Preface
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A mathematical representation can expose a structure that is expensive to see in another set of coordinates. It can also hide the cost of reaching those coordinates, discarding a residual, or reconstructing an answer. This volume studies both sides of that passage.
The first part follows an arithmetic selector through Fourier synthesis, rational projection, and a normalized complex state. The second asks when symmetry makes a quantum evolution accessible to classical computation. The third compares a restricted circuit with an ambient realization that may leave the restriction temporarily and return. The fourth studies restored reversible circuits, safe discrepancy covers, projection costs, and the limits of multiplicative summaries.
These treatments share the language of realizations, transfers, residuals, and obstructions. Their models remain distinct. A finite Fourier identity is not a gate synthesis algorithm. A small invariant sector is useful for simulation only when its representation, updates, and measurements are accessible. A catalytic saving depends on the chosen gate alphabet and cost. A lower bound for a faithful linear summary has the hypotheses of that summary model.
Each part gives its own definitions and proof conditions. Formal and computational companions follow the mathematical treatments. A shared glossary and collected references close the volume.
How to read the volume
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For a concrete entry point, begin with the twelve-dimensional arithmetic example and its visual study. The symmetry treatment develops invariant packets, moving frames, and compressed representations. The catalytic treatment begins with nested realization models before giving the complete linear core and its quantum and reversible extensions. The circuit treatment supplies restoration conventions before deriving its cover and projection results.
A proved statement has a written argument under explicit hypotheses. An exact finite computation concerns the declared bounded state space. A formal result identifies a specified Lean declaration and its proof record. A research proposal or conjecture remains open where its decisive obligation is not established. Panel titles state these distinctions directly.
The mathematical notation is local to each part. In particular, circuit obstructions use the binary field by default, while linear gate theory also works over general fields and rings. Formal coverage applies to the selected declarations, rather than to every written theorem in the volume.
Arithmetic realizations
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An arithmetic selector is followed through Fourier synthesis, exact rational projection, and a normalized complex state. The twelve-dimensional example makes each change of representation explicit.
Quantum–Arithmetic Bridge
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Arithmetic in Fourier space
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Symmetry collapse in quantum computation
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Invariant packets, moving frames, and compressed representations organize quantum evolution. The treatment keeps the cost of coordinates, updates, and measurement alongside the dimension of the retained state space.
Catalytic gate set distortion
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A restricted realization can be compared with an ambient implementation that leaves the restriction and returns. The resulting savings depend on the gate alphabet, resource convention, and cost assigned to each step.
Circuit obstructions
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Restored reversible circuits lead to cut comparisons, safe discrepancy covers, and projection costs. The later chapters examine faithful linear summaries and the obstruction to composing multiplicative descriptions.
Circuit examples in view
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Formal and computational companions
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The companions identify formal coverage, exact finite computations, and the obligations that remain open. Verification records establish the stated results within their declared models and bounds.
The formal arithmetic bridge
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Symmetry: supplementary proofs and computation
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Gate matrices, census data, and open obligations
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The formal circuit model
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Read The purpose and evidence contract Read Claim coverage and theorem index
Verification and proof boundaries
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Glossary
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Ambient realization
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A representation or implementation in a larger permitted model, with its maps and costs specified.
Arithmetic selector
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A diagonal choice of Fourier modes, transported to a signal projector by a unitary basis change.
Complete safe cover
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A family of detectors supported on the positive inputs and detecting every positive input.
Gate set defect
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The difference between minimum restricted and ambient implementation costs under a fixed gate alphabet and resource convention.
Residual
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The component not retained by a specified projection or coarse representation. Its recovery requires additional data.
Restoration
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The requirement that logical inputs and workspace return to their prescribed boundary values after computation.
Sources and references
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