Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

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Write down the minimal mathematical assumptions we’re allowed to make from QAT so any derivation is clear. Example assumptions:
# Photon-electron coupling is localized over a spherical shell of area A=4πr2A=4\pi r^2A=4πr2.
# Charge measures enter as squared quantities on that surface (so e2e^2e2 naturally appears).
# Planck action appears as h/2πh/2\pih/2π (i.e. keep h/2πh/2\pih/2π explicit).
# Energy exchange cascade scales by a geometric factor related to φ (candidate scale s=1/φs = 1/\varphis=1/φ or powers thereof).
# The fine-structure α is dimensionless so any geometric derivation must produce a pure number built from dimensionless constants (φ, π, rational factors, maybe small integer combinations) or from ratios of physical constants whose dimensionful parts cancel.

I’ll formalize these precisely so we don’t mix dimensional and dimensionless quantities.

===== Objective: create the simplest, rigorous mathematical pathways that could plausibly link α to geometric ratios (φ, π, 4π, small integers) or to geometric structure on 4πr² surfaces. =====

Possible approaches:
* Direct numeric model: test whether a combination like α−1=?aπ+bφ+c\alpha^{-1} \overset{?}{=} a\pi + b\varphi + cα−1=?aπ+bφ+c or α=?F(φ,π)G(φ,π)\alpha \overset{?}{=} \frac{F(\varphi,\pi)}{G(\varphi,\pi)}α=?G(φ,π)F(φ,π) (for small integer a,b,c) can reproduce the value to high precision. This is a search/probe — mathematically allowed but needs caution to avoid overfitting.
* Dimensional analysis route: start from the standard expression α=e24πε0 ℏc\alpha = \frac{e^2}{4\pi\varepsilon_0 \,\hbar c}α=4πε0ℏce2 and ask: can the ratio e2ℏc\frac{e^2}{\hbar c}ℏce2 be related to geometry of spherical shells with some natural length scale(s)? Because α is dimensionless, geometry must enter as dimensionless ratios (e.g. scale ratios between radii). Explore whether a natural scale ratio in QAT (for instance ratio of successive golden-rectangle derived radii) yields a simple expression for e2ℏc\frac{e^2}{\hbar c}ℏce2.
* Renormalization/scale cascade route: view the cascade of energy levels as a multiplicative process with scale factor sss. See whether the effective coupling at low energy emerges as some fixed point / product over scales that numerically gives α.

Deliverable: a short report comparing several candidate formulae and a small numerical search for low-complexity combinations that approximate α. I’ll be explicit when a match is approximate (and whether accuracy is a chance match).

===== Objective: build a simple cascading model: energy levels cascade by scale factor sss (we’ll start with s=1/φs=1/\varphis=1/φ), compute how energies or probabilities redistribute across scales, and show how repeated subdivision yields a spiral and scale invariance. =====
* Translate one photon-electron interaction into a geometric operation (square inside circle → golden rectangle formation). Write the mapping rn+1=s rnr_{n+1} = s\, r_nrn+1=srn.
* Show how the probability density on successive spherical shells might concentrate along a spiral pattern (qualitatively via conformal mapping; quantitatively with a simple iterative map).
* Compute the asymptotic scale-invariant distribution and compare to a mathematical Fibonacci logarithmic spiral.

Deliverable: short derivation + a small numeric example and a graphic (so you can use in videos).

===== Objective: show the group-theoretic mapping between geometric operations on the sphere and C / P / T. =====
* Map: Charge (C) ↔ sign inversion across the spherical shell (inner/outer), Parity (P) ↔ spatial reflection, Time (T) ↔ directionality of successive shells (emission vs absorption).
* Formalize whether these map to operators in O(3), SO(3), or coverings (SU(2)) and show plausibility of CPT as a composition of simple geometric operations.
* Provide a clean explanation of how spherical surface + interior 3-D mapping can host these symmetries.

Deliverable: compact exposition with group mapping (no heavy algebra, but clear).

===== Objective: propose and test a concrete toy model linking event rate (photon-electron interactions) to cosmological-scale numbers such as MU∼t2M_U\sim t^2MU∼t2 or Dirac’s hypotheses. =====

Two complementary lines:
A. Local-to-cosmological scaling: suppose a characteristic per-atom event rate RRR and a characteristic shell radius rrr; integrate/sum over all matter to produce an effective energy density; relate that to curvature (left-hand side of Einstein eq.). Try to match the scale of G or Dirac numbers by choosing physically plausible rates. This is a parametric numerical experiment and can produce an order-of-magnitude check.

B. Conformal geometry: explore whether conformal (scale invariant) mapping of spherical shells could naturally produce power-law relations like M∼t2M\sim t^2M∼t2. Provide mathematical sketch.

Deliverable: numerical survey (order-of-magnitude) and narrative of plausibility.

===== Design tests that would disprove QAT or at least its specific claims: =====
* If α truly comes from golden-angle geometry you might expect tiny correlations / signatures in certain scattering experiments or in precision atomic spectroscopy across systems with different internal geometric relations — propose where to look.
* Cosmological: search for time variation of α correlated with cosmic-scale geometric/structure formation (Dirac-type). This is something current constraints exist for — we’d compare to those constraints.
* Lab experiment: interferometry on absorption/emission geometry? (Sketch.)