Every long-context claim in the literature is an empirical one. A model is trained, a haystack is searched, a number is reported. Nobody derives the number first.
If context is carried by a resonance field rather than an attention matrix, the number stops being a choice. A field has modes. Modes have a spectrum. The distance between the lowest two eigenvalues, the spectral gap, sets how far a signal propagates before it is indistinguishable from the ground state.
Call the resulting length the Effective Context Bound. It is analytic. You can compute it from the field's parameters before you train anything.
Whether the bound is useful is a separate question, and an open one. A tight analytic bound on a bad field is still a bad field. We are checking.
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