108-record correspondence map (JSON)
Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates
COVERAGE_108.json
This is the original source text. Reading it does not run or verify the program.
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25 {
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27 "stable_id": "PSC-I-D-COEFFICIENT-REGIME",
28 "kind": "definition",
29 "title": "Coefficient and module regimes",
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46 "statement": "\\begin{definition}[Coefficient and module regimes]\n\\label{def:coefficient-regime}\nAll modules are left modules unless stated otherwise. The algebraic regime consists of a unital ring $A$ and a unital left module $M$ over $A$. When linear dimensions, tensor products, quivers, or operator blocks are used, $A$ is a unital algebra over $\\bbk$, where $\\bbk$ is a field, and $M$ is a vector space over $\\bbk$. The topological regime consists of a Hausdorff topological vector space over a complete valued field, together with a unital algebra of continuous linear operators. Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement.\n\\end{definition}",
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75 {
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77 "stable_id": "PSC-I-B-SYMBOLIC-SCOPE",
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96 "statement": "\\begin{boundary}[Symbolic scope]\n\\label{bnd:symbolic-scope}\nThis edition proves the algebraic and topological statements printed below. It does not assert historical novelty, independent external certification, automatic discovery of useful packet systems, nonlinear obstruction theory, multilevel refinement, arithmetic congruences, quantum gravity, automorphy, or local or global Langlands reciprocity.\n\\end{boundary}",
97 "source_proof": null,
98 "new_block_contribution": []
99 },
100 {
101 "ordinal": 4,
102 "stable_id": "PSC-I-D-TYPED-GRAMMAR",
103 "kind": "definition",
104 "title": "Typed operator grammar",
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108 "Komorebi.PaperIRelease.GrammarAlgebra.Grammar",
109 "Komorebi.PaperIRelease.GrammarAlgebra.TypedRealization",
110 "Komorebi.PaperIRelease.GrammarAlgebra.TypedHom",
111 "Komorebi.PaperIRelease.GrammarAlgebra.grammarFromPathRelations",
112 "Komorebi.PaperIRelease.GrammarAlgebra.idealGrammar"
113 ],
114 "scope_and_remaining_work": "Finite vertex set, arbitrary arrow set, source/target maps, independent vector spaces and typed arrow maps; relations are finite linear combinations of paths with fixed endpoints. The more general free-algebra relation format is specialized both to homogeneous typed path relations and to all representatives of an actual TwoSidedIdeal.",
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124 "proof_end_line": null,
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127 "statement": "\\begin{definition}[Typed operator grammar]\n\\label{def:typed-grammar}\nLet $Q=(Q_0,Q_1,s,t)$ be a quiver with finite vertex set. A typed operator grammar over $\\bbk$ is a pair\n\\[\n\\mathfrak G=(Q,I),\n\\]\nwhere $I\\triangleleft \\bbk Q$ is an ideal on both sides generated by linear combinations over $\\bbk$ of paths having common source and common target. Write\n\\[\n\\Lambda_{\\mathfrak G}:=\\bbk Q/I.\n\\]\nA realization of $\\mathfrak G$ is a family $(X_v)_{v\\in Q_0}$ and linear maps\n\\[\nX_a:X_{s(a)}\\longrightarrow X_{t(a)}\n\\qquad(a\\in Q_1)\n\\]\nthat annihilate every relation in $I$. A morphism $f:X\\to Y$ of realizations is a family of linear maps $f_v:X_v\\to Y_v$ satisfying\n\\[\nf_{t(a)}X_a=Y_af_{s(a)}\n\\qquad(a\\in Q_1).\n\\]\n\\end{definition}",
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130 "acceptance_provenance": [
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138 ],
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140 },
141 {
142 "ordinal": 5,
143 "stable_id": "PSC-I-D-COLLAPSED-REALIZATION",
144 "kind": "definition",
145 "title": "Collapsed one-space realization",
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151 "Komorebi.Foundation.LinearProjectorFamily.rangeSynthesis_rangeAnalysis",
152 "Komorebi.Foundation.LinearProjectorFamily.rangeAnalysis_rangeSynthesis",
153 "Komorebi.Foundation.LinearProjectorFamily.exactPacketCoordinates",
154 "Komorebi.PaperIRelease.GrammarAlgebra.TypedData.total",
155 "Komorebi.PaperIRelease.GrammarAlgebra.TypedRealization.collapse",
156 "Komorebi.PaperIRelease.GrammarAlgebra.TypedData.path_action",
157 "Komorebi.PaperIRelease.GrammarAlgebra.TypedData.pathCombination_action"
158 ],
159 "scope_and_remaining_work": "Total space is the finite product, canonically the finite direct sum. Vertex operators are single-after-projection maps and arrows are single-after-arrow-after-projection. Induction on the genuinely source/target-indexed TypedPath proves the actual block-composition action, including the zero-length vertex path.",
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169 "proof_end_line": null,
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171 },
172 "statement": "\\begin{definition}[Realization on a single space]\n\\label{def:collapsed-realization}\nFor a typed realization $X$, define its realization on a single space by\n\\[\n|X|:=\\bigoplus_{v\\in Q_0}X_v.\n\\]\nEvery vertex idempotent acts as a coordinate projector on $|X|$, and every path acts as a block operator from its source space to its target space.\n\\end{definition}",
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185 },
186 {
187 "ordinal": 6,
188 "stable_id": "PSC-I-T-TYPED-MODULE-EQUIVALENCE",
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190 "title": "Typed realization and module equivalence",
191 "status": "covered",
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194 "Komorebi.PaperIRelease.GrammarAlgebra.typedModuleEquivalence",
195 "Komorebi.PaperIRelease.GrammarAlgebra.typedIdealModuleEquivalence",
196 "Komorebi.PaperIRelease.GrammarAlgebra.typedPathRelationModuleEquivalence",
197 "Komorebi.PaperIRelease.GrammarAlgebra.idealPresentationEquiv",
198 "Komorebi.PaperIRelease.GrammarAlgebra.splitActionIso",
199 "Komorebi.PaperIRelease.GrammarModules.actionModuleEquivalence"
200 ],
201 "scope_and_remaining_work": "The grammar algebra is constructed by FreeAlgebra and RingQuot, with complete orthogonal vertex idempotents and exact source/target arrow corners. Collapse is full, faithful and essentially surjective, using actual vertex ranges and their analysis/synthesis equivalence. Action objects are then proved equivalent to ordinary ModuleCat for all unital left modules; the coefficient action is recovered through algebraMap. A separate quotient comparison uses the actual TwoSidedIdeal ring congruence and proves both inverse identities. No category equivalence, split decomposition, path action or module structure is supplied as an assumption.",
202 "current_source": {
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213 },
214 "statement": "\\begin{theorem}[Typed realization and module equivalence]\n\\label{thm:typed-module-equivalence}\nThe category of realizations of $\\mathfrak G$ is equivalent to the category of unital left modules over $\\Lambda_{\\mathfrak G}$. Explicitly, a realization $X$ determines the collapsed space $|X|$ of \\cref{def:collapsed-realization}\nwith the vertex idempotent $e_v$ acting as the coordinate projector onto $X_v$ and each arrow acting in the block between its declared source and target. Conversely, a unital module $M$ over $\\Lambda_{\\mathfrak G}$ determines the realization\n\\[\nM_v=e_vM,\n\\qquad\nM_a(x)=ax.\n\\]\nThe two constructions are inverse up to canonical natural isomorphism.\n\\end{theorem}",
215 "source_proof": "\\begin{proof}\nBecause $Q_0$ is finite, $1=\\sum_{v\\in Q_0}e_v$ in $\\bbk Q$, the $e_v$ are pairwise orthogonal, and the same identities hold in $\\Lambda_{\\mathfrak G}$. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of $I$ acts as zero, so the action factors through $\\Lambda_{\\mathfrak G}$. The vertex idempotents give\n\\[\n|X|=\\bigoplus_v e_v|X|=\\bigoplus_vX_v.\n\\]\nConversely, if $M$ is unital, then\n\\[\nM=1M=\\sum_ve_vM.\n\\]\nOrthogonality makes the sum direct. For an arrow $a:v\\to w$, the identities $a=e_wae_v$ imply\n\\[\na(e_vM)\\subseteq e_wM.\n\\]\nThe relations in $I$ act as zero because $M$ is a module over $\\Lambda_{\\mathfrak G}$. Applying both constructions returns the original vertex spaces and arrow maps, or the original module decomposition, through the evident identity maps. These identifications commute with morphisms, giving an equivalence of categories.\n\\end{proof}",
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217 "acceptance_provenance": [
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223 "verdict": "proved as written"
224 }
225 ],
226 "remaining_work": null
227 },
228 {
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230 "stable_id": "PSC-I-D-REALIZED-ALGEBRA",
231 "kind": "definition",
232 "title": "Realized operator algebra",
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239 "Komorebi.SeedRankOperations.seedRank_imageAlgebra_eq",
240 "Komorebi.SingleOperatorGraph.singleOperator_seedNumber_isLeast_of_primary_decomposition",
241 "Komorebi.SingleOperatorGraph.singleOperator_operatorAlgebra_seedRanks_eq_of_primary_decomposition",
242 "Komorebi.PaperIRelease.GrammarAlgebra.CollapsedRealization.action",
243 "Komorebi.PaperIRelease.GrammarAlgebra.CollapsedRealization.span_imageAlgebra_eq"
244 ],
245 "scope_and_remaining_work": "The realized algebra is literally the range of the constructed algebra homomorphism into Module.End. It is not a supplied equivalent algebra or an abstract surrogate.",
246 "current_source": {
247 "file": "editable/traditional/02_foundations.tex",
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255 "proof_end_line": null,
256 "proof_sha256": null
257 },
258 "statement": "\\begin{definition}[Realized operator algebra]\n\\label{def:realized-algebra}\nFor a realization $X$, the realized operator algebra is\n\\[\n\\A_X:=\\im\\!\\left(\\Lambda_{\\mathfrak G}\\longrightarrow\\End_{\\bbk}(|X|)\\right).\n\\]\nIt records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.\n\\end{definition}",
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268 }
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271 },
272 {
273 "ordinal": 8,
274 "stable_id": "PSC-I-P-ACTION-FACTORIZATION",
275 "kind": "proposition",
276 "title": "Action factorization and hull invariance",
277 "status": "covered",
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281 ],
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283 "Komorebi.SeedRankOperations.seedRank_imageAlgebra_eq",
284 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
285 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
286 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
287 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
288 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
289 "Komorebi.PaperIRelease.GrammarAlgebra.CollapsedRealization.span_imageAlgebra_eq",
290 "Komorebi.PaperIRelease.GrammarAlgebra.CollapsedRealization.seedRank_imageAlgebra_eq"
291 ],
292 "scope_and_remaining_work": "The two actual spans are compared by span induction and range-surjectivity. The cardinal seed ranks are equal via the identity semilinear bijection into the actual image algebra. Arbitrary seed sets and zero modules are included; there is no finite-generation assumption.",
293 "current_source": {
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303 "proof_sha256": "aef3690fcf4da8306ffb79444a8671fe430703b5acd910e4f1f17a7011b6d18d"
304 },
305 "statement": "\\begin{proposition}[Action factorization and hull invariance]\n\\label{prop:action-factorization}\nLet $S\\subseteq |X|$. The generated submodule under the grammar algebra equals the generated submodule under the realized operator algebra:\n\\[\n\\Lambda_{\\mathfrak G}S=\\A_XS.\n\\]\nConsequently, every statement about seed rank depends only on the realized action, not on redundant relations in a chosen presentation of the grammar.\n\\end{proposition}",
306 "source_proof": "\\begin{proof}\nThe action map $\\rho:\\Lambda_{\\mathfrak G}\\twoheadrightarrow\\A_X$ is surjective by definition and satisfies $\\lambda s=\\rho(\\lambda)s$. Hence every grammar orbit vector lies in $\\A_XS$. The reverse inclusion holds because every element of $\\A_X$ is $\\rho(\\lambda)$ for some $\\lambda$.\n\\end{proof}",
307 "new_block_contribution": [],
308 "acceptance_provenance": [
309 {
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313 "reviewed_proof_sha256": "aef3690fcf4da8306ffb79444a8671fe430703b5acd910e4f1f17a7011b6d18d",
314 "verdict": "proved as written"
315 }
316 ],
317 "remaining_work": null
318 },
319 {
320 "ordinal": 9,
321 "stable_id": "PSC-I-D-SEED-SYSTEM",
322 "kind": "definition",
323 "title": "Linear seed system",
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327 2,
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332 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange_apply_mk",
333 "Komorebi.SeedRankBasics.generates_singleton_iff_surjective",
334 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense",
335 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_intertwines",
336 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_unique",
337 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange",
338 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange_apply_mk",
339 "Komorebi.SeedRankBasics.multiSeedQuotientEquiv",
340 "Komorebi.SeedRankOperations.seedFinrank_isLeast_free_surjection",
341 "Komorebi.Foundation.LinearProjectorFamily.rangeSynthesis_rangeAnalysis",
342 "Komorebi.Foundation.LinearProjectorFamily.rangeAnalysis_rangeSynthesis",
343 "Komorebi.Foundation.LinearProjectorFamily.exactPacketCoordinates",
344 "Komorebi.PaperIRelease.CategoryPackets.SeedObject",
345 "Komorebi.PaperIRelease.TopologyCategories.SeedObject"
346 ],
347 "scope_and_remaining_work": "Algebraic and continuous packet cases are represented by concrete operator algebras, packet objects and arbitrary seed sets, which include every finite distinguished family after passage to its range. The record is a regime/data definition, not an assertion that one bundled structure covers the optional packet-free and topological regimes. Optional fields are represented in separate structures.",
348 "current_source": {
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354 "statement_end_line": 152,
355 "statement_sha256": "f1b9660868ad56e506bd33c2ab474312509199727cc8c23a64d4334641974e26",
356 "proof_start_line": null,
357 "proof_end_line": null,
358 "proof_sha256": null
359 },
360 "statement": "\\begin{definition}[Linear seed system]\n\\label{def:seed-system}\nA linear seed system is a tuple\n\\[\n\\mathfrak S=(X,\\A,S,\\Pcal,\\tau),\n\\]\nwhere $X$ is a unital left module over $\\A$, $\\A$ is fixed independently of the target element, $S=(s_1,\\ldots,s_r)$ is a finite distinguished seed family, $\\Pcal$ is either empty or a declared packet system, and $\\tau$ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.\n\\end{definition}",
361 "source_proof": null,
362 "new_block_contribution": [],
363 "acceptance_provenance": [
364 {
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366 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
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368 "reviewed_proof_sha256": null,
369 "verdict": "documentary definition; packet regimes represented"
370 }
371 ],
372 "remaining_work": null
373 },
374 {
375 "ordinal": 10,
376 "stable_id": "PSC-I-D-GENERATED-HULL",
377 "kind": "definition",
378 "title": "Generated hull",
379 "status": "definition_represented",
380 "old_contract_rows": [
381 2,
382 29
383 ],
384 "mapped_declarations": [
385 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange",
386 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange_apply_mk",
387 "Komorebi.SeedRankBasics.multiSeedQuotientEquiv",
388 "Komorebi.SeedRankOperations.seedFinrank_isLeast_free_surjection",
389 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.dense_operatorHull_image",
390 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.map_packetRange_eq",
391 "Komorebi.PaperIRelease.CategoryPackets.hull_eq_span",
392 "PaperIRelease.TopologyTransport.operatorHull",
393 "Komorebi.PaperIRelease.TopologyCategories.closedHull"
394 ],
395 "scope_and_remaining_work": "The scalar span of algebra-orbit vectors is proved equal as a set to the module span. The general topological operator hull is an actual Submodule and its topologicalClosure is used in the topological development.",
396 "current_source": {
397 "file": "editable/traditional/02_foundations.tex",
398 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
399 "kind": "definition",
400 "record_id_line": 167,
401 "statement_start_line": 155,
402 "statement_end_line": 166,
403 "statement_sha256": "859743fdac21ebb8ba120148cbf03f5511974f6e3356b5f733a3beea316b55f5",
404 "proof_start_line": null,
405 "proof_end_line": null,
406 "proof_sha256": null
407 },
408 "statement": "\\begin{definition}[Generated hull]\n\\label{def:generated-hull}\nFor $S\\subseteq X$, define\n\\[\n\\Hull_{\\A}(S):=\\sum_{s\\in S}\\A s.\n\\]\nIn the topological regime define\n\\[\n\\Hull_{\\A,\\tau}(S):=\\cl{\\Hull_{\\A}(S)}^{\\,\\tau}.\n\\]\nFor a singleton $S=\\{s\\}$, write $\\Hull_{\\A}(s)$.\n\\end{definition}",
409 "source_proof": null,
410 "new_block_contribution": [],
411 "acceptance_provenance": [
412 {
413 "semantic_review": "semantic_categories.json",
414 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
415 "reviewed_statement_sha256": "859743fdac21ebb8ba120148cbf03f5511974f6e3356b5f733a3beea316b55f5",
416 "reviewed_proof_sha256": null,
417 "verdict": "definition represented"
418 }
419 ],
420 "remaining_work": null
421 },
422 {
423 "ordinal": 11,
424 "stable_id": "PSC-I-D-NATURAL-PRESENTATION",
425 "kind": "definition",
426 "title": "Natural presentation",
427 "status": "documentary",
428 "old_contract_rows": [],
429 "mapped_declarations": [],
430 "scope_and_remaining_work": "A methodological criterion rather than a decidable Lean predicate.",
431 "current_source": {
432 "file": "editable/traditional/02_foundations.tex",
433 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
434 "kind": "definition",
435 "record_id_line": 175,
436 "statement_start_line": 171,
437 "statement_end_line": 174,
438 "statement_sha256": "94104ee5b6710359c3fa14a5febde34de8455668f83dfc5f49cf3e5028770033",
439 "proof_start_line": null,
440 "proof_end_line": null,
441 "proof_sha256": null
442 },
443 "statement": "\\begin{definition}[Natural presentation]\n\\label{def:natural-presentation}\nA seed presentation is natural relative to a declared grammar when the primitive operators and relations are fixed independently of the target, the relation system is finite or effectively enumerable, the topology is intrinsic to the realization, no primitive operator directly stores the complete target, and the presentation exposes a reusable invariant, theorem, or computation not encoded by a primitive chosen for the target.\n\\end{definition}",
444 "source_proof": null,
445 "new_block_contribution": []
446 },
447 {
448 "ordinal": 12,
449 "stable_id": "PSC-I-P-ALGEBRA-MODULE-GENERATION",
450 "kind": "proposition",
451 "title": "Algebra generation and module generation are distinct",
452 "status": "covered",
453 "old_contract_rows": [],
454 "mapped_declarations": [
455 "Komorebi.PaperIRelease.SeedOperations.regular_generates_one",
456 "Komorebi.PaperIRelease.SeedOperations.regular_seedRank"
457 ],
458 "scope_and_remaining_work": "The regular action is multiplication on A. The singleton unit is proved generating by surjectivity of scalar evaluation; the nontrivial regular module then has seedRank exactly one. No assertion about algebra presentations is used as a premise.",
459 "current_source": {
460 "file": "editable/traditional/02_foundations.tex",
461 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
462 "kind": "proposition",
463 "record_id_line": 185,
464 "statement_start_line": 177,
465 "statement_end_line": 184,
466 "statement_sha256": "9ba2f9c1f32694b86a7d9ac63ef11d2b3c46ee5c38307f3abcc3c12048cf91d0",
467 "proof_start_line": 188,
468 "proof_end_line": 190,
469 "proof_sha256": "d851abd6935b719f8efe162bc12db23f80fd3339751b68ec79ccb3a7e22517f8"
470 },
471 "statement": "\\begin{proposition}[Algebra generation and module generation are distinct]\n\\label{prop:algebra-module-generation}\nEvery nonzero unital algebra $A$, regarded as a left module over itself, has\n\\[\n\\srank_A(A)=1,\n\\]\nwith cyclic seed $1_A$. This conclusion is independent of the number of algebra generators required to present $A$ as an algebra.\n\\end{proposition}",
472 "source_proof": "\\begin{proof}\nFor every $a\\in A$, one has $a=a1_A$. Hence $A1_A=A$. The number of operations needed to generate $A$ as an algebra is a different invariant and does not enter the module calculation.\n\\end{proof}",
473 "new_block_contribution": [],
474 "acceptance_provenance": [
475 {
476 "semantic_review": "semantic_wave2.json",
477 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
478 "reviewed_statement_sha256": "9ba2f9c1f32694b86a7d9ac63ef11d2b3c46ee5c38307f3abcc3c12048cf91d0",
479 "reviewed_proof_sha256": "d851abd6935b719f8efe162bc12db23f80fd3339751b68ec79ccb3a7e22517f8",
480 "verdict": "proved as written"
481 }
482 ],
483 "remaining_work": null
484 },
485 {
486 "ordinal": 13,
487 "stable_id": "PSC-I-B-TARGET-DEPENDENT-OPERATORS",
488 "kind": "boundary",
489 "title": "Target-dependent operators",
490 "status": "scope_boundary",
491 "old_contract_rows": [],
492 "mapped_declarations": [],
493 "scope_and_remaining_work": "Target-independence restriction remains documentary.",
