Stratified Universality

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1 \section{Detailed claim-boundary ledger}
2 \label{app:block14-claim-ledger}
3 
4 \subsection{Claims supported at manuscript level}
5 
6 The following statements are now supportable under the printed hypotheses:
7 
8 \begin{itemize}
9 \item Komorebi Math has a substantial common resolved syntax and many proved local realization theories.
10 \item The finite additive-Karoubi resolved envelope exists and has the stated universal property.
11 \item Invertible representation changes and noninvertible transformations can be separated formally through representation groupoids, bridge passports, and a typed directed category.
12 \item Information loss can be recorded separately as morphism collapse, object collapse, essential-image restriction, and loss of declared constructors or certificates.
13 \item A finite profiled atlas has a descent, theorem-transport, and invariant-assembly theorem once its hypotheses are verified.
14 \item Exact finite atlas models exist for graph holonomy, semilinear Galois descent, affine Mahler chart composition, split Mahler packet cores, and the normalized C-finite recurrence-rational-state triangle.
15 \item Finite graph transport has a unique maximal path-flat core, while full frame trivialization is equivalent to identity fundamental holonomy.
16 \item A canonical obstruction object exists relative to declared correction data; coarse vanishing can erase a nonzero fine obstruction.
17 \item Under the stated classical hypotheses, Hilbert 90 gives effective vector descent and the Brauer class is the exact projective-to-linear lifting obstruction.
18 \item Restricted complete invariants exist in several strata. The persistent C-finite slice has a complete global-to-packet minimality transfer theorem.
19 \item The unit purely periodic atlas over \(\mathbb F_p\) is a verified heterogeneous atlas across linear, digital, and functional-Mahler strata. Its objects and morphisms descend effectively, and its periodic resolved invariant package is complete on the declared normalized semantic category.
20 \item The unit rational-mask atlas over \(\mathbb F_p\) is a verified heterogeneous atlas across reduced functional-Mahler masks, minimal Cartier modules, and minimal digital automata. Its exact collapse obstruction separates the ultimately periodic locus from the genuinely non-C-finite automatic locus, and its descended invariant package is complete on the declared semantic category.
21 \item For a fixed radix and field, every regular sequence has a canonical minimal section module, a strict shortlex coordinate record, and a word-Hankel rank certificate. These charts satisfy effective descent, and the pointed section module is complete on the declared regular category.
22 \item Over a finite field, regularity is equivalent to automaticity and the minimal automaton is the finite section orbit. Over characteristic-zero fields, finite regular rank can coexist with an infinite kernel orbit.
23 \item Every integer-valued fixed-radix sequence of finite rational section rank has a canonical finite free raw section lattice and a canonical finite free integer-valued saturation of the same rank.
24 \item The Smith invariants and Fitting ideals of the finite saturation quotient determine the ghost kernel, semantic image size, missing quotient, good moduli, and prime-power generator profile exactly.
25 \item Every prime-power coefficient reduction has a finite minimal semantic section automaton, and the compatible full module tower reconstructs the embedded \(p\)-adic raw lattice.
26 \item \(p\)-adic completion, profinite completion, ordinary integral lifting, and actual section realization are distinct effectivity levels with explicit obstruction quotients.
27 \item Every integral unit series has unique fixed Euler exponents and unique divisor ghosts, with exact M\"obius integrality conditions and coefficient recurrence.
28 \item Every integral multiplicative arithmetic function is reconstructed exactly from prime-power local series, and local Euler exponents and ghosts form complete additive coordinates for Dirichlet convolution.
29 \item On the multiplicative exponent locus, the divisor--ghost transform gives an effective four-chart atlas and a complete prime-power descended invariant.
30 \item Continuous \(\mathbb Z_p\)-valued functions, their canonical Mahler coefficients, complete clopen branch packets, and branchwise Mahler coefficients form an effective normalized local atlas.
31 \item Affine Mahler kernels have finite triangular support, compose by exact mixed-radix substitution, and satisfy strict Chinese-remainder coherence.
32 \item Finite-degree affine channels have distinct characteristic-zero, integral, modular, and collective-packet bridge passports, with sharp minimal packet sizes under the printed hypotheses.
33 \item Declared monotone Mahler valuation profiles survive affine polyphase refinement, and principal-unit state packets acquire a certified linearly improving precision slope under \(p\)-power refinement.
34 \item Every nonzero D-finite series and non-eventually-zero P-recursive sequence over a characteristic-zero field has a canonical monic minimal Ore annihilator; a finite transient ledger upgrades the tail operator to an exact semantic chart.
35 \item Arithmetic multisection preserves holonomicity, and every fixed closure pipeline has an exact excess-order certificate relative to the unique minimal packet operator.
36 \item In the marked binomial--Fuchsian family, nonintegral exponent gives minimal packet differential order \(N\), positive integral exponent gives rational order one, and nonpositive integral exponent gives the exact finite-support alias/cancellation phase diagram.
