Allen’s Impossible Number Theory (AINT)
AINT = also interpreted as Absolutes Injected Negating Truth
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.
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’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.
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.
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.
Two Modes of Collapse
- Overload collapse (gravity-driven): Mass-energy overwhelms outward support ⇒ lattice descends (3D→…→null).
- Stillness collapse (motion-loss): Anchoring points dissolve when circulation stops on the 2-D scaffold ⇒ inward collapse.
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.
Allenics — Foundational Truths of Continuity
AINT (Abstraction Injected Negating Truth), the Allen Orbital Lattice (AOL), and the Fractal Reset Law
Physics says: “Nothing disappears.”
Allenics adds: “Reality continues.”
Physics guards the substance; Allenics guards the process. Together they express how the universe behaves in practice: conserved existence and uninterrupted becoming.
Allenic Truth 1 — Reality continues
Reality flows on regardless of the abstractions we place upon it. In Pattern Field Theory, this continuity is encoded structurally by the Allen Orbital Lattice (AOL™): each resonance step is part of an unbroken, coherent traversal.
Allenic Truth 2 — Observations can inject assumptions, but the process is always true
This is the philosophical core of AINT — Abstraction Injected Negating Truth. Paradoxes are born when an assumption or abstraction is injected that negates the underlying coherence. The lattice exhibits that coherence directly: the process never breaks.
“So the key here is that the observations can inject assumptions, but the reality of the process is always true.”
— James Johan Sebastian Allen
Allenic Truth 3 — Every progression is the first step of a new race
From the observer’s local-time perspective, motion is not an infinite queue of subtasks. It is a perpetual first step into a fresh fractal window — always back to a new step. This is the Fractal Reset Law in Allenics.
Illustration: In Zeno’s Dichotomy, at 1/2 you are at the first step of a new race of length 1/2; at 3/4, the first step of a new race of length 1/4; and so on. The “infinite backlog” never exists for the observer.
Allenic Truth 4 — Paradoxes collapse under observation
If a paradox were literally true, its consequences would eliminate the possibility of observation. The fact that we observe motion, probability, and resonance confirms that paradoxes are artifacts of injected abstractions — not features of reality.
Allenic Law of Observability: A paradox that negates reality also negates the observer. Since observation persists, paradoxes cannot overwrite continuity.
AINT: Name and Formula
Name (Principle): AINT = Abstraction Injected Negating Truth — paradoxes are not genuine; they are created by injected assumptions that negate underlying coherence.
Formula (Operational): AINT = Abstraction Injection n times — count the layers of abstraction needed to flip truth into apparent contradiction. The AINT depth (n) diagnoses how the paradox was constructed.
Resolution Operator: Mapping the distorted model onto the lattice removes the injections: Resolution = 𝓛(Aⁿ(T)) = T, where 𝓛 is the AOL mapping and T is the underlying truth.
The Ultimate Truth
“Get up from your laptop. Go to the front door and open it. Look outside. If the world is still there, then the universe must still be ticking along like it always has done.”
— James Johan Sebastian Allen
This ties back to Allenic Truth #4: The world is there because reality continues — paradoxes cannot overwrite continuity.