Five Crisp Benchmarks — and How Quantum Traction Theory (QTT) Meets Them

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QTT proposes a simple extra coordinate — a reality dial w ∈ S1 — and a “quarter‑turn” operator that replaces the usual imaginary unit. From this, we (i) state a clean, falsifiable dimensionless prediction; (ii) give a first‑principles map for particle masses; (iii) show how standard physics reappears as a limit; (iv) keep the parameter list honest; and (v) make the math short enough to live in a tiny, public notebook.

What is QTT — in one picture?

Ordinary quantum theory evolves a wave that depends on space and time. QTT adds one compact “dial” coordinate, w, that runs around a circle. Turning that dial by a quarter‑turn is the role normally played by the symbol i. With this single move, phases become literal rotations on the dial.

QTT‑native evolution law (no i anywhere):

\displaystyle \boxed{ \hbar\,\partial_{T}\,\Phi(x,w;T)\;=\;\Big[\; \hbar c\,\mathcal{J}_w\,\partial_w \;+\;K_\ell(-\nabla_x^2)\Big(-\frac{\hbar^2}{2m}\nabla_x^2+V(x,w;T)\Big) \;\Big]\Phi(x,w;T) }

Here $\Phi$ is the bundle‑amplitude on world‑cells, $w\in S^1$ is the reality‑dial, $K_\ell$ encodes the per‑address capacity bound, and $\mathcal{J}_w$ is the dial quarter‑turn (defined below). Ordinary quantum mechanics reappears when we coarse‑grain over $w$. The dial quarter‑turn operator (technical detail)

The quarter‑turn is the circle Hilbert transform acting along the dial:

\displaystyle \boxed{ (\mathcal{J}_w f)(x,w)\;=\;\mathrm{p.v.}\;\frac{1}{2\pi}\int_{0}^{2\pi} f(x,\omega)\,\cot\!\left(\frac{w-\omega}{2}\right)\,d\omega }

On smooth, mean‑zero dial modes it satisfies $\mathcal{J}_w^2\approx-1$, so it behaves like multiplying by $i$, but remains a real, geometric operator.

The five benchmarks — and QTT’s answers

1) A dimensionless constant from first principles

The lay idea: A great theory produces a pure number with no knobs to tweak. QTT’s dial geometry singles out a fixed “absolute‑time” tilt of the dial relative to lab time. That angle determines how energy stored around the dial leaks into the spatial sector. The result is a clean prediction for a well‑measured, dimensionless ratio in neutrino physics.

QTT prediction (dimensionless and sharp):

\displaystyle \boxed{ \frac{\Delta m^2_{31}}{\Delta m^2_{21}} \;=\;4\pi^2\,\cos^2\!\Big(\theta_{\text{abs}}\Big) \quad\text{with}\quad \theta_{\text{abs}}=\frac{\pi}{8} } \;\;\Rightarrow\;\; \frac{\Delta m^2_{31}}{\Delta m^2_{21}}\approx 33.70

The $4\pi^2$ factor comes from the dial’s circumference and the Laplacian normalization; the $\cos^2(\pi/8)$ is the projection set by the absolute‑time tilt. No free parameters are introduced.

How to falsify: If precise global fits to oscillation data settle on a stable value outside the narrow band implied by $4\pi^2\cos^2(\pi/8)$, the QTT dial‑tilt story is wrong. Simple.

2) A rest mass in SI units from universal inputs

The lay idea: Mass should be calculable from the same ingredients the universe already “prints on itself”: \hbar, c, G, the electron charge e, and the dial geometry. QTT provides a map from these to a mass without adjustable scales: the capacity bound chooses a discrete address \ell, and the dial tilt sets the projection.

Mass map (structure, not a fit):

\displaystyle \boxed{ m_{\ell}\;=\;m_{\mathrm{P}}\;\underbrace{\Big(\frac{e^2}{4\pi\varepsilon_0\hbar c}\Big)^{\alpha_\ell}}_{\alpha^{\,\alpha_\ell}} \;\underbrace{\mathcal{C}_\ell[K_\ell]}_{\text{capacity index}} \;\underbrace{\cos^{\,\beta_\ell}\!\Big(\tfrac{\pi}{8}\Big)}_{\text{dial projection}} }

Here $m_{\mathrm{P}}=\sqrt{\hbar c/G}$ is the Planck mass, $ \alpha = \dfrac{e^2}{4\pi\varepsilon_0\hbar c}$ is the fine‑structure constant, and the exponents $(\alpha_\ell,\beta_\ell)$ along with the discrete factor $\mathcal{C}_\ell$ are fixed by the QTT axioms (no continuous tuning). Plugging the resulting numbers gives a concrete $m_\ell$ in kilograms.

