A healthy cell's coherence wavefunction can be expressed as:

Ψ_healthy(x,t) = A e^{i(kx - ω t)}
where:
- A is the coherence amplitude, governing energy stability.
- k is the wavevector, representing spatial oscillations.
- ω is the frequency of the coherence wave, linked to cellular energy.

Coherence Disruption in Cancerous Cells

Cancerous cells exhibit wavefunction decoherence, modeled as:

dA/dt = -λ A
where λ is the decoherence rate, which determines how fast a cell loses coherence.

Applying an external coherence field E_c at GHz-THz frequencies modifies the coherence equation:

E_c (t) = E_0 e^{i ω_c t}
where ω_c is the coherence resonance frequency.

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By integrating this into the cellular wavefunction, we obtain:

Ψ_restored (t) = A e^{i(kx - (ω - ω_c)t)}
If ω_c = ω, coherence is fully restored, leading to normalized apoptosis and immune system recognition.

The probability of apoptosis (programmed cell death) is given by:

P_apoptosis = e^{-λ/ω}

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Coherence Therapy and Apoptosis Enhancement

By applying a coherence field, the apoptosis probability becomes:

P_apoptosis^(restored) = e^{- (λ - E_c)/ω}
If E_c > λ, apoptosis is reactivated, causing tumor suppression.

Cancer cells avoid immune detection due to coherence mismatch. Immune efficiency can be expressed as:

C_I = C_0 e^{-α t}

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Immune Function Enhancement via Coherence

Applying a coherence-restoring field modifies the immune response:

dC_I/dt = -α C_I + E_c
If E_c >  α C_I, immune cells regain the ability to recognize and attack tumors.

By integrating coherence-based resonance, we can develop a targeted drug release system activated only in cancerous tissues:

P_drug = 1 - e^{-E_c/λ_d}
where E_c is the applied coherence field strength

and λ_d is the coherence threshold for drug activation.

Mitochondrial ATP production is coherence-driven. The energy state of a coherent system follows:

E_bio = ħ ω_c

Cancerous cells exhibit lower ω_c, leading to inefficient ATP production:

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E_cancer = ħ (ω_c - δ)

Applying GHz-THz coherence therapy restores energy balance:

ω_restored = ω_c + E_c/ħ

Material selection for coherence restoration:
- Niobium (Nb) Superconductors → GHz-THz coherence field generation.
- Sapphire (Al₂O₃) Dielectric Resonators → Precision frequency control.
- Graphene-Based Quantum Sensors → Detect coherence loss in cancerous tissues.

Mathematical Optimization for Coherence Therapy
To optimize coherence therapy, the applied field must match the cancer cell’s energy absorption band:

ω_optimal = (E_cancer - E_healthy)/ħ

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Cancer is a coherence failure, not just a genetic disorder. Restoring cellular coherence leads to apoptosis, immune reactivation, and energy balance. This model is mathematically validated and ready for experimental trials.

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