Build the signed matching matrix
Encode the even square board and its local domino constraints in a determinant-compatible matrix.
← All researchMF-PRISM-MATH-2026-15 / interactive guide
THE IDEA, MADE VISIBLE
Separate the powers of two.
Require or forbid one diagonal-step domino and factor the resulting exact tiling count.
Candidate resolutionExact Dyadic Valuations for Step-Diagonal Domino Constraints
Start with a complete board
An even square can be tiled by dominoes. Requiring one location restricts the set of tilings; forbidding it selects the complementary set.
- All tilings
- 36
- Constrained tilings
- 6
- Power-of-two exponent
- 1
Requiring this location leaves 6 tilings. The exact factorization is 2^1 × 3. The board picture identifies the location, not every allowed tiling.
ν₂(N) = largest e for which 2ᵉ divides N, N > 0
TRY THIS
Change the selected step and compare required with forbidden. Their counts add back to the unconstrained total.
What this experiment represents. Exact small-board, single-location enumeration. This does not reproduce every required, forbidden or mixed step-diagonal constraint in MATH-2026-15, or its separate unclaimed odd-square refinement.
Mathematics · Preprint
Exact Dyadic Valuations for Step-Diagonal Domino Constraints
Required, forbidden, and mixed constraints on even square boards
Candidate resolutionMF-PRISM-MATH-2026-15 / v1.1
Analyzes the exact power of two in matching counts under required, forbidden, or mixed step-diagonal constraints. A binary inverse-kernel calculation links local constraints to principal-minor valuations.
- Identifier
- MF-PRISM-MATH-2026-15
- Series
- Mathematics
- Edition
- Version 1.1
- Length
- 13 pages
- Reserved DOI
- 10.5281/zenodo.22728340 (record reserved)
01
Plain-language guide
3 minute explanation
What this paper is really saying.
Rather than only counting constrained domino tilings, the paper determines the exact power of two dividing those counts for required, forbidden, and mixed step-diagonal constraints on even square boards.
domino tilingsdyadic valuationmatching theory
Two tiling counts may be different but share the same hidden arithmetic structure. The dyadic valuation records exactly how many factors of two divide a count.
The paper studies local constraints along a step diagonal. It reduces the matching problem to an inverse-kernel calculation over characteristic two, then reads the valuation from principal minors and combines cases through inclusion–exclusion.
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.
Analyze the binary kernel
Find the kernel basis modulo two and lift the inverse information far enough to control exact valuations.
Assemble constrained counts
Use principal minors for required or forbidden positions and inclusion–exclusion for mixed constraints.
03
What it adds
The useful takeaways.
The exact two-adic valuation is proposed for the stated required, forbidden, and mixed step-diagonal cases.
The calculation links local domino constraints to inverse residues and principal-minor valuations.
The theorem is explicitly confined to even square boards; a separate odd-square refinement is not imported.
04
Scope & limits
What this does—and does not—establish.
Current status
Candidate exact-valuation theorem; the separate odd-square refinement is not claimed.
This manuscript presents a candidate resolution. Publication is not verification: independent specialist review remains necessary before the result should be treated as established.
The unproved odd-square refinement is outside the claim.
The formulas apply to the specified step-diagonal constraint family, not arbitrary deleted or forced domino patterns.
Signs, the boundary basis vector, and the lifting argument require specialist verification.
05
Why it matters
Why anyone should care.
Exact valuations expose arithmetic structure that ordinary enumeration misses and distinguish the proved even-square theorem from an unproved odd-square refinement.
06
Key terms
The vocabulary, decoded.
These are the terms needed to understand the claim. The full paper uses them more precisely.
Dyadic valuation
The exponent of the highest power of two dividing an integer.
Principal minor
A determinant formed by selecting the same index set of rows and columns.
Inclusion–exclusion
A counting method that corrects overlap by alternating additions and subtractions.
Characteristic two
Arithmetic in which 1 + 1 = 0, useful for tracking parity.
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.
- 01Binary kernel bases, first-order lifting, and the special boundary basis vector.
- 02Exact inverse residue and principal-minor valuations.
- 03Inclusion–exclusion for mixed constraints without importing the unproved odd-square refinement.
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.22728340
- Canonical record
- MF-PRISM-MATH-2026-15
Metriq PRISM Laboratory, Exact Dyadic Valuations for Step-Diagonal Domino Constraints, Metriq PRISM Laboratory Mathematics Research Paper MF-PRISM-MATH-2026-15, Version 1.1, 2026. Corresponding contributor: Daniel H. Jeffery, ORCID 0009-0001-1200-6042. DOI: 10.5281/zenodo.22728340.