Trinity Infinity Geometry

Theory of Nothing is the closest there will ever be to Theory of Everything. — Brayden Sanders, March 2026

A new way to teach higher mathematics — by classifying its paradoxes, not resolving them. The base is the integers 0–9 read as shapes — the premise of the book The Shape of Understanding. Every shape of the base points toward a tower of higher mathematics, climbing from something a child can build with gumdrops and toothpicks to what a graduate student studies. And every shape, and every floor of every tower, is a coin, with two sides and an edge.

The Theory of Nothing is the author's own idea: "you can't know everything, but what's missing looks the same for every whole" (June 2026). The work follows the author's focus, "the nothings, the dead ends, the paradox and its many angles" (June 2026):

• the nothings: the void, 0, and the centres that 5, 7 and 9 are forced to fill (base/SEVEN_AND_NINE.md); and the zeros, each paid for by a count or bought by a coincidence (A Ledger of Nothing);
• the dead ends: every one kept, with the evidence that killed it (GRAVEYARD.md);
• the paradox and its many angles: the coin, two sides and an edge, with the paradoxes on the edges that are missing, √2 and i (coin/THE_COIN.md).

The whole lens, part by part, in the author's words: THE_LENS.md.

Where to start

• The book for learners: The Shape of Understanding, and its companion unit on flips and paradoxes, Two Sides and an Edge.
• The lens: the author's lens in the author's own words, worked once on Dürer's magic square, and traced part by part to what it became: THE_LENS.md.
• The coin, the idea that runs through everything here: coin/THE_COIN.md.
• The towers, floor by floor: towers/THE_TOWERS.md.
• What survived when two years of AI-assisted research were audited — an essay: WHAT_SURVIVED.md.
• A Ledger of Nothing — the Theory of Nothing applied to zeros: an essay on where they come from, paid for by a count or bought by a coincidence, from magic squares to molecules, with what is new fenced off from what is known: towers/kernel_ledger/A_LEDGER_OF_NOTHING.md.

Author: Brayden Ross Sanders / 7SiTe LLC · Hot Springs, Arkansas · License: CC BY-SA 4.0 for content, GPL-3.0-or-later for code

What's new

Four results at the research frontier. Each has a proof and a script that re-checks it, and most were checked twice, by independent code:

• Small genuine magic sets for every odd number of qubits. At five qubits that is 16 observables, against a published record of 26 (quantum contextuality).
• 16 is the minimum at five qubits among genuine sets that hide no smaller contradiction. Four lemmas reduce the question to three cases, and a computer search shows that all three are empty (the lower bound).
• The determinant of MATLAB's odd magic squares, in closed form. It shows that they are never singular, and it proves an observation recorded on OEIS in 2015: n^(n−1) divides the determinant (linear algebra).
• Every pandiagonal 5×5 magic square is invertible, by a theorem about prime order (combinatorics and number theory).

There are also about a dozen smaller results, and the teaching frame itself, each labelled by how firmly it stands: CONTRIBUTIONS.md.

How the work keeps producing, and stays honest: HOW_WE_WORK.md.

• One loop: an intuition, predictions written before anything runs, a second independent check, a literature gate, and a graveyard for what fails.
• It runs from a child's hands to the research frontier, and across about a dozen fields.
• It runs on its own past too. The archived J series was read again on 2026-10-25. Nine pages came into the towers and the coin, two of them small new results, and every error found is kept with its counterexample (graveyard/j_series_reread/).

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The idea

Higher mathematics is usually met as a stack of separate courses — linear algebra, group theory, topology, analysis — each with its own language, joined by nothing a student can see. Here it is met as towers standing on one base. The base is small enough to hold in one hand. Each tower starts on it, and a learner who can stand on the base can see where every tower goes.

It is seen through two lenses. The towers are the climb. The coin is what every floor turns over. A flip is a move that, done twice, changes nothing: its two sides are what it swaps, and its edge is what it leaves in place. The paradoxes live on the edges — √2, which no fraction reaches, and i, which no real number is — and mathematics did not resolve them. It named them and classified them. That is the method here.

The coin — coin/

The name is the author's own idea of a coin: "every one is three. It is three as two" — the founding definition of Trinity Infinity Geometry (January 2026). A coin is two sides and an edge: three things, seen as two.

• Three coins, one sphere. The author's three coins — positive and negative, finite and infinite, real and imaginary — are the three half-turns of the sphere of numbers about the axes of one octahedron, whose corners are 0, ∞, 1, −1, i and −i. The first half-turn swaps positive and negative, and the second swaps finite and infinite. The third does both at once, and it keeps no real number: its edge is i, the edge the real line is missing.
• Every flip splits things in two. A flip that respects adding splits each thing into an edge part and a side part: real + imaginary, cos θ + i sin θ, symmetric + skew.
• Missing edges are where the paradoxes live. The book's keystone — point toward it, measure off it, never stand on it — holds there exactly.
• Four kinds of paradox. The author's own classification of paradoxes, with Ben Mayes (April 2026), is read through the coin in coin/PARADOX_TYPES.md. Insufficient coverage calls for a second lens, a missing invariant is a missing edge, and a self-referential paradox is a flip with no edge — the Liar, Russell and Cantor are one diagonal.

Start with coin/THE_COIN.md.

