The thesis
The Circular Mathematics thesis in six short parts. The full, citable text is the Codex; every line below cites it.
The idea on one page
Established mathematics takes the line as its reference: it measures lengths, counts steps and approximates curves by segments. Circular Mathematics takes the turn as its reference: it measures in turns, counts rotations and treats the line as the limit case of the circle, with infinite radius.
The change proposed is one of center. Established mathematics already contains the circle: complex numbers, Fourier series, topology, rotation groups. What changes is putting the turn at the center of notation, calculation and teaching, where the line and the radian stand today.
What it is
A mathematical language in which the turn is the primary object: notation, units, types and ways of seeing. Everything it states as mathematics must be true in established mathematics. It rewrites, organizes and reveals, without contradicting.
What it is not
Not a refutation of established mathematics. It does not change the value of any constant. It is not, by itself, a new physics.
§1
Four guiding theses
T1
The hidden integers
Measured in turns, the angular invariants of geometry become integers or simple fractions: the tangent of a simple closed curve, traversed counterclockwise, turns exactly 1 turn, a triangle's angles sum to half a turn, a polyhedron's angular defects sum to 2 turns.
C-6.1C-6.5C-6.6See area by the curve →T2
π lives on the border
In the domain of the turn, the half turn is exact: ½. The number π only appears when a turn is translated into length or area.
C-2.2C-7.1See π as half a turn →T3
Rationality is closure
Turning repeatedly by p/q of a turn, in lowest terms, closes the cycle in exactly q steps. Turning by an irrational fraction never closes.
C-5.1C-5.2See closing curves →T4
Every number is a turn with a radius
The spin number r⟳t makes multiplication a scaling followed by a turn, and reveals −1, i, roots and logarithms as turns.
C-4.1C-4.4See multiplication as turning →
Epistemic seals
Every entry in the Codex declares its state. Nothing changes seal without a proof, a test or a cited piece of evidence.
- intuition
- A generative perception of the author, not yet formalized. It is recorded with a date on the Frontier until it becomes a definition, conjecture or hypothesis.
- definition
- An adopted convention. Neither true nor false: it is chosen, and must be coherent.
- classical theorem
- A result proved in current mathematics or physics, rewritten in turns. It requires a reference.
- internal theorem
- A consequence proved from the definitions of the Codex.
- computed result
- The reproducible output of an identified algorithm or test.
- experimental evidence
- A documented observation or measurement, with a source.
- conjecture
- A mathematical statement not yet proved.
- hypothesis
- An empirical proposition (physical, biological, educational or computational) that needs testing.
- symbolic interpretation
- A philosophical, spiritual, artistic or historical reading. It is not a mathematical claim.
- revoked
- An old formulation that failed or was replaced. It stays on record so it does not return by mistake.
§0.2
The frontier
The Codex does not only translate what is known; it explores. The frontier is where the author's intuitions, open questions and hypotheses are kept, each with a path to test it. New axioms are allowed if they are consistent and fruitful, and a physical hypothesis only enters as science with dimensional analysis, a quantitative prediction, a refutation criterion and a test.
The author's intuitions (seal: intuition)
I-1Circular optics changes how speed, time, space and gravity are interpreted.
Known bridges: C-9.13C-9.10C-9.15C-9.16
I-2Rotating spirals generate forces tied to gravity.
Known bridges: C-9.17C-9.18
I-3In quantum gravity, space may be made of turns (spin networks, Penrose 1971; loop quantum gravity, a research program without experimental confirmation).
Known bridges: no established bridge recorded yet
I-4The brain is a "spin core": neurons compute with phases, geometry and spin.
Known bridges: C-9.21
I-5An AI built on circular mathematics would reason differently, perhaps closer to the human way.
Known bridges: C-9.22
I-6Phase hardware could surpass binary electronics.
Known bridges: C-9.7C-9.23
I-7Circular mathematics could feed other AIs, through an MCP server or a skill.
Known bridges: no established bridge recorded yet
I-8The circular view may reveal new axioms, constants and formulas.
Known bridges: no established bridge recorded yet
I-9The work of Walter Russell holds keys to Circular Mathematics.
Known bridges: no established bridge recorded yet
Hypotheses: revoked and open
Revoked H-1, H-2, H-3
"c = 2π", "c is the maximum angular frequency" and "E = m⟳ × (2⟳)²" failed: they equate a speed with an angle or are dimensionally invalid. The valid forms are v = λ·f, no known maximum frequency for light, and E = mc² with E = hf.
Open: physical hypotheses H-4, H-5, H-7
The smallest stable "Planck orbit", gravity as circular resonance, and mass as a phase delay have no operational definition or model yet, and must not be presented as technology.
Symbolic interpretation H-6, H-8
"Causality is phase resonance" is true only in part (coupled oscillators synchronize, but two can also synchronize through a common source). Consciousness, intention and healing by resonance stay in the poetic layer, never as science.
Open: empirical hypotheses H-9, H-10
"The brain computes with spin" has no evidence for electronic spin, and the nuclear-spin idea is speculative. "Phase hardware beats binary electronics" holds in niches (communications, optical linear algebra), not in general.
§10
Lineage
Mathematics and science
Results that were proved or measured, and that the Codex uses.
Pythagoras
Harmony as ratios of string lengths; the frequency ratios came in the 17th century.
C-5.5
Archimedes
The spiral and area by sectors.
C-7.2C-7.3
Euler
The complex exponential: a half turn takes 1 to −1.
C-2.4
Gauss, Bonnet
Total curvature as a whole number of turns.
C-6.6C-9.16
Ferraris, Tesla, Dolivo-Dobrovolsky
The rotating magnetic field and the three-phase system.
C-9.25
Planck
The constant per cycle.
C-9.2C-9.10
Minkowski, Varićak, Robb
Spacetime; speed as a hyperbolic angle.
C-9.13
Einstein
Gravity as the rate of clocks (gravitational redshift, 1907–1911).
C-9.15
Roger Penrose
Spin networks as candidates for the structure of space (a research program).
The author's inspirations
Sources of language, images and questions (symbolic interpretation). The Codex uses no results from them without verification.
Walter Russell
The author's main source of inspiration: waves, octaves, spirals, cube and sphere (dossier in preparation).
Buckminster Fuller
The name and reading of "Vector Equilibrium", and synergetics (the cuboctahedron's geometry is a computed result).
Srinivasa Ramanujan
The precedent of intuition and proof working together.
§11.3