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1 \clearpage
2 \section{What refinement establishes, and what remains}
3 \label{a-conclusion}
4 The finite construction establishes a concrete chain of statements. Unit coordinates factor a finite arithmetic register. Normalized Haar averages identify a coarse Hilbert space inside a fine one. A compatible Hermitian Hamiltonian then carries the same evolution on that range. This last step uses compatibility, rather than normalization alone. When a new coupling breaks the reducing-range property, the hidden sector produces leakage and a nontrivial self-energy.
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6 The spectral construction makes that distinction calculable. Polyphase coordinates display the fine operator's channels without removing any of them. Radial or residue elimination introduces an energy-dependent effective operator on a regular resolvent chart. The memory formulation retains the hidden initial state and the causal history. Chebyshev decimation describes an exact spectral relation where its Dirichlet inverse exists. Hypergeometric companion functions organize entire power-series identities in finite or bounded settings, with additional domains when unbounded operators enter.
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8 The infinite construction is a theorem for a declared hierarchical model. Probability Haar measure and normalized cylinders give a common Hilbert space. Orthogonal depth blocks and self-adjoint weights specify its maximal Hamiltonian domain. Finite compatible blocks then approximate the state evolution strongly. Graph-norm convergence concerns vectors in that domain. Operator-norm convergence and compact resolvent require stronger spectral conditions. A finite walk whose old vertices change degree under extension is outside this compatibility theorem until a suitable comparison map is supplied.
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10 \begin{readerroadmap}[title={The remaining research interfaces}]
11 The analytic adic hierarchy still needs an exact formal translation if it is to carry a Lean certificate. General depth-dependent walks need their own limit argument. A physical fiber theory needs a covariant coupling, a state-selection or production rule, and consistent energy accounting. Each interface is an additional claim with its own hypotheses.
12 \end{readerroadmap}
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14 The visual atlas and verification companion make the scope reviewable. Their finite examples test identities, illuminate domain distinctions, and retain negative controls. They do not extrapolate a bounded calculation into an infinite proof. Related archived formal results establish normalized cells, exact tower properties, and dyadic reconstruction statements. They are useful mathematical interfaces, but their theorem names and hypotheses do not certify every result in this volume.
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16 A candidate fiber interpretation must specify how refinement degrees of freedom couple to spacetime and how they are populated. The covariant fiber chapter addresses that question before the cosmological calculations are used. Its energy-budget obstruction remains controlling: downstream transport from a prescribed distribution is meaningful, but it is not evidence that the proposed production history generated that distribution.

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