Supplementary Note From Algorithmic π to Relation-First Information Ontology and the π Paradox in It from Bit and It from Qubit

This preprint investigates an overlooked ontological implication of the computability of π. In discussions surrounding discrete information ontology—most notably It from Bit and its quantum successor It from Qubit—the fact that πcan be generated to arbitrary precision by a finite algorithm is often treated as a decisive response to the “infinite digits” objection. The present work argues that this defense reverses the ontological order: algorithmic generability does not support the primacy of stored discrete states; rather, it foregrounds the primacy of an invariant generative relation (rule/constraint) that unfolds digit sequences only under a chosen resolution. To formalize this reversal, the paper distinguishes state-information (discrete stored configurations such as bit-strings and registers) from relation-information (invariant relational constraints that generate states when queried at finite resolution). From this perspective, πis paradigmatic not of “stored information” but of relation-first information: it functions as the mathematical signature of rotational and phase invariance, i.e., continuous transformation geometry. Building on this relation-first framework, the paper formulates the πparadox as an internal tension within discrete information-first programs: πis structurally indispensable to fundamental physics—wave dynamics, Fourier decomposition, rotational symmetry, Gaussian measures, quantum interference, and phase evolution—yet it is not finitely encodable in any discrete bit or qubit representation. While finite approximations suffice epistemically for prediction, treating discrete encodings as ontologically fundamental collapses the constitutive reading of It from Bit/Qubit into a merely instrumental one (“we model reality with bits”). The paradox deepens in It from Qubit, since qubit state space and quantum gates rely essentially on continuous phase parameters and group rotations (U(1), SU(2)), commonly expressed in multiples of π .Thus, qubits do not eliminate continuity; they formalize discrete outcomes on top of irreducibly continuous phase geometry. In this sense, πappears as an ontological residue of generative continuity that persists under any attempt to close reality within purely discrete informational states. The paper therefore motivates a conceptual reordering: rather than It from Bit or It from Qubit as ontological closure, a more consistent formulation is “Bits from Phase”—discrete encodings as derivative records produced by sampling and resolution constraints—pointing more broadly toward an It from Wave perspective. The analysis also connects to a “generative resolution / shutter-speed” interpretation, where discrete states correspond to finite observational sampling Δ t>0, and marks what remains irreducible under digitization. Version note (v2): This version updates formatting/typesetting only. The scientific content and claims are unchanged from v1.