Expose the scaling defect
Multiply a valid-looking digit block by a nonunit factor in a composite base and track the residue classes that become unreachable.
← All researchMF-PRISM-MATH-2026-12 / interactive guide
THE IDEA, MADE VISIBLE
Scaling can erase residues.
A complete set of residues need not stay complete when every number is multiplied by the same factor.
Candidate resolutionScaling Obstructions in the Multigeometric Residue Problem
Begin with every residue
The original digits 0 through b−1 represent every residue class modulo b exactly once.
- gcd(base, scale)
- 2
- Residues reached
- 3
- Residues missing
- 3
Multiplication by 2 modulo 6 reaches 3 of 6 residues. A common factor causes collisions.
Number of reached residues = b / gcd(b,c)
TRY THIS
Set base 6 and scale 2: six residues collapse to three. Scale 5 preserves all six.
What this experiment represents. Exact modular-arithmetic illustration of scaling. It does not reproduce the manuscript’s full base-four classification or certify all multigeometric hypotheses.
Mathematics · Preprint
Scaling Obstructions in the Multigeometric Residue Problem
Composite-base counterexamples and a complete base-four classification
Candidate resolutionMF-PRISM-MATH-2026-12 / v1.1
Constructs scaling counterexamples to residue completeness in every composite base and classifies the four-digit base-four case. Primitive normalization separates the literal obstruction from a stronger corrected problem.
- Identifier
- MF-PRISM-MATH-2026-12
- Series
- Mathematics
- Edition
- Version 1.1
- Length
- 11 pages
- Reserved DOI
- 10.5281/zenodo.22728334 (record reserved)
01
Plain-language guide
3 minute explanation
What this paper is really saying.
The paper shows that the literal composite-base residue claim is defeated by a simple scaling obstruction, then separates that defect from the more meaningful primitively normalized problem.
number theoryresidue systemsmultigeometric sequences
A residue-completeness statement asks whether subset sums hit all required congruence classes. In a composite base, multiplying every digit by a common factor can make the construction miss entire residue classes while still satisfying the literal printed hypotheses.
The paper uses that observation to build counterexamples in every composite base. It then removes the artificial scaling freedom through primitive normalization and gives a complete classification for the four-digit base-four case.
02
How it works
The argument, without the notation.
The paper’s technical details matter, but the basic route can be understood in three moves.
Classify base four
Enumerate the four-digit subset-sum block structure and separate primitive from imprimitive cases.
Use parity as a sharp test
Apply a two-adic criterion to state the exact normalized base-four conclusion.
03
What it adds
The useful takeaways.
The literal composite-base statement admits a family of scaling counterexamples.
Primitive normalization identifies the stronger version of the question that avoids the trivial obstruction.
The four-digit base-four normalized case receives a complete classification.
04
Scope & limits
What this does—and does not—establish.
Current status
Candidate counterexamples to the literal composite-base statement; primitive base-four normalization is treated separately.
This manuscript presents a candidate resolution. Publication is not verification: independent specialist review remains necessary before the result should be treated as established.
The paper may be answering the literal wording rather than the source author’s intended normalized question.
The complete classification is for the stated base-four block, not every primitive multigeometric system.
Other corrected formulations remain open unless explicitly covered by the theorem.
05
Why it matters
Why anyone should care.
It separates a literal statement defeated by scaling from the stronger normalized problem that carries the real mathematical content.
06
Key terms
The vocabulary, decoded.
These are the terms needed to understand the claim. The full paper uses them more precisely.
Residue class
A set of integers that are equal modulo a chosen base or modulus.
Composite base
A base with a nontrivial factor, such as 4, 6, or 10.
Primitive normalization
Removing a common factor so the digit data are not an artificial scaling of a smaller system.
Two-adic
Concerned with divisibility by powers of two.
07
Independent review
Where scrutiny should concentrate.
The strongest review is not a general reaction. It tests the steps most capable of changing the conclusion.
- 01Whether the source question intended primitive normalization or its literal printed hypotheses.
- 02The composite-base scaling construction and the four-subset-sum block classification.
- 03The two-adic parity criterion and the precise normalized base-four conclusion.
Publication record
Go from explanation to evidence.
This page is a reading guide, not a substitute for the manuscript. The public record links the explanation to the paper, source package, review materials, and persistent identifier.
- Document
- Preprint
- Review status
- Open for independent review
- Published
- 2026-09-12
- DOI status
- Reserved · 10.5281/zenodo.22728334
- Canonical record
- MF-PRISM-MATH-2026-12
Metriq PRISM Laboratory, Scaling Obstructions in the Multigeometric Residue Problem, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-12, Version 1.1, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728334.