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complete family of idempotents gives exact coordinates, but a single family does not record scale or integral visibility.  We study towers in which every coarse idempotent is the sum of its fine descendants.  The coordinate changes are invertible and coherent, and every operator block is the sum of its refined blocks.  Residue selectors give a model over arbitrary rings.  When roots of unity are available they agree with character averages, and coprime moduli factor through the Chinese remainder theorem.  In a complete filtered algebra, a diagonal operator plus positive weight transfers has an inverse obtained by summing admissible weighted paths.  The formula truncates exactly at finite precision, while finite transfer graphs have rational path series.  For a rational spectral decomposition over a discrete valuation ring, resultants give denominator bounds.  The exact denominator is the largest Smith exponent of the lattice gluing module.  This yields a necessary and sufficient precision delay, monotonicity under refinement, and a full congruence solving formula.  We close with exact formulas for the propagation of a coarsening defect through powers, resolvents, holomorphic functions, exponentials, and multilevel towers."},"title":"Fourier hypotheses","finding":"The abstract abbreviates the conditions for the character-average formula. The later theorem states the finite Fourier certificate and the following paragraph gives a zero-divisor counterexample.","text":"The character-average formula uses the finite Fourier orthogonality hypotheses stated in the linked theorem. Coefficient selectors themselves require no division. A primitive root of unity alone is insufficient over a ring with zero divisors.","references":[{"path":"papers/paper-ii/current/editable/traditional/chapters/02_coordinates.tex","line":404,"expected":"\\begin{theorem}[Root-of-unity realization]\\label{thm:root-of-unity}","label":"Root-of-unity realization","target":"read/paper-ii/refinement-and-residue-coordinates.html#L404"},{"path":"papers/paper-ii/current/editable/traditional/chapters/02_coordinates.tex","line":441,"expected":"The field or orthogonality condition is essential. A primitive element of","label":"Why the hypotheses matter","target":"read/paper-ii/refinement-and-residue-coordinates.html#L441"}]},{"id":"paper-ii-walk-series-variable","work":"paper-ii","page":"read/paper-ii/weighted-transfer.html","placement":"L473","source":{"path":"papers/paper-ii/current/editable/traditional/chapters/03_weighted.tex","line":484,"expected":"For fixed start and target vertices $s,t$, define"},"title":"Two roles for t","finding":"The target vertex and the formal series variable are both denoted t. A notation change is not uniquely forced, so the original notation is retained and its two roles are explained.","text":"In this subsection, s and t in subscripts label the start and target vertices. The t used in the power series is a formal variable, separate from the target-vertex label.","references":[{"path":"papers/paper-ii/current/editable/traditional/chapters/03_weighted.tex","line":484,"expected":"For fixed start and target vertices $s,t$, define","label":"Original walk-series definition","target":"read/paper-ii/weighted-transfer.html#L475"}]},{"id":"kcos-local-krylov-reference","work":"monographs-cos-method","page":"read/monographs-cos-method/05-spectrum.html","placement":"L166","source":{"path":"monographs/cos-method/book/chapters/05_spectrum.tex","line":166,"expected":"The local Krylov certificate from the previous section and the global spectral"},"title":"Location of the local certificate","finding":"The phrase “the previous section” is stale after book assembly. The local Krylov certificate is developed in Chapter 3, not the immediately preceding section of Chapter 5.","text":"The local Krylov certificate is developed in Chapter 3, in the sections on approximate light cones and phase scheduling. Those are the earlier results intended by “the previous section.”","references":[{"path":"monographs/cos-method/book/chapters/03_phases.tex","line":72,"expected":"\\section{Certified approximate light cones}\\label{sec:final-15}","label":"Approximate light cones","target":"read/monographs-cos-method/03-phases.html#L72"},{"path":"monographs/cos-method/book/chapters/03_phases.tex","line":312,"expected":"\\section{Certified budget-optimal phase scheduling}\\label{sec:final-16}","label":"Phase scheduling","target":"read/monographs-cos-method/03-phases.html#L312"}]},{"id":"kcos-full-range-reference-precision","work":"monographs-cos-method","page":"read/monographs-cos-method/05-spectrum.html","placement":"L285","source":{"path":"monographs/cos-method/book/chapters/05_spectrum.tex","line":292,"expected":"\\(1.2\\times10^{-15}\\), and the tail interval collapsed to the exact value."},"title":"Precision of the numerical reference","finding":"The retained full-dimensional numerical run has distinct floating endpoints and a nonzero bracket width. It does not establish an exact zero-width probability value.","text":"The retained full-range run reports a bracket about 9.50 × 10⁻¹⁴ decades wide. Read “collapsed to the exact value” as “narrowed around the full-spectrum reference.” This numerical observation is distinct from the preceding exact algebraic theorem.","references":[{"path":"monographs/cos-method/book/chapters/05_spectrum.tex","line":292,"expected":"\\(1.2\\times10^{-15}\\), and the tail interval collapsed to the exact value.","label":"Retained numerical description","target":"read/monographs-cos-method/05-spectrum.html#L285"}]},{"id":"kcos-python-failure-status","work":"monographs-cos-method","page":"read/monographs-cos-method/10-accuracy.html","placement":"L109","source":{"path":"monographs/cos-method/book/chapters/10_accuracy.tex","line":115,"expected":"\\boxed{\\texttt{certified = false}}"},"title":"Python failure status","finding":"The display uses a lowercase boolean word. The contract and reproduction code use the Python spelling certified=False.","text":"In the Python interface, the failure flag is certified=False, with a capital F. The lowercase word in the displayed summary denotes the same false status.","references":[{"path":"monographs/cos-method/book/chapters/10_accuracy.tex","line":70,"expected":"controller returns \\texttt{certified=False} rather than promoting the interval","label":"Accuracy contract","target":"read/monographs-cos-method/10-accuracy.html#L46"},{"path":"monographs/cos-method/book/chapters/C_reproduction.tex","line":24,"expected":"    # Returns certified=False if the requested accuracy is not proved.","label":"Python reproduction example","target":"read/monographs-cos-method/c-reproduction.html#L11"}]},{"id":"kcos-benchmark-error-range","work":"monographs-cos-method","page":"read/monographs-cos-method/11-evidence.html","placement":"L22","source":{"path":"monographs/cos-method/book/chapters/11_evidence.tex","line":34,"expected":"& $-16$ & $2.88$--$6.52\\times10^{-15}$ decades & independent integral converged & ranks 80--120\\\\"},"title":"Shared scale in the error range","finding":"The summary’s compact range typography leaves the common power of ten implicit. The detailed four-row benchmark table explicitly gives both endpoints.","text":"The error range in the heterogeneous inversion row is 2.88 × 10⁻¹⁵ to 6.52 × 10⁻¹⁵ decades. The power of ten applies to both endpoints, as shown by the detailed benchmark table.","references":[{"path":"monographs/cos-method/book/chapters/11_evidence.tex","line":156,"expected":"balanced & 80 & $6.52\\times10^{-15}$ & $1.50\\times10^{-14}$ & $6.84\\times10^{-20}$\\\\","label":"Detailed inversion benchmark","target":"read/monographs-cos-method/11-evidence.html#L152"}]}],"collection_revision":"2f443f44c190a64c713d6ea71c3d99ce48f78252"}