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Shadow Theory
Paper 01Canonicalv3

The Readout Non-Equivalence Theorem for Bounded Realized Domains

Bounded readouts are not complete realization packages when no admissible faithful recovery exists

DOI: 10.5281/zenodo.21184299View on Zenodo

Authority role

Establishes the non-equivalence principle: an exact readout, quotient, or bounded presentation does not by itself recover the realization-relevant structure of the domain it summarizes.

Summary

Distinguishes quotient/readout presentation from realization-structure equivalence and proves they come apart: a readout domain can be an exact quotient of the space it summarizes while failing to be realization-structure equivalent to it. Includes an equivariant no-selector theorem and a worked finite statistical-mechanics model, with explicit exception classes. This is the foundational principle of Shadow Theory: a shadow is an exact projected interface, not the source-level structure itself.

Notes

Reading notes

The paper's central move is to separate two relations that are usually conflated. For a realization/readout map ρ:C→S\rho: \mathcal{C} \to S:

  • Quotient presentation: S≅C/∼ρS \cong \mathcal{C}/\sim_\rho — always holds when S=Img(ρ)S = \mathrm{Img}(\rho). The readout is exact.
  • Realization-structure equivalence: S≃realCS \simeq_{\mathrm{real}} \mathcal{C} — requires the readout domain to recover the full realization-relevant structure via admissible recovery maps.

Theorem — Domain-level non-equivalence (informal)

If no admissible realization-faithful section of ρ\rho exists, then S̸≃realCS \not\simeq_{\mathrm{real}} \mathcal{C} even though S≅C/∼ρS \cong \mathcal{C}/\sim_\rho as a quotient.

The statistical-mechanics instance is handled with care: the decisive obstruction is not fiber multiplicity but the absence of an admissible invariant package-role class in some macro-fiber under the macro-indiscernibility symmetry group — proved via an equivariant no-selector theorem and verified in a concrete finite occupation-number model.

Cite this paper

Rodgers, Jeremy. (2026). The Readout Non-Equivalence Theorem for Bounded Realized Domains (v3). https://doi.org/10.5281/zenodo.21184299