494 "current_source": {
495 "file": "editable/traditional/02_foundations.tex",
496 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
497 "kind": "boundary",
498 "record_id_line": 196,
499 "statement_start_line": 192,
500 "statement_end_line": 195,
501 "statement_sha256": "aece204c76d6b378ec937375eebd02a43d0a48dbed1f94054b4ed514ef573dd1",
502 "proof_start_line": null,
503 "proof_end_line": null,
504 "proof_sha256": null
505 },
506 "statement": "\\begin{boundary}[Operators chosen for a target]\n\\label{bnd:target-dependent-operators}\nFor a fixed seed $s$ and target $x$, the existence of a specially defined operator $A_x$ satisfying $A_xs=x$ does not establish a natural seed calculus unless $A_x$ belongs to a grammar fixed independently of the target satisfying \\cref{def:natural-presentation}.\n\\end{boundary}",
507 "source_proof": null,
508 "new_block_contribution": []
509 },
510 {
511 "ordinal": 14,
512 "stable_id": "PSC-I-D-SEED-EVALUATION",
513 "kind": "definition",
514 "title": "Seed evaluation map",
515 "status": "definition_represented",
516 "old_contract_rows": [
517 2
518 ],
519 "mapped_declarations": [
520 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange",
521 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange_apply_mk",
522 "Komorebi.SeedRankBasics.multiSeedQuotientEquiv",
523 "Komorebi.SeedRankOperations.seedFinrank_isLeast_free_surjection"
524 ],
525 "scope_and_remaining_work": "multiSeedMap is the actual Finsupp linear combination map; finite families give the stated finite free module.",
526 "current_source": {
527 "file": "editable/traditional/02_foundations.tex",
528 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
529 "kind": "definition",
530 "record_id_line": 224,
531 "statement_start_line": 215,
532 "statement_end_line": 223,
533 "statement_sha256": "db3e882c18dfdf5770d2779811295f6d287e9ffa95dbfb79bec513f96cba185d",
534 "proof_start_line": null,
535 "proof_end_line": null,
536 "proof_sha256": null
537 },
538 "statement": "\\begin{definition}[Seed evaluation map]\n\\label{def:seed-evaluation}\nLet $A$ be a unital ring, let $M$ be a left module over $A$, and let $S=(s_1,\\ldots,s_r)\\in M^r$. Define the linear map over $A$\n\\[\n\\varepsilon_S:A^{\\oplus r}\\longrightarrow M,\n\\qquad\n(a_1,\\ldots,a_r)\\longmapsto\\sum_{j=1}^{r}a_js_j.\n\\]\n\\end{definition}",
539 "source_proof": null,
540 "new_block_contribution": []
541 },
542 {
543 "ordinal": 15,
544 "stable_id": "PSC-I-T-FREE-PRESENTATION",
545 "kind": "theorem",
546 "title": "Finite free presentation",
547 "status": "covered",
548 "old_contract_rows": [
549 2
550 ],
551 "mapped_declarations": [
552 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange",
553 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange_apply_mk",
554 "Komorebi.SeedRankBasics.multiSeedQuotientEquiv",
555 "Komorebi.SeedRankOperations.seedFinrank_isLeast_free_surjection"
556 ],
557 "scope_and_remaining_work": "Actual range/span equality, quotient equivalence and generation/surjectivity equivalence are present.",
558 "current_source": {
559 "file": "editable/traditional/02_foundations.tex",
560 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
561 "kind": "theorem",
562 "record_id_line": 240,
563 "statement_start_line": 226,
564 "statement_end_line": 239,
565 "statement_sha256": "6e2db91e9df6b69a04e531d9b552fe7148f878663bcd3600c36c94ac62592ef8",
566 "proof_start_line": 244,
567 "proof_end_line": 250,
568 "proof_sha256": "5ba6a1b0db2637582535075c52bf19281e300cd73b8fd873b34dafe1fac381a2"
569 },
570 "statement": "\\begin{theorem}[Finite free presentation]\n\\label{thm:free-presentation}\nFor the map in \\cref{def:seed-evaluation},\n\\[\n\\im\\varepsilon_S=\\Hull_A(S),\n\\qquad\nN_S:=\\Ker\\varepsilon_S\\leq A^{\\oplus r},\n\\]\nand the induced map is an isomorphism of left modules over $A$\n\\[\n\\boxed{A^{\\oplus r}/N_S\\cong\\Hull_A(S).}\n\\]\nConsequently, $S$ generates $M$ if and only if $\\varepsilon_S$ is surjective.\n\\end{theorem}",
571 "source_proof": "\\begin{proof}\nThe image consists exactly of finite sums $\\sum_ja_js_j$, which is $\\Hull_A(S)$. Because $\\varepsilon_S$ is linear over $A$, its kernel is a left submodule. The first isomorphism theorem gives\n\\[\nA^{\\oplus r}/N_S\\cong\\im\\varepsilon_S=\\Hull_A(S).\n\\]\nThe final equivalence is the equality $\\im\\varepsilon_S=M$.\n\\end{proof}",
572 "new_block_contribution": [],
573 "acceptance_provenance": [
574 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
575 ]
576 },
577 {
578 "ordinal": 16,
579 "stable_id": "PSC-I-C-MINIMAL-FREE-COVER",
580 "kind": "corollary",
581 "title": "Minimal free-cover characterization",
582 "status": "covered",
583 "old_contract_rows": [
584 2
585 ],
586 "mapped_declarations": [
587 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange",
588 "Komorebi.SeedRankBasics.multiSeedQuotientEquivRange_apply_mk",
589 "Komorebi.SeedRankBasics.multiSeedQuotientEquiv",
590 "Komorebi.SeedRankOperations.seedFinrank_isLeast_free_surjection"
591 ],
592 "scope_and_remaining_work": "seedFinrank_isLeast_free_surjection proves the finite minimum, not an executable basis algorithm.",
593 "current_source": {
594 "file": "editable/traditional/02_foundations.tex",
595 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
596 "kind": "corollary",
597 "record_id_line": 276,
598 "statement_start_line": 267,
599 "statement_end_line": 275,
600 "statement_sha256": "8e4ae31565265679e9c8bf397291baf65b396ad21bb88e7f00b8d17328b1cb00",
601 "proof_start_line": 279,
602 "proof_end_line": 281,
603 "proof_sha256": "01ff58e723f01c2be6bfe1e953be4ac9b433a33ed6d75fcf954a7fc15fc33900"
604 },
605 "statement": "\\begin{corollary}[Characterization by minimal free covers]\n\\label{cor:minimal-free-cover}\nFor every finitely generated left module $M$ over $A$,\n\\[\n\\boxed{\n\\srank_A(M)=\\min\\{r\\geq0:\\text{there exists a surjection }A^{\\oplus r}\\twoheadrightarrow M\\}.\n}\n\\]\n\\end{corollary}",
606 "source_proof": "\\begin{proof}\nA seed family of size $r$ gives a surjection by \\cref{thm:free-presentation}. Conversely, the images of the standard basis vectors under any surjection $A^{\\oplus r}\\twoheadrightarrow M$ form a generating family of size at most $r$.\n\\end{proof}",
607 "new_block_contribution": [],
608 "acceptance_provenance": [
609 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
610 ]
611 },
612 {
613 "ordinal": 17,
614 "stable_id": "PSC-I-C-ONE-SEED-PRESENTATION",
615 "kind": "corollary",
616 "title": "One-seed presentation",
617 "status": "covered",
618 "old_contract_rows": [
619 1
620 ],
621 "mapped_declarations": [
622 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange",
623 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange_apply_mk",
624 "Komorebi.SeedRankBasics.generates_singleton_iff_surjective",
625 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense",
626 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_intertwines",
627 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_unique"
628 ],
629 "scope_and_remaining_work": "oneSeedMap has the stated annihilator kernel and its quotient is the literal cyclic range.",
630 "current_source": {
631 "file": "editable/traditional/02_foundations.tex",
632 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
633 "kind": "corollary",
634 "record_id_line": 294,
635 "statement_start_line": 283,
636 "statement_end_line": 293,
637 "statement_sha256": "92a43462490ec1465fe4129738284f9c3f53e2d8bb7b63f980c1d03a0fbbb2ed",
638 "proof_start_line": 297,
639 "proof_end_line": 299,
640 "proof_sha256": "7818e5d9477115ff5fdf80e981911f508e575ca86f51600d3f696aae7a952cec"
641 },
642 "statement": "\\begin{corollary}[Presentation with one seed]\n\\label{cor:one-seed-presentation}\nFor $s\\in M$, the annihilator\n\\[\n\\operatorname{Ann}_A(s):=\\{a\\in A:as=0\\}\n\\]\nis a left ideal and\n\\[\n\\boxed{A/\\operatorname{Ann}_A(s)\\cong\\Hull_A(s).}\n\\]\n\\end{corollary}",
643 "source_proof": "\\begin{proof}\nThe evaluation map for one seed is $a\\mapsto as$. Its kernel is $\\operatorname{Ann}_A(s)$, which is therefore a left ideal, and \\cref{thm:free-presentation} gives the quotient isomorphism.\n\\end{proof}",
644 "new_block_contribution": [],
645 "acceptance_provenance": [
646 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
647 ]
648 },
649 {
650 "ordinal": 18,
651 "stable_id": "PSC-I-P-ANNIHILATOR-TWO-SIDED",
652 "kind": "proposition",
653 "title": "Two-sided annihilator criterion",
654 "status": "covered",
655 "old_contract_rows": [],
656 "mapped_declarations": [
657 "Komorebi.PaperICompletion.CyclicClassification.orbitMap_ker_eq_annihilator_of_surjective",
658 "Komorebi.PaperIRelease.SeedOperations.vectorAnnihilator",
659 "Komorebi.PaperIRelease.SeedOperations.annihilator_isTwoSided_iff",
660 "Komorebi.PaperIRelease.SeedOperations.cyclic_annihilator_eq_module_iff"
661 ],
662 "scope_and_remaining_work": "vectorAnnihilator is the actual kernel of a↦a•s. Right multiplication closure is equivalent to annihilating Submodule.span A {s}; Ideal.IsTwoSided supplies the standard two-sided-ideal interface. Under singleton generation, the actual ideal equals Module.annihilator A M. Noncommutative order is a*b and (a*b)•s=a•(b•s).",
663 "current_source": {
664 "file": "editable/traditional/02_foundations.tex",
665 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
666 "kind": "proposition",
667 "record_id_line": 313,
668 "statement_start_line": 302,
669 "statement_end_line": 312,
670 "statement_sha256": "dd991bd6ce28ee7f94302ed9dd3fb00719cf26de3b6ccf740f441b118bdd8e54",
671 "proof_start_line": 316,
672 "proof_end_line": 325,
673 "proof_sha256": "fa4d07649341df5b1b843588b8b759632a004d48a33365a9d64c82b41ba54796"
674 },
675 "statement": "\\begin{proposition}[Criterion for the annihilator to be an ideal on both sides]\n\\label{prop:two-sided-annihilator}\nFor $s\\in M$, the left ideal $L=\\operatorname{Ann}_A(s)$ is an ideal on both sides if and only if\n\\[\nL\\,\\Hull_A(s)=0.\n\\]\nIf $s$ is cyclic, these conditions are equivalent to\n\\[\n\\operatorname{Ann}_A(s)=\\operatorname{Ann}_A(M).\n\\]\n\\end{proposition}",
676 "source_proof": "\\begin{proof}\nThe left ideal $L$ is an ideal on both sides exactly when $ab\\in L$ for every $a\\in L$ and $b\\in A$. This is equivalent to\n\\[\n0=(ab)s=a(bs)\n\\qquad(a\\in L,\\ b\\in A).\n\\]\nBy additivity, the last identity is equivalent to $L\\Hull_A(s)=0$.\n\nIf $s$ is cyclic, then $\\Hull_A(s)=M$. Hence the preceding condition is $L\\subseteq\\operatorname{Ann}_A(M)$. The reverse inclusion always holds because an operator annihilating all of $M$ annihilates $s$. Thus equality follows. Conversely, $\\operatorname{Ann}_A(M)$ is an ideal on both sides, so equality implies that $L$ is an ideal on both sides.\n\\end{proof}",
677 "new_block_contribution": [
678 "Komorebi.PaperICompletion.CyclicClassification.orbitMap_ker_eq_annihilator_of_surjective"
679 ],
680 "acceptance_provenance": [
681 {
682 "semantic_review": "semantic_wave2.json",
683 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
684 "reviewed_statement_sha256": "dd991bd6ce28ee7f94302ed9dd3fb00719cf26de3b6ccf740f441b118bdd8e54",
685 "reviewed_proof_sha256": "fa4d07649341df5b1b843588b8b759632a004d48a33365a9d64c82b41ba54796",
686 "verdict": "proved as written"
687 }
688 ],
689 "remaining_work": null
690 },
691 {
692 "ordinal": 19,
693 "stable_id": "PSC-I-T-COMPLETED-PRESENTATION",
694 "kind": "theorem",
695 "title": "Completion of a dense free presentation",
696 "status": "covered",
697 "old_contract_rows": [
698 1
699 ],
700 "mapped_declarations": [
701 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange",
702 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange_apply_mk",
703 "Komorebi.SeedRankBasics.generates_singleton_iff_surjective",
704 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense",
705 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_intertwines",
706 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_unique",
707 "PaperIRelease.TopologyDenseCompletion.transportedUniformity",
708 "PaperIRelease.TopologyDenseCompletion.transported_isUniformInducing",
709 "PaperIRelease.TopologyDenseCompletion.transported_locallyConvex",
710 "PaperIRelease.TopologyDenseCompletion.transported_uniformAddGroup",
711 "PaperIRelease.TopologyDenseCompletion.transported_uniformConstSMul",
712 "PaperIRelease.TopologyDenseCompletion.completionLinearEquiv",
713 "PaperIRelease.TopologyDenseCompletion.completionLinearEquiv_unique",
714 "PaperIRelease.TopologyDenseCompletion.presentation_action_uniformContinuous",
715 "PaperIRelease.TopologyDenseCompletion.completedAction_apply_coe",
716 "PaperIRelease.TopologyDenseCompletion.completion_intertwines",
717 "PaperIRelease.TopologyDenseCompletion.quotientEmbedding_injective",
718 "PaperIRelease.TopologyDenseCompletion.completedQuotientEquiv",
719 "PaperIRelease.TopologyDenseCompletion.completedQuotientEquiv_apply_mk",
720 "PaperIRelease.TopologyDenseCompletion.dense_free_evaluation",
721 "PaperIRelease.TopologyDenseCompletion.completedFreePresentation"
722 ],
723 "scope_and_remaining_work": "The actual finite free evaluation A^(Fin r)→X is used, its actual kernel is quotiented, and UniformSpace.comap pulls back the target uniformity to that quotient. Dense range is derived from dense module span. AbstractCompletion.compareEquiv supplies a uniform equivalence, linearity over A is proved by completion density, and uniqueness follows from equality on the dense image. Constant action uniform continuity is inherited through the actual quotient embedding; scalar action on Completion is the usual extension. The local-convexity pullback lemma uses ordered scalars (in particular ℝ, including underlying real spaces for complex spaces); the stronger uniform-module completion proof does not require local convexity.",
724 "current_source": {
725 "file": "editable/traditional/02_foundations.tex",
726 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
727 "kind": "theorem",
728 "record_id_line": 348,
729 "statement_start_line": 333,
730 "statement_end_line": 347,
731 "statement_sha256": "23708a85485e7162e31b64ee0a8fb316bec231a1aadaa0e00f0a08f70dfe1721",
732 "proof_start_line": 351,
733 "proof_end_line": 353,
734 "proof_sha256": "e4a08162def2200ac98935c9f8649156b6c41bfe942d833276464f0602af4983"
735 },
736 "statement": "\\begin{theorem}[Completion of a dense free presentation]\n\\label{thm:completed-presentation}\nLet $X$ be a complete Hausdorff locally convex space, let $\\A\\subseteq\\End(X)$ act continuously, and let $S=(s_1,\\ldots,s_r)$. Put\n\\[\nE:=\\A^{\\oplus r}/N_S\n\\]\nalgebraically and let $j:E\\to X$ be the injective map induced by $\\varepsilon_S$. Give $E$ the locally convex topology transported from $j(E)$, equivalently the topology generated by\n\\[\n\\widetilde p(e):=p(j(e))\n\\]\nfor the continuous seminorms $p$ on $X$. Every $a\\in\\A$ acts continuously on $E$ in the transported topology and therefore extends to $\\widehat E$. If $\\Hull_{\\A}(S)$ is dense in $X$, then $j$ extends uniquely to a topological isomorphism of modules over $\\A$\n\\[\n\\boxed{\\widehat E\\cong X.}\n\\]\n\\end{theorem}",
737 "source_proof": "\\begin{proof}\nBy \\cref{thm:free-presentation}, $j$ is an algebraic isomorphism of modules over $\\A$ from $E$ onto $\\Hull_{\\A}(S)$. The transported seminorms make it a topological isomorphism onto that subspace. For $a\\in\\A$ and a continuous seminorm $p$ on $X$, continuity of $a:X\\to X$ supplies continuous seminorms controlling $p(aj(e))$, so the induced action on $E$ is continuous and extends uniquely to $\\widehat E$. The subspace is dense by hypothesis. A Cauchy net in $E$ is mapped to a Cauchy net in $X$, which converges because $X$ is complete. This defines a continuous extension $\\widehat j:\\widehat E\\to X$. For any $x\\in X$, choose a net $(e_\\lambda)$ in $E$ with $j(e_\\lambda)\\to x$. The transported topology makes $(e_\\lambda)$ Cauchy, so it converges to some $\\xi\\in\\widehat E$, and continuity gives $\\widehat j(\\xi)=x$. Thus $\\widehat j$ is surjective. If $\\widehat j(\\xi)=0$, represent $\\xi$ by a Cauchy net $(e_\\lambda)$. Then $j(e_\\lambda)\\to0$, so every defining seminorm $\\widetilde p(e_\\lambda)$ tends to zero. Thus $e_\\lambda\\to0$ in the completion and $\\xi=0$. Hence $\\widehat j$ is bijective. Its inverse is continuous by the universal property of completion, or directly because the topology on $E$ was transported from $X$.\n\\end{proof}",
738 "new_block_contribution": [],
739 "acceptance_provenance": [
740 {
741 "semantic_review": "semantic_analytic.json",
742 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
743 "reviewed_statement_sha256": "23708a85485e7162e31b64ee0a8fb316bec231a1aadaa0e00f0a08f70dfe1721",
744 "reviewed_proof_sha256": "e4a08162def2200ac98935c9f8649156b6c41bfe942d833276464f0602af4983",
745 "verdict": "proved as written"
746 }
747 ],
748 "remaining_work": null
749 },
750 {
751 "ordinal": 20,
752 "stable_id": "PSC-I-D-SEED-RANK",
753 "kind": "definition",
754 "title": "Seed rank",
755 "status": "definition_represented",
756 "old_contract_rows": [
757 3,
758 31
759 ],
760 "mapped_declarations": [
761 "Komorebi.SeedRankOperations.cyclic_iff_seedRank_eq_one",
762 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.BilateralShiftRepresentation.concretePower_apply",
763 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.concreteBilateralShift_topologicallyCyclic",
764 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.concreteBilateralShift_seedCardinality_isLeast",
765 "PaperIRelease.TopologyTransport.topologicalSeedRank",
766 "PaperIRelease.TopologyTransport.exists_generating_set_card_eq_seedRank",
767 "PaperIRelease.TopologySemilinearRank.Generates",
768 "PaperIRelease.TopologySemilinearRank.moduleSeedRank",
769 "PaperIRelease.TopologySemilinearRank.rank_attained",
770 "PaperIRelease.TopologySemilinearRank.moduleSeedRank_eq_operatorSeedRank"
771 ],
772 "scope_and_remaining_work": "The operator hull is the scalar span of actual action values; for an acting algebra this is AS. The cardinal infimum ranges over all densely generating subsets, and existence of a minimum is proved from identity membership and cardinal well ordering. Empty generators and infinite cardinals are included. The direct moduleSeedRank definition uses Dense of the literal acting-ring module span, and a proved equality identifies it with the integer-linear operator-family definition even for a nonfaithful action.",
773 "current_source": {
774 "file": "editable/traditional/02_foundations.tex",
775 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
776 "kind": "definition",
777 "record_id_line": 382,
778 "statement_start_line": 370,
779 "statement_end_line": 381,
780 "statement_sha256": "81a23b39cf2b6194ae8b0bc84ac270ee5ef86e4703b5fb5411f5fa9519aafae9",
781 "proof_start_line": null,
782 "proof_end_line": null,
783 "proof_sha256": null
784 },
785 "statement": "\\begin{definition}[Seed rank]\n\\label{def:seed-rank}\nFor a unital left module $M$ over $A$, define\n\\[\n\\srank_A(M):=\\min\\{|S|:AS=M\\}.\n\\]\nFor a topological module $(X,\\tau)$, define\n\\[\n\\srank_{\\A,\\tau}(X):=\\min\\{|S|:\\cl{\\A S}^{\\,\\tau}=X\\}.\n\\]\nSet $\\srank_A(0)=0$. When the value is finite, it is the minimal number of module generators.\n\\end{definition}",
786 "source_proof": null,
787 "new_block_contribution": [],
788 "acceptance_provenance": [
789 {
790 "semantic_review": "semantic_analytic.json",
791 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
792 "reviewed_statement_sha256": "81a23b39cf2b6194ae8b0bc84ac270ee5ef86e4703b5fb5411f5fa9519aafae9",
793 "reviewed_proof_sha256": null,
794 "verdict": "proved as written"
795 }
796 ],
797 "remaining_work": null
798 },
799 {
800 "ordinal": 21,
801 "stable_id": "PSC-I-P-CYCLIC-CHARACTERIZATION",
802 "kind": "proposition",
803 "title": "Cyclic characterization",
804 "status": "covered",
805 "old_contract_rows": [
806 3
807 ],
808 "mapped_declarations": [
809 "Komorebi.SeedRankOperations.cyclic_iff_seedRank_eq_one"
810 ],
811 "scope_and_remaining_work": "The nontriviality hypothesis agrees with the current publication.",
812 "current_source": {
813 "file": "editable/traditional/02_foundations.tex",
814 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
815 "kind": "proposition",
816 "record_id_line": 393,
817 "statement_start_line": 384,
818 "statement_end_line": 392,
819 "statement_sha256": "de62f3fddcf5cefb429536723a8933a119585faa47b6f2e51745ea055ae6a9cc",
820 "proof_start_line": 396,
821 "proof_end_line": 398,
822 "proof_sha256": "abd8197daa4a9754c47420ab264bb5d650e3a85cda91f5ec42cf0e20c81a5c5d"
823 },
824 "statement": "\\begin{proposition}[Cyclic characterization]\n\\label{prop:cyclic-characterization}\nFor $M\\neq0$,\n\\[\n\\srank_A(M)=1\n\\quad\\Longleftrightarrow\\quad\nM\\text{ contains a cyclic vector}.\n\\]\n\\end{proposition}",
825 "source_proof": "\\begin{proof}\nBoth conditions say that there is $m\\in M$ with $Am=M$.\n\\end{proof}",
826 "new_block_contribution": [],
827 "acceptance_provenance": [
828 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
829 ]
830 },
831 {
832 "ordinal": 22,
833 "stable_id": "PSC-I-P-ACTING-ALGEBRA-MONOTONICITY",
834 "kind": "proposition",
835 "title": "Monotonicity under enlargement of the acting algebra",
836 "status": "covered",
837 "old_contract_rows": [
838 4
839 ],
840 "mapped_declarations": [
841 "Komorebi.SeedRankOperations.seedRank_le_of_surjective_semilinear",
842 "PaperIRelease.TopologyTransport.topologicalSeedRank",
843 "PaperIRelease.TopologyTransport.exists_generating_set_card_eq_seedRank",
844 "PaperIRelease.TopologyTransport.seedRank_mono_algebra"
845 ],
