Independent Research · Paper II
Geometry, Escape, and Phase Transitions in Violation Sets for the Generalized Damascus Inequality
A structural classification of higher-dimensional Damascus violation sets, their boundedness, separation, exponent dynamics, and asymptotic geometry.
Abstract / research note
What this work establishes
Extends the Damascus inequality program into higher dimensions, locating sharp transitions in escape, nesting, topology, and continuous-exponent behavior. Finite lower-cell inclusions are established with deterministic exact rational Bernstein subdivision.
In plain English
What the paper is really doing
This paper studies where a family of mathematical inequalities fails. Each possible failure can be treated as a point in a geometric space; together, those points form a violation set. The paper maps how that shape changes as the exponent and the number of variables increase.
The object
Instead of finding one counterexample, the paper studies every counterexample at once. This turns a question about formulas into a question about geometry: what does the full region of failure look like?
The questions
Does the region stay confined or escape toward infinity? Does it keep away from critical coordinate planes? As the exponent changes, does each violation remain a violation—or can the sets stop nesting?
The method
Exact formulas control what happens far from the centre, local expansions describe how violations first appear, and difficult finite cases are certified by deterministic rational computation rather than floating-point evidence.
Headline findings
The results worth carrying away
- Three variables are the last dimension in which violation sets are nested for every exponent.
- With four variables, one-step nesting holds only through exponent 6; after that, the clean growth pattern breaks.
- The paper gives the exact first dimension where nesting fails for each exponent, revealing a sharp dimensional staircase.
- It classifies when violation sets remain bounded, when they escape, and when they stay separated from the coordinate planes.
- A key four-variable set is shown to have exactly three connected pieces, each without holes.
- The theory is extended from integer exponents to continuously varying exponents, exposing another precise onset threshold.