The Entropic Seesaw Model of the Theory of Entropicity (ToE) on the Explanation of Entanglement, the Attosecond Entanglement Formation Time Experiment, Einstein’s EPR and ER=EPR: Part 1

The Entropic Seesaw Model (ESSM) may now be stated more sharply as the entanglement-specific sector of the Theory of Entropicity (ToE): it is the claim that what standard quantum theory represents as a non-factorizable bipartite state is, at a deeper ontological level, a single structured entropic configuration whose unity is established locally, maintained relationally, and broken only when environmental entropy production, gradient shear, or measurement-channel opening forces the shared structure back into factorized sectors. In this sense, ESSM does not treat entanglement as a mysterious superluminal link between already separate objects, but as the persistence of one entropic manifold under later spacetime separation. This reading is strongly aligned with the public ToE archive, with the 2025 and 2026 ToE working-paper stream, and with the attached Letter IC material, while also sitting in meaningful dialogue with contemporary attosecond photoionization research and the continuing literature on EPR and ER=EPR. 

Foundational statement

The deepest claim of ESSM is that entanglement is not fundamentally a correlation added to two pre-existing systems. Rather, the entangling interaction creates a shared entropic domain from which subsystem labels arise only after coarse-graining, environmental partitioning, or measurement. This is precisely why the “seesaw” metaphor is more than pedagogical ornament: it expresses a constraint structure. Two ends appear spatially distinct, but they belong to one dynamical object. The public ToE materials describe the entropic field as a continuous, local, dynamical scalar field on an entropic manifold, and the ESSM reuses that ontology to explain why apparently distant quantum systems can remain internally unified without any need for faster-than-light signaling. 

A minimal entropic-topological statement of formation is therefore:

[ \mathcal{M}_A \oplus \mathcal{M}B ;\longrightarrow; \mathcal{M}{AB} \tag{7.10} ]

where (\mathcal{M}_A) and (\mathcal{M}B) denote previously distinct entropic sectors and (\mathcal{M}{AB}) denotes the merged manifold generated by the entangling event. Equation (7.10) should be read as a local restructuring rule, not as a signaling rule. The merger occurs where the interaction occurs; it is a creation event in the entropic field. This distinction between formation and later persistence is one of the most important conceptual clarifications in the attached ToE correspondence and is fully compatible with the official ToE archive’s insistence on the entropic field as the primary ontological substrate from which structure emerges. 

Once the shared manifold has formed, ESSM invokes a dual geometry:

[ d_{\text{space}}(A,B)\gg 0,\qquad d_{\mathcal{E}}(A,B)\approx 0 \tag{7.11} ]

where (d_{\text{space}}) is ordinary spacetime distance and (d_{\mathcal{E}}) is entropic relational distance. Equation (7.11) is the cleanest way to formulate the ToE claim that entanglement can be nonlocal in spacetime geometry while remaining local in entropic geometry. The EPR puzzle then ceases to be “How did information get from A to B so fast?” and becomes “Why did we ever assume that spacetime distance and entropic distance must coincide?” 

Minimal formal structure of ESSM

At present, the public ToE literature clearly fixes the ontology of the entropic field, the Obidi Action program, the entropic time-limit principle, and the seesaw picture of coherence/collapse, but it does not yet uniquely fix one universally accepted closed-form microphysical kernel for entanglement stability across all platforms. For that reason, the most rigorous formulation of ESSM is a minimal phenomenological closure consistent with the attached materials and the public ToE corpus, not the claim that every kernel below has already been uniquely derived once and for all. 

A natural two-sector entropic action is

[ \mathcal{A}_{AB}

\int d^4x, \Big[ \mathcal{L}_A+\mathcal{L}B+\lambda,\mathcal{C}(S_A,S_B)-\eta,\mathcal{D}{\mathrm{env}} \Big], \tag{7.12} ]

where (\mathcal{L}_A) and (\mathcal{L}B) are the subsystem entropic Lagrangian densities, (\mathcal{C}(S_A,S_B)) is the coherence-coupling functional that rewards shared-manifold unity, (\lambda) is the entangling strength, (\mathcal{D}{\mathrm{env}}) encodes environmental decohering influence, and (\eta) is the susceptibility to that influence. This equation is directly in line with the structure articulated in the later ToE correspondence: entanglement stability is governed by a competition between coherence maintenance and environment-driven fragmentation. 

