Natural-density logarithmic Collatz descent | ProofAtlasSkip to content<br>The theorem at a glance<br>Two logarithmic-time density conclusions at a glance<br>The proof uses a global power-law phase gap and fixed quantitative rate, turns fixed-target control into an odd-relative Syracuse conclusion, then uses the two-adic lift to obtain the ordinary-density raw Collatz conclusion.<br>Read the exact theorem and checked proof Main theorem · expanded proposition · proof walkthrough · Lean evidence→<br>Enlarge<br>Theorem schematic<br>Two density domains within two logarithmic clocks
EnlargeThe Syracuse endpoint uses odd-relative density among odd starts; the raw Collatz endpoint uses ordinary density among positive starts.odd-relative density(Syracuse hit) = 1 with C_syr<br>For thresholds growing along odd inputs, odd-relative-density-one many odd starts have a Syracuse hit before 145 · log N odd-to-odd steps. For thresholds growing on all positive inputs, ordinary-natural-density-one many positive starts have a raw Collatz hit before 436 · log N individual steps.
The theorem at a glance<br>Square-root logarithmic time window at a glance<br>The natural-density theorem supplies the upper clock for the square-root threshold. A deterministic halving argument supplies the strict lower clock, and the checked corollary retains one witness satisfying both inequalities.
Enlarge<br>Theorem schematic<br>A square-root hit inside a logarithmic window
EnlargeFor natural-density-one many starts, a raw Collatz hit below √N occurs inside a genuine logarithmic time window.log N/(2 log 2)<br>The same witness time is strictly greater than log N divided by 2 log 2 and no greater than the checked raw Collatz clock, which is below 436 · log N. The lower bound is special to the square-root target.
About these visual explanationsThese AI-generated visuals explain the theorem and proof route; they are not proof evidence. Their publication review was completed separately from review of the formal result. The exact Lean proposition and checked source remain authoritative.<br>Open PDF ↗Companion research paper<br>Natural-Density Almost-Bounded Collatz Orbits in Logarithmic Time ↗<br>Lech Mazur · 2026-07-16 · 27 pages<br>A mathematical paper developing the natural-density-one logarithmic-time Collatz descent theorem, its Rhin phase-gap input, the quantitative rate architecture, the Syracuse-to-raw-time bridge, and the square-root time-window corollary.<br>Read the paper (PDF) ↗Download PDF<br>Paper, rights, and source relationshipHosting authorized by the rightsholder. Original ProofAtlas metadata cover; not a reproduction of a paper page.<br>Lech Mazur, “Natural-Density Almost-Bounded Collatz Orbits in Logarithmic Time,” version 1, 16 July 2026.<br>The paper is mathematical exposition linked to the same theorem family; the exact checked Lean declarations and pinned source remain authoritative if wording differs.<br>The paper discusses a theorem that permits a density-zero exceptional set and does not claim the full Collatz conjecture or convergence of every orbit.
Collatz results landscape<br>How these results relate
depends onstrengthenscomparison only
The atlas keeps proof dependencies separate from stronger sibling results and useful comparisons. The two lanes below are disconnected: neither predecessor-count family is an input to the density-and-time family.<br>Predecessor-count bounds<br>Three exact 0.90 theorem variants<br>Real cutoffx0.90 Natural cutoffN0.90 Positive scalectx0.901
Density and logarithmic time<br>Rhin feeds the quantitative rate engine<br>Rhin phase gap→Quantitative rate engine→Natural-density theoremSquare-root time window
separate companion · not a dependencyTerras finite power-saving bound
Exact relationship evidenceReviewed dependency path: Rhin phase gap → ND31 main → ND31 bounds → same-exponent rate → fixed rate → the two sibling density-family endpoints. Six retained depends_on edges support this contracted path.<br>Comparison only: the Terras result is explicitly recorded as a separate companion, not an input to the natural-density proof.<br>No inferred edge: shared source files, a common subject, or historical background do not create a theorem dependency.
Companion result · not a dependency<br>Compare the finite power-saving stopping bound<br>The Terras-style theorem gives an explicit exceptional-proportion bound for descent below the starting value. It is a separate checked route, not an input to this natural-density proof.
Open the companion theoremScope limits<br>What this formalization does not claim<br>This does not prove the Collatz conjecture, convergence for every start, or arrival at 1; a density-zero exceptional set may remain.<br>The threshold must tend to infinity but need not be monotone, and the hit below it is strict.<br>The odd-relative Syracuse conclusion and ordinary-density raw Collatz conclusion have different domains and must not be collapsed into one claim about all positive starts.<br>The 145 bound counts odd-to-odd Syracuse steps, while the 436 bound counts raw Collatz steps...