Research

Projection Algebra & Universality Thesis

41 papers. 7 irreducible axioms. 375:1 compression. The mathematics that generates the system.

Projection Algebra

Three axioms generate a tripartite structure: empty, partial, full. Division by zero becomes maximal informational potential.

Three axioms (idempotency, bounded magnitude, compositional closure) generate a tripartite structure.

Tripartite Structure

∅

magnitude: 0

Null projection — maximal potential. 0/0 resolved.

☽

magnitude: (0,1)

Proper subspace — the coherence band.

•

magnitude: 1

Identity projection — transmission without loss.

GiROS Mapping

. = ∅ (isolated, maximal potential), . . : = ☽ (partial, binding), | = • (full, transmission)

Division by zero is not undefined but the empty projection: maximal informational potential.

Every Function Is a Projection

Every function decomposes into three projections: collapse, rename, select. Projection is not a part of mathematics. Projection IS mathematics.

Decomposition: Every function = collapse × rename × select (three projections)

Extension: Every function embeds as an idempotent on a direct sum

Generality: Lifts to arbitrary categories via the Karoubi envelope

Projection is not a part of mathematics. Projection IS mathematics.

Paper Corpus

41 papers distilled into 7 axioms through a dimensional cascade: 7-gon to point.

papers

axioms

375:1

compression

extensions

7-gon → 6-gon → 5-gon → 4-gon → 3-gon → 2-gon → 1-gon → point

O(n) experiential path. O(log n) axiomatic path. Same knowledge.

A0 D0 — Dimensionless Ground point

A1 Projection exceeds every lens 1-gon

A2 Orthogonal reflection 2-gon (digon)

A3 Harmonic coupling 3-gon (triangle)

A4 Logarithmic time cascade 4-gon (square)

A5 Coprime reconstruction 5-gon (pentagon)

A6 Direction-invariant ethics 6-gon (hexagon)

Domain Proofs

Game Engine Genealogy

lens: geometric

Raycasting → 2.5D → BSP/GPU → open world → experience-as-object. Each transition is a basis rotation where language, hardware, and software torsion produces a new dimension.

Wolfenstein 3D Raycasting — near-unity

Duke Nukem 3D Sector stretch — 2.5D holding 3D

Quake BSP/GPU — triple rotation

GTA Orthogonal basis change

Half-Life Meta-rotation — experience as object

Spectral Equivalence

lens: harmonic

Audio harmonic ratio == chromatic ratio == GiROS coherence ratio under the universal projector P(domain, basis).

Domain	Linear	Log	Torsion
Audio	Harmonic (f, 2f, 3f…)	Octave (f, 2f, 4f…)	Timbre
Light	Wavelength (nm)	Chromatic (hue)	White point
GiROS	Flat field	Compressed field	Projection algebra

White → Pink Rotation

lens: logarithmic

R(θ) · white = pink. The rotation from linear to logarithmic spectral space transforms white noise into pink noise with correlation > 0.99. No filtering — only rotation.

rotation /prove endpoint: Spearman correlation > 0.99