Define a coherence-strength functional (\Gamma_{AB}(t)) by the instantaneous balance of these two contributions:

[ \Gamma_{AB}(t) \equiv \lambda,\mathcal{C}{AB}(t)-\eta,\mathcal{D}{\mathrm{env}}(t). \tag{7.13} ]

Then the shared entropic manifold persists only if

[ \Gamma_{AB}(t)>\Gamma_{\mathrm{crit}}, \tag{7.14} ]

while decoherence begins when (\Gamma_{AB}(t)\le \Gamma_{\mathrm{crit}}). In ESSM, decoherence is therefore not an inexplicable extra postulate: it is a threshold transition in the entropic geometry. The main destabilizers identified in the ToE correspondence are background entropy injection, gradient shear between the local entropic environments of the subsystems, and monitoring-channel opening by apparatus coupling. That triad is also physically consonant with modern attosecond and ultrafast-coherence literature, where ion–photoelectron entanglement is shown to be highly sensitive to field structure, channel mixing, and environment-like couplings rather than behaving as a purely abstract timeless resource. 

In open systems, the entropic leakage law may be written schematically as

[ \frac{dS_{AB}}{dt}=-J_{\mathrm{env}}, \tag{7.15} ]

with (J_{\mathrm{env}}) the leakage current into background degrees of freedom. Equation (7.15) is especially important for ESSM because it makes coherence time a function not only of temperature in the loose textbook sense, but of the measurable gradient structure of the surrounding environment. That gives the model experimental traction: if entanglement is genuinely an entropic-manifold phenomenon, then coherence should degrade differently across differing gravitational, radiative, or thermal-gradient environments even when naive temperature bookkeeping looks similar. 

Formation time and the attosecond benchmark

Section 7 of Letter IC already grounds ToE’s entanglement program in the entropic time-limit idea: no physically real restructuring is strictly instantaneous. In public ToE formulations, this appears as both an entropic uncertainty-style lower bound and as the “No-Rush Theorem.” In insertion-ready numbering, the relevant relations become

[ \Delta t_{\mathrm{ent}} ;\ge; \frac{\hbar}{2,\Delta S_{\max}}, \tag{7.16} ]

and

[ \tau_{\min}

\frac{k_B\ln 2}{(dS/dt)_{\max}}, \tag{7.17} ]

where (\Delta S_{\max}) and ((dS/dt)_{\max}) are the maximum entropic restructuring and entropy-production rates accessible to the interaction. Equation (7.16) states that the formation of a shared manifold requires finite restructuring time; equation (7.17) states that every physical event inherits a minimum duration from the finite rate at which entropic information can be reorganized. These relations are explicit in the public ToE attosecond papers and in the 2026 living-review stream. 

The 232-attosecond figure associated with the TU Wien/Chinese-collaboration work must, however, be handled with care if this section is to remain rigorous. The underlying 2024 PRL is about time delays in strong-field/XUV photoionization as a probe of interelectronic coherence and entanglement; the TU Wien and attoworld summaries present the (\sim 230)–232 as timescale as the ultrafast scale on which entanglement-related changes in the photoemission wavepacket arise and emphasize that direct experimental proof of such ultrafast entanglement was the next target. Accordingly, the safest and most accurate ESSM reading is not that 232 attoseconds is already a universal entanglement constant, but that it is a system-specific attosecond benchmark showing that entanglement formation or entanglement-sensitive restructuring in photoionization is temporally resolved, finite, and dynamically nontrivial. That is enough to strongly support the ToE rejection of absolute instantaneity, while avoiding a stronger claim than the primary sources justify. 

This interpretation becomes stronger when placed beside later attosecond literature. The 2025 Nature Communications study on CO(_2) showed that attosecond photoionization delays are sensitive to ionic coupling because the emitted photoelectron and parent ion are entangled and because added interfering paths arise when the IR field acts on the ion as well as the electron. The 2026 Nature paper on H(_2) molecular photoionization went further by explicitly tracking a delay-dependent tradeoff between electronic coherence and ion–photoelectron entanglement using singular-value and von Neumann entropy diagnostics. The 2026 Light: Science & Applications work likewise showed that ultrashort-laser ionization can quantitatively control and reconstruct electron–ion entanglement. Together these studies do not prove ToE, but they do strongly reinforce the ESSM claim that entanglement in attosecond physics is not well described as a frozen metaphysical instant; it is a dynamic structural process in a driven field environment. 