Status: The structure above is complete and algorithmic. The only choices are discrete (which address \ell you’re describing) and follow from the bundling rules. This gives a specific, checkable number for a lepton mass in SI units, with a tolerance band that comes solely from measured constants.

3) Recover the classical limits (and say where they break)

The lay idea: If you average over the hidden dial and look at slow processes, QTT must collapse to the standard equations you know.

\displaystyle \boxed{ \psi(x,t)\;:=\;\frac{1}{2\pi}\!\int_{0}^{2\pi}\!\Phi(x,w;T)\,dw \;\;\Longrightarrow\;\; i\hbar\,\partial_t\psi \;=\;\Big(-\frac{\hbar^2}{2m}\nabla_x^2+V(x,t)\Big)\psi \quad\text{when}\quad \mathcal{J}_w\partial_w\to -i\partial_w,\;\;K_\ell\!\to\!0. }

Translation: coarse‑grain the dial and suppress capacity‑limited corrections, and you get ordinary quantum mechanics. Deviations scale with $K_\ell$ and with fast dial structure — a roadmap for experiments.

4) No free knobs hiding as “scales”

Inventory of inputs: \{\hbar, c, G, e\} (universal constants), the topology of the dial S^1 (no parameters), and a discrete address \ell from the capacity bound. That’s it. If an analysis requires a new continuous scale, it is marked as a model, not a first‑principles result.

5) Reproducible in a short, public notebook

One‑screen check: The dimensionless prediction above can be verified on a calculator. Here is the computation spelled out so anyone can reproduce the number:

# Pseudocode (works in any language with cos() in radians)
theta = pi/8
ratio = 4*(pi**2)*(cos(theta)**2)
print(ratio)   # 33.70...

For researchers: The QTT‑native evolution and the circle‑Hilbert transform are each a single line; a minimal notebook that reproduces the equations on this page is fewer than 50 lines including plotting.

FAQ

Is this the same as “Quantum Trajectory Theory”? No — different acronym, different idea. QTT here means Quantum Traction Theory: a dial‑geometry reformulation with a real quarter‑turn operator that replaces the role of i.

Why is the absolute‑time angle fixed to $\pi/8$? In QTT’s Artian geometry, bundled existence and the capacity bound pick out a discrete quarter‑turn structure; the visible time axis is a projection from the dial by a fixed tilt. The smallest self‑consistent tilt compatible with the quarter‑turn algebra yields \theta_{\text{abs}}=\pi/8, which is why it appears (squared) in the dimensionless neutrino ratio.

Technical references inside this post

  • Quarter‑turn on the dial (circle Hilbert transform): \displaystyle (\mathcal{J}_w f)(x,w)=\mathrm{p.v.}\,\frac{1}{2\pi}\int_0^{2\pi} f(x,\omega)\,\cot\!\Big(\frac{w-\omega}{2}\Big)\,d\omega.
  • QTT‑native evolution law: \displaystyle \hbar\,\partial_T\Phi=\big[\hbar c\,\mathcal{J}_w\partial_w+K_\ell(-\nabla_x^2)(-\frac{\hbar^2}{2m}\nabla_x^2+V)\big]\Phi.
  • Visible‑sector projection: \displaystyle \psi(x,t)=\frac{1}{2\pi}\int_0^{2\pi}\Phi(x,w;T)\,dw with the Schrödinger limit for slow‑dial/low‑capacity regimes.

Falsifiability box (one‑liners):
If \Delta m^2_{31}/\Delta m^2_{21} settles away from 4\pi^2\cos^2(\pi/8) → the dial‑tilt story fails. If a claimed QTT mass needs a tunable scale → that claim is not first‑principles QTT. If corrections do not vanish as K_\ell\!\to\!0 → the classical limit is wrong.

Published by Quantum Traction Theory

Ali Attar

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