846 "scope_and_remaining_work": "For unital operator algebras, actual family inclusion implies operator-hull inclusion. Closure monotonicity transports a minimum generating subset. This exactly supplies the topological clause, combined with the previously mapped algebraic semilinear rank theorem.",
847 "current_source": {
848 "file": "editable/traditional/02_foundations.tex",
849 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
850 "kind": "proposition",
851 "record_id_line": 408,
852 "statement_start_line": 400,
853 "statement_end_line": 407,
854 "statement_sha256": "115d0049078122c1d6c43bb8ec53237a6efae32ef13330b8e31d8bf77e68eb56",
855 "proof_start_line": 411,
856 "proof_end_line": 413,
857 "proof_sha256": "07d505d074ec7b49eb4dc8a0ac2f3c3f87863da5df5aec1941cadda3a243b199"
858 },
859 "statement": "\\begin{proposition}[Monotonicity under enlargement of the acting algebra]\n\\label{prop:acting-algebra-monotonicity}\nLet $B\\subseteq A\\subseteq\\End_{\\bbk}(X)$ be unital operator algebras with the same identity. Then\n\\[\n\\boxed{\\srank_A(X)\\leq\\srank_B(X).}\n\\]\nThe same statement holds for topological seed rank when both algebras act continuously.\n\\end{proposition}",
860 "source_proof": "\\begin{proof}\nFor every $S\\subseteq X$, one has $BS\\subseteq AS$. Hence every generating family over $B$ is also a generating family over $A$. Taking closures gives the topological statement.\n\\end{proof}",
861 "new_block_contribution": [],
862 "acceptance_provenance": [
863 {
864 "semantic_review": "semantic_analytic.json",
865 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
866 "reviewed_statement_sha256": "115d0049078122c1d6c43bb8ec53237a6efae32ef13330b8e31d8bf77e68eb56",
867 "reviewed_proof_sha256": "07d505d074ec7b49eb4dc8a0ac2f3c3f87863da5df5aec1941cadda3a243b199",
868 "verdict": "proved as written"
869 }
870 ],
871 "remaining_work": null
872 },
873 {
874 "ordinal": 23,
875 "stable_id": "PSC-I-T-QUOTIENT-MONOTONICITY",
876 "kind": "theorem",
877 "title": "Quotient monotonicity",
878 "status": "covered",
879 "old_contract_rows": [
880 4,
881 29
882 ],
883 "mapped_declarations": [
884 "Komorebi.SeedRankOperations.seedRank_le_of_surjective_semilinear",
885 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.dense_operatorHull_image",
886 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.map_packetRange_eq",
887 "PaperIRelease.TopologySemilinearRank.rank_le_of_continuous_additive_surjective",
888 "PaperIRelease.TopologySemilinearRank.rank_le_of_continuous_semilinear_surjective",
889 "PaperIRelease.TopologySemilinearRank.generates_image",
890 "PaperIRelease.TopologySemilinearRank.moduleSeedRank_eq_operatorSeedRank"
891 ],
892 "scope_and_remaining_work": "The new theorem starts with the printed arbitrary unital ring homomorphism, additive map, scalar compatibility, continuity, and surjectivity. Span induction maps the actual source module span into the target module span; continuous surjectivity transports density. An attained minimum generating set maps to an image of no larger cardinality. No ring-homomorphism surjectivity or faithfulness is assumed. The direct module-span rank is proved equal to the earlier operator-family rank using integer-linear acting operators. This supplies the exact topological clause alongside the separately accepted algebraic theorem.",
893 "current_source": {
894 "file": "editable/traditional/02_foundations.tex",
895 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
896 "kind": "theorem",
897 "record_id_line": 441,
898 "statement_start_line": 421,
899 "statement_end_line": 440,
900 "statement_sha256": "ccde1353d26e9f7bc7aef341c35be5304729daa69f8f42274ab70e2f9d7016a7",
901 "proof_start_line": 445,
902 "proof_end_line": 457,
903 "proof_sha256": "cfe9e670e142ca03f0f871004e40135e5df870db4c1f14b2579e0e1c893b56ed"
904 },
905 "statement": "\\begin{theorem}[Quotient monotonicity]\n\\label{thm:quotient-monotonicity}\nLet $\\varphi:A\\to B$ be a unital ring homomorphism and let\n\\[\n\\Phi:M\\twoheadrightarrow N\n\\]\nbe a surjective additive map satisfying\n\\[\n\\Phi(am)=\\varphi(a)\\Phi(m)\n\\qquad(a\\in A,\\ m\\in M).\n\\]\nThen\n\\[\n\\boxed{\\srank_B(N)\\leq\\srank_A(M).}\n\\]\nIf $M$ and $N$ are topological modules, $\\Phi$ is continuous and surjective, and the actions are continuous, then\n\\[\n\\srank_{B,\\tau_N}(N)\\leq\\srank_{A,\\tau_M}(M).\n\\]\n\\end{theorem}",
906 "source_proof": "\\begin{proof}\nLet $S$ generate $M$. For every $n\\in N$, choose $m\\in M$ with $\\Phi(m)=n$ and write $m=\\sum_ia_is_i$. Then\n\\[\nn=\\sum_i\\varphi(a_i)\\Phi(s_i),\n\\]\nso $\\Phi(S)$ generates $N$.\n\nFor the topological statement, let $S$ generate $M$ densely. Given $n=\\Phi(m)$, choose a net $m_\\lambda\\in AS$ converging to $m$. Continuity gives\n\\[\n\\Phi(m_\\lambda)\\longrightarrow n,\n\\]\nand every $\\Phi(m_\\lambda)$ lies in $B\\Phi(S)$. Thus $B\\Phi(S)$ is dense in $N$.\n\\end{proof}",
907 "new_block_contribution": [],
908 "acceptance_provenance": [
909 {
910 "semantic_review": "semantic_analytic.json",
911 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
912 "reviewed_statement_sha256": "ccde1353d26e9f7bc7aef341c35be5304729daa69f8f42274ab70e2f9d7016a7",
913 "reviewed_proof_sha256": "cfe9e670e142ca03f0f871004e40135e5df870db4c1f14b2579e0e1c893b56ed",
914 "verdict": "proved as written"
915 }
916 ],
917 "remaining_work": null
918 },
919 {
920 "ordinal": 24,
921 "stable_id": "PSC-I-C-ISOMORPHISM-INVARIANCE",
922 "kind": "corollary",
923 "title": "Isomorphism invariance",
924 "status": "covered",
925 "old_contract_rows": [
926 5,
927 28
928 ],
929 "mapped_declarations": [
930 "Komorebi.SeedRankOperations.seedRank_imageAlgebra_eq",
931 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
932 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
933 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
934 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
935 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
936 "PaperIRelease.TopologySemilinearRank.rank_eq_of_continuous_semilinear_equiv",
937 "PaperIRelease.TopologySemilinearRank.moduleSeedRank_eq_operatorSeedRank"
938 ],
939 "scope_and_remaining_work": "A continuous semilinear module equivalence with continuous inverse preserves the direct module-span cardinal rank, by applying the exact semilinear surjection inequality in both directions. The same-ring case uses identity ring maps. The earlier bidirectional operator-family theorem covers intertwining operator-algebra isomorphisms, and the new rank equality bridges these definitions. This supplies the exact topological clause alongside separately accepted algebraic invariance.",
940 "current_source": {
941 "file": "editable/traditional/02_foundations.tex",
942 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
943 "kind": "corollary",
944 "record_id_line": 468,
945 "statement_start_line": 464,
946 "statement_end_line": 467,
947 "statement_sha256": "fc01d1773cdeebcaba8346b323df6df33d48e0f6f7adda763b14307b2e902ca1",
948 "proof_start_line": 471,
949 "proof_end_line": 473,
950 "proof_sha256": "0d13556e1e0dbbb063292ae89f682ad97f5d48a0df5375c4e8f108ea40f83253"
951 },
952 "statement": "\\begin{corollary}[Isomorphism invariance]\n\\label{cor:isomorphism-invariance}\nAn intertwining module isomorphism preserves algebraic seed rank. A topological intertwining isomorphism preserves topological seed rank.\n\\end{corollary}",
953 "source_proof": "\\begin{proof}\nApply \\cref{thm:quotient-monotonicity} to the isomorphism and its inverse.\n\\end{proof}",
954 "new_block_contribution": [],
955 "acceptance_provenance": [
956 {
957 "semantic_review": "semantic_analytic.json",
958 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
959 "reviewed_statement_sha256": "fc01d1773cdeebcaba8346b323df6df33d48e0f6f7adda763b14307b2e902ca1",
960 "reviewed_proof_sha256": "0d13556e1e0dbbb063292ae89f682ad97f5d48a0df5375c4e8f108ea40f83253",
961 "verdict": "proved as written"
962 }
963 ],
964 "remaining_work": null
965 },
966 {
967 "ordinal": 25,
968 "stable_id": "PSC-I-T-EXACT-SEQUENCE-BOUNDS",
969 "kind": "theorem",
970 "title": "Short exact sequence bounds",
971 "status": "covered",
972 "old_contract_rows": [
973 4
974 ],
975 "mapped_declarations": [
976 "Komorebi.SeedRankOperations.seedRank_le_of_surjective_semilinear",
977 "Komorebi.PaperIRelease.shortExact_seedRank_bounds",
978 "Komorebi.PaperIRelease.seedRank_le_add_of_exact"
979 ],
980 "scope_and_remaining_work": "Actual linear f,g with injectivity, range f=ker g and surjectivity; canonical seedRank=top.spanRank. Minimum generators of N are lifted and combined with images of minimum generators of L. The proof derives generation of M and the cardinal upper bound; the lower bound is surjective monotonicity. No splitting, finite generation or desired bound is assumed.",
981 "current_source": {
982 "file": "editable/traditional/02_foundations.tex",
983 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
984 "kind": "theorem",
985 "record_id_line": 489,
986 "statement_start_line": 475,
987 "statement_end_line": 488,
988 "statement_sha256": "2e4e5af8593f49860709b8eb0b2d2b88577c0bc765a8b9e0c086441aaf142348",
989 "proof_start_line": 492,
990 "proof_end_line": 498,
991 "proof_sha256": "af99949af408c8245814df04aaae9190c5bb4acd7504aca665d7a2728e461377"
992 },
993 "statement": "\\begin{theorem}[Short exact sequence bounds]\n\\label{thm:exact-sequence-bounds}\nLet\n\\[\n0\\longrightarrow L\\xrightarrow{\\iota}M\\xrightarrow{\\pi}N\\longrightarrow0\n\\]\nbe a short exact sequence of left modules over $A$. Then\n\\[\n\\boxed{\n\\srank_A(N)\\leq\\srank_A(M)\n\\leq\\srank_A(L)+\\srank_A(N).\n}\n\\]\n\\end{theorem}",
994 "source_proof": "\\begin{proof}\nThe lower inequality is quotient monotonicity. For the upper inequality, choose generating families $(\\ell_i)_{i\\in I}$ of $L$ and $(n_j)_{j\\in J}$ of $N$ with the least possible cardinalities. Choose a lift $m_j\\in M$ of each $n_j$. For any $m\\in M$, a finite subset $F\\subseteq J$ and coefficients $a_j\\in A$ give\n\\[\n\\pi(m)=\\sum_{j\\in F}a_jn_j.\n\\]\nThe vector $m-\\sum_{j\\in F}a_jm_j$ belongs to $\\iota(L)$, so it is a finite linear combination of the $\\iota(\\ell_i)$. The union of the lifted family and the image of the generating family of $L$ therefore generates $M$. Its cardinality is at most $|I|+|J|$, giving the upper bound. This argument uses finite sums for each vector even when either generating family is infinite.\n\\end{proof}",
995 "new_block_contribution": [],
996 "acceptance_provenance": [
997 {
998 "semantic_review": "semantic_wave1.json",
999 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
1000 "reviewed_statement_sha256": "2e4e5af8593f49860709b8eb0b2d2b88577c0bc765a8b9e0c086441aaf142348",
1001 "reviewed_proof_sha256": "af99949af408c8245814df04aaae9190c5bb4acd7504aca665d7a2728e461377",
1002 "verdict": "proved as written"
1003 }
1004 ],
1005 "remaining_work": null
1006 },
1007 {
1008 "ordinal": 26,
1009 "stable_id": "PSC-I-P-SHARED-DIRECT-SUM",
1010 "kind": "proposition",
1011 "title": "Shared direct-sum bounds",
1012 "status": "covered",
1013 "old_contract_rows": [
1014 4,
1015 6
1016 ],
1017 "mapped_declarations": [
1018 "Komorebi.SeedRankOperations.seedRank_le_of_surjective_semilinear",
1019 "Komorebi.SeedRankOperations.seedRank_prod_le_add",
1020 "Komorebi.PaperIRelease.SeedOperations.shared_direct_sum_bounds"
1021 ],
1022 "scope_and_remaining_work": "The actual diagonal A-action on M×N realizes the binary direct sum. Projection surjectivity gives both lower bounds; inclusion of generating sets in the two factors yields cardinal addition for the upper bound.",
1023 "current_source": {
1024 "file": "editable/traditional/02_foundations.tex",
1025 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1026 "kind": "proposition",
1027 "record_id_line": 517,
1028 "statement_start_line": 506,
1029 "statement_end_line": 516,
1030 "statement_sha256": "613d0bed5e9f4890c069d8d37670d0b8c9cb3a981df5bceb95686c08dc5638c6",
1031 "proof_start_line": 520,
1032 "proof_end_line": 522,
1033 "proof_sha256": "0ca72f2b94c20762dfe579abf8fa4a9c8e1138408ff54b2b7920d01fe3afecb0"
1034 },
1035 "statement": "\\begin{proposition}[Direct sum bounds for a shared action]\n\\label{prop:shared-direct-sum}\nIf one algebra $A$ acts diagonally on $M\\oplus N$, then\n\\[\n\\boxed{\n\\max\\{\\srank_A(M),\\srank_A(N)\\}\n\\leq\\srank_A(M\\oplus N)\n\\leq\\srank_A(M)+\\srank_A(N).\n}\n\\]\n\\end{proposition}",
1036 "source_proof": "\\begin{proof}\nThe coordinate projections are intertwining quotients and give the lower bound. For the upper bound, embed minimal generating families of $M$ as $(m_i,0)$ and of $N$ as $(0,n_j)$. Their union generates by finite sums and has cardinality at most the sum of the two seed ranks.\n\\end{proof}",
1037 "new_block_contribution": [],
1038 "acceptance_provenance": [
1039 {
1040 "semantic_review": "semantic_wave2.json",
1041 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
1042 "reviewed_statement_sha256": "613d0bed5e9f4890c069d8d37670d0b8c9cb3a981df5bceb95686c08dc5638c6",
1043 "reviewed_proof_sha256": "0ca72f2b94c20762dfe579abf8fa4a9c8e1138408ff54b2b7920d01fe3afecb0",
1044 "verdict": "proved as written"
1045 }
1046 ],
1047 "remaining_work": null
1048 },
1049 {
1050 "ordinal": 27,
1051 "stable_id": "PSC-I-T-INDEPENDENT-DIRECT-SUM",
1052 "kind": "theorem",
1053 "title": "Independent direct-sum formula",
1054 "status": "covered",
1055 "old_contract_rows": [
1056 7
1057 ],
1058 "mapped_declarations": [
1059 "Komorebi.SeedRankOperations.seedRank_prod_productAlgebra"
1060 ],
1061 "scope_and_remaining_work": "Actual product-algebra action and the maximum formula are proved.",
1062 "current_source": {
1063 "file": "editable/traditional/02_foundations.tex",
1064 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1065 "kind": "theorem",
1066 "record_id_line": 539,
1067 "statement_start_line": 524,
1068 "statement_end_line": 538,
1069 "statement_sha256": "b6f497941ccc0be69e3780d70cd1d79cb39363dedb591b81527183724fc58748",
1070 "proof_start_line": 542,
1071 "proof_end_line": 554,
1072 "proof_sha256": "4487867c7f9d8f9712056afdb827df33137d9ec6ebb3c32a14360d1e3b5cb194"
1073 },
1074 "statement": "\\begin{theorem}[Direct sum formula for independent actions]\n\\label{thm:independent-direct-sum}\nLet $M$ be a left module over $A$ and $N$ a left module over $B$. Under the action of the product algebra\n\\[\n(a,b)(m,n)=(am,bn),\n\\]\none has\n\\[\n\\boxed{\n\\srank_{A\\times B}(M\\oplus N)\n=\n\\max\\{\\srank_A(M),\\srank_B(N)\\}.\n}\n\\]\n\\end{theorem}",
1075 "source_proof": "\\begin{proof}\nThe coordinate projections give the lower bound. Let $r$ be the maximum on the right, regarded as a cardinal, and choose an index set $I$ of cardinality $r$. Choose minimal generating families for $M$ and $N$, index them by subsets of $I$, and fill any unused positions with zeros. Write the resulting families as $(m_i)_{i\\in I}$ and $(n_i)_{i\\in I}$. For any $m\\in M$ and $n\\in N$, their finite expressions involve a common finite subset $F\\subseteq I$ after zero coefficients have been added. Thus\n\\[\nm=\\sum_{i\\in F}a_im_i,\n\\qquad\nn=\\sum_{i\\in F}b_in_i,\n\\]\nand\n\\[\n(m,n)=\\sum_{i\\in F}(a_i,b_i)(m_i,n_i).\n\\]\nThe family of paired vectors generates and has cardinality at most $r$. The empty family handles the case in which both modules are zero.\n\\end{proof}",
1076 "new_block_contribution": [],
1077 "acceptance_provenance": [
1078 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1079 ]
1080 },
1081 {
1082 "ordinal": 28,
1083 "stable_id": "PSC-I-P-TENSOR-BOUND",
1084 "kind": "proposition",
1085 "title": "Tensor-product bound",
1086 "status": "covered",
1087 "old_contract_rows": [
1088 8
1089 ],
1090 "mapped_declarations": [
1091 "Komorebi.SeedRankOperations.tensorOperator_smul_tmul",
1092 "Komorebi.SeedRankOperations.seedRank_tensorProduct_le_mul"
1093 ],
1094 "scope_and_remaining_work": "The genuine tensor-product action and cardinal product upper bound are proved.",
1095 "current_source": {
1096 "file": "editable/traditional/02_foundations.tex",
1097 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1098 "kind": "proposition",
1099 "record_id_line": 568,
1100 "statement_start_line": 557,
1101 "statement_end_line": 567,
1102 "statement_sha256": "b1b31d79b3ffc3d73319922c4a3f2f1dde12a0f774ceda1ae29227e9c63a6d4e",
1103 "proof_start_line": 571,
1104 "proof_end_line": 577,
1105 "proof_sha256": "7e578a5bdea52bf725e05edb164502795257a676b5f8ecedf5703cb0fe104f64"
1106 },
1107 "statement": "\\begin{proposition}[Tensor product bound]\n\\label{prop:tensor-bound}\nLet $A$ and $B$ be unital algebras over $\\bbk$, let $M$ be a left module over $A$, and let $N$ be a left module over $B$. Under the natural $A\\otimes_{\\bbk}B$ action,\n\\[\n\\boxed{\n\\srank_{A\\otimes B}(M\\otimes_{\\bbk}N)\n\\leq\n\\srank_A(M)\\srank_B(N).\n}\n\\]\n\\end{proposition}",
1108 "source_proof": "\\begin{proof}\nChoose minimal generating families $(m_i)_{i\\in I}$ for $M$ and $(n_j)_{j\\in J}$ for $N$. Each vector in either module is a finite linear combination of its generators. Bilinearity therefore expresses every simple tensor as a finite sum of terms\n\\[\n(am_i)\\otimes(bn_j)=(a\\otimes b)(m_i\\otimes n_j).\n\\]\nSimple tensors span the algebraic tensor product, so the family $(m_i\\otimes n_j)_{(i,j)\\in I\\times J}$ generates it. This family has cardinality at most $|I||J|$, which is the claimed bound. The proof also covers an empty generating family, when the corresponding module and tensor product are zero.\n\\end{proof}",
1109 "new_block_contribution": [],
1110 "acceptance_provenance": [
1111 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1112 ]
1113 },
1114 {
1115 "ordinal": 29,
1116 "stable_id": "PSC-I-B-TENSOR-NONEQUALITY",
1117 "kind": "boundary",
1118 "title": "Tensor rank is not generally multiplicative",
1119 "status": "scope_boundary",
1120 "old_contract_rows": [
1121 8
1122 ],
1123 "mapped_declarations": [
1124 "Komorebi.SeedRankOperations.tensorOperator_smul_tmul",
1125 "Komorebi.SeedRankOperations.seedRank_tensorProduct_le_mul"
1126 ],
1127 "scope_and_remaining_work": "No universal tensor equality is exported. This boundary is not counted as a positive theorem.",
1128 "current_source": {
1129 "file": "editable/traditional/02_foundations.tex",
1130 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1131 "kind": "boundary",
1132 "record_id_line": 588,
1133 "statement_start_line": 584,
1134 "statement_end_line": 587,
1135 "statement_sha256": "9328501ddd219b9e0defabc4233a3e0c6437a8890f6920385de876bc5223e204",
1136 "proof_start_line": null,
1137 "proof_end_line": null,
1138 "proof_sha256": null
1139 },
1140 "statement": "\\begin{boundary}[Tensor rank is not generally multiplicative]\n\\label{bnd:tensor-nonequality}\nThe inequality in \\cref{prop:tensor-bound} can be strict. Equality requires additional hypotheses on the acting algebras and their module multiplicities and is not exported as a universal law.\n\\end{boundary}",
1141 "source_proof": null,
1142 "new_block_contribution": []
1143 },
1144 {
1145 "ordinal": 30,
1146 "stable_id": "PSC-I-P-SCALAR-EXTENSION-MONOTONICITY",
1147 "kind": "proposition",
1148 "title": "Scalar-extension monotonicity",
1149 "status": "covered",
1150 "old_contract_rows": [
1151 8
1152 ],
1153 "mapped_declarations": [
1154 "Komorebi.SeedRankOperations.tensorOperator_smul_tmul",
1155 "Komorebi.SeedRankOperations.seedRank_tensorProduct_le_mul",
1156 "Komorebi.PaperIRelease.ScalarExtension.tensorOperator_smul_tmul",
1157 "Komorebi.PaperIRelease.ScalarExtension.one_tmul_generates",
1158 "Komorebi.PaperIRelease.ScalarExtension.scalar_extension_seedRank_le"
1159 ],
1160 "scope_and_remaining_work": "The source constructs the natural (K⊗k A)-action on K⊗k M using commuting tensor-factor actions and proves (c⊗a)•(d⊗m)=(c*d)⊗(a•m). Tensor and span induction prove the literal image (1⊗-) of any generating set generates. Minimum-cardinality generation plus cardinality of an image gives the rank inequality.",
1161 "current_source": {
1162 "file": "editable/traditional/02_foundations.tex",
1163 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1164 "kind": "proposition",
1165 "record_id_line": 608,
1166 "statement_start_line": 592,
1167 "statement_end_line": 607,
1168 "statement_sha256": "89013f407adf17424cd55db8a6f59a4a9141e34d39a1d350731d9e18d542a3e4",
1169 "proof_start_line": 612,
1170 "proof_end_line": 624,
1171 "proof_sha256": "696ebaeacd42328424056660c8afe25c3886d9df4b0b2e093ad4b995357c50a1"
1172 },
1173 "statement": "\\begin{proposition}[Monotonicity under extension of scalars]\n\\label{prop:scalar-extension}\nLet \\(K/\\bbk\\) be a field extension, let \\(A\\) be a unital algebra over \\(\\bbk\\), and let \\(M\\) be a finitely generated left module over \\(A\\). Set\n\\[\nA_K:=K\\otimes_{\\bbk}A,\n\\qquad\nM_K:=K\\otimes_{\\bbk}M.\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{A_K}(M_K)\\leq\\srank_A(M).\n}\n\\]\nEvery generating family \\(S\\subseteq M\\) extends to the generating family \\(1\\otimes S\\subseteq M_K\\).\n\\end{proposition}",
1174 "source_proof": "\\begin{proof}\nLet \\(S=\\{s_1,\\ldots,s_r\\}\\) generate \\(M\\). Every pure tensor \\(\\lambda\\otimes m\\in M_K\\) has an expression\n\\[\nm=\\sum_{i=1}^r a_is_i,\n\\]\nso\n\\[\n\\lambda\\otimes m\n=\n\\sum_{i=1}^r(\\lambda\\otimes a_i)(1\\otimes s_i).\n\\]\nPure tensors span \\(M_K\\), hence \\(1\\otimes S\\) generates \\(M_K\\) over \\(A_K\\). Taking the minimum over \\(S\\) proves the inequality.\n\\end{proof}",
1175 "new_block_contribution": [],
1176 "acceptance_provenance": [
1177 {
1178 "semantic_review": "semantic_wave2.json",
1179 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
1180 "reviewed_statement_sha256": "89013f407adf17424cd55db8a6f59a4a9141e34d39a1d350731d9e18d542a3e4",
1181 "reviewed_proof_sha256": "696ebaeacd42328424056660c8afe25c3886d9df4b0b2e093ad4b995357c50a1",
