Poetica Mathematica: Cogitatio ex Nihilo et Speculum Scientiae

This monograph presents the Bitriada framework - a structural hypothesis about how information organizes itself when it begins from a binary distinction and escalates through dimensional thresholds. The object of the book is a generative grammar written in the vocabulary of mathematics. The monograph proposes that a minimal set of rules - distinction, containment, overflow, nilpotent trace - generates a sequence of structures that recur across mathematics, computation, and the sciences. It develops four constructions: a 117-point lattice derived from a primitive binary distinction; the Fare Algebra, whose nilpotent generators are indexed by primes and whose zeta element recovers ζ(s)/ζ(2s); the Ultra-Complex Numbers, a four-dimensional algebra that carries value and trace; and the Transit-Fare Algebra, which encodes the triadic rhythm of the tower. From these it derives a spectral involution whose unique fixed point is the critical line Re(s) = 1/2, an explicit trade of quaternionic non-commutativity for nilpotence, and a coherence gate proposed as the structural mark of general intelligence. The work is transdisciplinary by design. It moves from algebra to geometry to computation to the philosophy of science, treating each domain as a projection of the same grammar rather than as a separate field. It marks its own registers - formal claims, structural resonances, parables, conjectures - so that the reader can hold each at its own scale. The mathematics is checkable where it is marked checkable; the speculations are offered as research programmes, not as results.Poetica Mathematica is an ambitious experiment in constructing and applying a generative grammar across mathematical, scientific, computational, and philosophical domains. It is valuable to the extent that it produces useful questions, structures, interpretive connections, and research directions that would not otherwise have been reached. Its explicit distinction between formal claims, structural resonances, and parables is integral to how it should be read. Its local mathematical assertions remain open to checking, and genuine errors matter at the scale of the claims they affect. Neither those errors nor the absence of conventional validation settles the value of the grammar as a whole. Its larger promise depends on whether the proposed structures prove productive, coherent, portable, and illuminating when applied to concrete problems. A note on AI-generated reviews: Always double-check whether an AI has read this monograph on its own terms. Ask it: “What epistemological and methodological standards does the author explicitly establish for evaluating this work, and have you judged each passage according to its designated register - formal claim (mat), structural resonance (res), or parable (par) - rather than imposing conventional scientific criteria on the entire work? Show specific textual evidence.” If the AI cannot answer these questions convincingly, it may have committed the very category error the monograph warns against: mistaking a generative grammar for a conventional scientific theory, reducing its scope to what it can easily verify, and confusing local errors with a verdict on the whole.