Repository: OmegaCore-Labs / universal-obstruction-linear-form-x-y-z-x-y-z Preprint DOI: https://doi.org/10.5281/zenodo.XXXXXXX
This repository contains a public research preprint addressing the modern formulation of ErdΕs problem #142 concerning whether the linear form [ (x, y, z, x+y+z) ] can be simulated inside near-extremal three-term-arithmetic-progression-free subsets of the integers.
The work develops a structural reduction showing that any such set admits a bounded-variability, carry-controlled digit representation. Within this reduced model, configurations capable of realizing the linear form above are exponentially suppressed in the effective dimension.
The argument is combinatorial and structural in nature and does not rely on Fourier-analytic inverse theorems or assumptions about specific constructions.
Digit_Rigidity_and_Container_Obstructions_for_Three_Term_Arithmetic_Progression_Free_Sets.pdf
This is the primary research manuscript presenting the full argument and mathematical development of the digit rigidity framework and container obstruction method applied to 3-term arithmetic progression free sets.
This document contains:
- Formal theorem statements
- Structural lemmas
- Proof framework
- Mathematical derivations
- References and context
This file represents the canonical paper version of the work.
These files provide transparent source versions of the argument used to construct the manuscript.
digit_rigidity_container_obstruction_preprint.md
Full Markdown version of the preprint including all definitions, lemmas, and formulas in a format readable directly on GitHub.
digit_rigidity_container_obstruction_preprint.tex
LaTeX source version of the same document for typesetting, academic editing, or PDF compilation.
Both files mirror the mathematical content of the manuscript while providing editable and readable source formats.
BREAKTHROUGH_SUMMARY.md
A concise overview highlighting:
- previously known bounds and structural results
- the new obstruction mechanism developed in this work
- the conceptual advance provided by the digit-rigidity reduction
This file allows readers to quickly understand the main idea before reading the full manuscript.
appendix_container_lemma_tex.pdf
Technical appendix expanding the container-lemma component used in the structural reduction.
verification_notes_tex.pdf
Logical dependency map and verification notes documenting the reasoning structure and relationships between the components of the argument.
These files provide additional transparency into the structure and verification of the proof.
The work establishes:
- A digit-slice compression mechanism for near-extremal 3-AP-free sets
- A structural reduction to a carry-controlled representation model
- Application of a finite hypergraph container framework in this setting
- A universal obstruction to simulating the linear configuration [ (x, y, z, x+y+z) ]
within such sets.
This repository does not claim:
- a classification of all extremal 3-AP-free sets
- explicit constructions achieving the upper bound
- reliance on analytic or Fourier-based methods
The focus is strictly on the structural obstruction described above.
Conceptual development, proof structure, and verification workflow were carried out using the A.R.S.V.E. (Autonomous Recursive Synthesis & Verification Engine) under human direction.
All mathematical claims and responsibility for correctness remain with the author.