1182 "verdict": "proved as written"
1183 }
1184 ],
1185 "remaining_work": null
1186 },
1187 {
1188 "ordinal": 31,
1189 "stable_id": "PSC-I-D-SEED-RANK-WITNESS",
1190 "kind": "definition",
1191 "title": "Seed-rank witness",
1192 "status": "documentary",
1193 "old_contract_rows": [
1194 4
1195 ],
1196 "mapped_declarations": [
1197 "Komorebi.SeedRankOperations.seedRank_le_of_surjective_semilinear"
1198 ],
1199 "scope_and_remaining_work": "A named use of the quotient lower bound, not a new proof target.",
1200 "current_source": {
1201 "file": "editable/traditional/02_foundations.tex",
1202 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1203 "kind": "definition",
1204 "record_id_line": 636,
1205 "statement_start_line": 632,
1206 "statement_end_line": 635,
1207 "statement_sha256": "81381ceea3d420ff6ba2f0b9378863598308dc392cf597cc1dc0b4d7f84f7cc0",
1208 "proof_start_line": null,
1209 "proof_end_line": null,
1210 "proof_sha256": null
1211 },
1212 "statement": "\\begin{definition}[Witness for seed rank]\n\\label{def:seed-rank-witness}\nA surjective intertwiner $\\Phi:M\\twoheadrightarrow N$ is a witness for seed rank of order $r$ when $\\srank(N)=r$. It certifies $\\srank(M)\\geq r$.\n\\end{definition}",
1213 "source_proof": null,
1214 "new_block_contribution": []
1215 },
1216 {
1217 "ordinal": 32,
1218 "stable_id": "PSC-I-P-SCALAR-QUOTIENT-WITNESS",
1219 "kind": "proposition",
1220 "title": "Scalar quotient witness",
1221 "status": "covered",
1222 "old_contract_rows": [
1223 9
1224 ],
1225 "mapped_declarations": [
1226 "Komorebi.SeedRankOperations.finrank_le_seedRank_of_scalar_quotient"
1227 ],
1228 "scope_and_remaining_work": "The character, surjective semilinear map and finite scalar dimension match the printed hypotheses.",
1229 "current_source": {
1230 "file": "editable/traditional/02_foundations.tex",
1231 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1232 "kind": "proposition",
1233 "record_id_line": 649,
1234 "statement_start_line": 638,
1235 "statement_end_line": 648,
1236 "statement_sha256": "92e35ac7e226b34fce4aa0e1e40a198c1273cc439ac42a8ea2f437f75965a632",
1237 "proof_start_line": 652,
1238 "proof_end_line": 654,
1239 "proof_sha256": "5277fd937ec1d401ea9faee19d494b8eac9baa40a4b74f73b4ca3d7c8835d35c"
1240 },
1241 "statement": "\\begin{proposition}[Scalar quotient witness]\n\\label{prop:scalar-quotient}\nLet $M$ be a module over $A$ with coefficient field $\\bbk$. Suppose $M$ has a quotient $N$ of dimension $r$ on which $A$ acts through a character\n\\[\n\\chi:A\\longrightarrow\\bbk.\n\\]\nThen\n\\[\n\\boxed{\\srank_A(M)\\geq r.}\n\\]\n\\end{proposition}",
1242 "source_proof": "\\begin{proof}\nFor every $n\\in N$, the cyclic submodule $An$ is contained in the scalar span $\\bbk n$. Therefore $q$ seed images span a subspace of dimension at most $q$. Generating the quotient of dimension $r$ requires at least $r$ seeds. Quotient monotonicity transfers the lower bound to $M$.\n\\end{proof}",
1243 "new_block_contribution": [],
1244 "acceptance_provenance": [
1245 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1246 ]
1247 },
1248 {
1249 "ordinal": 33,
1250 "stable_id": "PSC-I-CI-NAKAYAMA",
1251 "kind": "classicalinput",
1252 "title": "Nakayama lemma",
1253 "status": "classical_input_integrated",
1254 "old_contract_rows": [],
1255 "mapped_declarations": [
1256 "Komorebi.PaperICompletion7.Nakayama.eq_top_of_sup_module_jacobson",
1257 "Komorebi.PaperICompletion7.Nakayama.eq_top_of_sup_ideal_smul",
1258 "Komorebi.PaperICompletion7.Nakayama.eq_top_of_sup_jacobson_smul"
1259 ],
1260 "scope_and_remaining_work": "The noncommutative input is integrated at the printed finitely generated left-module scope. For any left ideal I contained in Ring.jacobson R, the actual submodule equality N ⊔ I • top = top implies N = top. No commutativity, Noetherian, semisimplicity, finite length, or two-sidedness of I is required for this lifting theorem. The proof derives the radical lifting result from maximal submodules and the ring-radical action inclusion, rather than assuming the desired conclusion.",
1261 "current_source": {
1262 "file": "editable/traditional/02_foundations.tex",
1263 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1264 "kind": "classicalinput",
1265 "record_id_line": 673,
1266 "statement_start_line": 669,
1267 "statement_end_line": 672,
1268 "statement_sha256": "68901a557c34762568711f0e6c8b400485a0c4b66477b8f231e899d9a48f5878",
1269 "proof_start_line": null,
1270 "proof_end_line": null,
1271 "proof_sha256": null
1272 },
1273 "statement": "\\begin{classicalinput}[Nakayama lemma]\n\\label{ci:nakayama}\nLet $A$ be a unital ring, let $J\\subseteq\\Jac(A)$, and let $M$ be a finitely generated left module over $A$. If $M=N+JM$ for a submodule $N\\leq M$, then $M=N$.\n\\end{classicalinput}",
1274 "source_proof": null,
1275 "new_block_contribution": [],
1276 "block07_contribution": [
1277 "Komorebi.PaperICompletion7.Nakayama.eq_top_of_sup_module_jacobson",
1278 "Komorebi.PaperICompletion7.Nakayama.eq_top_of_sup_ideal_smul",
1279 "Komorebi.PaperICompletion7.Nakayama.eq_top_of_sup_jacobson_smul"
1280 ]
1281 },
1282 {
1283 "ordinal": 34,
1284 "stable_id": "PSC-I-T-MODULE-TOP",
1285 "kind": "theorem",
1286 "title": "Generator number is detected modulo the Jacobson radical",
1287 "status": "covered",
1288 "old_contract_rows": [],
1289 "mapped_declarations": [
1290 "Komorebi.PaperICompletion7.Nakayama.IdealTop",
1291 "Komorebi.PaperICompletion7.Nakayama.idealTopMap",
1292 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_apply",
1293 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_smul",
1294 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_surjective",
1295 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_ker",
1296 "Komorebi.PaperICompletion7.Nakayama.map_eq_top_iff",
1297 "Komorebi.PaperICompletion7.ModuleTop.image_generates_iff",
1298 "Komorebi.PaperICompletion7.ModuleTop.family_generates_iff",
1299 "Komorebi.PaperICompletion7.ModuleTop.lifts_generate",
1300 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_idealTop",
1301 "Komorebi.PaperICompletion7.ModuleTop.seedFinrank_eq_idealTop",
1302 "Komorebi.PaperICompletion7.ModuleTop.idealTop_finite",
1303 "Komorebi.PaperICompletion7.ModuleTop.RadicalTop",
1304 "Komorebi.PaperICompletion7.ModuleTop.generates_iff_radicalTop",
1305 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_radicalTop",
1306 "Komorebi.PaperICompletion7.ModuleTop.seedFinrank_eq_radicalTop",
1307 "Komorebi.PaperICompletion7.ModuleTop.quotient_basis_lifts_generate",
1308 "IsSemisimpleRing.isSemisimpleModule"
1309 ],
1310 "scope_and_remaining_work": "The complete generating-set/family equivalence and canonical cardinal seedRank equality are proved for arbitrary unital rings and finitely generated left modules using the literal M / (Jac(R) • top), its actual scalar action over R / Jac(R), and an exact quotient-map kernel. Any lifts of a generating quotient family generate upstairs, and both minimum-cardinality inequalities are established. The natural seedFinrank equality is proved as well. No semisimplicity hypothesis is needed. The descriptive semisimple-top sentence is covered by the pinned general IsSemisimpleRing.isSemisimpleModule instance applied to this actual quotient module. This is not an algorithm for computing the radical or finding generators.",
1311 "current_source": {
1312 "file": "editable/traditional/02_foundations.tex",
1313 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1314 "kind": "theorem",
1315 "record_id_line": 684,
1316 "statement_start_line": 675,
1317 "statement_end_line": 683,
1318 "statement_sha256": "8e3170bbaf10331d7d774260fbdf4b7df0ca59f1c23116f008a1aa5ac2bf07a8",
1319 "proof_start_line": 688,
1320 "proof_end_line": 694,
1321 "proof_sha256": "363690b09ad663ae4020f84117b7f40c45e16fbe87abf4d31b9169eff14a5da3"
1322 },
1323 "statement": "\\begin{theorem}[Generator number is detected modulo the Jacobson radical]\n\\label{thm:module-top}\nLet $A$ be a unital ring, let $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$. A family $S\\subseteq M$ generates $M$ if and only if its image generates $M/JM$ as a module over $A/J$. When $A/J$ is semisimple Artinian, this quotient is the usual semisimple top. No semisimplicity of $A/J$ is required for the general equality. Consequently,\n\\[\n\\boxed{\n\\srank_A(M)=\\srank_{A/J}(M/JM).\n}\n\\]\n\\end{theorem}",
1324 "source_proof": "\\begin{proof}\nIf $S$ generates $M$, its image generates every quotient. Conversely, suppose the image of $S$ generates $M/JM$ and put $N=AS$. Then\n\\[\nM=N+JM.\n\\]\nNakayama's lemma gives $M=N$. Thus the same finite families generate before and after passage to the top, and the minimal cardinalities agree.\n\\end{proof}",
1325 "new_block_contribution": [],
1326 "block07_contribution": [
1327 "Komorebi.PaperICompletion7.Nakayama.IdealTop",
1328 "Komorebi.PaperICompletion7.Nakayama.idealTopMap",
1329 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_apply",
1330 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_smul",
1331 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_surjective",
1332 "Komorebi.PaperICompletion7.Nakayama.idealTopMap_ker",
1333 "Komorebi.PaperICompletion7.Nakayama.map_eq_top_iff",
1334 "Komorebi.PaperICompletion7.ModuleTop.image_generates_iff",
1335 "Komorebi.PaperICompletion7.ModuleTop.family_generates_iff",
1336 "Komorebi.PaperICompletion7.ModuleTop.lifts_generate",
1337 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_idealTop",
1338 "Komorebi.PaperICompletion7.ModuleTop.seedFinrank_eq_idealTop",
1339 "Komorebi.PaperICompletion7.ModuleTop.idealTop_finite",
1340 "Komorebi.PaperICompletion7.ModuleTop.RadicalTop",
1341 "Komorebi.PaperICompletion7.ModuleTop.generates_iff_radicalTop",
1342 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_radicalTop",
1343 "Komorebi.PaperICompletion7.ModuleTop.seedFinrank_eq_radicalTop",
1344 "Komorebi.PaperICompletion7.ModuleTop.quotient_basis_lifts_generate",
1345 "IsSemisimpleRing.isSemisimpleModule"
1346 ],
1347 "acceptance_provenance": [
1348 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1349 ]
1350 },
1351 {
1352 "ordinal": 35,
1353 "stable_id": "PSC-I-C-LOCAL-ALGEBRA-FORMULA",
1354 "kind": "corollary",
1355 "title": "Local algebra formula",
1356 "status": "covered",
1357 "old_contract_rows": [],
1358 "mapped_declarations": [
1359 "Komorebi.PaperICompletion7.LocalResidue.proper_sup_ne_top",
1360 "Komorebi.PaperICompletion7.LocalResidue.maximal_left_ideal_unique",
1361 "Komorebi.PaperICompletion7.LocalResidue.jacobson_isMaximal",
1362 "Komorebi.PaperICompletion7.LocalResidue.residueDivisionRing",
1363 "Komorebi.PaperICompletion7.LocalResidue.seedRank_eq_residue_dimension",
1364 "Komorebi.PaperICompletion7.LocalResidue.seedFinrank_eq_residue_finrank"
1365 ],
1366 "scope_and_remaining_work": "The local formula is proved for noncommutative local rings and finitely generated left modules. From IsLocalRing R the source proves uniqueness of the maximal left ideal, identifies the actual Jacobson radical as maximal, constructs DivisionRing (R / Jac(R)) internally, and proves the canonical seedRank equals the left vector-space Module.rank of the actual radical quotient. The natural finite-rank equality is also proved. No commutative residue-field surrogate or caller-supplied division-ring certificate is used.",
1367 "current_source": {
1368 "file": "editable/traditional/02_foundations.tex",
1369 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1370 "kind": "corollary",
1371 "record_id_line": 719,
1372 "statement_start_line": 709,
1373 "statement_end_line": 718,
1374 "statement_sha256": "50bd885a16ef40862a42e9ddd15b7dd03758b5ee514ee98aab28acf22c1e26ed",
1375 "proof_start_line": 722,
1376 "proof_end_line": 724,
1377 "proof_sha256": "6f2113931446b1f64db3f48e7208ef273d4e308233b9269838ae686cd94b5c8b"
1378 },
1379 "statement": "\\begin{corollary}[Local algebra formula]\n\\label{cor:local-algebra-formula}\nLet $A$ be a local ring with Jacobson radical $J$ and residue division ring $D=A/J$. For every finitely generated left module $M$ over $A$,\n\\[\n\\boxed{\n\\srank_A(M)=\\dim_D(M/JM),\n}\n\\]\nwhere the dimension is that of a left vector space over $D$.\n\\end{corollary}",
1380 "source_proof": "\\begin{proof}\nThe module $M/JM$ over $A/J$ is a left vector space over $D$. Its minimal number of generators is its dimension over $D$.\n\\end{proof}",
1381 "new_block_contribution": [],
1382 "block07_contribution": [
1383 "Komorebi.PaperICompletion7.LocalResidue.proper_sup_ne_top",
1384 "Komorebi.PaperICompletion7.LocalResidue.maximal_left_ideal_unique",
1385 "Komorebi.PaperICompletion7.LocalResidue.jacobson_isMaximal",
1386 "Komorebi.PaperICompletion7.LocalResidue.residueDivisionRing",
1387 "Komorebi.PaperICompletion7.LocalResidue.seedRank_eq_residue_dimension",
1388 "Komorebi.PaperICompletion7.LocalResidue.seedFinrank_eq_residue_finrank"
1389 ],
1390 "acceptance_provenance": [
1391 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1392 ]
1393 },
1394 {
1395 "ordinal": 36,
1396 "stable_id": "PSC-I-CI-WEDDERBURN-ARTIN",
1397 "kind": "classicalinput",
1398 "title": "Wedderburn--Artin decomposition",
1399 "status": "classical_input_integrated",
1400 "old_contract_rows": [],
1401 "mapped_declarations": [
1402 "Komorebi.PaperICompletion8.SemisimpleFormula.exists_wedderburn_decomposition",
1403 "Komorebi.PaperICompletion8.SemisimpleFormula.finiteAlgebra_radicalQuotient_isSemisimple",
1404 "Komorebi.PaperICompletion8.SemisimpleFormula.finiteAlgebra_radicalQuotient_decomposition"
1405 ],
1406 "scope_and_remaining_work": "The actual arbitrary-division-ring Wedderburn-Artin existence theorem is now integrated at its semisimple-ring scope. The finite-algebra radical quotient is proved semisimple and its actual decomposition is constructed. These depend on proved pinned Mathlib results, not a custom existence axiom. Compatible finite module decompositions are separately constructed in ProductPresentation and SemisimpleExistence.",
1407 "current_source": {
1408 "file": "editable/traditional/02_foundations.tex",
1409 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1410 "kind": "classicalinput",
1411 "record_id_line": 735,
1412 "statement_start_line": 727,
1413 "statement_end_line": 734,
1414 "statement_sha256": "effdfe288b991fcc5cdf7549800403bbb6fd698be57f017880428042cc23a058",
1415 "proof_start_line": null,
1416 "proof_end_line": null,
1417 "proof_sha256": null
1418 },
1419 "statement": "\\begin{classicalinput}[Wedderburn and Artin decomposition]\n\\label{ci:wedderburn}\nEvery semisimple Artinian ring is isomorphic to a finite product\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nof full matrix rings over division rings.\n\\end{classicalinput}",
1420 "source_proof": null,
1421 "new_block_contribution": [],
1422 "block08_contribution": [
1423 "Komorebi.PaperICompletion8.SemisimpleFormula.exists_wedderburn_decomposition",
1424 "Komorebi.PaperICompletion8.SemisimpleFormula.finiteAlgebra_radicalQuotient_isSemisimple",
1425 "Komorebi.PaperICompletion8.SemisimpleFormula.finiteAlgebra_radicalQuotient_decomposition"
1426 ]
1427 },
1428 {
1429 "ordinal": 37,
1430 "stable_id": "PSC-I-L-REGULAR-MODULE-MULTIPLICITIES",
1431 "kind": "lemma",
1432 "title": "Regular-module multiplicities",
1433 "status": "covered",
1434 "old_contract_rows": [
1435 10
1436 ],
1437 "mapped_declarations": [
1438 "Komorebi.SemisimpleCompletion.semisimple_seedRank_eq_maxCeilDiv_of_equivalences",
1439 "Komorebi.SemisimpleCompletion.splitProduct_seedRank_eq_maxCeilDiv",
1440 "Komorebi.PaperICompletion8.RegularMultiplicity.simpleModule",
1441 "Komorebi.PaperICompletion8.RegularMultiplicity.simple_smul",
1442 "Komorebi.PaperICompletion8.RegularMultiplicity.simple_isSimple",
1443 "Komorebi.PaperICompletion8.RegularMultiplicity.other_factor_smul_zero",
1444 "Komorebi.PaperICompletion8.RegularMultiplicity.regularColumnsEquiv",
1445 "Komorebi.PaperICompletion8.RegularMultiplicity.regularColumnsEquiv_apply",
1446 "Komorebi.PaperICompletion8.RegularMultiplicity.multiplicityDirectSumEquiv",
1447 "Komorebi.PaperICompletion8.RegularMultiplicity.regularDirectSumEquiv"
1448 ],
1449 "scope_and_remaining_work": "The actual left regular module is explicitly equivalent to its natural columns, then to the displayed finite direct sum of natural-module copies over arbitrary division rings. Coordinate ring homomorphisms define the inflated actions, other factors act by zero, and positive-dimensional natural factors are proved simple. No transpose-as-ring-map or commutative-scalar shortcut is used. Finite products for the inner repeated copies are the canonical finite direct-sum model.",
1450 "current_source": {
1451 "file": "editable/traditional/02_foundations.tex",
1452 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1453 "kind": "lemma",
1454 "record_id_line": 748,
1455 "statement_start_line": 737,
1456 "statement_end_line": 747,
1457 "statement_sha256": "cd1bf9aa7ec7ad71f24491e1969c318f330a268fb73a29873ff2e9494eb91799",
1458 "proof_start_line": 751,
1459 "proof_end_line": 757,
1460 "proof_sha256": "a9140c814e2a6625f80dfb5541adc9e0a342404f224d6e3b84dad75f95182992"
1461 },
1462 "statement": "\\begin{lemma}[Multiplicities in the regular module]\n\\label{lem:regular-multiplicities}\nLet\n\\[\nA=\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nand let $S_i=D_i^{d_i}$ denote the natural simple left module for the $i$th factor, extended by zero across the other factors. Then, as a left module over $A$,\n\\[\n\\boxed{A\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus d_i}.}\n\\]\n\\end{lemma}",
1463 "source_proof": "\\begin{proof}\nFor one factor $M_d(D)$, let $e_{jj}$ be the diagonal matrix units. The left regular module decomposes as\n\\[\nM_d(D)=\\bigoplus_{j=1}^{d}M_d(D)e_{jj}.\n\\]\nEach minimal left ideal $M_d(D)e_{jj}$ is isomorphic to the natural column module $D^d$. Taking the direct product of factors gives the stated decomposition.\n\\end{proof}",
1464 "new_block_contribution": [],
1465 "block08_contribution": [
1466 "Komorebi.PaperICompletion8.RegularMultiplicity.simpleModule",
1467 "Komorebi.PaperICompletion8.RegularMultiplicity.simple_smul",
1468 "Komorebi.PaperICompletion8.RegularMultiplicity.simple_isSimple",
1469 "Komorebi.PaperICompletion8.RegularMultiplicity.other_factor_smul_zero",
1470 "Komorebi.PaperICompletion8.RegularMultiplicity.regularColumnsEquiv",
1471 "Komorebi.PaperICompletion8.RegularMultiplicity.regularColumnsEquiv_apply",
1472 "Komorebi.PaperICompletion8.RegularMultiplicity.multiplicityDirectSumEquiv",
1473 "Komorebi.PaperICompletion8.RegularMultiplicity.regularDirectSumEquiv"
1474 ],
1475 "acceptance_provenance": [
1476 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1477 ]
1478 },
1479 {
1480 "ordinal": 38,
1481 "stable_id": "PSC-I-T-SEMISIMPLE-SEED-RANK",
1482 "kind": "theorem",
1483 "title": "Semisimple seed-rank formula over arbitrary division rings",
1484 "status": "covered",
1485 "old_contract_rows": [
1486 10
1487 ],
1488 "mapped_declarations": [
1489 "Komorebi.SemisimpleCompletion.semisimple_seedRank_eq_maxCeilDiv_of_equivalences",
1490 "Komorebi.SemisimpleCompletion.splitProduct_seedRank_eq_maxCeilDiv",
1491 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_radicalTop",
1492 "Komorebi.PaperICompletion7.ModuleTop.seedFinrank_eq_radicalTop",
1493 "Komorebi.PaperICompletion8.DivisionRows.leftGenerates_iff_rowsSpan",
1494 "Komorebi.PaperICompletion8.DivisionRows.multiplicity_le_of_leftGenerates",
1495 "Komorebi.PaperICompletion8.DivisionRows.exists_leftGenerators_of_le",
1496 "Komorebi.PaperICompletion8.DivisionProduct.exists_divisionProduct_generators_iff_coordinate_bounds",
1497 "Komorebi.PaperICompletion8.DivisionProduct.divisionProduct_seedRank_eq_maxCeilDiv",
1498 "Komorebi.PaperICompletion8.RegularMultiplicity.directSum_seedRank",
1499 "Komorebi.PaperICompletion8.SemisimpleFormula.exists_finite_simple_decomposition",
1500 "Komorebi.PaperICompletion8.ProductPresentation.simple_equiv_natural",
1501 "Komorebi.PaperICompletion8.ProductPresentation.regroupSimpleFactors",
1502 "Komorebi.PaperICompletion8.ProductPresentation.exists_multiplicity_presentation",
1503 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_compatible_presentation",
1504 "Komorebi.PaperICompletion8.SemisimpleFormula.semisimple_seedRank_of_decompositions",
1505 "Komorebi.PaperICompletion8.SemisimpleFormula.semisimple_seedFinrank_of_decompositions",
1506 "Komorebi.PaperICompletion8.SemisimpleFormula.radicalTop_seedRank_of_decompositions",
1507 "Komorebi.PaperICompletion8.SemisimpleFormula.radicalTop_seedFinrank_of_decompositions",
1508 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_semisimple_seed_formula",
1509 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_artinian_radicalTop_seed_formula",
1510 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_finiteAlgebra_seed_formula"
1511 ],
1512 "scope_and_remaining_work": "Actual product matrix ring with left multiplication and natural column modules acted on through the matching factor. Semilinear eM along eR enforces compatibility. DivisionRows proves left generation iff row spanning and m<=r*d; standard rows prove sufficiency. Product assembly, column rearrangement and seed-rank invariance prove the formula. RadicalTop is literally M/(Jac(R) smul top); existence constructs decompositions.",
1513 "current_source": {
1514 "file": "editable/traditional/02_foundations.tex",
1515 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1516 "kind": "theorem",
1517 "record_id_line": 780,
1518 "statement_start_line": 759,
1519 "statement_end_line": 779,
1520 "statement_sha256": "c604c6473ceeb6b651f37301bd3e719d69a555f04c9e4aea5fa6ad49bcd35de4",
1521 "proof_start_line": 784,
1522 "proof_end_line": 795,
1523 "proof_sha256": "bdc16a3f1ddb94b37e8199caa8f1ee7a343533455311bbaf4d814b5234993512"
