ALLENIX Allen’s Interdisciplinary Logic of the Universe

Allen’s Impossible Number Theory (AINT)

AINT = also interpreted as Absolutes Injected Negating Truth

Contents
  1. Definition
  2. Why AINT matters
  3. Paradoxes & counter-paradoxes (examples)
  4. Parallel lines & dimensional collapse
  5. ALLENIX — Embracing the logic of the universe
  6. This page will grow

Definition

Allen’s Impossible Number Theory (AINT) proposes that paradoxes in mathematics and logic persist not because reality is contradictory, but because an injected assumption is slipped into the reasoning. This injected assumption masquerades as an absolute and negates the truth by blocking the natural resolution of the pattern.

The “impossible number” is a placeholder for that injection — a hypothetical quantity or condition that the argument quietly relies on but cannot justify within the system itself. Expose or remove the injection and the paradox dissolves; otherwise the same injection must be applied consistently to parallel cases, creating a counter-paradox.

AINT, in a line: Paradoxes persist because of Absolutes Injected Negating Truth. Identify the injection; restore coherence.

Why AINT matters

  • Resolution by exposure: When the injected assumption is made explicit, many paradoxes simply dissolve.
  • Fairness by extension: If the injection is kept, it must apply equally across domains — revealing counter-paradoxes and double standards.
  • Unity of math & logic: AINT encourages reasoning about the rules (logic) alongside the rules themselves (mathematics).
Allen (AINT): “Parallel lines never meet” is true in our frame. Change the frame — down (projection), in (curvature), or up (dimension) — and the outcome changes.

Allen’s Counter-Paradoxes

Each item follows AINT — Absolutes Injected Negating Truth. Expose the injected assumption and the paradox dissolves or must be applied consistently across domains.

1) Collatz Counter-Paradox

Injected assumption: “The conjecture is unsolved until every integer is tested.”

Counter: By that standard, Euclidean parallel lines would also remain unproven unless checked at every centimeter to infinity. Structure, not exhaustion, establishes truth. See the 3n+1 page for Allen’s treatment.

2) Parallel Lines Counter-Paradox

Injected assumption: “Parallel lines never meet, universally.”

Counter: In 2-D projection they converge at a vanishing point; in curved 3-D spacetime geodesics can focus; in higher-D embeddings they may separate again. “Never” is only frame-true.

3) Addition Counter-Paradox

Injected assumption: “We must brute-check that 1 + 1 + 1 + … never produces a negative.”

Counter: Monotonic increase is axiomatic for standard addition; no infinite testing is required. The paradox denies the rule it relies on.

4) Equality Counter-Paradox (ACER)

Injected assumption: “Equality between groups must mean identity.”

Counter: No two people are equal; each life is unique in gifts, flaws, and battles. Equality is dignity and rights, not sameness. Claims of superiority collapse to uniqueness.

5) Singularity Counter-Paradox

Injected assumption: “At a singularity, density becomes infinite.”

Counter: In a dimensional cascade (3-D → 2-D → 1-D → null), extent vanishes; with zero volume, “density” is undefined, not infinite. The paradox injected the persistence of extent.

AINT Rule of Thumb: If a claim depends on a universal “never/always,” identify the frame and the injection. Change the frame → change the outcome.

Parallel Lines & Dimensional Collapse (ALLENIX · AINT)

In our 3-D Euclidean frame, “parallel lines never meet” is true by axiom. But this is a frame statement. When the frame changes — by projection to 2-D, by curvature (near black holes), or by embedding in higher dimension — the outcome changes.

Allen (projection law): Same 3-D direction → same 2-D vanishing point.
Allen (parallelity): Increasing curvature forces parallels together; the singularity is the loss of distinctness.
Allen (dimensional rule): Lowering dimension enforces convergence; raising dimension permits separation.
Parallels across frames: 3D→2D vanishing, singularity convergence, higher-D divergence (1) Projection — 3D parallels meet at a 2D vanishing point image plane (z=f) eye V (2) Curvature — geodesics converge at a singularity singularity (3) Higher-D — added freedom allows separation lift to higher-D
Left: 3D parallels converge at a 2D vanishing point under projection. Center: in curved space, initially parallel geodesics focus into a singularity (loss of distinctness). Right: lifting to higher dimension adds freedom and separates apparent crossings.

Dimensional Cascade to the Metacontinuum

A black hole is a dimensional cascade: 3-D → 2-D → 1-D → null. At the endpoint (metacontinuum) there is no measurable extent, so classical quantities like “density” cease to apply.

Singularity Density Paradox (Allen): Not “infinite” density, but no density — because the carrier of extent vanishes. (Density is undefined when volume → null.)

Two Modes of Collapse

  1. Overload collapse (gravity-driven): Mass-energy overwhelms outward support ⇒ lattice descends (3D→…→null).
  2. Stillness collapse (motion-loss): Anchoring points dissolve when circulation stops on the 2-D scaffold ⇒ inward collapse.
Allen (anchor law): Without movement, anchors fail; without anchors, lattices fall.

ALLENIX — Embracing the logic of the universe

ALLENIX (Allen’s Interdisciplinary Logic of the Universe) does not claim to create the universe’s logic; it seeks to embrace it. By cataloguing paradoxes, counter-paradoxes, and their injected assumptions, ALLENIX aligns reasoning with observed coherence rather than with unearned absolutes.

This page will grow

This is a living reference. More paradoxes and counter-paradoxes will be added as they are articulated under AINT and ALLENIX. If a paradox dissolves, we will note how; if it remains, we will apply its injection consistently across comparable domains.