Reading map
Five families, one mathematical question
This monograph investigates structural invariants in five families of special functions. Feedback scales, arithmetic frequencies, signed prime masses, rotation orbits, and coupled channels provide distinct settings. Their connections are established through specific formulas, invariants, and obstructions, not assumed from a shared vocabulary.
The argument starts with entire functions defined by contracted nonlinear feedback. It then studies arithmetic generating functions and prime-sieve measures, before passing to orbit spectral zeta functions and exact coefficient laboratories. Coupled entire systems connect several of these ideas, while a resolution atlas measures the difference between finite observation and global mathematical structure.
The main chapters give defining formulas, stated hypotheses, mathematical consequences, and computational figures. Extended proofs follow as technical appendices. The final notes distinguish analytic proofs, finite checks, classical inputs, and questions that remain open.
Four starting constructions
The first four families lead to distinct results: an entire-growth law, an obstruction to finite differential equations, a positivity boundary, and a rotation-invariant spectral coordinate. A coupled fifth family is developed in Part IV.
| Family | Structure worked out | Status of the construction |
|---|---|---|
| Asymmetric dilation | Entire solutions; exact growth balance; off-diagonal nonrescaling; intrinsic feedback banks | Candidate extension of a known one-dilation tangent analog |
| Complex arithmetic depth | Exact exponential type; coupled shift calculus; strict positive kernel; exact parameter recovery; no polynomial-coefficient linear ODE | Candidate extension of the project's integer arithmetic energy |
| Prime-sieve packets | Exact moment field; real Hankel singularities; positivity-margin collapse; multiple-prime geometry; arithmetic resolvent | Candidate arithmetic specialization of classical orthogonal-polynomial machinery |
| Orbit spectral zeta | Global meromorphic continuation; rational integer values; regularized product; multiplicity-aware orbit quotient | Worked specialization of known Mittag–Leffler seeds and spectral-zeta methods |
An established diagonal antecedent
The equation \(f_q'(z)=1+f_q(qz)^2\), \(f_q(0)=0\), is already present in Méndez [Mendez2024], Example 2.4. It remains an important baseline, but its definition is not a discovery of this project. The asymmetric extension keeps two separate contractions and permits a proved distinction from every amplitude/argument rescaling of that baseline.
How to read the evidence
Theorem statements below concern exact infinite objects under the displayed hypotheses. Plots and decimal roots illustrate them. Finite assertions check implementations within recorded ranges. None of these three forms of evidence establishes historical priority by itself.
The source archive preserves extended arguments, earlier review records, negative controls, and open questions. The evidence register distinguishes those records from the checks performed for this edition.
Part I: Entire functions and their growth
Contractions, coefficients and nonlinear feedback.