A conceptual preface to the demonstrations. Plain-language, but precise. The framework is in progress; this describes what it is, not a claim that it is finished or unique.
Mathematically, TFT is the theory of one circle-valued field:
\[\theta:\ \mathbb{R}^3\times S^1 \longrightarrow S^1, \qquad \text{equivalently a complex field } \psi = \rho\, e^{i\theta}.\]A field whose value is a phase. That is the whole ontology. Almost everything demonstrated in this repository is not extra machinery bolted on — it is the geometry and topology that any such field carries automatically. Four pieces of standard mathematics come for free the moment you commit to that one object.
The circle has fundamental group $\pi_1(S^1)=\mathbb{Z}$: a phase field can wind an integer number of times, and that integer cannot change continuously. That single topological fact is charge quantization, is baryon number — the “why is charge an integer?” that the Standard Model inserts by hand. It is topology, not dynamics. (The Skyrme / topological-soliton lineage.)
→ verify_charge_quantization.py
The gradient $\nabla\theta$ behaves as a gauge potential $A$; its curl is a magnetic flux; the flux is quantized in units of $2\pi\times(\text{winding})$. This is the mathematics of a $U(1)$ fiber bundle and its curvature — the same math as flux quantization in a superconductor. Electromagnetism is not added; it is the geometry of a phase.
→ verify_goldstone_1r2.py, verify_force_sign.py
A field with these symmetries yields two conserved quantities: the internal-symmetry current $j^\mu=\rho^2\partial^\mu\theta$ (from $\theta\to\theta+\text{const}$), and the stress–energy $T^{\mu\nu}$ (from spacetime translations).
So the two long-range forces having opposite personalities is not a choice — Noether’s theorem forces it. This is perhaps the cleanest single “backdoor” in the framework.
→ verify_poisson_metric.py, verify_gravity_coupling.py
Reading $mc^2 = hf$ literally — the clock is the turning of the phase, $d\tau = \hbar\,d\theta/E$ — takes the phase of a quantum state seriously as time (the de Broglie reading). And letting energy density source a metric through a Poisson equation, $\nabla^2\Phi = (\text{positive source}) \to 1/r$, is the Sakharov–Jacobson “gravity as emergent elasticity of a substrate” mathematics: you obtain gravity’s shape and sign without choosing a force law.
→ verify_G_as_rate.py, tft_solar_system.py
The step is a change of variables plus a single postulate: assume the fundamental object is one $U(1)$ phase, and refuse to insert anything by hand. When you do that, the parts of physics that are secretly “phase geometry” — integer quantization, gauge structure, the two Noether currents, emergent-metric gravity, phase-as-time — reassemble themselves without being separately assumed.
Even the galaxy result is “the same equation, one length moved”: the phase’s mass gap $m=\sqrt{\Lambda}/\ell_0$ sits either at the microscopic scale (ordinary screening) or at the Hubble scale (→ a MOND-like law with $a_0 \propto cH_0$). One knob, two regimes. (The coefficient in $a_0 = cH_0/2\pi$ is selected by the data, not derived — see A0_STATUS.md.)
This is why it can feel like a backdoor into what was already there — because it largely is what was already there: Noether’s theorem, the homotopy of $S^1$, $U(1)$ bundles, Sakharov’s emergent gravity — re-derived from a single doorway instead of assembled from separate postulates. Finding that many things are one thing in disguise is what unification means.
Because it is a re-encoding, it inherits both sides — stated plainly so the claim is defensible:
| Limit 1 — a re-description does not compute what the original could not. The numbers that remain open (Newton’s $G$, $ | \Lambda | $, the lepton masses, the baryon asymmetry $\eta$) are exactly the ones this doorway cannot fix — they were never encoded in the topology of a single circle to begin with. Each is a problem open in every framework, not a gap unique to this one. |
The real test is not the re-organization, however economical — it is whether the doorway reveals a room you could not see from the front. A re-encoding that only re-labels known physics is elegant but not yet new physics; one that forces a connection nobody built into the originals is.
The candidate here is the $a_0$–dark-energy link: because the same ultralight phase sets both the galactic acceleration scale and the dark-energy density, galaxy rotation curves and the dark-energy equation of state are forced to be the same field — a cross-observable prediction (matching DESI’s $w_0$, falsifiable in $w_a$) that neither MOND nor $\Lambda$CDM makes. See PREDICTIONS.md.
In one line: A remarkable amount of physics is the differential geometry and topology of one circle-valued field — and TFT is the single door you walk through to see that.