1524 },
1525 "statement": "\\begin{theorem}[Semisimple seed rank formula over arbitrary division rings]\n\\label{thm:semisimple-seed-rank}\nLet\n\\[\nA\\cong\\prod_{i=1}^{q}M_{d_i}(D_i)\n\\]\nbe semisimple Artinian and let\n\\[\nM\\cong\\bigoplus_{i=1}^{q}S_i^{\\oplus m_i}\n\\]\nbe a finitely generated left module over $A$. Then\n\\[\n\\boxed{\n\\srank_A(M)\n=\n\\max_{1\\leq i\\leq q}\n\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil.\n}\n\\]\nIf $A$ is any algebra of finite dimension over a field, the same formula applied to the decomposition of $M/\\Jac(A)M$ over $A/\\Jac(A)$ computes $\\srank_A(M)$.\n\\end{theorem}",
1526 "source_proof": "\\begin{proof}\nBy \\cref{cor:minimal-free-cover}, $M$ is generated by $r$ elements exactly when it is a quotient of $A^{\\oplus r}$. By \\cref{lem:regular-multiplicities},\n\\[\nA^{\\oplus r}\\cong\\bigoplus_iS_i^{\\oplus rd_i}.\n\\]\nEvery short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to $M$ exists if and only if\n\\[\nm_i\\leq rd_i\n\\qquad\\text{for every }i.\n\\]\nThe smallest such $r$ is the displayed maximum. For a nonsemisimple algebra of finite dimension, \\cref{thm:module-top} reduces the calculation to the semisimple module $M/\\Jac(A)M$ over $A/\\Jac(A)$.\n\\end{proof}",
1527 "new_block_contribution": [],
1528 "block07_contribution": [
1529 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_radicalTop",
1530 "Komorebi.PaperICompletion7.ModuleTop.seedFinrank_eq_radicalTop"
1531 ],
1532 "block08_contribution": [
1533 "Komorebi.PaperICompletion8.DivisionRows.leftGenerates_iff_rowsSpan",
1534 "Komorebi.PaperICompletion8.DivisionRows.multiplicity_le_of_leftGenerates",
1535 "Komorebi.PaperICompletion8.DivisionRows.exists_leftGenerators_of_le",
1536 "Komorebi.PaperICompletion8.DivisionProduct.exists_divisionProduct_generators_iff_coordinate_bounds",
1537 "Komorebi.PaperICompletion8.DivisionProduct.divisionProduct_seedRank_eq_maxCeilDiv",
1538 "Komorebi.PaperICompletion8.RegularMultiplicity.directSum_seedRank",
1539 "Komorebi.PaperICompletion8.SemisimpleFormula.exists_finite_simple_decomposition",
1540 "Komorebi.PaperICompletion8.ProductPresentation.simple_equiv_natural",
1541 "Komorebi.PaperICompletion8.ProductPresentation.regroupSimpleFactors",
1542 "Komorebi.PaperICompletion8.ProductPresentation.exists_multiplicity_presentation",
1543 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_compatible_presentation",
1544 "Komorebi.PaperICompletion8.SemisimpleFormula.semisimple_seedRank_of_decompositions",
1545 "Komorebi.PaperICompletion8.SemisimpleFormula.semisimple_seedFinrank_of_decompositions",
1546 "Komorebi.PaperICompletion8.SemisimpleFormula.radicalTop_seedRank_of_decompositions",
1547 "Komorebi.PaperICompletion8.SemisimpleFormula.radicalTop_seedFinrank_of_decompositions",
1548 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_semisimple_seed_formula",
1549 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_artinian_radicalTop_seed_formula",
1550 "Komorebi.PaperICompletion8.SemisimpleExistence.exists_finiteAlgebra_seed_formula"
1551 ],
1552 "acceptance_provenance": [
1553 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below.",
1554 {
1555 "semantic_review": "semantic_wave1.json",
1556 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
1557 "reviewed_statement_sha256": "c604c6473ceeb6b651f37301bd3e719d69a555f04c9e4aea5fa6ad49bcd35de4",
1558 "reviewed_proof_sha256": "bdc16a3f1ddb94b37e8199caa8f1ee7a343533455311bbaf4d814b5234993512",
1559 "verdict": "proved as written"
1560 }
1561 ],
1562 "remaining_work": null
1563 },
1564 {
1565 "ordinal": 39,
1566 "stable_id": "PSC-I-C-ALGEBRAICALLY-CLOSED-FORMULA",
1567 "kind": "corollary",
1568 "title": "Algebraically closed multiplicity formula",
1569 "status": "covered",
1570 "old_contract_rows": [
1571 10
1572 ],
1573 "mapped_declarations": [
1574 "Komorebi.SemisimpleCompletion.semisimple_seedRank_eq_maxCeilDiv_of_equivalences",
1575 "Komorebi.SemisimpleCompletion.splitProduct_seedRank_eq_maxCeilDiv",
1576 "Komorebi.PaperICompletion8.SemisimpleFormula.seedRank_eq_radicalTop_restricted",
1577 "Komorebi.PaperICompletion8.SemisimpleFormula.split_originalRing_top_seedRank",
1578 "Komorebi.PaperICompletion8.SemisimpleFormula.split_radicalTop_seedRank",
1579 "Komorebi.PaperICompletion8.SemisimpleFormula.radicalTop_seedRank_of_decompositions",
1580 "Komorebi.PaperICompletion7.ModuleTop.seedRank_eq_radicalTop"
1581 ],
1582 "scope_and_remaining_work": "Correct public map uses decomposition of A/J, not A. The supplied algebra equivalence forgets to a ring equivalence, and the module decomposition stays compatible. Specializing the exact radical-top theorem to all D_i=k gives the current statement.",
1583 "current_source": {
1584 "file": "editable/traditional/02_foundations.tex",
1585 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1586 "kind": "corollary",
1587 "record_id_line": 823,
1588 "statement_start_line": 809,
1589 "statement_end_line": 822,
1590 "statement_sha256": "20c07c2db5ab4c4a956355fe5e45008edff100a4b279dd27a37ddff90871fcbf",
1591 "proof_start_line": 826,
1592 "proof_end_line": 828,
1593 "proof_sha256": "eb2640a6fc843f84453fa9168e1b85830258b30c56a1e1987c1608c4898a6117"
1594 },
1595 "statement": "\\begin{corollary}[Algebraically closed multiplicity formula]\n\\label{cor:algebraically-closed-formula}\nLet $\\bbk$ be algebraically closed, let $A$ be a unital algebra of finite dimension over $\\bbk$, put $J=\\Jac(A)$, and let $M$ be a finitely generated left module over $A$. Assume compatible decompositions indexed by an integer $q\\geq0$:\n\\[\nA/J\\cong\\prod_{i=1}^{q}M_{d_i}(\\bbk),\n\\qquad\nM/JM\\cong\\bigoplus_{i=1}^{q}(\\bbk^{d_i})^{\\oplus m_i},\n\\]\nwhere the $d_i$ are positive integers and the $m_i$ are nonnegative integers. The first isomorphism is an isomorphism of algebras over $\\bbk$. The second is an isomorphism of left modules over $A/J$, using the natural column module for each matrix factor and zero action from the other factors. Then\n\\[\n\\srank_A(M)=\\max_{1\\leq i\\leq q}\\left\\lceil\\frac{m_i}{d_i}\\right\\rceil,\n\\]\nwith $\\max\\varnothing=0$ when $q=0$.\n\\end{corollary}",
1596 "source_proof": "\\begin{proof}\nBy \\cref{thm:module-top}, $\\srank_A(M)=\\srank_{A/J}(M/JM)$. Apply \\cref{thm:semisimple-seed-rank} to the displayed decompositions of $A/J$ and $M/JM$, with every $D_i=\\bbk$. This gives the stated maximum. If $q=0$, the unital $A/J$-module $M/JM$ is zero, and \\cref{thm:module-top} gives seed rank zero.\n\\end{proof}",
1597 "new_block_contribution": [],
1598 "block08_contribution": [
1599 "Komorebi.PaperICompletion8.SemisimpleFormula.seedRank_eq_radicalTop_restricted",
1600 "Komorebi.PaperICompletion8.SemisimpleFormula.split_originalRing_top_seedRank",
1601 "Komorebi.PaperICompletion8.SemisimpleFormula.split_radicalTop_seedRank"
1602 ],
1603 "acceptance_provenance": [
1604 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below.",
1605 {
1606 "semantic_review": "semantic_wave1.json",
1607 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
1608 "reviewed_statement_sha256": "20c07c2db5ab4c4a956355fe5e45008edff100a4b279dd27a37ddff90871fcbf",
1609 "reviewed_proof_sha256": "eb2640a6fc843f84453fa9168e1b85830258b30c56a1e1987c1608c4898a6117",
1610 "verdict": "proved as written"
1611 }
1612 ],
1613 "remaining_work": null
1614 },
1615 {
1616 "ordinal": 40,
1617 "stable_id": "PSC-I-C-FULL-MATRIX",
1618 "kind": "corollary",
1619 "title": "Full matrix algebra",
1620 "status": "covered",
1621 "old_contract_rows": [
1622 11
1623 ],
1624 "mapped_declarations": [
1625 "Komorebi.SemisimpleCompletion.fullMatrix_every_nonzero_vector_generates",
1626 "Komorebi.SemisimpleCompletion.fullMatrix_seedRank_eq_one",
1627 "Komorebi.PaperICompletion8.RegularMultiplicity.natural_nonzero_generates",
1628 "Komorebi.PaperICompletion8.RegularMultiplicity.natural_seedRank_eq_one"
1629 ],
1630 "scope_and_remaining_work": "For every division ring and positive matrix size, the natural module has canonical seed rank one and every nonzero vector generates. The proof uses entries w_i / v_j with the inverse on the correct right side. The positive-size convention of the surrounding natural-simple-module discussion is essential: size zero has seed rank zero. This guard remains explicit in Lean and is tested, not silently discarded.",
1631 "current_source": {
1632 "file": "editable/traditional/02_foundations.tex",
1633 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1634 "kind": "corollary",
1635 "record_id_line": 838,
1636 "statement_start_line": 830,
1637 "statement_end_line": 837,
1638 "statement_sha256": "f22f88ae7b9fe79b50f6c82420d301d784c941d6340fc700723841d724826055",
1639 "proof_start_line": 841,
1640 "proof_end_line": 843,
1641 "proof_sha256": "d60bfb368c6307507a4f2b56ece1e7dc9f0aa14ac0c113a46cfddb3cff7a0b94"
1642 },
1643 "statement": "\\begin{corollary}[Full matrix algebra]\n\\label{cor:full-matrix}\nFor the natural module $S=D^d$ over $M_d(D)$,\n\\[\n\\srank_{M_d(D)}(S)=1.\n\\]\nMore strongly, every nonzero vector of $S$ is cyclic.\n\\end{corollary}",
1644 "source_proof": "\\begin{proof}\nThe formula gives $\\lceil1/d\\rceil=1$. Directly, let $v\\neq0$ and choose a coordinate $v_j\\neq0$. For any target $w=(w_i)$, the matrix whose $j$th column has entries $w_iv_j^{-1}$ and whose other columns vanish sends $v$ to $w$.\n\\end{proof}",
1645 "new_block_contribution": [],
1646 "block08_contribution": [
1647 "Komorebi.PaperICompletion8.RegularMultiplicity.natural_nonzero_generates",
1648 "Komorebi.PaperICompletion8.RegularMultiplicity.natural_seedRank_eq_one"
1649 ],
1650 "acceptance_provenance": [
1651 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1652 ]
1653 },
1654 {
1655 "ordinal": 41,
1656 "stable_id": "PSC-I-C-MULTIPLICITY-OBSTRUCTION",
1657 "kind": "corollary",
1658 "title": "Multiplicity obstruction",
1659 "status": "covered",
1660 "old_contract_rows": [
1661 12
1662 ],
1663 "mapped_declarations": [
1664 "Komorebi.SemisimpleCompletion.multiplicity_obstruction",
1665 "Komorebi.SemisimpleCompletion.multiplicity_seedRank_formula",
1666 "Komorebi.PaperICompletion8.DivisionBlock.oneBlock_seedRank_eq_ceilDiv",
1667 "Komorebi.PaperICompletion8.RegularMultiplicity.singleColumnsEquiv",
1668 "Komorebi.PaperICompletion8.RegularMultiplicity.naturalCopies_seedRank",
1669 "Komorebi.PaperICompletion8.RegularMultiplicity.naturalDirectSum_seedRank"
1670 ],
1671 "scope_and_remaining_work": "For arbitrary division rings and positive block size, the actual shared matrix action on m natural-module copies has canonical rank ceil(m/d). The rectangular-matrix, natural-column-product and finite-direct-sum formulations are connected by actual module equivalences. Zero multiplicity is included. This is not the independent-factor action, whose bound is a maximum across distinct factors.",
1672 "current_source": {
1673 "file": "editable/traditional/02_foundations.tex",
1674 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1675 "kind": "corollary",
1676 "record_id_line": 856,
1677 "statement_start_line": 845,
1678 "statement_end_line": 855,
1679 "statement_sha256": "df87d9465fd05e359ca1a96ba5b69e501f5e6b9b93b2d4e1b555a84476ec84ef",
1680 "proof_start_line": 859,
1681 "proof_end_line": 861,
1682 "proof_sha256": "e9d037de6537f5d184efa3035c2ddb24d629a42b305da4e39d5166e6a998b6ec"
1683 },
1684 "statement": "\\begin{corollary}[Multiplicity obstruction]\n\\label{cor:multiplicity-obstruction}\nFor the diagonal action of $M_d(D)$ on $S^{\\oplus m}$,\n\\[\n\\boxed{\n\\srank_{M_d(D)}(S^{\\oplus m})\n=\n\\left\\lceil\\frac{m}{d}\\right\\rceil.\n}\n\\]\n\\end{corollary}",
1685 "source_proof": "\\begin{proof}\nApply \\cref{thm:semisimple-seed-rank} to the single simple type with multiplicity $m$.\n\\end{proof}",
1686 "new_block_contribution": [],
1687 "block08_contribution": [
1688 "Komorebi.PaperICompletion8.DivisionBlock.oneBlock_seedRank_eq_ceilDiv",
1689 "Komorebi.PaperICompletion8.RegularMultiplicity.singleColumnsEquiv",
1690 "Komorebi.PaperICompletion8.RegularMultiplicity.naturalCopies_seedRank",
1691 "Komorebi.PaperICompletion8.RegularMultiplicity.naturalDirectSum_seedRank"
1692 ],
1693 "acceptance_provenance": [
1694 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1695 ]
1696 },
1697 {
1698 "ordinal": 42,
1699 "stable_id": "PSC-I-CI-PID-PRIMARY-DECOMPOSITION",
1700 "kind": "classicalinput",
1701 "title": "Primary decomposition for polynomial modules",
1702 "status": "classical_input_integrated",
1703 "old_contract_rows": [],
1704 "mapped_declarations": [
1705 "Komorebi.PaperICompletion.CyclicClassification.exists_full_annihilator_vector",
1706 "Komorebi.PaperICompletion3.PrimarySeedFormula.exists_positive_primaryProduct_equiv",
1707 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.primaryPresentation_nonempty",
1708 "Komorebi.PaperIRelease.PolynomialModule.polynomialModule_isTorsion",
1709 "Komorebi.PaperIRelease.PolynomialModule.primary_internal_decomposition",
1710 "Komorebi.PaperIRelease.PolynomialModule.finitely_many_primary_components",
1711 "Komorebi.PaperIRelease.PolynomialModule.exists_positive_primary_decomposition",
1712 "Komorebi.PaperIRelease.PolynomialModule.intrinsic_primary_finite_power_sum",
1713 "Komorebi.PaperIRelease.PolynomialModule.monicSpectrumEquiv",
1714 "Komorebi.PaperIRelease.PolynomialModule.monic_primary_internal_decomposition",
1715 "Komorebi.PaperIRelease.PolynomialModule.finitely_many_monic_primary_components"
1716 ],
1717 "scope_and_remaining_work": "For an existing polynomial module, multiplication by X constructs its evaluation operator and an actual polynomial-linear equivalence; finite coefficient dimension gives torsion. PID decomposition constructs positive elementary divisors, while actual intrinsic primary submodules are proved equal to ideal primary components. PolynomialMonic proves a bijection between monic irreducibles and nonzero prime ideals via p↦(p), deriving injectivity from monicity and surjectivity from a normalized PID prime generator. It reindexes the internal sum and finite nonzero support to precisely the manuscript index set. intrinsic_primary_finite_power_sum is a literal finite DirectSum of k[X]/(p^e), with positive e, for each specified irreducible p.",
1718 "current_source": {
1719 "file": "editable/traditional/02_foundations.tex",
1720 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1721 "kind": "classicalinput",
1722 "record_id_line": 891,
1723 "statement_start_line": 883,
1724 "statement_end_line": 890,
1725 "statement_sha256": "446dc677476b213cc4daac554de2b3f15ab816c9d0d6eeff98a67965eea843a9",
1726 "proof_start_line": null,
1727 "proof_end_line": null,
1728 "proof_sha256": null
1729 },
1730 "statement": "\\begin{classicalinput}[Primary decomposition for polynomial modules]\n\\label{ci:primary-decomposition}\nEvery module $X$ over $\\bbk[t]$ with finite dimension over the coefficient field decomposes as\n\\[\nX=\\bigoplus_{p\\in\\Irr(\\bbk[t])}X_p,\n\\]\nwhere $\\Irr(\\bbk[t])$ denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero $X_p$ is a finite direct sum of modules $\\bbk[t]/(p^e)$.\n\\end{classicalinput}",
1731 "source_proof": null,
1732 "new_block_contribution": [
1733 "Komorebi.PaperICompletion.CyclicClassification.exists_full_annihilator_vector"
1734 ],
1735 "block03_contribution": [
1736 "Komorebi.PaperICompletion3.PrimarySeedFormula.exists_positive_primaryProduct_equiv",
1737 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.primaryPresentation_nonempty"
1738 ],
1739 "acceptance_provenance": [
1740 {
1741 "semantic_review": "semantic_wave2.json",
1742 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
1743 "reviewed_statement_sha256": "446dc677476b213cc4daac554de2b3f15ab816c9d0d6eeff98a67965eea843a9",
1744 "reviewed_proof_sha256": null,
1745 "verdict": "proved as written"
1746 }
1747 ],
1748 "remaining_work": null
1749 },
1750 {
1751 "ordinal": 43,
1752 "stable_id": "PSC-I-T-SINGLE-OPERATOR",
1753 "kind": "theorem",
1754 "title": "Single-operator seed rank",
1755 "status": "covered",
1756 "old_contract_rows": [
1757 13
1758 ],
1759 "mapped_declarations": [
1760 "Komorebi.SingleOperatorGraph.singleOperator_seedNumber_isLeast_of_primary_decomposition",
1761 "Komorebi.SingleOperatorGraph.singleOperator_operatorAlgebra_seedRanks_eq_of_primary_decomposition",
1762 "Komorebi.PaperICompletion.SpectralCertificates.selfAdjoint_operatorAlgebra_seedRanks",
1763 "Komorebi.PaperICompletion3.PrimarySeedFormula.exists_operator_primary_seed_formula",
1764 "Komorebi.PaperICompletion3.PrimarySeedFormula.residueFiberMap_surjective",
1765 "Komorebi.PaperICompletion3.PrimarySeedFormula.primaryProduct_seedNumber_isLeast",
1766 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.operator_seedRank_eq_primaryGeneratorCount",
1767 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.PrimaryPresentation.maxCount_eq",
1768 "Komorebi.PaperICompletion4.IntrinsicPrimary.primarySubmodule_eq_idealPrimary",
1769 "Komorebi.PaperICompletion4.IntrinsicPrimary.primaryToGlobal_ker",
1770 "Komorebi.PaperICompletion4.IntrinsicPrimary.primaryToGlobal_surjective",
1771 "Komorebi.PaperICompletion4.IntrinsicPrimary.primaryQuotientEquivGlobal",
1772 "Komorebi.PaperICompletion4.GroupedPrimary.groupedPrimaryEquiv",
1773 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_primary_membership",
1774 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_scalarImage_membership",
1775 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operatorPrimaryComponentEquiv",
1776 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operatorPrimaryQuotientEquiv",
1777 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicMultiplicity_eq_fiber",
1778 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicMultiplicity_eq_card_of_component_decomposition",
1779 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.primary_fiber_presentation_invariant",
1780 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicSeedMaximum_attained",
1781 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_seedFinrank_eq_intrinsicSeedMaximum",
1782 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_seedRank_eq_intrinsicSeedMaximum"
1783 ],
1784 "scope_and_remaining_work": "The full printed statement is now mapped. The actual power-annihilation submodule is proved equal to the ideal primary component. Its inclusion into V/pV has exact kernel pX_p and is onto from the internally discharged Artinian hypothesis. The induced quotient equivalence is residue-field linear. Every intrinsic multiplicity is identified with every actual presentation fibre. The grouped factors are proved to be the literal X_p, and any positive primary decomposition of X_p has the same intrinsic count. The presentation-free maximum over all monic irreducibles is proved bounded and attained, and equals both natural-valued and cardinal-valued seed rank. No caller-supplied decomposition or seed-count conclusion is needed by the public maximum theorem. These are noncomputable mathematical constructions. The printed proof uses a different route through local-module/Nakayama machinery; that separately listed machinery is not thereby marked complete.",
1785 "current_source": {
1786 "file": "editable/traditional/02_foundations.tex",
1787 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1788 "kind": "theorem",
1789 "record_id_line": 908,
1790 "statement_start_line": 893,
1791 "statement_end_line": 907,
1792 "statement_sha256": "e96051e5d8333ea1ba1efe94f70b260ad356a41bc5ecad49525ec1df9c759967",
1793 "proof_start_line": 914,
1794 "proof_end_line": 939,
1795 "proof_sha256": "6a3a27a42e85efa603e05da2c67252eedcf2c3dc1439e1e66b2dc21fed2b3f83"
1796 },
1797 "statement": "\\begin{theorem}[Seed rank for a single operator]\n\\label{thm:single-operator}\nLet $X$ have finite dimension over $\\bbk$ and let $T\\in\\End_{\\bbk}(X)$. Regard $X$ as a module over $\\bbk[t]$ by $t\\cdot x=Tx$. For every monic irreducible $p$, let $X_p$ be the primary component for $p$ and set\n\\[\n\\mu_p(T):=\n\\dim_{\\bbk[t]/(p)}\\bigl(X_p/p(T)X_p\\bigr).\n\\]\nThen\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)=\\max_p\\mu_p(T).\n}\n\\]\nMoreover, $\\mu_p(T)$ is the number of cyclic primary summands for $p$ in the decomposition into elementary divisors of $X_p$.\n\\end{theorem}",
1798 "source_proof": "\\begin{proof}\nLet the minimal polynomial be\n\\[\nm_T(t)=\\prod_{p}p(t)^{e_p}.\n\\]\nThe acting algebra is\n\\[\n\\bbk[T]\\cong\\bbk[t]/(m_T)\n\\cong\\prod_pR_p,\n\\qquad\nR_p:=\\bbk[t]/(p^{e_p}),\n\\]\nby the Chinese remainder theorem. The factor $R_p$ acts on $X_p$ and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives\n\\[\n\\srank_{\\bbk[T]}(X)=\\max_p\\srank_{R_p}(X_p).\n\\]\nEach $R_p$ is local with maximal ideal generated by the class of $p$ and residue division field $K_p=\\bbk[t]/(p)$. By \\cref{cor:local-algebra-formula},\n\\[\n\\srank_{R_p}(X_p)\n=\n\\dim_{K_p}X_p/p(T)X_p\n=\n\\mu_p(T).\n\\]\nFinally, every summand $\\bbk[t]/(p^e)$ contributes one copy of $K_p$ after quotienting by $p$, so the dimension equals the number of summands in the elementary divisor decomposition.\n\\end{proof}",
1799 "new_block_contribution": [
1800 "Komorebi.PaperICompletion.SpectralCertificates.selfAdjoint_operatorAlgebra_seedRanks"
1801 ],
1802 "block03_contribution": [
1803 "Komorebi.PaperICompletion3.PrimarySeedFormula.exists_operator_primary_seed_formula",
1804 "Komorebi.PaperICompletion3.PrimarySeedFormula.residueFiberMap_surjective",
1805 "Komorebi.PaperICompletion3.PrimarySeedFormula.primaryProduct_seedNumber_isLeast",
1806 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.operator_seedRank_eq_primaryGeneratorCount",
1807 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.PrimaryPresentation.maxCount_eq"
1808 ],
1809 "block04_contribution": [
1810 "Komorebi.PaperICompletion4.IntrinsicPrimary.primarySubmodule_eq_idealPrimary",
