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. The site distinguishes mathematical results presented with proofs from developing research frameworks and philosophical interpretation.
Mathematical preprint
Adjoint transfer, congruence lifting, order defects, exact branching laws, and higher-order Jordan reconstruction depth.
Reconstruction Theory
Reconstruction signatures, observable and invisible directions, response families, reconstruction depth and reconstruction time.
Reconstruction Ontology
A developing interpretation of identity, persistence and change through reconstruction and structured response.
Finite periodic structure
Fibonacci-derived modular structures, period towers, return laws, degeneracies and ternary organisation.
Transformations and lifts
Matrix groups, congruence levels, tangent lifts, order defects and adjoint transfer.
Identity through response
A developing conceptual framework asking what can be reconstructed from invariants and structured responses to change.
Current paper
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
DR(X) = { (p − 1)/2 outside HU; p − 1 on HU \ <U>; ∞ on <U> }.
The middle layer has the invariant leading coefficient
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
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
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
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, reconstruction signatures, observable and invisible directions, reconstruction depth, and reconstruction time.
Ω is used as a conceptual symbol for a structured terrain of possible states, transformations, observations and reconstruction relations. This is presently a research framework and philosophical interpretation—not a claim that a new physical field has been experimentally established.
Methods, verification & AI
Mathematical claims are developed through algebraic derivation, finite computational checks and comparison with relevant literature. Computational verification is treated as an independent discovery and error-detection layer, not as a substitute for proof.
Use of artificial intelligence
Artificial intelligence, including ChatGPT, is used as a research-assistance tool in the project: for exploratory dialogue, formulation of hypotheses, code drafting and review, organisation of computational experiments, exposition, language editing, and literature-search assistance. Mathematical claims intended as results are not accepted solely because an AI system proposes them; they are subjected to explicit derivation, computational checks where applicable, and literature review. Responsibility for the published content and claims remains with the author.