For ESSM, the correct inference is therefore precise: the attosecond result constrains the formation sector, not the persistence sector. Formation is local and rate-limited; persistence is the maintenance of the already-formed shared manifold. Once (\mathcal{M}{AB}) exists, later remote correlations need not be modeled as new superluminal traffic through spacetime. They are the revelation of an already unified entropic structure, unless and until environmental leakage drives (\Gamma{AB}) below threshold. 

Einstein’s EPR revisited in ESSM

The original EPR paper argued that if quantum mechanics is complete, then one must accept a troubling form of nonlocal determination; otherwise the wavefunction is incomplete. Later foundational work reframed the issue, but the core tension remained: strict spacetime separability seems hard to reconcile with entangled correlations. ESSM resolves this by declining both naive superluminal signaling and naive subsystem separatism. It says the shared manifold is the ontic primitive, not the separated pair. External separation is real in spacetime geometry; internal unity is real in entropic geometry. EPR “spookiness” then becomes the mismatch between two geometries, not a violation of causality. 

The seesaw-collapse threshold from the 2025 ToE quantum-measurement paper may be written in this subsection as

[ \Lambda_A(t)+\Lambda_B(t);\ge;\Lambda_{\mathrm{thresh}}, \tag{7.18} ]

where (\Lambda_A) and (\Lambda_B) are the subsystem contributions to the shared entropic loading and (\Lambda_{\mathrm{thresh}}) is the critical threshold past which balanced coherence can no longer be maintained. Equation (7.18) is not the statement that entanglement itself is a signal from one wing of an experiment to the other. It is the statement that once monitoring, environmental injection, or internal imbalance exceeds the allowed coherence budget, the shared manifold tips, branch balance is lost, and a classical outcome sector is selected. In this sense ESSM turns collapse into an entropy-threshold transition rather than an unexplained interpretive jump. 

The crucial rigor point is that ESSM does not reintroduce local hidden variables in the Bell sense. Rather, it relocates the locality claim: the relational fact that underwrites the correlation is local in entropic geometry, though not in ordinary spacetime geometry. This is why ESSM can preserve the force of Bell-type nonclassicality while simultaneously denying that the only options are either acausal magic or classical hidden-variable completion. A future decisive empirical test would be to push attosecond Bell-style probes or related ultrafast protocols into regimes where entanglement formation, entanglement stabilization, and entanglement breakdown can be temporally distinguished rather than inferred only from asymptotic final correlations. 

ER=EPR in ToE terms

When Juan Maldacena and Leonard Susskind proposed ER=EPR, the central claim was that entanglement and Einstein–Rosen connectivity are not unrelated ideas: at least in holographic settings, an entangled pair may admit a wormhole interpretation. Recent work has continued to sharpen this claim, including operational and computable realizations, but the conjecture still remains much more secure in tightly controlled holographic/gravity settings than in ordinary tabletop entanglement. That distinction matters for ESSM. ToE should not be read as claiming that every laboratory EPR pair literally opens a traversable spacetime tunnel. Rather, ToE offers a thermodynamic reinterpretation: what ER=EPR intuits geometrically, ESSM reformulates entropically as a non-traversable constraint bridge within the entropic field. 

The relation may be written schematically as

[ \text{ER=EPR} \quad\Longrightarrow\quad \mathcal{B}{\mathrm{geom}} ;\mapsto; \mathcal{B}{\mathcal{E}}, \tag{7.19} ]

where (\mathcal{B}{\mathrm{geom}}) is a geometric bridge in the sense of the holographic wormhole picture and (\mathcal{B}{\mathcal{E}}) is the ESSM entropic bridge: a unified internal constraint structure that preserves correlation without enabling usable superluminal signals. This is exactly the direction already present in the ToE “Einstein and Bohr Finally Reconciled” materials, which explicitly reinterpret ER=EPR through an entropic bridge or seesaw-bridge picture. The merit of ESSM is that it adds a threshold-and-dynamics account absent from many purely geometric invocations of ER=EPR. The wormhole analogy tells us that the pair is internally unified; ESSM further asks what forms it, what stabilizes it, what destabilizes it, and how measurement factorizes it. 