1811 "Komorebi.PaperICompletion4.IntrinsicPrimary.primaryToGlobal_ker",
1812 "Komorebi.PaperICompletion4.IntrinsicPrimary.primaryToGlobal_surjective",
1813 "Komorebi.PaperICompletion4.IntrinsicPrimary.primaryQuotientEquivGlobal",
1814 "Komorebi.PaperICompletion4.GroupedPrimary.groupedPrimaryEquiv",
1815 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_primary_membership",
1816 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_scalarImage_membership",
1817 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operatorPrimaryComponentEquiv",
1818 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operatorPrimaryQuotientEquiv",
1819 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicMultiplicity_eq_fiber",
1820 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicMultiplicity_eq_card_of_component_decomposition",
1821 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.primary_fiber_presentation_invariant",
1822 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicSeedMaximum_attained",
1823 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_seedFinrank_eq_intrinsicSeedMaximum",
1824 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operator_seedRank_eq_intrinsicSeedMaximum"
1825 ],
1826 "acceptance_provenance": [
1827 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1828 ]
1829 },
1830 {
1831 "ordinal": 44,
1832 "stable_id": "PSC-I-C-INVARIANT-FACTOR-COUNT",
1833 "kind": "corollary",
1834 "title": "Invariant-factor count",
1835 "status": "covered",
1836 "old_contract_rows": [
1837 13
1838 ],
1839 "mapped_declarations": [
1840 "Komorebi.SingleOperatorGraph.singleOperator_seedNumber_isLeast_of_primary_decomposition",
1841 "Komorebi.SingleOperatorGraph.singleOperator_operatorAlgebra_seedRanks_eq_of_primary_decomposition",
1842 "Komorebi.PaperICompletion.CyclicClassification.minpolyQuotientEquivOrbit",
1843 "Komorebi.PaperICompletion3.InvariantFactorCount.invariantFactorProduct_seedNumber_isLeast",
1844 "Komorebi.PaperICompletion3.InvariantFactorCount.invariantFactor_operator_seedRanks"
1845 ],
1846 "scope_and_remaining_work": "Complete at the displayed conditional scope: an actual polynomial-module equivalence to the finite product of nonconstant factors and their divisibility chain imply least seed number, natural seed rank and cardinal seed rank equal the factor count. The theorem also handles the empty product. Finite products and finite direct sums represent the same finite decomposition. Existence of an ordered invariant-factor presentation is not asserted by this declaration, and is not an additional hypothesis-free claim of the printed conditional corollary.",
1847 "current_source": {
1848 "file": "editable/traditional/02_foundations.tex",
1849 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1850 "kind": "corollary",
1851 "record_id_line": 964,
1852 "statement_start_line": 951,
1853 "statement_end_line": 963,
1854 "statement_sha256": "c26c3a48531f1cf7f79090c82c8db185af548fc1137f1c427ac815d1d8120e89",
1855 "proof_start_line": 967,
1856 "proof_end_line": 969,
1857 "proof_sha256": "533a439d53e34d64a146bd022b3edcb661344334f9f535506462cd427e2151e2"
1858 },
1859 "statement": "\\begin{corollary}[Invariant factor count]\n\\label{cor:invariant-factor-count}\nIf $X\\neq0$ and the rational canonical decomposition is written with nonconstant invariant factors\n\\[\nX\\cong\\bigoplus_{j=1}^{r}\\bbk[t]/(f_j),\n\\qquad\nf_1\\mid f_2\\mid\\cdots\\mid f_r,\n\\]\nthen\n\\[\n\\boxed{\\srank_{\\bbk[T]}(X)=r.}\n\\]\n\\end{corollary}",
1860 "source_proof": "\\begin{proof}\nFor each irreducible $p$, the number of invariant factors divisible by $p$ equals the number of elementary divisors for $p$. Because $f_1\\mid\\cdots\\mid f_r$ and $f_1\\neq1$ after zero summands are removed, some irreducible divisor of $f_1$ divides all $r$ invariant factors. Hence $\\max_p\\mu_p(T)=r$.\n\\end{proof}",
1861 "new_block_contribution": [
1862 "Komorebi.PaperICompletion.CyclicClassification.minpolyQuotientEquivOrbit"
1863 ],
1864 "block03_contribution": [
1865 "Komorebi.PaperICompletion3.InvariantFactorCount.invariantFactorProduct_seedNumber_isLeast",
1866 "Komorebi.PaperICompletion3.InvariantFactorCount.invariantFactor_operator_seedRanks"
1867 ],
1868 "acceptance_provenance": [
1869 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1870 ]
1871 },
1872 {
1873 "ordinal": 45,
1874 "stable_id": "PSC-I-C-ALGEBRAICALLY-CLOSED-SINGLE-OPERATOR",
1875 "kind": "corollary",
1876 "title": "Geometric-multiplicity formula",
1877 "status": "covered",
1878 "old_contract_rows": [
1879 13,
1880 32
1881 ],
1882 "mapped_declarations": [
1883 "Komorebi.SingleOperatorGraph.singleOperator_seedNumber_isLeast_of_primary_decomposition",
1884 "Komorebi.SingleOperatorGraph.singleOperator_operatorAlgebra_seedRanks_eq_of_primary_decomposition",
1885 "Komorebi.SingleOperatorGraph.graph_seedNumber_eq_max_eigenspace_multiplicity",
1886 "Komorebi.SingleOperatorGraph.graph_seedNumber_eq_max_eigenspace_multiplicity_of_distinct",
1887 "Komorebi.SingleOperatorGraph.operatorAlgebra_seedFinrank_eq_of_seedNumber_isLeast",
1888 "Komorebi.SingleOperatorGraph.operatorAlgebra_seedRank_eq_of_seedNumber_isLeast",
1889 "Komorebi.SingleOperatorGraph.path_endpoint_seedNumber_isLeast_one",
1890 "Komorebi.SingleOperatorGraph.complex_cycle_seedNumber_isLeast_two",
1891 "Komorebi.PaperICompletion.SpectralCertificates.selfAdjoint_seedRank_eq_max_eigenspace_finrank",
1892 "Komorebi.PaperICompletion3.PrimarySeedFormula.exists_operator_primary_seed_formula",
1893 "Komorebi.PaperICompletion5.LinearPrimaryEigenspace.intrinsicMultiplicity_X_sub_C_eq_eigenspace",
1894 "Komorebi.PaperICompletion5.GeometricSeedFormula.operator_seedFinrank_eq_spectralSeedMaximum",
1895 "Komorebi.PaperICompletion5.GeometricSeedFormula.operator_seedRank_eq_spectralSeedMaximum",
1896 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.spectralSeedMaximum_attained"
1897 ],
1898 "scope_and_remaining_work": "Complete at the finite-dimensional algebraically closed scope inherited from the single-operator theorem. The actual intrinsic multiplicity for X-C(a) is identified with the dimension of the actual eigenspace, without diagonalizability. The canonical natural and cardinal seed ranks equal the maximum over the actual spectrum. Off-spectrum dimensions are zero, the empty-spectrum/zero-space maximum is defined as zero, and the spectral maximum is attained on nonzero spaces. No executable spectral algorithm is claimed.",
1899 "current_source": {
1900 "file": "editable/traditional/02_foundations.tex",
1901 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1902 "kind": "corollary",
1903 "record_id_line": 983,
1904 "statement_start_line": 971,
1905 "statement_end_line": 982,
1906 "statement_sha256": "babe43b45ebd6abb23a0e31ed41c0918a4babb97578847c852f2c9d11e69515f",
1907 "proof_start_line": 986,
1908 "proof_end_line": 988,
1909 "proof_sha256": "9947d9f2d5cf652a3c057bfd66d47e3ce17e7596981b6bfc1e7476acdaa57d31"
1910 },
1911 "statement": "\\begin{corollary}[Formula for geometric multiplicity]\n\\label{cor:geometric-multiplicity}\nIf $\\bbk$ is algebraically closed, then\n\\[\n\\boxed{\n\\srank_{\\bbk[T]}(X)\n=\n\\max_{\\lambda\\in\\operatorname{Spec}(T)}\n\\dim_{\\bbk}\\Ker(T-\\lambda I).\n}\n\\]\n\\end{corollary}",
1912 "source_proof": "\\begin{proof}\nThe irreducibles are $p=t-\\lambda$. On the primary component for $\\lambda$, the quotient by $(T-\\lambda I)X_\\lambda$ has dimension equal to the number of Jordan blocks, which is also $\\dim\\Ker(T-\\lambda I)$.\n\\end{proof}",
1913 "new_block_contribution": [
1914 "Komorebi.PaperICompletion.SpectralCertificates.selfAdjoint_seedRank_eq_max_eigenspace_finrank"
1915 ],
1916 "block03_contribution": [
1917 "Komorebi.PaperICompletion3.PrimarySeedFormula.exists_operator_primary_seed_formula"
1918 ],
1919 "block05_contribution": [
1920 "Komorebi.PaperICompletion5.LinearPrimaryEigenspace.intrinsicMultiplicity_X_sub_C_eq_eigenspace",
1921 "Komorebi.PaperICompletion5.GeometricSeedFormula.operator_seedFinrank_eq_spectralSeedMaximum",
1922 "Komorebi.PaperICompletion5.GeometricSeedFormula.operator_seedRank_eq_spectralSeedMaximum",
1923 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.spectralSeedMaximum_attained"
1924 ],
1925 "acceptance_provenance": [
1926 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
1927 ]
1928 },
1929 {
1930 "ordinal": 46,
1931 "stable_id": "PSC-I-C-CYCLIC-MATRIX",
1932 "kind": "corollary",
1933 "title": "Cyclic-matrix criterion",
1934 "status": "covered",
1935 "old_contract_rows": [
1936 14
1937 ],
1938 "mapped_declarations": [
1939 "Komorebi.SingleOperatorGraph.cyclic_matrix_criterion_of_primary_decomposition",
1940 "Komorebi.SingleOperatorGraph.cyclic_matrix_four_way_iff_of_primary_decomposition",
1941 "Komorebi.PaperICompletion.CyclicClassification.hasCyclicVector_iff_minpoly_eq_charpoly",
1942 "Komorebi.PaperICompletion.CyclicClassification.operatorAlgebra_seedRank_eq_one_iff_minpoly_eq_charpoly",
1943 "Komorebi.PaperICompletion.CyclicClassification.cyclic_four_way_without_polynomial_certificate",
1944 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.PrimaryPresentation.cyclic_iff_label_injective",
1945 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.PrimaryPresentation.label_injective_iff_minpoly_eq_charpoly",
1946 "Komorebi.PaperICompletion3.InvariantFactorCount.invariantFactor_operator_seedRanks",
1947 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operatorPrimaryComponentEquiv",
1948 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicMultiplicity_eq_fiber",
1949 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.primary_fiber_presentation_invariant",
1950 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.hasCyclicVector_iff_minpoly_quotient_presentation",
1951 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.cyclicMinpolyQuotientEquiv",
1952 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.hasCyclicVector_iff_all_spectral_multiplicities_one",
1953 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.minpoly_eq_charpoly_iff_all_spectral_multiplicities_one",
1954 "Komorebi.PaperICompletion6.ScalarExtension.minpoly_baseChange",
1955 "Komorebi.PaperICompletion6.ScalarExtension.minpoly_natDegree_baseChange",
1956 "Komorebi.PaperICompletion6.ScalarExtension.minpoly_eq_charpoly_baseChange_iff",
1957 "Komorebi.PaperICompletion6.ScalarExtension.hasCyclicVector_baseChange_iff",
1958 "Komorebi.PaperICompletion6.ScalarExtension.hasCyclicVector_iff_extended_spectral_multiplicities_one",
1959 "Komorebi.PaperICompletion6.PrimaryCyclicCriterion.hasCyclicVector_iff_polynomialModule_cyclic",
1960 "Komorebi.PaperICompletion6.PrimaryCyclicCriterion.submodule_cyclic_of_hasCyclicVector",
1961 "Komorebi.PaperICompletion6.PrimaryCyclicCriterion.hasCyclicVector_iff_nonzero_primary_cyclic",
1962 "Komorebi.PaperICompletion6.OneInvariantFactor.HasOneInvariantFactor",
1963 "Komorebi.PaperICompletion6.OneInvariantFactor.factor_eq_minpoly",
1964 "Komorebi.PaperICompletion6.OneInvariantFactor.hasCyclicVector_iff_oneInvariantFactor",
1965 "Komorebi.PaperICompletion6.OneInvariantFactor.quotientVectorEquiv_intertwines",
1966 "Komorebi.PaperICompletion6.OneInvariantFactor.toMatrix_quotientCompanionBasis",
1967 "Komorebi.PaperICompletion6.OneInvariantFactor.exists_companion_basis_of_cyclic",
1968 "Komorebi.PaperICompletion6.SixWayCyclicCriterion.cyclic_iff_invariantFactor_count_one",
1969 "Komorebi.PaperICompletion6.SixWayCyclicCriterion.cyclic_six_way",
1970 "Komorebi.PaperICompletion6.SixWayCyclicCriterion.cyclic_six_way_algebraicClosure"
1971 ],
1972 "scope_and_remaining_work": "Block 06 closes the exact six-way criterion for a nonzero finite-dimensional operator over an arbitrary field. The final List.TFAE includes actual cyclicity, canonical seed rank one, cyclicity of every nonzero intrinsic primary component, an actual monic positive-degree one-invariant-factor presentation, minpoly=charpoly, and one-dimensional eigenspaces at every spectral value of the literal tensor extension to AlgebraicClosure k. Minimal-polynomial preservation and cyclicity preservation/descent are proved, not supplied as hypotheses. The one-factor presentation has a constructed basis in which the original operator is the standard companion matrix; its factor is proved to be the minimal polynomial. A separate theorem proves that any actual nonconstant invariant-factor chain has exactly one factor iff the operator is cyclic. This is a mathematical existence and equivalence theorem, not an executable normal-form algorithm or a full sorted invariant-factor construction for arbitrary noncyclic operators. The nonzero-space guard is retained in the six-way statement. Legacy conditional wrappers are preserved unchanged for compatibility, but the new public criterion needs no decomposition, multiplicity, polynomial-preservation, spectral, or seed-number certificates.",
1973 "current_source": {
1974 "file": "editable/traditional/02_foundations.tex",
1975 "file_sha256": "2754157dbfbcf1a8558084956b9a64798fae346b60bbf640a7e513c020b571c5",
1976 "kind": "corollary",
1977 "record_id_line": 1002,
1978 "statement_start_line": 990,
1979 "statement_end_line": 1001,
1980 "statement_sha256": "d418194e254211c8f964adf5796418ad7aaff14e630caffc05cb8d935de9803a",
1981 "proof_start_line": 1005,
1982 "proof_end_line": 1007,
1983 "proof_sha256": "09b4987a4d021df774a715f3371cd0b255dcd7c3d5e3587dfed92be6dccf1988"
1984 },
1985 "statement": "\\begin{corollary}[Criterion for a cyclic matrix]\n\\label{cor:cyclic-matrix}\nFor $X\\ne0$ of finite dimension, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $T$ has a cyclic vector.\n\\item $\\srank_{\\bbk[T]}(X)=1$.\n\\item Every nonzero primary component for $p$ is cyclic.\n\\item The rational canonical form has one invariant factor.\n\\item The minimal polynomial equals the characteristic polynomial.\n\\item After scalar extension to an algebraic closure, every eigenvalue has geometric multiplicity one.\n\\end{enumerate}\n\\end{corollary}",
1986 "source_proof": "\\begin{proof}\nThe equivalence of (i) and (ii) is \\cref{prop:cyclic-characterization}. By \\cref{thm:single-operator}, (ii) holds exactly when every $\\mu_p(T)$ belonging to a nonzero primary component equals one, which is (iii). Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv). In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)$\\Leftrightarrow$(v). Scalar extension preserves the equality of minimal and characteristic polynomials. Over an algebraic closure, that equality is equivalent to one Jordan block for each eigenvalue, which is (vi).\n\\end{proof}",
1987 "new_block_contribution": [
1988 "Komorebi.PaperICompletion.CyclicClassification.hasCyclicVector_iff_minpoly_eq_charpoly",
1989 "Komorebi.PaperICompletion.CyclicClassification.operatorAlgebra_seedRank_eq_one_iff_minpoly_eq_charpoly",
1990 "Komorebi.PaperICompletion.CyclicClassification.cyclic_four_way_without_polynomial_certificate"
1991 ],
1992 "block03_contribution": [
1993 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.PrimaryPresentation.cyclic_iff_label_injective",
1994 "Komorebi.PaperICompletion3.OperatorPrimaryAPI.PrimaryPresentation.label_injective_iff_minpoly_eq_charpoly",
1995 "Komorebi.PaperICompletion3.InvariantFactorCount.invariantFactor_operator_seedRanks"
1996 ],
1997 "block04_contribution": [
1998 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.operatorPrimaryComponentEquiv",
1999 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.intrinsicMultiplicity_eq_fiber",
2000 "Komorebi.PaperICompletion4.OperatorIntrinsicFormula.primary_fiber_presentation_invariant"
2001 ],
2002 "block05_contribution": [
2003 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.hasCyclicVector_iff_minpoly_quotient_presentation",
2004 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.cyclicMinpolyQuotientEquiv",
2005 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.hasCyclicVector_iff_all_spectral_multiplicities_one",
2006 "Komorebi.PaperICompletion5.CyclicGeometricCriterion.minpoly_eq_charpoly_iff_all_spectral_multiplicities_one"
2007 ],
2008 "block06_contribution": [
2009 "Komorebi.PaperICompletion6.ScalarExtension.minpoly_baseChange",
2010 "Komorebi.PaperICompletion6.ScalarExtension.minpoly_natDegree_baseChange",
2011 "Komorebi.PaperICompletion6.ScalarExtension.minpoly_eq_charpoly_baseChange_iff",
2012 "Komorebi.PaperICompletion6.ScalarExtension.hasCyclicVector_baseChange_iff",
2013 "Komorebi.PaperICompletion6.ScalarExtension.hasCyclicVector_iff_extended_spectral_multiplicities_one",
2014 "Komorebi.PaperICompletion6.PrimaryCyclicCriterion.hasCyclicVector_iff_polynomialModule_cyclic",
2015 "Komorebi.PaperICompletion6.PrimaryCyclicCriterion.submodule_cyclic_of_hasCyclicVector",
2016 "Komorebi.PaperICompletion6.PrimaryCyclicCriterion.hasCyclicVector_iff_nonzero_primary_cyclic",
2017 "Komorebi.PaperICompletion6.OneInvariantFactor.HasOneInvariantFactor",
2018 "Komorebi.PaperICompletion6.OneInvariantFactor.factor_eq_minpoly",
2019 "Komorebi.PaperICompletion6.OneInvariantFactor.hasCyclicVector_iff_oneInvariantFactor",
2020 "Komorebi.PaperICompletion6.OneInvariantFactor.quotientVectorEquiv_intertwines",
2021 "Komorebi.PaperICompletion6.OneInvariantFactor.toMatrix_quotientCompanionBasis",
2022 "Komorebi.PaperICompletion6.OneInvariantFactor.exists_companion_basis_of_cyclic",
2023 "Komorebi.PaperICompletion6.SixWayCyclicCriterion.cyclic_iff_invariantFactor_count_one",
2024 "Komorebi.PaperICompletion6.SixWayCyclicCriterion.cyclic_six_way",
2025 "Komorebi.PaperICompletion6.SixWayCyclicCriterion.cyclic_six_way_algebraicClosure"
2026 ],
2027 "acceptance_provenance": [
2028 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
2029 ]
2030 },
2031 {
2032 "ordinal": 47,
2033 "stable_id": "PSC-I-D-PACKET-SYSTEM",
2034 "kind": "definition",
2035 "title": "Finite packet system",
2036 "status": "definition_represented",
2037 "old_contract_rows": [
2038 15
2039 ],
2040 "mapped_declarations": [
2041 "Komorebi.Foundation.LinearProjectorFamily.rangeSynthesis_rangeAnalysis",
2042 "Komorebi.Foundation.LinearProjectorFamily.rangeAnalysis_rangeSynthesis",
2043 "Komorebi.Foundation.LinearProjectorFamily.exactPacketCoordinates"
2044 ],
2045 "scope_and_remaining_work": "LinearProjectorFamily encodes the actual finite completeness, idempotence and annihilation laws.",
2046 "current_source": {
2047 "file": "editable/traditional/03_packets_and_graphs.tex",
2048 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2049 "kind": "definition",
2050 "record_id_line": 26,
2051 "statement_start_line": 10,
2052 "statement_end_line": 25,
2053 "statement_sha256": "251dfa1ba22ba22eaf53fdcc7880aa1fe05933738f362ef3f6ae72583f539459",
2054 "proof_start_line": null,
2055 "proof_end_line": null,
2056 "proof_sha256": null
2057 },
2058 "statement": "\\begin{definition}[Finite packet system]\n\\label{def:packet-system}\nA finite packet system on a vector space $X$ over $\\bbk$ is a family\n\\[\n\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\subseteq\\End_{\\bbk}(X)\n\\]\nsatisfying\n\\[\nP_\\alpha^2=P_\\alpha,\n\\qquad\nP_\\alpha P_\\beta=0\\quad(\\alpha\\neq\\beta),\n\\qquad\n\\sum_{\\alpha\\in I}P_\\alpha=I_X.\n\\]\nSet $X_\\alpha=P_\\alpha X$. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.\n\\end{definition}",
2059 "source_proof": null,
2060 "new_block_contribution": []
2061 },
2062 {
2063 "ordinal": 48,
2064 "stable_id": "PSC-I-D-PACKET-SUPPORT",
2065 "kind": "definition",
2066 "title": "Packet support and packet-pure vectors",
2067 "status": "definition_represented",
2068 "old_contract_rows": [
2069 26
2070 ],
2071 "mapped_declarations": [
2072 "Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support",
2073 "Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction",
2074 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.closure_le_ker_of_le_ker",
2075 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.topological_reachability_obstruction",
2076 "Komorebi.PaperIRelease.packetSupport",
2077 "Komorebi.PaperIRelease.PacketPure",
2078 "Komorebi.PaperIRelease.packetPure_iff_zero_or_range"
2079 ],
2080 "scope_and_remaining_work": "packetSupport is exactly {i | P_i x≠0}. PacketPure is Set.Subsingleton of that support, equivalent for finite support to cardinality at most one. packetPure_iff_zero_or_range identifies purity with zero or membership in one actual packet range; a pure family is pointwise this predicate.",
2081 "current_source": {
2082 "file": "editable/traditional/03_packets_and_graphs.tex",
2083 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2084 "kind": "definition",
2085 "record_id_line": 38,
2086 "statement_start_line": 28,
2087 "statement_end_line": 37,
2088 "statement_sha256": "d95377c31eb89901afc468fd8f0ba293fee6865b0ae3b22eff99df183cc9eb9e",
2089 "proof_start_line": null,
2090 "proof_end_line": null,
2091 "proof_sha256": null
2092 },
2093 "statement": "\\begin{definition}[Packet support and packet pure vectors]\n\\label{def:packet-support}\nFor a finite packet system \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(x\\in X\\), define\n\\[\n\\Supp_{\\Pcal}(x)\n:=\n\\{\\alpha\\in I:P_\\alpha x\\neq0\\}.\n\\]\nThe vector \\(x\\) is \\emph{packet pure} if \\(|\\Supp_{\\Pcal}(x)|\\leq1\\). A finite family is packet pure if each of its members is packet pure, although different members may lie in different packets.\n\\end{definition}",
2094 "source_proof": null,
2095 "new_block_contribution": [],
2096 "acceptance_provenance": [
2097 {
2098 "semantic_review": "semantic_wave2.json",
2099 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
2100 "reviewed_statement_sha256": "d95377c31eb89901afc468fd8f0ba293fee6865b0ae3b22eff99df183cc9eb9e",
2101 "reviewed_proof_sha256": null,
2102 "verdict": "represented as written"
2103 }
2104 ],
2105 "remaining_work": null
