Reading orientation
Part I. The complete released 33-result formal core, including nonnormal memory and canonical finite holomorphic naturality.
Part II. The complete strengthened mathematical development: bounded refinement towers, Bayesian information, Hermite transfer, quantitative memory compression, and competing risks.
Part III. Distinct retained results from the earlier human edition, exact examples, and the bounded-operator reciprocal calculus received from Paper IV.
Part IV. Forty-two visual pages work through exact finite models, analytical examples, and the boundaries of the formal contract.
Reading this edition
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Paper VI has several substantial source branches. This edition preserves the latest formal-core manuscript in Part I and the complete strengthened standalone manuscript in Part II. Part III retains twenty-four earlier results with distinct statements or hypotheses, five exact examples, and the holomorphic operator-substitution section migrated from Paper IV. Overlap is intentional and is not a claim of independent novelty for each proof environment.
The formal denominator is exactly the thirty-three scoped results of Part I. The inherited September 6, 2026 release records 341 explicit Lean theorem declarations across 47 proof modules and an audit of 782 namespace declarations. These are different units. No fresh Lean compilation was available for this October edition, and the formal count does not automatically include Parts II or III. The Lean guide gives the exact targets and specialization boundaries.
Part I includes three explicitly open conjectures. Part II supplies bounded or finite theorems for several related questions; it does not settle arbitrary continuum limits, unbounded-generator convergence, general statistical sufficiency, or globally optimal nonnormal model reduction. The reader should keep each result's declared space, topology, positivity, and stability hypotheses. Part I's mass-only Bayesian commutation criterion assumes at least two coarse blocks; Part II's stronger sufficiency convention also preserves within-block conditionals, so its one-block case still requires a constant likelihood.
The current checks, historical compiler evidence, and numerical regressions are recorded separately in the accompanying package. Finite checks and illustrations have bounded scopes. They do not replace proofs. The September 12 worked visual appendix, its exact models, and additional guides to the strengthened and retained material are supplied in the visual companion.
Read Part I for the common operator language and the exact released formal contract. Read Part II for bounded Banach-space development and the broader probabilistic and analytical realizations. Read Part III for product-topology limits, reversible coarse dynamics, prior-restricted and coarse filtering, killed-transfer perturbations, rational realization, and the transferred operator reciprocal family. Section numbers in Part II have prefix E; those in Part III have prefix R. Original labels and statement locations are mapped in the editable sources.
The final mathematical audit makes explicit the hypotheses needed for contour winding, positive memory decay, finite moment closure, and eventual absorption. It also clarifies summability across varying spaces. Every substantive change is recorded against preserved sources. Historical source test counts retain their historical dates.
Part I The released formal core
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Projector-Resolved Seed Calculus VI Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations
Abstract
This paper develops the transfer-semigroup realization of the projector-resolved seed calculus. A seed is propagated by a one-parameter semigroup, a Markov kernel, a normalized Bayesian transfer, a fractional spectral power, a Green operator, or a potential-weighted Feynman–Kac operator. Exact packet projectors resolve every transfer into input-output blocks, and the semigroup law becomes block convolution over intermediate packets. Coarse and fine generators are compared by a refinement defect. A Duhamel identity proves that this generator-level defect is exactly the source of the corresponding semigroup defect and gives a quantitative stability bound. In the stochastic realization, point-mass seeds generate random walks and continuous-time Markov evolution; exact lumpability is the vanishing of the same refinement defect. Conditional expectation is treated as a weighted orthogonal packet projector, with nested partitions producing martingale-difference packets. Bayesian conditioning is a normalized nonlinear operator. Its independent-data composition law, packet-evidence decomposition, stability bound, conjugate updates, and continuous likelihood flow are proved explicitly. The Bayesian refinement conjecture is resolved in finite dimensions: for strictly positive likelihoods and at least two coarse packets, exact commutation for every prior is equivalent to blockwise-constant likelihoods up to global scale, and every failure factors through an explicit information-loss packet. Fractional powers are constructed packetwise through the spectral projectors of a positive operator, and the square-root semigroup is recovered by subordination. The fractional-naturality conjecture is also resolved for bounded nonnegative self-adjoint refinement systems: the fractional intertwining defect has an exact resolvent integral, a gap-free Hölder certificate, a sharper common-gap estimate, and a Duhamel semigroup bound. Green kernels, finite-difference Poisson problems, and second-order low-mode convergence provide the PDE realization. A finite-state Feynman–Kac formula is proved by path enumeration. Exact Schur–Feshbach elimination shows that discarded latent packets return as a frequency-dependent self-energy and, in time coordinates, as a memory kernel. Survival and hazard functions are generated by transient semigroups. Nineteen code-generated figures, machine-readable results, the original thirty-seven automated tests, and seventeen added boundary-case checks accompany this reviewed edition. The inherited September 6 release records five checked Lean blocks covering all 33 scoped theorem-like results in Appendix Section A. Finite-dimensional theorems, classical infinite-dimensional extensions, numerical certificates, and open continuum claims are kept separate.
Reader roadmap
Scope and claim boundary. The paper proves the stated finite-dimensional semigroup, packet, Markov, Bayesian, conditional-expectation, fractional spectral, Green-kernel, Feynman–Kac, Schur–Feshbach, and survival identities. It also proves quantitative refinement-defect formulas under explicit norm hypotheses, including exact transfer of a bounded generator defect to every fractional power \(0<\alpha<1\) for bounded nonnegative self-adjoint generators. It does not claim that an arbitrary infinite-dimensional operator generates a semigroup, that every stochastic process admits a finite packet representation, that every posterior family is conjugate, that finite Bayesian sufficiency automatically extends to infinite disintegrations, or that every PDE continuum limit follows from a finite test. Domain, closability, sectoriality, measurability, tightness, and compactness hypotheses remain necessary in their respective infinite-dimensional settings. Classical sources are retained in the separate end-of-series reference ledger rather than reproduced in this paper.