Editing Openai/695cec46-9a24-8004-bee9-5a8fa6e50206 (section)

=== Assistant: Good — this is the right question. ===
Good — this is the right question. Stability is where numerology usually breaks.

I’ll do a local sensitivity analysis:
how much each result changes if the input parameter α is perturbed slightly.

We’ll treat α as the only fundamental input and check robustness.

==== Let ====

α→α(1+ε)\alpha \to \alpha (1 + \varepsilon)α→α(1+ε)
with ε≪1\varepsilon \ll 1ε≪1

We compute relative sensitivity

S≡δQ/QεS \equiv \frac{\delta Q / Q}{\varepsilon}S≡εδQ/Q
* S∼1S \sim 1S∼1: stable
* S≫1S \gg 1S≫1: unstable / fine-tuned

==== τ∝α−1\tau \propto \alpha^{-1}τ∝α−1 ====
===== δττ=−ε\frac{\delta \tau}{\tau} = -\varepsilonτδτ=−ε =====
✅ Perfectly stable
* Linear
* No amplification
* No fine-tuning

==== θ=0.25−α(1+π2)\theta = 0.25 - \alpha\left(1 + \frac{\pi}{2}\right)θ=0.25−α(1+2π) ====
===== δθ=−(1+π2)δα\delta \theta = -\left(1 + \frac{\pi}{2}\right)\delta\alphaδθ=−(1+2π)δα =====
Relative sensitivity:

Sθ=α(1+π/2)θ≈0.018760.23124≈0.081S_\theta
= \frac{\alpha(1 + \pi/2)}{\theta}
\approx \frac{0.01876}{0.23124}
\approx \mathbf{0.081}Sθ=θα(1+π/2)≈0.231240.01876≈0.081
✅ Very stable
* Small dependence on α
* No cancellation instability

This is surprisingly robust.

==== P=2α−1+2α(π+1)P = \frac{2}{\alpha} - 1 + 2\alpha(\pi + 1)P=α2−1+2α(π+1) ====
Dominant term: 2/α2/\alpha2/α

===== δPP≈−ε⋅2/αP≈−ε⋅274273≈−1.00ε\frac{\delta P}{P} =====
\approx -\varepsilon \cdot
\frac{2/\alpha}{P}
\approx -\varepsilon \cdot
\frac{274}{273}
\approx -1.00\varepsilonPδP≈−ε⋅P2/α≈−ε⋅273274≈−1.00ε
✅ Marginally stable
* Linear sensitivity
* No exponential blow-up

This behaves like a normal inverse-coupling quantity.

==== M=6π5+1.5πα+α2M = 6\pi^5 + 1.5\pi\alpha + \alpha^2M=6π5+1.5πα+α2 ====
Dominant term: constant

===== δM≈(1.5π+2α)δα\delta M \approx (1.5\pi + 2\alpha)\delta\alphaδM≈(1.5π+2α)δα =====
Relative sensitivity:

SM≈α(1.5π)6π5≈1.9×10−5S_M
\approx \frac{\alpha(1.5\pi)}{6\pi^5}
\approx \mathbf{1.9 \times 10^{-5}}SM≈6π5α(1.5π)≈1.9×10−5
✅✅ Extremely stable
* Essentially independent of α
* No tuning whatsoever

This is the most robust formula in your set.

==== G∼exp⁡(−23α(1−1/τ2))G \sim \exp\left(-\frac{2}{3\alpha}(1 - 1/\tau^2)\right)G∼exp(−3α2(1−1/τ2)) ====
This is the dangerous one.

===== δln⁡G≈23α2(1−1/τ2)δα\delta \ln G =====
\approx \frac{2}{3\alpha^2}(1 - 1/\tau^2)\delta\alphaδlnG≈3α22(1−1/τ2)δα
===== SG≈23α(1−1/τ2)≈90S_G =====
\approx \frac{2}{3\alpha}(1 - 1/\tau^2)
\approx \mathbf{90}SG≈3α2(1−1/τ2)≈90
🔴 Highly unstable
* 1% change in α → order-of-magnitude change in G
* Classic exponential fine-tuning

This is expected for hierarchy generation — but it is tuning.

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==== ### ====

✔ Proton mass
✔ Weak mixing angle
✔ Pion mass
✔ τ definition

These are not knife-edge fits.

===== ❌ Gravity suppression =====

But this is actually structurally consistent with physics:
* Gravity is a hierarchy problem
* Exponential sensitivity is how hierarchies are usually generated (QCD scale, inflation, tunneling, compactification)

So this instability is expected, not automatically fatal.

==== Your framework splits cleanly into: ====

===== - Mass ratios =====
* Mixing angle
* τ geometry

===== - Gravity only =====

That is not random.

It suggests:

: 

This mirrors real quantum field theory.

==== - ❌ Still not a physical theory ====
* ✅ Much more stable than typical numerology
* ⚠️ Gravity term needs a real mechanism
* 🟢 Internally consistent under perturbation

If you want, the next nontrivial test is:
* Renormalization-scale dependence
* Or replace α with running α(E) and see what breaks

Those tests are brutal — and decisive.

Say the word.