exact destructive cancellation — a calibrated negative result on this range, not a claim of non-existence at every length.

Why this matters

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. This note identifies exactly where, in the H2/Parry/Pell chain, that choice becomes unavoidable, without taking either option itself.