2026-10-17 · Zenodo · Jordan 5 × 5
Research programme
The project began with periodic Fibonacci structures modulo 9 and developed into a broader investigation of lifting, invariance, reconstruction and observable response under transformations. Mathematical results, exploratory frameworks and philosophical interpretations are distinguished.
Mathematical results
Published preprints
Finite fields, matrix order lifting, Jordan reconstruction, and discrete dynamics.
Research framework
Reconstruction Theory
Reconstruction signatures, observable and invisible directions, response families and reconstruction depth.
Conceptual interpretation
Reconstruction Ontology
Identity, persistence and change understood through reconstruction and structured response.
FPMT-24
Finite periodic structures
Fibonacci-derived modular structures, period towers, return laws and ternary organisation.
Reconstruction algebra
Transformations and lifts
Matrix groups, congruence levels, tangent lifts, order defects and adjoint transfer.
Reconstruction ontology
Identity through response
A developing conceptual framework, not an experimentally established physical field.
Publications
Six open Zenodo preprints and one arXiv record. Titles below are descriptive where the Zenodo listing truncated them; the DOI links lead to the authoritative records.
2026-10-17 · Zenodo · Fibonacci / 3-adic
Universal Classification of Primitive Scalar Witnesses in Fibonacci Dynamics Modulo Powers of Three
View on Zenodo ↗2026-10-16 · Zenodo · Discrete dynamics
Exact Periods, Defect Observability, and Optimal Ternary Reconstruction
View on Zenodo ↗2026-10-14 · Zenodo · FPMT-24
Symmetry Groups and Reconstruction Codes in 24-Step Fibonacci Dynamics
View on Zenodo ↗2026-10-14 · Zenodo · Jordan 3 × 3
A Three-Layer Reconstruction-Depth Theorem for the Regular 3 × 3 Jordan System
View on Zenodo ↗2026-10-10 · Zenodo · Matrix lifting
Adjoint Transfer and Order-Lifting Dichotomies for 2 × 2 Matrices over Prime-Power Rings
View on Zenodo ↗Local PDF · PDF2026-10-11 · arXiv · Matrix lifting
Adjoint Transfer and Order-Lifting Dichotomies for 2 × 2 Matrices over Prime-Power Rings
Read on arXiv ↗The arXiv record below concerns the 2 × 2 matrix paper; it is not the newly submitted 5 × 5 paper.
Read the original detailed mathematical overview (English)
The original English technical explanation and formulas from the previous website are preserved below.
Independent research preprint · Version 1.19 · 11 October 2026
Adjoint Transfer and Order-Lifting Dichotomies for 2 × 2 Matrices over Prime-Power Rings
Ivo Vajc · Independent researcher · ORCID 0009-0005-5145-6491
The paper develops an affine order-defect approach to matrix-order lifting over prime-power rings. The first variation of the order defect is governed by a cyclic adjoint transfer on the traceless tangent space.
For odd characteristic, the full-order adjoint transfer is classified across scalar, nonscalar semisimple and nonscalar Jordan states. Nonscalar semisimple states have a rank-one channel on the traceless centralizer axis. In the Jordan sector, characteristic 3 has a rank-one first-order channel, whereas for odd primes p > 3 the first-order transfer vanishes.
Higher-order Jordan structure. The vanishing of this particular first-order response does not imply that all higher-order responses vanish. For a nonscalar Jordan state, with
HU = { X ∈ sl2(Fp) : tr(UX) = 0 },
DR(X) = { (p − 1)/2 outside HU; p − 1 on HU \ <U>; ∞ on <U> }.
DR(X) = { (p − 1)/2 outside HU; p − 1 on HU \ <U>; ∞ on <U> }.
The middle layer has the invariant leading coefficient
[εp−1] [q(I + εX)]mp
= m (−det X)(p−1)/2 U.
On the final nilpotent axis, the full-order perturbative response vanishes identically. This is a statement about this particular response, not about arbitrary observables.
Exact Jordan response flag. The three response strata form the invariant flag
sl2(Fp) ⊃ HU ⊃ <U>,
with stratum sizes p2(p − 1), p(p − 1), p
and reconstruction depths (p − 1)/2, p − 1, ∞.
with stratum sizes p2(p − 1), p(p − 1), p
and reconstruction depths (p − 1)/2, p − 1, ∞.
The second finite depth is twice the first. Here ∞ means exact polynomial vanishing of this full-order perturbative response, not merely failure to detect a response up to some finite order.
Within this 2 × 2 Jordan perturbation problem, the three layers are equivalently encoded by the exact discriminant-contact law
DR(X) = ((p − 1) / 2) vε(ΔX).
Characteristic 3 is the depth-one case of the Jordan perturbation formula. For determinant-compatible lifts with nonscalar reduction modulo 3, the order-lifting analysis gives the recursive law
PRE → 9 PRE + 18 ACT ACT → 27 ACT
Read preprint v1.19 · PDF
Reconstruction Theory
A central working idea is that an object need not be characterized by a single observed state. It may instead be characterized by a sufficiently rich family of responses to transformations. This motivates reconstruction classes, signatures, observable and invisible directions, reconstruction depth, and reconstruction time.
Ω is a conceptual symbol for a structured terrain of possible states, transformations, observations and reconstruction relations. It is a research framework and philosophical interpretation, not evidence of a newly established physical field.
Methods, verification & AI
Mathematical claims are developed through algebraic derivation, finite computational checks and comparison with relevant literature. Computations help detect errors but are not substitutes for mathematical proofs.
Artificial intelligence, including ChatGPT, supports exploratory dialogue, hypothesis formulation, code drafting, experiment design, exposition, language editing and literature-search assistance. The author remains responsible for published content and claims.