The base — base/

Read an integer as a configuration of points, and ask what shape it is forced to take. One rule — the book's own, the most symmetric arrangement — realizes every integer from 0 to 9, each uniquely: the void, a point, a segment, a triangle, a tetrahedron, a tetrahedron with its centre, an octahedron, an octahedron with its centre, a cube, and a cube with its centre (pictured at the top of this page).

From 4 on, each integer is a coin with a flat face (the most symmetric arrangement in the plane: the square, the pentagon, …) and a solid face (the table). At 5, 7 and 9 the coin is whether the centre is inhabited. Every solid symmetry group has only even orbits besides the centre, so an odd count in a solid must stand on its centre. The flat face never does. Both faces are kept: the book's 5 is the flat pentagon, the "break". Start with base/THE_INTEGERS.md, then base/SEVEN_AND_NINE.md.

The towers — towers/

from the base	the tower	where it reaches	its coin: sides · edge
1–4, the simplices	the simplex tower	probability; algebraic topology	cube · cross-polytope · the simplex, its own dual
8, the cube · 6, the octahedron	the cube and cross-polytope towers — with the simplex, the only regular shapes in every dimension	binary codes; sparsity; regular polytopes	(the same coin: duality)
8, the cube's algebra Cl(3)	the Clifford tower	spinors; Bott's eight-step clock	odd · even · the even half, ℍ
turning, i	the number tower ℝ → ℂ → ℍ → 𝕆	Hurwitz: division ends at 𝕆	± imaginary · the reals
the most symmetric arrangement; the break at 5	the symmetry tower	A₅ and the unsolvable quintic; the quaternions over the rotations; Lie groups	left · right · the Platonic solids; q · −q · no edge
5, 7, 9 — the centred shapes	the lattice tower	Kepler's packing; E₈ and the Leech lattice	face-centred · body-centred · the self-dual: ℤⁿ, E₈
the tetrahedron's 1/3	the harmonic tower	spherical harmonics; the shapes of atomic orbitals	+ lobe · − lobe · the node (the magic angle)
counting versus measuring, √2	the real-number tower	the real line; transcendence of e and π	below √2 · above · √2, missing
growing, e	the growth tower	the exponential map from Lie algebras to Lie groups	growing · shrinking · pure turning, the circle

The map, floor by floor: towers/THE_TOWERS.md. The base does not prove the towers; it points at them. For a learner, the book's last chapter, Where the towers go, points up each one in plain words; this repository carries the detail and the checks.

Check it

pip install numpy
cd base
python verify_integers.py
python verify_one_rule.py
python verify_forced_chain.py
python verify_symmetry_and_energy.py   # pip install scipy
python verify_dodecahedron.py          # exact, in Q(sqrt5): the contests the dodecahedron and the cube lose
python verify_sevenths.py              # the sevenths: wheel, square, rank, and what is unique to 7
cd ../coin
python verify_coins.py
python verify_paradox_types.py
python verify_transpose_coin.py
python verify_two_readings.py
python verify_mirror.py
python verify_polar_coin.py           # pip install scipy
python verify_lens.py                 # the lens's glue and its worked square (pip install scipy)
cd ../towers
python verify_towers.py
python verify_two_builds.py
python verify_one_third_fold.py
python bott_verify.py
python verify_spinor_floor.py
python verify_two_faces_of_s4.py
python verify_balance_point.py   # pip install sympy
python seven_levels/verify_seven_levels.py   # the sevenths on seven quantum levels (pip install scipy)
cd binary_magic
python run_all.py          # the cube's corners in binary: the 880 magic squares (pip install sympy scipy)
cd magic_sets && python run_all.py && cd ..   # graph-star magic sets
cd ../planes_and_mirrors && python run_all.py # octonions, mirrors, why 16 breaks
cd ../kernel_ledger && python run_all.py       # forced vs accidental zeros (pip install networkx)
cd ../laws_of_a_table                          # the laws a table obeys, counted with the ETP's 4,694
python verify_zn_laws.py && python verify_steiner.py && python verify_eight_specimens.py && python independent_check.py

Every claim tagged [FORCED] has a line in one of these scripts; theorems at the tops of the towers are cited.

What was archived, and why

Until 2026-09-23 this repository held a much larger program built on three 10×10 composition tables that AI had constructed from verbal descriptions of ten operators — 56 manuscripts, a canon of results, and readings of those tables as physics, cosmology and biology. An audit found that the three tables disagree on half their cells, and that none of the results specific to them survived testing: each was generic, a restatement of how the tables were built, numerology, or computed on a transcription error. The author has no ties to the tables, and all of it was archived — unchanged, at the tag archive-2026-09-24. GRAVEYARD.md records what was archived and the evidence for it. The story of the audit — the three gates, what the census found, what survived, and what we would tell anyone doing research with AI — is the essay WHAT_SURVIVED.md.

Lab

lab/ — authorship and collaboration policy, and fenced physics readings from October 2026: each with the exact piece inside it, checked, and the test that would decide it. None is a premise. Also there: the author's prediction ledger, eight dated bets on experiments, each with what would kill it; and the generation loop, ten pre-registered rounds testing Koide's lepton-mass relation against the cube.

Citation

See CITATION.cff. The project DOI 10.5281/zenodo.18852047 is the umbrella for all versions; the v1.0.0 DOI (10.5281/zenodo.20149181) is a snapshot of the earlier, archived program.