37 \item For a finite Galois extension in the semisimple setting, subgroups, fixed fields, and faithful Reynolds projectors form an order-reversing normalized atlas.
38 \item Nonnormal intermediate fields form conjugate projector atlases with canonical core and normal-closure envelopes, while Mackey double cosets give exact induction--restriction transition packets.
39 \item Ramification groups form a nested projector resolution, and the Artin and Swan conductors are weighted unresolved-rank sums.
40 \item Finite Galois vector descent is effective on objects and morphisms; projective descent lifts to an ordinary vector packet exactly when its relative Brauer class vanishes.
41 \item Every continuous finite-dimensional complex representation of a profinite group factors through a finite quotient, and shrinking open normal-subgroup projectors converge strongly to the identity.
42 \item A faithful exact tensor fiber functor reconstructs an affine group scheme by neutral Tannakian duality; faithful corner realization alone does not cross this tensor boundary.
43 \item The certified global-synthesis bridge matrix distinguishes reversible named overlaps from fully faithful inclusions, directional loss, partial constructors, and absent bridges.
44 \item On the five declared connected components, local complete invariants have coherent comparison maps and descend to a complete tagged invariant that reflects isomorphisms.
45 \item The marked weighted Fuss family gives a genuine reversible overlap among tree, selected algebraic, and P-recursive presentations, classified by the finite pair \((q,\alpha)\).
46 \item At every declared complete finite resolution in an additive idempotent-complete category, analysis and synthesis are inverse, morphisms have unique lane-transfer matrices, composition is block contraction, and compatible refinement aggregates those matrices functorially.
47 \item Finite lane selectors split retained transport from an exact residual. Under the separately printed abelian or strict-fold hypotheses, the discarded or collided part is measured by the corresponding kernel-exact sequence.
48 \item Factorial residue lanes realize the formal theorem exactly, and the stated polyphase, Cartier, holonomic, local, nonlinear-tree, algebraic, and Galois cases are valid only with their declared native hypotheses.
49 \item Finite nontrivial address holonomy does not obstruct complete lane transport when the address and splitting cocycles remain coherent: lane matrices descend as compatible sections of a matrix local system, and composition remains exact path contraction.
50 \item Monodromy-orbit sums are the finest ordinary globally labeled coarsening obtainable from the original object-lane projectors, while simultaneous source-target pair orbits are the minimal nonempty holonomy-compatible Boolean transport supports; their selectors are nonzero only on orbits supporting nonzero compatible sections.
51 \item For finite permutation modules, pair-orbit indicator matrices form an exact intertwiner basis with nonnegative integer path-count structure constants. For cyclic Fourier idempotents, rational Galois orbit sums are indexed by divisors and have exact Ramanujan-sum coefficients.
52 \end{itemize}
53 
54 \subsection{Claims not established}
55 
56 The following statements are not proved here:
57 
58 \begin{itemize}
59 \item every mathematical object has a finite Komorebi presentation;
60 \item all strata are equivalent categories;
61 \item every bridge is invertible;
62 \item one global canonical minimal presentation exists across arbitrary grammars;
63 \item one finite invariant classifies every stratum;
64 \item equivalence of arbitrary presentations is decidable;
65 \item the full eight-stratum heterogeneous atlas already satisfies effective descent, or the five certified global-synthesis components are reversibly connected to one another;
66 \item every automatic sequence has a rational functional-Mahler mask, or functional Mahler equations and Newton-Mahler interpolation are unrestrictedly equivalent;
67 \item one base-independent or all-ring canonical regular normal form, one minimal semiring realization theorem, or one cross-radix invariant compatible with every base;
68 \item the integral lattice theorem automatically extends to arbitrary coefficient rings, non-Noetherian modules, semirings, all radices at once, or finite-data recognition of rational regularity;
69 \item unique Euler coordinates imply finite structural complexity, a minimal product compiler, or a canonical factorization across arbitrary product grammars;
70 \item ordinary Cauchy convolution and Dirichlet convolution are globally isomorphic, or prime-power seeds classify arithmetic functions without the multiplicativity axiom;
71 \item the multiplicative Euler-exponent locus is closed under ordinary Euler-product multiplication;
72 \item functional-Mahler coefficient equations and Newton--Mahler \(p\)-adic interpolation are canonically or unrestrictedly equivalent;
73 \item every continuous \(p\)-adic function is locally analytic, every affine-kernel valuation lower bound is always sharp, or one optimal mixed-radix refinement tree has been constructed;
74 \item the local atlas already carries a complete combined spectral--local invariant or a universal state-recognition theorem;
75 \item arbitrary D-finite multisections have a proved finite torsion signature, a closed minimal-order formula, or a canonical apparent-singularity removal algorithm;
76 \item local exponent eigenvalues alone determine the full monodromy representation, logarithmic extension data, or global differential Galois group;
