Information geometry and reduced quantum description

Amari, Shun-ichi, Nagaoka, Hiroshi

Jun Lucis

BIG-B15 presents a reduced operator representation for comparing observation traces and boundary history within Boundary Information Geometry (BIG). BIG-B13 and BIG-B13.1 studied observation-like updates in hidden-depth states, including basin selection, Born-like behavior, an empirical logistic selection kernel, cloud averaging, and sequential consistency. BIG-B14 introduced boundary history as a trace retained on the active boundary, with reduced operations such as writing, retention, fusion, division, and lineage-level persistence. The present paper asks whether these two forms of trace retention can be described within a common state-update language. The goal is not to derive quantum measurement, the Born rule, biological memory, heredity, or standard quantum channels. Rather, the paper provides a reduced representation framework in which hidden-depth clouds, boundary histories, kernel responses, and boundary-history update maps can be compared as history-dependent transformations of state. The paper first embeds hidden-depth states into a real two-dimensional Hilbert-like representation and shows that finite hidden-depth clouds can be represented by density-like mixtures. Increasing cloud width lowers purity and smooths the projective response curve. It then compares the empirical BIG logistic kernel with the Born cos² reference and affine qubit-like probability forms. The results show that the BIG logistic kernel is not identical to the Born cos² reference. An unconstrained affine fit improves the error slightly but violates probability bounds, while the constrained affine probability form returns to the Born cos² reference. Higher-feature representations approximate the BIG kernel much more accurately. The paper then lifts B14 boundary history profiles HΣ(s) into density-like mixture representations. This representation captures normalized trace geometry, localization, and spread, while absolute history strength must still be retained separately. Finally, representative BIG update maps are classified as linear, affine, nonlinear, or state-dependent. Forgetting/smoothing and fixed-partition division are linear under fixed parameters, writing with a fixed stimulus is affine, normalization into density-like weights is nonlinear, fusion addition is linear, and fusion compression is structurally overlap-dependent. B15 should therefore be read as a representation and comparison paper. It does not identify BIG with standard quantum theory. Instead, it clarifies which parts of BIG admit Hilbert-like, density-like, or operator-like representations, and which parts require nonlinear, normalized, state-dependent, coarse-grained, or higher-feature descriptions. In this sense, B15 provides a bridge between observation traces, boundary history, and a possible future boundary-history operator framework.

Marcel Reginatto

It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of probabilities (information geometry), the description of dynamics using a Hamiltonian formalism (symplectic geometry), and requirements of consistency (Kähler geometry). The theory that results is standard quantum mechanics, but in a geometrical formulation that includes also a particular case of a family of nonlinear gauge transformations introduced by Doebner and Goldin. The analysis is carried out for the case of discrete quantum mechanics. The work presented here relies heavily on, and extends, previous work done in collaboration with M. J. W. Hall.

Jason Padgett

This paper develops a mathematical dual description between photoelectric detection and a Quantum Information Holography (QIH)-inspired model of microtubule readout. After reducing each system to a complex input field, substrate response, and two-outcome transition, the corresponding readout operators are shown to be mathematically equivalent under an explicit mapping of states, fields, frequencies, and response functions. The result does not claim that microtubules and photoelectric surfaces are physically identical, nor that microtubules have been experimentally demonstrated to implement this mechanism. Rather, it establishes a common operator-level architecture linking field amplitude, phase, interference, threshold-dependent transitions, and information readout. The framework is then extended computationally to a matched complex decoder and a time-resolved information geometry. This makes it possible to simulate and render the internal field, interference structure, state transitions, relational strengths, and reconstructed geometry frame by frame. The resulting architecture provides a directly testable software implementation independent of whether the broader QIH interpretation ultimately describes the physical universe.

Masahito Hayashi