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| Format: | Recurso digital |
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Zenodo
2026
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| Accès en ligne: | https://doi.org/10.5281/zenodo.18830404 |
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- <p>The ER = EPR conjecture (Maldacena & Susskind, 2013) asserts that Einstein–Rosen bridges (wormholes) and Einstein–Podolsky–Rosen correlations (quantum entanglement) are manifestations of the same underlying structure. The conjecture is qualitative: it identifies the two phenomena but does not provide a continuous, bounded measure of the degree of equivalence for a given system.</p> <p>This technical note shows that the Thales partition framework supplies this measure. For any bipartite entangled system with Schmidt coefficients x and y = 1 − x, the Thales altitude h = √(xy) simultaneously encodes the wormhole throat width (ER side) and the entanglement coupling strength (EPR side) on a single bounded scale h ∈ [0, 1/2]. The deficit δ = 1 − 2h quantifies the departure from exact ER = EPR equivalence. The variance–deficit exchange identity ∂L²/∂D = 2h provides the marginal cost of converting entanglement loss into geometric separation.</p> <p>The mean-independent point—where AM = GM = HM and δ = 0—is the unique configuration at which the geometric, coupling, and quantum-correlation descriptions coincide without translation. This point corresponds to known instability thresholds in every domain tested: the massless Weyl limit (β = 1) for Dirac fermions, the supercritical nuclear charge (Z = 1/α ≈ 137) for hydrogen, and the unstable circular orbit at r = 5M for Schwarzschild black holes. Stable systems carry mandatory nonzero deficit: exact ER = EPR is a boundary condition, not an accessible state.</p> <p>The framework provides five tools the conjecture currently lacks: (1) a bounded measure of equivalence degree (the deficit δ); (2) identification of exact equivalence with the mean-independent point and its instability thresholds; (3) a cost function for trading entanglement and geometry (the exchange rate 2h); (4) a regime classification for entangled black hole pairs via the mean differential AM − HM, applicable to every event in the LIGO–Virgo–KAGRA gravitational-wave catalogs; and (5) a three-layer dynamical hierarchy connecting the second-derivative equations of the three classical means (AM″ = 0, GM″ = −GM, HM″ = 1 − 4·HM) to the background, bridge, and correlation layers of the ER = EPR system.</p> <p>The connection to Ramanujan's modular programme is traced through three channels: the harmonic equation h″ = −h generating the function space of modular forms governing bridge geometry; the Casimir constant 1/12 appearing both as the zero-point energy of bridge vibrational modes and as the harmonic mean at the Schwarzschild ISCO; and the Hardy–Ramanujan partition function providing the microstate counting for bridge entropy.</p> <p>The framework is purely diagnostic. No modification of general relativity, quantum mechanics, or the ER = EPR conjecture is proposed.</p> <p><strong>Related work: De Jesus, Elias. (2026). The Thales Partition Toolkit for ER = EPR: Established Results, Cross-Domain Synthesis, and Speculative Extensions. Zenodo. https://doi.org/10.5281/zenodo.18890164</strong></p> <p> </p>