AUTHOR: Evan T. Kotler
Modern physical theories rest on a network of interdependent primitives: probability, information, identity, dynamics, and representation. These notions are typically introduced together, often justified operationally or empirically, and then refined within specific theoretical frameworks. While successful, this practice obscures a prior question that is rarely addressed directly:
Which of these structures are genuinely fundamental, and which are forced consequences of more primitive constraints?
The four papers collected here constitute a single research program aimed at answering that question in a deliberately austere setting. The guiding strategy is not to modify existing physical theories, but to step back and ask what must already be true before such theories can be meaningfully formulated at all.
The program proceeds by imposing only one standing structural requirement: stability under admissible extension. Informally, a distinction is regarded as meaningful only if it survives arbitrary admissible enlargement of context. This requirement replaces time, dynamics, observers, and measurement as the primary test of robustness. It is motivated not by physical intuition, but by the minimal demand that structure not collapse when its context is extended.
Working within this constraint, the program develops four results, each isolated in scope and explicitly layered.
The first paper establishes the structural foundation. Within finite relational systems subject to admissible extension, it shows that:
distinctions that are not stable under extension must be compressed,
primitive identity is incompatible with non-collapse,
only a restricted class of equivalence structures (“survivors”) can persist,
and when multiple survivors coexist, probability emerges necessarily as a public, compression-invariant frequency.
No probability measures, randomness assumptions, observers, or dynamics are presupposed. Probability is not interpreted epistemically or operationally; it is derived as a structural consequence of non-collapse alone. The paper is logically autonomous and introduces no strengthening assumptions.
The second paper addresses a common objection to structural derivations of probability, namely that information-theoretic concepts are already implicitly probabilistic.
It shows that this is not the case.
Starting from the same compression framework as Paper I, but without invoking probability at any stage, the paper demonstrates that:
irreversible compression induces loss of distinguishability,
entropy arises as a canonical numerical invariant of this loss,
channels are structural witness protocols mediating compression between relational regimes,
and standard information-theoretic inequalities follow from monotonicity under admissible extension.
Crucially, the distinction between classical and quantum information is shown to arise from the commutativity properties of admissible refinement, not from amplitudes, Hilbert spaces, or probability amplitudes. Probability, when it appears at all, enters only later as a derived frequency and is not required for the definition of information or entropy.
The third paper addresses universality. It begins by showing that, within the core framework, universality claims are ill-typed: admissibility thresholds possess only internal meaning, and cross-regime comparison is not licensed by default.
The paper then introduces a single, explicitly labeled strengthening—protocol comparability—which permits comparison within a restricted equivalence class without introducing metrics, scales, dynamics, or probability measures.
Under this assumption, admissible coarse-graining induces a discrete self-map on dimensionless threshold ratios. The paper proves the existence and uniqueness of a structural fixed point, yielding a form of universality that is:
non-numerical,
dimensionless,
independent of physical realization,
and explicitly conditional on the stated strengthening.
Universality thus appears as a fixed-point property of admissible structure, not as a postulated constant or law.
The fourth paper turns to representation. Having established that probability, information, and certain universal structures can arise without amplitudes or Hilbert spaces, it treats representation as an optional, explicitly licensed extension rather than a foundational assumption.
The key result is that admissible composition in finite relational systems is generically non-functorial, obstructing straightforward numerical or linear representations of probability. When this obstruction is respected rather than idealized away, any linear representation capable of encoding structural probability must be projective and phase-tolerant.
Introducing a minimal public coherence requirement—additivity on mutually exclusive contexts—the paper shows, via a Gleason-class theorem, that compatible probability assignments must factor through a positive quadratic form. The familiar Born rule thus appears not as a postulate of physics, but as the minimal consistent resolution of a representational obstruction.
No claim is made that amplitudes or Hilbert spaces are fundamental, nor that quantum mechanics is derived. The result is conditional and concerns representation, not ontology.
The four papers form a linear but non-circular dependency chain:
Paper I is logically autonomous.
Paper II assumes only the compression framework of Paper I.
Paper III assumes Paper I and introduces one explicit strengthening.
Paper IV assumes Papers I and II and introduces representational assumptions explicitly.
Rejecting any strengthening or representational assumption leaves all prior results intact.
This program does not propose new physical laws, constants, or interpretations. Its aim is more modest and more general: to clarify which conceptual structures are forced by minimal non-collapse requirements, and which enter only when additional assumptions are made explicit.
In doing so, it seeks to reposition probability, information, universality, and representation within a single structural hierarchy—one in which none are assumed prematurely, and all are earned.
First paper drop tomorrow. Title: Non-Collapse and the Forced Emergence of Probability.