2106 },
2107 {
2108 "ordinal": 49,
2109 "stable_id": "PSC-I-L-PACKET-AGGREGATION",
2110 "kind": "lemma",
2111 "title": "Finite packet aggregation",
2112 "status": "covered",
2113 "old_contract_rows": [],
2114 "mapped_declarations": [
2115 "Komorebi.Packet.fibreLinearProj_refinement",
2116 "Komorebi.PaperIRelease.PacketAggregation.groupedProjection",
2117 "Komorebi.PaperIRelease.PacketAggregation.groupedFamily",
2118 "Komorebi.PaperIRelease.PacketAggregation.groupedRangeDirectSumEquiv"
2119 ],
2120 "scope_and_remaining_work": "Q_j is the actual filtered sum of P_i. Computing P_a Q_j proves the laws. Actual range analysis/synthesis over the literal fibre subtype are inverse, and the finite product/direct-sum equivalence gives the printed conclusion.",
2121 "current_source": {
2122 "file": "editable/traditional/03_packets_and_graphs.tex",
2123 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2124 "kind": "lemma",
2125 "record_id_line": 56,
2126 "statement_start_line": 45,
2127 "statement_end_line": 55,
2128 "statement_sha256": "504cbd4eec7a5b5e416f91dcfd13e06586e5c1360990ed480424f4a85dfb9195",
2129 "proof_start_line": 59,
2130 "proof_end_line": 67,
2131 "proof_sha256": "9f75e67b18fd41fdeec9b9c6993c24c95926b9e2ea61e450978392cf8515ed72"
2132 },
2133 "statement": "\\begin{lemma}[Finite packet aggregation]\n\\label{lem:packet-aggregation}\nLet $\\sigma:I\\twoheadrightarrow J$ be a surjection and define\n\\[\nQ_j:=\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nThen $(Q_j)_{j\\in J}$ is a finite packet system and\n\\[\nQ_jX=\\bigoplus_{\\sigma(\\alpha)=j}X_\\alpha.\n\\]\n\\end{lemma}",
2134 "source_proof": "\\begin{proof}\nPairwise annihilation of the $P_\\alpha$ gives\n\\[\nQ_j^2=Q_j,\n\\qquad\nQ_jQ_k=0\\quad(j\\neq k).\n\\]\nSumming over $j$ recovers $\\sum_\\alpha P_\\alpha=I$. The range statement follows from the directness forced by pairwise annihilation.\n\\end{proof}",
2135 "new_block_contribution": [
2136 "Komorebi.Packet.fibreLinearProj_refinement"
2137 ],
2138 "acceptance_provenance": [
2139 {
2140 "semantic_review": "semantic_wave1.json",
2141 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
2142 "reviewed_statement_sha256": "504cbd4eec7a5b5e416f91dcfd13e06586e5c1360990ed480424f4a85dfb9195",
2143 "reviewed_proof_sha256": "9f75e67b18fd41fdeec9b9c6993c24c95926b9e2ea61e450978392cf8515ed72",
2144 "verdict": "proved as written"
2145 }
2146 ],
2147 "remaining_work": null
2148 },
2149 {
2150 "ordinal": 50,
2151 "stable_id": "PSC-I-D-ANALYSIS-SYNTHESIS",
2152 "kind": "definition",
2153 "title": "Packet analysis and synthesis",
2154 "status": "definition_represented",
2155 "old_contract_rows": [
2156 15
2157 ],
2158 "mapped_declarations": [
2159 "Komorebi.Foundation.LinearProjectorFamily.rangeSynthesis_rangeAnalysis",
2160 "Komorebi.Foundation.LinearProjectorFamily.rangeAnalysis_rangeSynthesis",
2161 "Komorebi.Foundation.LinearProjectorFamily.exactPacketCoordinates"
2162 ],
2163 "scope_and_remaining_work": "Analysis and synthesis use genuine projector ranges, not unconstrained packet slots.",
2164 "current_source": {
2165 "file": "editable/traditional/03_packets_and_graphs.tex",
2166 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2167 "kind": "definition",
2168 "record_id_line": 84,
2169 "statement_start_line": 69,
2170 "statement_end_line": 83,
2171 "statement_sha256": "f364fcc94b357d761a0bb880e5a2d5761fe7bfc687b768b0f4f7cc261a5ce820",
2172 "proof_start_line": null,
2173 "proof_end_line": null,
2174 "proof_sha256": null
2175 },
2176 "statement": "\\begin{definition}[Packet analysis and synthesis]\n\\label{def:analysis-synthesis}\nFor a finite packet system, define\n\\[\n\\mathcal A_{\\Pcal}:X\\longrightarrow\\bigoplus_{\\alpha\\in I}X_\\alpha,\n\\qquad\nx\\longmapsto(P_\\alpha x)_\\alpha,\n\\]\nand\n\\[\n\\mathcal S_{\\Pcal}:\\bigoplus_{\\alpha\\in I}X_\\alpha\\longrightarrow X,\n\\qquad\n(x_\\alpha)_\\alpha\\longmapsto\\sum_\\alpha x_\\alpha.\n\\]\n\\end{definition}",
2177 "source_proof": null,
2178 "new_block_contribution": []
2179 },
2180 {
2181 "ordinal": 51,
2182 "stable_id": "PSC-I-T-EXACT-PACKET-COORDINATES",
2183 "kind": "theorem",
2184 "title": "Exact packet-coordinate isomorphism",
2185 "status": "covered",
2186 "old_contract_rows": [
2187 15
2188 ],
2189 "mapped_declarations": [
2190 "Komorebi.Foundation.LinearProjectorFamily.rangeSynthesis_rangeAnalysis",
2191 "Komorebi.Foundation.LinearProjectorFamily.rangeAnalysis_rangeSynthesis",
2192 "Komorebi.Foundation.LinearProjectorFamily.exactPacketCoordinates"
2193 ],
2194 "scope_and_remaining_work": "Both inverse identities and the linear equivalence are proved.",
2195 "current_source": {
2196 "file": "editable/traditional/03_packets_and_graphs.tex",
2197 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2198 "kind": "theorem",
2199 "record_id_line": 101,
2200 "statement_start_line": 86,
2201 "statement_end_line": 100,
2202 "statement_sha256": "d50988e4b39d3f37422e1d0c2704aca3b8acd89beca5afd9d07ad6bc326fc24f",
2203 "proof_start_line": 105,
2204 "proof_end_line": 121,
2205 "proof_sha256": "87dea1a42da3309fb42ca456003864370bacc7be0fe6b6b83841bb2c8a188029"
2206 },
2207 "statement": "\\begin{theorem}[Exact packet coordinate isomorphism]\n\\label{thm:exact-packet-coordinates}\nFor every finite packet system,\n\\[\n\\boxed{\n\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}=I_X,\n\\qquad\n\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I_{\\oplus X_\\alpha}.\n}\n\\]\nHence\n\\[\n\\boxed{X\\cong\\bigoplus_{\\alpha\\in I}X_\\alpha.}\n\\]\n\\end{theorem}",
2208 "source_proof": "\\begin{proof}\nFor $x\\in X$,\n\\[\n\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x\n=\n\\sum_\\alpha P_\\alpha x\n=x.\n\\]\nIf $x_\\alpha\\in X_\\alpha$, then $P_\\beta x_\\alpha=\\delta_{\\alpha\\beta}x_\\alpha$, so\n\\[\n\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}(x_\\alpha)_\\alpha\n=\n\\left(P_\\beta\\sum_\\alpha x_\\alpha\\right)_\\beta\n=(x_\\beta)_\\beta.\n\\]\nThus analysis and synthesis are inverse linear maps.\n\\end{proof}",
2209 "new_block_contribution": [],
2210 "acceptance_provenance": [
2211 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
2212 ]
2213 },
2214 {
2215 "ordinal": 52,
2216 "stable_id": "PSC-I-C-PACKETWISE-EQUALITY",
2217 "kind": "corollary",
2218 "title": "Packetwise equality",
2219 "status": "covered",
2220 "old_contract_rows": [
2221 16
2222 ],
2223 "mapped_declarations": [
2224 "Komorebi.PacketNaturalityCompletion.Packets.eq_iff_projectors_eq"
2225 ],
2226 "scope_and_remaining_work": "Exact equality iff every projected coordinate agrees.",
2227 "current_source": {
2228 "file": "editable/traditional/03_packets_and_graphs.tex",
2229 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2230 "kind": "corollary",
2231 "record_id_line": 137,
2232 "statement_start_line": 128,
2233 "statement_end_line": 136,
2234 "statement_sha256": "61919a93863e4ac0a41653a394e44cd8f701eac05335283ac8cf73983da43859",
2235 "proof_start_line": 140,
2236 "proof_end_line": 142,
2237 "proof_sha256": "4f6a8e5c3548e4c5d8c3e8ad694f8ec7937a7d2a16374574dc893f9a2b0decbb"
2238 },
2239 "statement": "\\begin{corollary}[Packetwise equality]\n\\label{cor:packetwise-equality}\nFor $x,y\\in X$,\n\\[\nx=y\n\\quad\\Longleftrightarrow\\quad\nP_\\alpha x=P_\\alpha y\\text{ for every }\\alpha.\n\\]\n\\end{corollary}",
2240 "source_proof": "\\begin{proof}\nThe forward implication is immediate. For the reverse implication, sum the packet equalities and use $\\sum_\\alpha P_\\alpha=I$.\n\\end{proof}",
2241 "new_block_contribution": [],
2242 "acceptance_provenance": [
2243 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
2244 ]
2245 },
2246 {
2247 "ordinal": 53,
2248 "stable_id": "PSC-I-C-PACKET-UNIVERSAL-PROPERTY",
2249 "kind": "corollary",
2250 "title": "Universal mapping property of finite packet coordinates",
2251 "status": "covered",
2252 "old_contract_rows": [
2253 15,
2254 23
2255 ],
2256 "mapped_declarations": [
2257 "Komorebi.Foundation.LinearProjectorFamily.rangeSynthesis_rangeAnalysis",
2258 "Komorebi.Foundation.LinearProjectorFamily.rangeAnalysis_rangeSynthesis",
2259 "Komorebi.Foundation.LinearProjectorFamily.exactPacketCoordinates",
2260 "Komorebi.PacketNaturalityCompletion.FullBlocks.fullBlockAlgEquiv",
2261 "Komorebi.PacketNaturalityCompletion.FullBlocks.restrictBlocks_synthesizeBlocks",
2262 "Komorebi.PacketNaturalityCompletion.FullBlocks.synthesizeBlocks_restrictBlocks",
2263 "Komorebi.PacketNaturalityCompletion.FullBlocks.packetBlockAlgebra_mul_eq_blockMul",
2264 "Komorebi.PaperIRelease.PacketUniversal.assembleOut",
2265 "Komorebi.PaperIRelease.PacketUniversal.unique_packet_extension",
2266 "Komorebi.PaperIRelease.PacketUniversal.assembleIn",
2267 "Komorebi.PaperIRelease.PacketUniversal.unique_packet_lift"
2268 ],
2269 "scope_and_remaining_work": "Outgoing extension is the finite sum of f_i after actual rangeRestrict projections. Incoming assembly is actual rangeSynthesis after LinearMap.pi. Packet laws give restrictions/projections and completeness gives uniqueness. These universal properties induce the displayed Hom equivalences.",
2270 "current_source": {
2271 "file": "editable/traditional/03_packets_and_graphs.tex",
2272 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2273 "kind": "corollary",
2274 "record_id_line": 167,
2275 "statement_start_line": 146,
2276 "statement_end_line": 166,
2277 "statement_sha256": "3136b91ef330400f72205441080d679179476d4620287e1bd3658a36dca7a4be",
2278 "proof_start_line": 171,
2279 "proof_end_line": 183,
2280 "proof_sha256": "0253b889d211af1847f8b7641a205eff5a00a0b60ac7663c436bbdaf38abb4f6"
2281 },
2282 "statement": "\\begin{corollary}[Universal mapping property of finite packet coordinates]\n\\label{cor:packet-universal-property}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(Y\\) be a vector space over \\(\\bbk\\).\n\\begin{enumerate}\n\\item Every family of linear maps \\(f_\\alpha:X_\\alpha\\to Y\\) determines a unique linear map \\(f:X\\to Y\\) with \\(f|_{X_\\alpha}=f_\\alpha\\), namely\n\\[\nf=\\sum_\\alpha f_\\alpha P_\\alpha.\n\\]\n\\item Every family of linear maps \\(g_\\alpha:Y\\to X_\\alpha\\) determines a unique linear map \\(g:Y\\to X\\) with \\(P_\\alpha g=g_\\alpha\\), namely\n\\[\ng=\\sum_\\alpha g_\\alpha.\n\\]\n\\end{enumerate}\nEquivalently,\n\\[\n\\Hom(X,Y)\\cong\\prod_\\alpha\\Hom(X_\\alpha,Y),\n\\qquad\n\\Hom(Y,X)\\cong\\prod_\\alpha\\Hom(Y,X_\\alpha),\n\\]\nwhere the products are finite.\n\\end{corollary}",
2283 "source_proof": "\\begin{proof}\nFor \\(x=\\sum_\\alpha P_\\alpha x\\), any extension of the \\(f_\\alpha\\) must satisfy\n\\[\nf(x)=\\sum_\\alpha f_\\alpha(P_\\alpha x),\n\\]\nwhich proves both existence and uniqueness of the first formula. For the second, the sum \\(g=\\sum_\\alpha g_\\alpha\\) satisfies\n\\[\nP_\\beta g\n=\\sum_\\alpha P_\\beta g_\\alpha\n=g_\\beta\n\\]\nbecause \\(g_\\alpha(Y)\\subseteq X_\\alpha\\). Uniqueness follows from \\(\\sum_\\alpha P_\\alpha=I\\).\n\\end{proof}",
2284 "new_block_contribution": [],
2285 "acceptance_provenance": [
2286 {
2287 "semantic_review": "semantic_wave1.json",
2288 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
2289 "reviewed_statement_sha256": "3136b91ef330400f72205441080d679179476d4620287e1bd3658a36dca7a4be",
2290 "reviewed_proof_sha256": "0253b889d211af1847f8b7641a205eff5a00a0b60ac7663c436bbdaf38abb4f6",
2291 "verdict": "proved as written"
2292 }
2293 ],
2294 "remaining_work": null
2295 },
2296 {
2297 "ordinal": 54,
2298 "stable_id": "PSC-I-T-TENSOR-PACKET-SYSTEM",
2299 "kind": "theorem",
2300 "title": "Tensor-product packet systems",
2301 "status": "covered",
2302 "old_contract_rows": [],
2303 "mapped_declarations": [
2304 "Komorebi.PaperIRelease.tensorPackets",
2305 "Komorebi.PaperIRelease.tensor_packet_range",
2306 "Komorebi.PaperIRelease.tensorPacketRangeEquiv",
2307 "Komorebi.PaperIRelease.tensorPacketDirectSumCoordinates"
2308 ],
2309 "scope_and_remaining_work": "Actual TensorProduct.map projectors satisfy all laws by tensor induction. Tensor products of range inclusions have the tensor products of projections as left inverses; range_map identifies the actual packet range. This proves the canonical embedded interpretation of the printed range equality. Exact coordinates then give the literal DirectSum.",
2310 "current_source": {
2311 "file": "editable/traditional/03_packets_and_graphs.tex",
2312 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2313 "kind": "theorem",
2314 "record_id_line": 211,
2315 "statement_start_line": 188,
2316 "statement_end_line": 210,
2317 "statement_sha256": "afea7128594f24b1249382b028d823051a35d9bd3a3473c247a7b9bd63c5ba47",
2318 "proof_start_line": 215,
2319 "proof_end_line": 232,
2320 "proof_sha256": "a1a8250dc2284fb83d49185fdb87dedc5744f633335861227d38b439dc950ca0"
2321 },
2322 "statement": "\\begin{theorem}[Tensor product packet systems]\n\\label{thm:tensor-packet-system}\nLet \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) be a finite packet system on \\(X\\), and let \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be a finite packet system on \\(Y\\). Then\n\\[\n\\Pcal\\boxtimes\\Qcal\n:=\n(P_\\alpha\\otimes Q_\\gamma)_{(\\alpha,\\gamma)\\in I\\times J}\n\\]\nis a finite packet system on the algebraic tensor product \\(X\\otimes_{\\bbk}Y\\), with packet spaces\n\\[\n(P_\\alpha\\otimes Q_\\gamma)(X\\otimes Y)\n=X_\\alpha\\otimes Y_\\gamma.\n\\]\nHence\n\\[\n\\boxed{\nX\\otimes Y\n\\cong\n\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}\nX_\\alpha\\otimes Y_\\gamma.\n}\n\\]\n\\end{theorem}",
2323 "source_proof": "\\begin{proof}\nFor every pair of packet indices,\n\\[\n(P_\\alpha\\otimes Q_\\gamma)^2\n=P_\\alpha^2\\otimes Q_\\gamma^2\n=P_\\alpha\\otimes Q_\\gamma.\n\\]\nIf \\((\\alpha,\\gamma)\\neq(\\beta,\\delta)\\), then either \\(P_\\alpha P_\\beta=0\\) or \\(Q_\\gamma Q_\\delta=0\\), so the corresponding tensor idempotents annihilate each other. Finally,\n\\[\n\\sum_{\\alpha,\\gamma}P_\\alpha\\otimes Q_\\gamma\n=\n\\left(\\sum_\\alpha P_\\alpha\\right)\n\\otimes\n\\left(\\sum_\\gamma Q_\\gamma\\right)\n=I_X\\otimes I_Y.\n\\]\nThe range is generated by pure tensors \\(P_\\alpha x\\otimes Q_\\gamma y\\), hence equals \\(X_\\alpha\\otimes Y_\\gamma\\). Exact reconstruction follows from \\cref{thm:exact-packet-coordinates}.\n\\end{proof}",
2324 "new_block_contribution": [],
2325 "acceptance_provenance": [
2326 {
2327 "semantic_review": "semantic_wave1.json",
2328 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
2329 "reviewed_statement_sha256": "afea7128594f24b1249382b028d823051a35d9bd3a3473c247a7b9bd63c5ba47",
2330 "reviewed_proof_sha256": "a1a8250dc2284fb83d49185fdb87dedc5744f633335861227d38b439dc950ca0",
2331 "verdict": "proved as written"
2332 }
2333 ],
2334 "remaining_work": null
2335 },
2336 {
2337 "ordinal": 55,
2338 "stable_id": "PSC-I-P-DUAL-PACKET-SYSTEM",
2339 "kind": "proposition",
2340 "title": "Dual packet systems",
2341 "status": "covered",
2342 "old_contract_rows": [],
2343 "mapped_declarations": [
2344 "Komorebi.PaperIRelease.dualPackets",
2345 "Komorebi.PaperIRelease.dualPacketRangeEquiv",
2346 "Komorebi.PaperIRelease.dualPacketDirectSumCoordinates"
2347 ],
2348 "scope_and_remaining_work": "Actual dualMap means precomposition. Composition reverses indices and orthogonality uses the reversed inequality correctly. Restriction and precomposition with the actual packet projection are inverse, giving the stated packet duals and literal finite DirectSum.",
2349 "current_source": {
2350 "file": "editable/traditional/03_packets_and_graphs.tex",
2351 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2352 "kind": "proposition",
2353 "record_id_line": 247,
2354 "statement_start_line": 234,
2355 "statement_end_line": 246,
2356 "statement_sha256": "ade34fd8142ece280ab2ae85913a943cb28a3661978659fde4a644eb5018d119",
2357 "proof_start_line": 251,
2358 "proof_end_line": 258,
2359 "proof_sha256": "918bd1d6e0cd78673db96438ef5abe5189c8a0f136b9b4a9dcad0a813294c04d"
2360 },
2361 "statement": "\\begin{proposition}[Dual packet systems]\n\\label{prop:dual-packet-system}\nLet \\(X\\) be finite dimensional and let \\(\\Pcal=(P_\\alpha)\\) be a finite packet system. The algebraic dual maps\n\\[\nP_\\alpha^*:X^*\\to X^*,\n\\qquad\nP_\\alpha^*(\\varphi)=\\varphi\\circ P_\\alpha,\n\\]\nform a finite packet system on \\(X^*\\). Its packet with index \\(\\alpha\\) is canonically isomorphic to \\(X_\\alpha^*\\), and\n\\[\nX^*\\cong\\bigoplus_\\alpha X_\\alpha^*.\n\\]\n\\end{proposition}",
2362 "source_proof": "\\begin{proof}\nBecause dualization reverses products,\n\\[\nP_\\alpha^*P_\\beta^*=(P_\\beta P_\\alpha)^*\n=\\delta_{\\alpha\\beta}P_\\alpha^*.\n\\]\nAlso \\(\\sum_\\alpha P_\\alpha^*=I_{X^*}\\). Thus the dual maps form a packet system. Restriction gives a map from \\(P_\\alpha^*X^*\\) to \\(X_\\alpha^*\\). It is injective because a covector in the range of \\(P_\\alpha^*\\) vanishes on every other packet, and it is surjective because any covector on \\(X_\\alpha\\) extends to \\(X\\) through the direct sum decomposition.\n\\end{proof}",
2363 "new_block_contribution": [],
2364 "acceptance_provenance": [
2365 {
2366 "semantic_review": "semantic_wave1.json",
2367 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
2368 "reviewed_statement_sha256": "ade34fd8142ece280ab2ae85913a943cb28a3661978659fde4a644eb5018d119",
2369 "reviewed_proof_sha256": "918bd1d6e0cd78673db96438ef5abe5189c8a0f136b9b4a9dcad0a813294c04d",
2370 "verdict": "proved as written"
2371 }
2372 ],
2373 "remaining_work": null
2374 },
2375 {
2376 "ordinal": 56,
2377 "stable_id": "PSC-I-T-BANACH-PACKET-ISOMORPHISM",
2378 "kind": "theorem",
2379 "title": "Finite Banach packet isomorphism",
2380 "status": "covered",
2381 "old_contract_rows": [
2382 17
2383 ],
2384 "mapped_declarations": [
2385 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.topological_packet_isomorphism",
2386 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.packetEquiv",
2387 "Komorebi.PaperIRelease.GeneralNormedPackets.ContinuousProjectorFamily.equivalent_norm_packet_bounds",
2388 "Komorebi.PaperIRelease.GeneralNormedPackets.ContinuousProjectorFamily.analysis_isBounded",
2389 "Komorebi.PaperIRelease.GeneralNormedPackets.ContinuousProjectorFamily.synthesis_isBounded",
2390 "Komorebi.PaperIRelease.GeneralNormedPackets.ContinuousProjectorFamily.isClosed_packet"
2391 ],
2392 "scope_and_remaining_work": "The actual finite projectors are continuous and explicitly bounded as IsBoundedLinearMap over an arbitrary NormedField, including a trivial valuation. Individual bounds produce one finite uniform bound; analysis has that bound and synthesis has the finite-cardinality bound. Any coordinate norm equivalent to the canonical maximum norm is represented by a linear identification and its literal two-sided norm inequalities. Composition gives bounded actual analysis and synthesis, both inverse identities, and closed packet ranges. No conclusion-shaped hypotheses occur.",
2393 "current_source": {
2394 "file": "editable/traditional/03_packets_and_graphs.tex",
2395 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2396 "kind": "theorem",
2397 "record_id_line": 271,
2398 "statement_start_line": 267,
2399 "statement_end_line": 270,
2400 "statement_sha256": "3fd1f470c1a5dcd6273739c6f6cb95e5fbb8f8c95e4fc4c76e010430a72afa3e",
2401 "proof_start_line": 274,
2402 "proof_end_line": 280,
2403 "proof_sha256": "aed3b5725625c1c7f464c2a0fbe6ff126d1ea214ad009a38e98d3f4a9e0955fd"
2404 },
2405 "statement": "\\begin{theorem}[Finite Banach packet isomorphism]\n\\label{thm:banach-packets}\nLet $X$ be Banach and let $(P_\\alpha)_{\\alpha\\in I}$ be a finite family of bounded pairwise annihilating idempotents with sum $I_X$. Then every range $X_\\alpha=P_\\alpha X$ is closed. For any finite product norm equivalent to the maximum norm on $\\bigoplus_\\alpha X_\\alpha$, packet analysis and synthesis are bounded inverse operators.\n\\end{theorem}",
2406 "source_proof": "\\begin{proof}\nThe range of a bounded idempotent is its space of fixed points,\n\\[\nP_\\alpha X=\\Ker(I-P_\\alpha),\n\\]\nwhich is closed. Because $I$ is finite, analysis and synthesis are bounded for the maximum and sum norms, respectively, and therefore for every equivalent finite product norm. Their inverse identities are algebraic and were proved in \\cref{thm:exact-packet-coordinates}.\n\\end{proof}",
2407 "new_block_contribution": [],
2408 "acceptance_provenance": [
2409 {
2410 "semantic_review": "semantic_analytic.json",
2411 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2412 "reviewed_statement_sha256": "3fd1f470c1a5dcd6273739c6f6cb95e5fbb8f8c95e4fc4c76e010430a72afa3e",
2413 "reviewed_proof_sha256": "aed3b5725625c1c7f464c2a0fbe6ff126d1ea214ad009a38e98d3f4a9e0955fd",
2414 "verdict": "proved as written"
2415 }
2416 ],
2417 "remaining_work": null
2418 },
2419 {
2420 "ordinal": 57,
2421 "stable_id": "PSC-I-D-PACKET-CONDITION-NUMBER",
2422 "kind": "definition",
2423 "title": "Packet-coordinate condition number",
2424 "status": "definition_represented",
2425 "old_contract_rows": [],
2426 "mapped_declarations": [
2427 "Komorebi.PaperICompletion.ConditionedCoordinates.conditionNumber",
2428 "Komorebi.PaperICompletion.ConditionedCoordinates.packetConditionNumber_eq",
2429 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.conditionNumber",
2430 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.ofBoundedLinear",
2431 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.packetAnalysis",
2432 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.packetSynthesis",
2433 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.equivalent_norm_conditioned_coordinates"
2434 ],
2435 "scope_and_remaining_work": "The definition is the product of the ordinary operator norms of the actual bounded analysis and synthesis. The generic bounded additive homomorphism norm is defined as the infimum of nonnegative valid operator bounds; linearity is retained by ofBoundedLinear. Equivalent-norm coordinates have explicit actual-map equalities.",