77 \item the marked binomial--Fuchsian invariant classifies unmarked holonomic series or recovers an algebraic branch from an annihilator;
78 \item the $d^N$ algebraic packet bound is always minimal, or every resultant automatically selects the intended branch factor;
79 \item selected algebraic branches admit one canonical minimal multivariate system dimension across arbitrary polynomial grammars;
80 \item algebraic and D-finite presentations are equivalent without retained branch data;
81 \item Reynolds averaging remains exact when the group order is not invertible, or finite semisimple Galois charts automatically extend to modular and wild integral representations;
82 \item every projective cocycle descends to a vector space, or the Brauer class alone classifies the full central simple algebra without its remaining algebra data;
83 \item every tensor category has a neutral fiber functor, or a faithful additive or corner representation automatically reconstructs a Tannakian group scheme;
84 \item finite Galois effectivity proves effective descent for the unrestricted heterogeneous Komorebi atlas;
85 \item internal manuscript proof status implies independent peer review;
86 \item the name Universal Lane Transport establishes historical priority or novelty for classical residue filters, polyphase decompositions, Peirce block matrices, Cartier sections, or other imported ingredients;
87 \item every object admits a complete finite split lane system, every useful resolution is additive, or the finite theorem and its finite holonomy-twisted extension automatically extend to infinite, completed, nonsplit, noninvertible, higher-stack, or arbitrary nonlinear settings;
88 \item the number of lane addresses equals dimension, rank, minimal state size, entropy, independent information, or computational complexity without a separate theorem;
89 \item an incomplete selector or noninjective fold is reversible, or its residual and collision kernel can be discarded without changing the represented transport.
90 \end{itemize}
91 
92 \begin{resultbox}
93 \textbf{Final status of the present paper.}
94 The project now has an explicit typed and packet-resolved common foundation, a proved additive-Karoubi envelope, profiled stratum and bridge passports, a directed information-loss calculus, a typed comparison of four obstruction theories, a proved finite assembly theorem, two exact affine-Mahler mechanisms, one fully effective normalized C-finite presentation atlas, and two fully effective heterogeneous digital/Mahler atlases. The fixed-radix regular stage adds the full fixed-radix field-valued regular atlas, a canonical word-Hankel and shortlex normal form, and the automatic finite-orbit theorem. The integral prime-power stage adds the canonical integral raw/saturated lattice pair, the exact Smith--Fitting reduction defect, finite prime-power semantic automata, and the \(p\)-adic/profinite effectivity hierarchy. The product-multiplicative stage adds canonical fixed-basis Euler and ghost coordinates, prime-power effective descent for multiplicative functions, and the complete restricted divisor--ghost atlas. The local Mahler-polyphase stage adds the normalized continuous Newton--Mahler atlas, strict complete polyphase descent, sharp finite-degree channel packets, valuation-profile transport, and principal-unit precision amplification. The holonomic stage adds canonical minimal Ore charts with finite transients, the general multisection alias/cancellation ledger, and an exact marked binomial--Fuchsian packet-minimality theorem. The algebraic stage adds finite selected-branch normalization, exact packet polynomial systems, a generic $d^N$ packet-degree theorem, and collision signatures with sharp Catalan and radical model cases. The nonlinear tree stage adds a marked polynomial fixed-point atlas, exact diagonal packet Fa\`a di Bruno compression, an address-preserving cyclic-core dimension, and finite simply-generated-tree support and singularity signatures. The Galois-Tannakian stage adds the normalized finite subgroup--fixed-field--projector atlas, conjugate and Mackey transition packets, ramification details, effective Hilbert-90 descent, the exact relative Brauer obstruction, profinite finite-level resolution, and the neutral Tannakian tensor-fiber gate. The global synthesis stage adds the authoritative bridge matrix, normalizer-induced higher coherence, a five-component certified invariant atlas, the marked Fuss nonlinear--algebraic--holonomic triangle, and a complete tagged global invariant on the declared semantic union. The finite lane-transport stage adds the complete finite-resolution analysis, synthesis, block-composition, refinement, selector, residual, conditional fold-loss, factorial-realization, and certified-descent theorem family. The holonomy-twisted stage then removes global address triviality as a transport requirement, replacing fixed labels by matrix-local-system sections and recovering canonical ordinary lanes at monodromy-orbit resolution, with pair-orbit composition and rational cyclic orbit descent as exact realizations. The broad heterogeneous claim remains conditional on cross-radix and semiring compatibility, modular and wild symmetry, nonneutral tensor descent, unrestricted holonomic and multivariate algebraic normalization, general nonlinear normal forms and mixed-direction compression, stronger local valuation and optimal mixed-radix theory, unrestricted product grammars, and effective invariant comparison beyond the normalized domains proved here.
95 \end{resultbox}

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