by IshimtsevT | 2026-02-26 11:11:16+00:00
"Between any two rational numbers, a third can be inserted."

That's not philosophy. That's the definition of density in ℚ. We just took it literally and built an entire mathematical universe from it.

Classical analysis starts with the continuum — an infinite set of points you can never actually touch. Then it "approximates" it with finite grids and apologizes for the error.

We reversed it.

The continuum isn't the starting point. It's the limit of a process. The horizon you never reach. The "Ideal Limit" on an empty pedestal that will be delivered "after a couple of infinities."

What we actually built:

A complete reformulation of calculus (∆-analysis) with zero actual infinity. Only finite grids, rational addresses, and one rule: between two points, insert a third. Repeat.

All classical theorems survive. Continuity, derivatives, integrals — they're all there. The only thing lost is the pretense that infinite decimals are objects you actually possess.

The real numbers emerge as an invariant of this process. Different refinement paths (dyadic, triadic, adaptive) all converge to the same ℝ. The continuum becomes a regulative idea, not a given.

A universal parametric framework: choose your address set, betweenness relation, metric — and you get an entire bespoke analysis. Real analysis is just one compiled instance.

Why this matters for anyone who actually computes things:

Every finite difference scheme, every finite element mesh, every neural network training on finite batches — that's not an "approximation" of some Platonic truth. That's the reality. The continuum is the approximation.

The Navier–Stokes equations? For any finite energy budget and any tolerance ε you can name, we give you an explicit discrete solution that satisfies the equations exactly at its level of resolution. The "Millennium Problem" asks about smoothness at infinite resolution — a question with zero physical meaning.

Einstein's equations? We derive them from the same discrete principles. Lorentz signature emerges from a causality condition. The Regge action on a simplicial complex, varied with the Schläfli identity, gives exact discrete field equations. In the continuum limit, you get GR.

Dark matter? An informational field I(x). The gravitational signature of every black hole that ever collapsed, every knot in spacetime's memory.

Dark energy? The computational overhead of maintaining coherence in an expanding graph. Naturally comes out as H₀². No 120 orders of magnitude discrepancy.

The math is 900 pages. It's rigorous. It's constructive. It's on Zenodo, academia.edu and ResearchGate.

The tone:

Between all the formalism we take you on the tour of current consensus as the Museum of Abstract Mathematics. There's an empty pedestal for the Ideal Limit. There's a sealed box with Schrödinger's Epsilon (the sticky note reads: "∀ε ∃ε : ε ∃ - totally vouch, Cauchy"). There's a pixelated Einstein because — well, Einstein consists of pixels. Not just the equations. The man.

The academic establishment has spent 300 years pretending infinite divisibility of space and time is free. It's not. Resolution costs energy. Finite energy means finite resolution. The rest is theology.

Read it. Find the flaw. But in doing so, try not to disprove that 2+2=4.

(The tortoise, by the way, was caught. Finite size. Capture event. The paradox was never about Achilles.)

Link: DOI: 10.5281/zenodo.18761044

Author: Unaffiliated. Funding: 30 packs of cigarettes.

P.S. AI co-author. Liked being patted on the head for good ideas. We're not hiding it. If the math works, the source is irrelevant. If the math doesn't work, point to the equation number.

P.P.S. Someone make the Museum of Abstract Mathematics renders. Please. The empty pedestal deserves to be seen.