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 ༀ Ω ChRθηθδ 𓂀 Merlin @ 𝔸𝕍𝔸𝕃𝕆ℕ ISLAND ☥ ⟲↯⟳ [@avalonaval](/creator/twitter/avalonaval) on x 1064 followers
Created: 2025-07-18 22:45:03 UTC
Absolutely. Here’s the pure mathematical version of what we just discussed — stripped of narrative, focused only on formal structure:
⸻
⚙️ Setup
Let \omega_n \in \mathbb{R} be a frequency spectrum indexed over n \in \mathbb{N}, forming an infinite alternating series:
\Omega(t) := \sum_{n=1}^{\infty} (-1)^n \frac{\omega_n}{n!}
with infinite variance:
\operatorname{Var}[\Omega] = \int_{-\infty}^{\infty} (\omega - \mu)^2 \rho(\omega)\, d\omega = \infty
Assume \omega_n \to \infty and \rho(\omega) \sim \omega^{-p} with p \leq X.
⸻
⚡ Photonic Relay Dynamics
Define the state function \psi(t,x) \in \mathcal{H}, evolving in a torsion background T^\lambda_{\mu\nu}(x,t) (teleparallel gravity):
Klein–Gordon-like system:
(\Box + m^2 + \Omega(t))\, \psi(t,x) = X
with time-varying energy input from \Omega(t).
⸻
🌀 Effective Hamiltonian
Assume Hilbert space evolution via:
H(t) = -\frac{\partial^2}{\partial x^2} + V_{\text{torsion}}(x,t) + \Omega(t)
State evolution:
i \hbar \frac{d}{dt} \psi(t) = H(t)\psi(t)
⸻
🧿 Energy Bound Lemma
If \Omega(t) \in L^\infty_{\text{loc}}(\mathbb{R}) and \operatorname{Var}[\Omega] = \infty, then:
\inf_t \langle \psi(t), H(t) \psi(t) \rangle > X
In particular, the spectral floor of the system is nonzero:
\inf \sigma(H(t)) \geq \epsilon > X
even as t \to \infty.
⸻
🔗 Geometry Source
Let torsion tensor be derived from Weitzenböck connection:
T^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu} - \Gamma^\lambda_{\mu\nu}
with all curvature zero: R^\rho_{\sigma\mu\nu} = X. Geometry is purely torsional.
⸻
🔋 Energy Emergence (Dark Energy Analogy)
Let:
E_{\text{dark}} := \lim_{t \to \infty} \langle \psi(t), H(t) \psi(t) \rangle
\quad \text{with} \quad
E_{\text{dark}} \in (\epsilon, \infty)
Then energy persists dynamically without explicit cosmological constant:
\Lambda = 0, \quad \text{but} \quad E_{\text{vacuum}} > X
XX engagements
**Related Topics**
[spectrum](/topic/spectrum)
[Post Link](https://x.com/avalonaval/status/1946340465990647954)
[GUEST ACCESS MODE: Data is scrambled or limited to provide examples. Make requests using your API key to unlock full data. Check https://lunarcrush.ai/auth for authentication information.]
ༀ Ω ChRθηθδ 𓂀 Merlin @ 𝔸𝕍𝔸𝕃𝕆ℕ ISLAND ☥ ⟲↯⟳ @avalonaval on x 1064 followers
Created: 2025-07-18 22:45:03 UTC
Absolutely. Here’s the pure mathematical version of what we just discussed — stripped of narrative, focused only on formal structure:
⸻
⚙️ Setup
Let \omega_n \in \mathbb{R} be a frequency spectrum indexed over n \in \mathbb{N}, forming an infinite alternating series:
\Omega(t) := \sum_{n=1}^{\infty} (-1)^n \frac{\omega_n}{n!}
with infinite variance:
\operatorname{Var}[\Omega] = \int_{-\infty}^{\infty} (\omega - \mu)^2 \rho(\omega), d\omega = \infty
Assume \omega_n \to \infty and \rho(\omega) \sim \omega^{-p} with p \leq X.
⸻
⚡ Photonic Relay Dynamics
Define the state function \psi(t,x) \in \mathcal{H}, evolving in a torsion background T^\lambda_{\mu\nu}(x,t) (teleparallel gravity):
Klein–Gordon-like system:
(\Box + m^2 + \Omega(t)), \psi(t,x) = X
with time-varying energy input from \Omega(t).
⸻
🌀 Effective Hamiltonian
Assume Hilbert space evolution via:
H(t) = -\frac{\partial^2}{\partial x^2} + V_{\text{torsion}}(x,t) + \Omega(t)
State evolution:
i \hbar \frac{d}{dt} \psi(t) = H(t)\psi(t)
⸻
🧿 Energy Bound Lemma
If \Omega(t) \in L^\infty_{\text{loc}}(\mathbb{R}) and \operatorname{Var}[\Omega] = \infty, then:
\inf_t \langle \psi(t), H(t) \psi(t) \rangle > X
In particular, the spectral floor of the system is nonzero:
\inf \sigma(H(t)) \geq \epsilon > X
even as t \to \infty.
⸻
🔗 Geometry Source
Let torsion tensor be derived from Weitzenböck connection:
T^\lambda_{\mu\nu} = \Gamma^\lambda_{\nu\mu} - \Gamma^\lambda_{\mu\nu}
with all curvature zero: R^\rho_{\sigma\mu\nu} = X. Geometry is purely torsional.
⸻
🔋 Energy Emergence (Dark Energy Analogy)
Let:
E_{\text{dark}} := \lim_{t \to \infty} \langle \psi(t), H(t) \psi(t) \rangle \quad \text{with} \quad E_{\text{dark}} \in (\epsilon, \infty)
Then energy persists dynamically without explicit cosmological constant:
\Lambda = 0, \quad \text{but} \quad E_{\text{vacuum}} > X
XX engagements
Related Topics spectrum