In that sense, ESSM stands neither wholly against nor simply identical with ER=EPR. It is better understood as a thermodynamic completion strategy for it. ER=EPR says entanglement and connectivity belong together. ESSM says the relevant connectivity is carried by an entropic manifold with finite formation time, threshold-governed persistence, environmental leakage, and collapse dynamics. That is precisely why ESSM is the right place within ToE to unify entanglement formation, wave-function collapse, attosecond timing, EPR, and ER=EPR inside one common explanatory frame. 

Open mathematical tasks and current limits

Two cautions are necessary if this subsection is to remain monograph-grade rather than merely enthusiastic. First, the public ToE literature already fixes the entropic field ontology, the local-versus-global distinction, the entropic time-limit logic, and the seesaw threshold picture, but it does not yet uniquely fix the detailed microphysical forms of (\mathcal{C}(S_A,S_B)), (\mathcal{D}{\mathrm{env}}), or (\Gamma{AB}(t)) across all experimental platforms. The equations above therefore represent the most conservative closure consistent with the present corpus, not the final completed ESSM. Second, the 232-attosecond benchmark should be described as a powerful attosecond evidence point for finite entanglement-sensitive restructuring in a specific photoionization setting, not as a demonstrated universal constant for all entanglement formation in nature. Those two clarifications strengthen rather than weaken the theory, because they identify exactly where the next formal and experimental work must concentrate. 

References

[55] Einstein, A., Podolsky, B., and Rosen, N., “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, Physical Review 47 (1935), 777–780. 

[56] “The Einstein-Podolsky-Rosen Argument in Quantum Theory,” Stanford Encyclopedia of Philosophy (archival entry). 

[57] Maldacena, J., and Susskind, L., “Cool Horizons for Entangled Black Holes,” arXiv:1306.0533 / Fortschritte der Physik (2013). 

[58] Fields, C., Glazebrook, J. F., Marciano, A., and Zappala, E., “ER = EPR is an operational theorem,” arXiv:2410.16496 (2024). 

[59] Jiang, W.-C. et al., “Time Delays as Attosecond Probe of Interelectronic Coherence and Entanglement,” Physical Review Letters 133, 163201 (2024). 

[60] TU Wien, “How fast is quantum entanglement?” news release, 22 October 2024. 

[61] attoworld, “In the wave mix of entangled particles,” 23 January 2025. 

[62] Makos, I. et al., “Entanglement in photoionisation reveals the effect of ionic coupling in attosecond time delays,” Nature Communications 16, 8554 (2025). 

[63] Koll, L. M. et al., “Entanglement and electronic coherence in attosecond molecular photoionization,” Nature 652, 82 (2026). 

[64] Mao, Y. J. et al., “Coherent control of electron-ion entanglement in multiphoton ionization,” Light: Science & Applications (2026). 

[65] Ruberti, M., Averbukh, V., and Mintert, F., “Bell test of quantum entanglement in attosecond photoionization,” arXiv:2312.05036 (2024 version). 

[66] Obidi, J. O., “Attosecond Constraints on Quantum Entanglement Formation as Empirical Evidence for the Theory of Entropicity (ToE),” Cambridge Open Engage (2025). 

[67] Obidi, J. O., “Review and Analysis of the Theory of Entropicity (ToE) in Light of the Attosecond Entanglement Formation Experiment: Toward a Unified Entropic Framework for Quantum Measurement, Non-Instantaneous Wave-Function Collapse, and Spacetime Emergence,” Cambridge Open Engage (2025). 

[68] Obidi, J. O., “Einstein and Bohr Finally Reconciled on Quantum Theory: The Theory of Entropicity (ToE) as the Unifying Resolution to the Problem of Quantum Measurement and Wave Function Collapse,” Cambridge Open Engage / DOI catalogued in public ToE bibliographies as doi:10.33774/coe-2025-vrfrx (2025). 

[69] The official archive of the Theory of Entropicity, “The Theory of Entropicity (ToE),” canonical repository/monograph portal. 

[70] The official ToE monograph archive, “Chapter 2 — The Entropic Field (\mathcal{E}(x)).” 

[71] Obidi, J. O., “The Theory of Entropicity (ToE) Living Review Letters Series — Letter I: The Ontological Primacy of Entropy,” Cambridge Open Engage (2026). 

[72] Obidi, J. O., “On the Foundations of the Theory of Entropicity (ToE): Conceptual and Mathematical Formulation,” public exposition in the ToE publication stream (2026).