2436 "current_source": {
2437 "file": "editable/traditional/03_packets_and_graphs.tex",
2438 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2439 "kind": "definition",
2440 "record_id_line": 291,
2441 "statement_start_line": 282,
2442 "statement_end_line": 290,
2443 "statement_sha256": "7c67c989c536ab3eecad8fdf8c19b0ed6eef7c164c569d69a0e04de101b2460d",
2444 "proof_start_line": null,
2445 "proof_end_line": null,
2446 "proof_sha256": null
2447 },
2448 "statement": "\\begin{definition}[Packet coordinate condition number]\n\\label{def:packet-condition-number}\nFix a finite product norm on $\\bigoplus_\\alpha X_\\alpha$ equivalent to the maximum norm, as in \\cref{thm:banach-packets}. Define\n\\[\n\\kappa(\\Pcal):=\n\\|\\mathcal A_{\\Pcal}\\|\\,\n\\|\\mathcal S_{\\Pcal}\\|.\n\\]\n\\end{definition}",
2449 "source_proof": null,
2450 "new_block_contribution": [
2451 "Komorebi.PaperICompletion.ConditionedCoordinates.conditionNumber",
2452 "Komorebi.PaperICompletion.ConditionedCoordinates.packetConditionNumber_eq"
2453 ],
2454 "acceptance_provenance": [
2455 {
2456 "semantic_review": "semantic_analytic.json",
2457 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2458 "reviewed_statement_sha256": "7c67c989c536ab3eecad8fdf8c19b0ed6eef7c164c569d69a0e04de101b2460d",
2459 "reviewed_proof_sha256": null,
2460 "verdict": "proved as written"
2461 }
2462 ],
2463 "remaining_work": null
2464 },
2465 {
2466 "ordinal": 58,
2467 "stable_id": "PSC-I-P-CONDITIONED-COORDINATES",
2468 "kind": "proposition",
2469 "title": "Conditioned coordinate transport",
2470 "status": "covered",
2471 "old_contract_rows": [],
2472 "mapped_declarations": [
2473 "Komorebi.PaperICompletion.ConditionedCoordinates.one_le_conditionNumber",
2474 "Komorebi.PaperICompletion.ConditionedCoordinates.norm_coordinateOperator_le",
2475 "Komorebi.PaperICompletion.ConditionedCoordinates.norm_le_conditionNumber_mul_coordinateOperator",
2476 "Komorebi.PaperICompletion.ConditionedCoordinates.packet_conditioned_coordinate_bounds",
2477 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.one_le_conditionNumber",
2478 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.conditioned_coordinate_bounds",
2479 "Komorebi.PaperIRelease.GeneralConditionedCoordinates.equivalent_norm_conditioned_coordinates"
2480 ],
2481 "scope_and_remaining_work": "For nonzero X the identity has norm one. The actual inverse relation S∘A=id and operator norm submultiplicativity give κ≥1. The coordinate operator is literally A∘T∘S, and S∘[T]∘A=T gives the reverse estimate. The final wrapper constructs both bounded actual coordinate maps for every equivalent finite-product norm and proves both inequalities for every bounded linear T over any NormedField.",
2482 "current_source": {
2483 "file": "editable/traditional/03_packets_and_graphs.tex",
2484 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2485 "kind": "proposition",
2486 "record_id_line": 314,
2487 "statement_start_line": 293,
2488 "statement_end_line": 313,
2489 "statement_sha256": "6799946dd2c88a65a635b418c7d0a74e94a9af4d1dc88df66cdd3b05c4ebc180",
2490 "proof_start_line": 317,
2491 "proof_end_line": 327,
2492 "proof_sha256": "e0567e2c20c0dd98382dc7e094fbc4a844f45565c9ac34a507901466436e67d6"
2493 },
2494 "statement": "\\begin{proposition}[Conditioned coordinate transport]\n\\label{prop:conditioned-coordinates}\nFor a finite Banach packet system with $X\\ne0$,\n\\[\n\\kappa(\\Pcal)\\geq1.\n\\]\nIf $T\\in\\mathcal B(X)$ and\n\\[\n[T]_{\\Pcal}:=\\mathcal A_{\\Pcal}T\\mathcal S_{\\Pcal},\n\\]\nthen\n\\[\n\\boxed{\n\\|[T]_{\\Pcal}\\|\n\\leq\\kappa(\\Pcal)\\|T\\|,\n\\qquad\n\\|T\\|\n\\leq\\kappa(\\Pcal)\\|[T]_{\\Pcal}\\|.\n}\n\\]\n\\end{proposition}",
2495 "source_proof": "\\begin{proof}\nSince $I=\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}$,\n\\[\n1=\\|I\\|\\leq\\|\\mathcal S_{\\Pcal}\\|\\|\\mathcal A_{\\Pcal}\\|.\n\\]\nThe first displayed estimate follows from submultiplicativity. The inverse identity\n\\[\nT=\\mathcal S_{\\Pcal}[T]_{\\Pcal}\\mathcal A_{\\Pcal}\n\\]\ngives the second estimate.\n\\end{proof}",
2496 "new_block_contribution": [
2497 "Komorebi.PaperICompletion.ConditionedCoordinates.one_le_conditionNumber",
2498 "Komorebi.PaperICompletion.ConditionedCoordinates.norm_coordinateOperator_le",
2499 "Komorebi.PaperICompletion.ConditionedCoordinates.norm_le_conditionNumber_mul_coordinateOperator",
2500 "Komorebi.PaperICompletion.ConditionedCoordinates.packet_conditioned_coordinate_bounds"
2501 ],
2502 "acceptance_provenance": [
2503 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below.",
2504 {
2505 "semantic_review": "semantic_analytic.json",
2506 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2507 "reviewed_statement_sha256": "6799946dd2c88a65a635b418c7d0a74e94a9af4d1dc88df66cdd3b05c4ebc180",
2508 "reviewed_proof_sha256": "e0567e2c20c0dd98382dc7e094fbc4a844f45565c9ac34a507901466436e67d6",
2509 "verdict": "proved as written"
2510 }
2511 ],
2512 "remaining_work": null
2513 },
2514 {
2515 "ordinal": 59,
2516 "stable_id": "PSC-I-C-COMPLETION-PACKET-SPLIT",
2517 "kind": "corollary",
2518 "title": "Completion preserves a finite packet split",
2519 "status": "covered",
2520 "old_contract_rows": [
2521 18
2522 ],
2523 "mapped_declarations": [
2524 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.idempotentRange_eq_closure_map",
2525 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.packetRange_eq_closure_map",
2526 "Komorebi.PaperIRelease.GeneralFiniteCompletion.extendPacketFamily",
2527 "Komorebi.PaperIRelease.GeneralFiniteCompletion.extended_packet_range",
2528 "Komorebi.PaperIRelease.GeneralFiniteCompletion.extension_unique"
2529 ],
2530 "scope_and_remaining_work": "Over an arbitrary NormedField, the actual dense-submodule restrictions determine the ambient continuous projector identities by equality on a dense set. Each ambient range equals closure of the corresponding dense-submodule image, and two continuous ambient projector extensions agreeing on the dense submodule coincide. This is stronger than the bounded-input manuscript statement; boundedness of the given ambient projectors is available from the printed premise and is not needed for the density/uniqueness argument.",
2531 "current_source": {
2532 "file": "editable/traditional/03_packets_and_graphs.tex",
2533 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2534 "kind": "corollary",
2535 "record_id_line": 342,
2536 "statement_start_line": 334,
2537 "statement_end_line": 341,
2538 "statement_sha256": "67202410fc4580e865ea492dba6cf8dff7e0f296d89ac886c96188a0169a6b70",
2539 "proof_start_line": 345,
2540 "proof_end_line": 347,
2541 "proof_sha256": "5b5ce6344d6d25c724e1e2c8d1ebf24c672e713d5cb47843a92abf33c4c7a913"
2542 },
2543 "statement": "\\begin{corollary}[Completion preserves a finite packet split]\n\\label{cor:completion-packet-split}\nLet $M$ be a dense normed subspace of a Banach space $X$. Suppose every bounded $P_\\alpha$ leaves $M$ invariant and the restricted projectors form a packet system on $M$. Then\n\\[\nP_\\alpha X=\\cl{P_\\alpha M}\n\\]\nand the packet decomposition of $M$ extends uniquely to the completion $X$.\n\\end{corollary}",
2544 "source_proof": "\\begin{proof}\nFor $x\\in X$, choose $m_n\\in M$ with $m_n\\to x$. Continuity gives $P_\\alpha m_n\\to P_\\alpha x$, so $P_\\alpha X\\subseteq\\cl{P_\\alpha M}$. The reverse inclusion follows because $P_\\alpha X$ is closed and contains $P_\\alpha M$. The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.\n\\end{proof}",
2545 "new_block_contribution": [],
2546 "acceptance_provenance": [
2547 {
2548 "semantic_review": "semantic_analytic.json",
2549 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2550 "reviewed_statement_sha256": "67202410fc4580e865ea492dba6cf8dff7e0f296d89ac886c96188a0169a6b70",
2551 "reviewed_proof_sha256": "5b5ce6344d6d25c724e1e2c8d1ebf24c672e713d5cb47843a92abf33c4c7a913",
2552 "verdict": "proved as written"
2553 }
2554 ],
2555 "remaining_work": null
2556 },
2557 {
2558 "ordinal": 60,
2559 "stable_id": "PSC-I-D-UNCONDITIONAL-PACKET-SPACE",
2560 "kind": "definition",
2561 "title": "Unconditional packet sum space",
2562 "status": "definition_represented",
2563 "old_contract_rows": [],
2564 "mapped_declarations": [
2565 "PaperIRelease.TopologyUnconditional.ClosedPackets.sequences",
2566 "PaperIRelease.TopologyUnconditional.ClosedPackets.Space",
2567 "PaperIRelease.TopologyUnconditional.ClosedPackets.unconditionalNorm",
2568 "PaperIRelease.TopologyUnconditional.ClosedPackets.spaceNormedAddCommGroup",
2569 "PaperIRelease.TopologyUnconditional.ClosedPackets.spaceNormedSpace"
2570 ],
2571 "scope_and_remaining_work": "The carrier is exactly sequences x_n in the specified closed subspaces for which Summable x holds; mathlib Summable is the finite-subset net (unconditional summation). The norm is exactly sSup of norms of finite sums with coefficients in the closed scalar unit ball. Type synonyms avoid inherited product topology. Finiteness is derived, not assumed. RCLike covers real and complex scalars as stated.",
2572 "current_source": {
2573 "file": "editable/traditional/03_packets_and_graphs.tex",
2574 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2575 "kind": "definition",
2576 "record_id_line": 367,
2577 "statement_start_line": 352,
2578 "statement_end_line": 366,
2579 "statement_sha256": "bcbeba58302101658c66251d7b338a6a3932fb07b059e77931ffd063b5303b4f",
2580 "proof_start_line": null,
2581 "proof_end_line": null,
2582 "proof_sha256": null
2583 },
2584 "statement": "\\begin{definition}[Unconditional packet sum space]\n\\label{def:unconditional-packet-space}\nLet $(X_n)_{n\\geq1}$ be closed subspaces of a real or complex Banach space $X$. Define\n\\[\n\\bigoplus_{n\\geq1}^{u}X_n\n\\]\nto be the space of sequences $x=(x_n)$ with $x_n\\in X_n$ for which $\\sum_nx_n$ converges unconditionally in $X$. Equip it with\n\\[\n\\|x\\|_u\n:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|.\n\\]\n\\end{definition}",
2585 "source_proof": null,
2586 "new_block_contribution": [],
2587 "acceptance_provenance": [
2588 {
2589 "semantic_review": "semantic_analytic.json",
2590 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2591 "reviewed_statement_sha256": "bcbeba58302101658c66251d7b338a6a3932fb07b059e77931ffd063b5303b4f",
2592 "reviewed_proof_sha256": null,
2593 "verdict": "proved as written"
2594 }
2595 ],
2596 "remaining_work": null
2597 },
2598 {
2599 "ordinal": 61,
2600 "stable_id": "PSC-I-L-UNCONDITIONAL-COMPLETENESS",
2601 "kind": "lemma",
2602 "title": "Completeness of the unconditional packet sum",
2603 "status": "covered",
2604 "old_contract_rows": [],
2605 "mapped_declarations": [
2606 "PaperIRelease.TopologyUnconditional.summable_finiteSum_bounded",
2607 "PaperIRelease.TopologyUnconditional.ClosedPackets.coordinate_cauchy",
2608 "PaperIRelease.TopologyUnconditional.ClosedPackets.finiteSum_limit_bound",
2609 "PaperIRelease.TopologyUnconditional.ClosedPackets.finiteSum_uniform_limit",
2610 "PaperIRelease.TopologyUnconditional.ClosedPackets.summable_coordinate_limit",
2611 "PaperIRelease.TopologyUnconditional.ClosedPackets.spaceCompleteSpace"
2612 ],
2613 "scope_and_remaining_work": "Uniform boundedness on the complete continuous dual is justified by finite-dimensional real absolute summability of each scalar image; the isometric bidual inclusion yields finite multiplier bounds. The signed norm controls each coordinate; coordinate limits stay in closed subspaces. Each fixed finite multiplier estimate passes to the limit, uniformly in the finite set and coefficients. Unweighted tails prove Summable for the limit, and the same uniform estimate gives convergence in the signed norm. Metric completeness is then obtained without a projector hypothesis.",
2614 "current_source": {
2615 "file": "editable/traditional/03_packets_and_graphs.tex",
2616 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2617 "kind": "lemma",
2618 "record_id_line": 373,
2619 "statement_start_line": 369,
2620 "statement_end_line": 372,
2621 "statement_sha256": "339adef1c7737cb8adc8a4bc76b3a5a2f95a575ffb576a34404d8da60bfe0b62",
2622 "proof_start_line": 376,
2623 "proof_end_line": 405,
2624 "proof_sha256": "00c7e6d72274659463aa16a6166bd0168512ff5127b67b7019fa14bd64ae2fb5"
2625 },
2626 "statement": "\\begin{lemma}[Completeness of the unconditional packet sum]\n\\label{lem:unconditional-completeness}\nThe normed space $\\bigoplus_{n\\geq1}^{u}X_n$ in \\cref{def:unconditional-packet-space} is Banach.\n\\end{lemma}",
2627 "source_proof": "\\begin{proof}\nLet $x^{(k)}=(x_n^{(k)})$ be Cauchy in $\\|\\cdot\\|_u$. For every fixed $n$,\n\\[\n\\|x_n^{(k)}-x_n^{(\\ell)}\\|\n\\leq\n\\|x^{(k)}-x^{(\\ell)}\\|_u,\n\\]\nso $x_n^{(k)}$ converges to some $x_n\\in X_n$ because $X_n$ is closed. Given $\\varepsilon>0$, choose $k_0$ such that\n\\[\n\\|x^{(k)}-x^{(\\ell)}\\|_u<\\varepsilon\n\\qquad(k,\\ell\\geq k_0).\n\\]\nFor finite $F$ and $|a_n|\\leq1$, pass $\\ell\\to\\infty$ in the finite sum to obtain\n\\[\n\\left\\|\\sum_{n\\in F}a_n(x_n^{(k)}-x_n)\\right\\|\n\\leq\\varepsilon\n\\qquad(k\\geq k_0).\n\\]\nHence $\\|x^{(k)}-x\\|_u\\leq\\varepsilon$ once $x$ is known to be unconditionally summable.\n\nChoose such a $k\\geq k_0$. Since $x^{(k)}$ is unconditionally summable, there is a finite $F_0$ such that every finite $F\\subseteq\\mathbb N\\setminus F_0$ and every $|a_n|\\leq1$ satisfy\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n^{(k)}\\right\\|<\\varepsilon.\n\\]\nThe preceding uniform estimate then gives\n\\[\n\\left\\|\\sum_{n\\in F}a_nx_n\\right\\|<2\\varepsilon.\n\\]\nThus the tails of $\\sum_nx_n$ are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore $x\\in\\bigoplus^uX_n$, and the earlier estimate proves $x^{(k)}\\to x$ in $\\|\\cdot\\|_u$.\n\\end{proof}",
2628 "new_block_contribution": [],
2629 "acceptance_provenance": [
2630 {
2631 "semantic_review": "semantic_analytic.json",
2632 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2633 "reviewed_statement_sha256": "339adef1c7737cb8adc8a4bc76b3a5a2f95a575ffb576a34404d8da60bfe0b62",
2634 "reviewed_proof_sha256": "00c7e6d72274659463aa16a6166bd0168512ff5127b67b7019fa14bd64ae2fb5",
2635 "verdict": "proved as written"
2636 }
2637 ],
2638 "remaining_work": null
2639 },
2640 {
2641 "ordinal": 62,
2642 "stable_id": "PSC-I-T-COUNTABLE-UNCONDITIONAL-PACKETS",
2643 "kind": "theorem",
2644 "title": "Countable unconditional packet isomorphism",
2645 "status": "covered",
2646 "old_contract_rows": [],
2647 "mapped_declarations": [
2648 "PaperIRelease.TopologyCountablePackets.Projectors.multiplierNorms",
2649 "PaperIRelease.TopologyCountablePackets.Projectors.ofSupBound",
2650 "PaperIRelease.TopologyCountablePackets.Projectors.analysis",
2651 "PaperIRelease.TopologyCountablePackets.Projectors.synthesis",
2652 "PaperIRelease.TopologyCountablePackets.Projectors.synthesis_analysis",
2653 "PaperIRelease.TopologyCountablePackets.Projectors.analysis_synthesis",
2654 "PaperIRelease.TopologyCountablePackets.Projectors.packetEquiv",
2655 "PaperIRelease.TopologyCountablePackets.Projectors.packetCompleteSpace",
2656 "PaperIRelease.TopologyCountablePackets.Projectors.generalSpaceIsometry",
2657 "PaperIRelease.TopologyCountablePackets.Projectors.countableUnconditionalPacketEquiv",
2658 "PaperIRelease.TopologyCountablePackets.Projectors.countable_analysis_norm_le",
2659 "PaperIRelease.TopologyCountablePackets.Projectors.countable_synthesis_norm_le"
2660 ],
2661 "scope_and_remaining_work": "ofSupBound constructs K as the literal supremum of finite multiplier operator norms from BddAbove. The carriers use Summable sequences in actual projector ranges. Continuous projectors commute with the sum, so annihilation proves both inverse identities. The literal signed norm yields analysis bound K and synthesis bound 1. A proved linear isometry identifies the projector-specific carrier and norm with the general ClosedPackets.Space. Completeness follows either from the general theorem or from the constructed uniform equivalence with Banach X.",
2662 "current_source": {
2663 "file": "editable/traditional/03_packets_and_graphs.tex",
2664 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2665 "kind": "theorem",
2666 "record_id_line": 437,
2667 "statement_start_line": 407,
2668 "statement_end_line": 436,
2669 "statement_sha256": "7579699c5a34468a6f9df136e7c897641c5409e5f5e597cd849348bddc92a469",
2670 "proof_start_line": 441,
2671 "proof_end_line": 467,
2672 "proof_sha256": "e04fd9c7ae9dfbdb6d664aebcdc012228bd7d1702fbc2515101ad6adcd535182"
2673 },
2674 "statement": "\\begin{theorem}[Countable unconditional packet isomorphism]\n\\label{thm:countable-packets}\nLet $X$ be a real or complex Banach space and let $(P_n)_{n\\geq1}\\subseteq\\mathcal B(X)$ satisfy\n\\[\nP_nP_m=\\delta_{nm}P_n\n\\]\nand\n\\[\nx=\\sum_{n\\geq1}P_nx\n\\]\nunconditionally for every $x\\in X$. Assume\n\\[\nK:=\n\\sup_{F\\Subset\\mathbb N}\n\\sup_{|a_n|\\leq1}\n\\left\\|\\sum_{n\\in F}a_nP_n\\right\\|<\\infty.\n\\]\nThen analysis and synthesis define inverse Banach space isomorphisms\n\\[\n\\boxed{\nX\\cong\\bigoplus_{n\\geq1}^{u}P_nX.\n}\n\\]\nMore precisely,\n\\[\n\\|\\mathcal A_{\\Pcal}\\|\\leq K,\n\\qquad\n\\|\\mathcal S_{\\Pcal}\\|\\leq1.\n\\]\n\\end{theorem}",
2675 "source_proof": "\\begin{proof}\nEach $P_nX=\\Ker(I-P_n)$ is closed. For $x\\in X$,\n\\[\n\\|\\mathcal A_{\\Pcal}x\\|_u\n=\n\\sup_{F,a}\n\\left\\|\\sum_{n\\in F}a_nP_nx\\right\\|\n\\leq K\\|x\\|.\n\\]\nFor $y=(y_n)\\in\\bigoplus^uP_nX$, define\n\\[\n\\mathcal S_{\\Pcal}y=\\sum_ny_n.\n\\]\nSince the finite partial sums with every coefficient equal to one are among those used in $\\|y\\|_u$,\n\\[\n\\|\\mathcal S_{\\Pcal}y\\|\n\\leq\\|y\\|_u.\n\\]\nThe assumed expansion gives $\\mathcal S_{\\Pcal}\\mathcal A_{\\Pcal}x=x$. Conversely, boundedness of $P_m$ permits termwise application to the unconditionally convergent series:\n\\[\nP_m\\mathcal S_{\\Pcal}y\n=\n\\sum_nP_my_n\n=y_m.\n\\]\nThus $\\mathcal A_{\\Pcal}\\mathcal S_{\\Pcal}=I$. Completeness of the coordinate space is \\cref{lem:unconditional-completeness}.\n\\end{proof}",
2676 "new_block_contribution": [],
2677 "acceptance_provenance": [
2678 {
2679 "semantic_review": "semantic_analytic.json",
2680 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
2681 "reviewed_statement_sha256": "7579699c5a34468a6f9df136e7c897641c5409e5f5e597cd849348bddc92a469",
2682 "reviewed_proof_sha256": "e04fd9c7ae9dfbdb6d664aebcdc012228bd7d1702fbc2515101ad6adcd535182",
2683 "verdict": "proved as written"
2684 }
2685 ],
2686 "remaining_work": null
2687 },
2688 {
2689 "ordinal": 63,
2690 "stable_id": "PSC-I-R-UNIFORM-BOUNDEDNESS",
2691 "kind": "remark",
2692 "title": "Uniform boundedness",
2693 "status": "documentary",
2694 "old_contract_rows": [],
2695 "mapped_declarations": [],
2696 "scope_and_remaining_work": "A uniform-boundedness explanation. No instance for the signed packet partial-sum family is mapped.",
2697 "current_source": {
2698 "file": "editable/traditional/03_packets_and_graphs.tex",
2699 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2700 "kind": "remark",
2701 "record_id_line": 473,
2702 "statement_start_line": 469,
2703 "statement_end_line": 472,
2704 "statement_sha256": "2ae822d9b50496c6462b94007b5ff1a8d083e7109e6da7d0c2453ca40b80373c",
2705 "proof_start_line": null,
2706 "proof_end_line": null,
2707 "proof_sha256": null
2708 },
2709 "statement": "\\begin{remark}[Uniform boundedness]\n\\label{rem:uniform-boundedness}\nIf every expansion $\\sum_nP_nx$ is unconditional, the operators obtained from finite partial sums with signs form a pointwise bounded family. The uniform boundedness principle therefore yields a finite constant $K$. We retain this constant explicitly in \\cref{thm:countable-packets} because later estimates use it to bound the coordinate map.\n\\end{remark}",
2710 "source_proof": null,
2711 "new_block_contribution": []
2712 },
2713 {
2714 "ordinal": 64,
2715 "stable_id": "PSC-I-B-INFINITE-PACKET-TOPOLOGY",
2716 "kind": "boundary",
2717 "title": "Infinite packet sums",
2718 "status": "scope_boundary",
2719 "old_contract_rows": [],
2720 "mapped_declarations": [],
2721 "scope_and_remaining_work": "Convergence modes remain separate; finite identities are not used as infinite sums.",
2722 "current_source": {
2723 "file": "editable/traditional/03_packets_and_graphs.tex",
2724 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2725 "kind": "boundary",
2726 "record_id_line": 479,
2727 "statement_start_line": 475,
2728 "statement_end_line": 478,
2729 "statement_sha256": "305cbc8493dffcc2407698c4df7a3f54fecc76bb57e10c75390b0cedc5bce53d",
2730 "proof_start_line": null,
2731 "proof_end_line": null,
2732 "proof_sha256": null
2733 },
2734 "statement": "\\begin{boundary}[Infinite packet sums]\n\\label{bnd:infinite-packet-topology}\nFor an infinite family, an identity such as $\\sum_\\alpha P_\\alpha=I$ requires a declared convergence mode and the corresponding uniform boundedness statement. Pointwise, strong, norm, weak, and unconditional convergence are not interchangeable.\n\\end{boundary}",
2735 "source_proof": null,
2736 "new_block_contribution": []
2737 },
2738 {
2739 "ordinal": 65,
2740 "stable_id": "PSC-I-D-TRANSFER-BLOCK",
2741 "kind": "definition",
2742 "title": "Transfer block",
2743 "status": "definition_represented",
2744 "old_contract_rows": [
2745 19,
2746 23
2747 ],
2748 "mapped_declarations": [
2749 "Komorebi.Foundation.transferBlock_reconstruction_linearMap",
2750 "Komorebi.Foundation.transferBlock_reconstruction_end",
2751 "Komorebi.PacketNaturalityCompletion.FullBlocks.fullBlockAlgEquiv",
2752 "Komorebi.PacketNaturalityCompletion.FullBlocks.restrictBlocks_synthesizeBlocks",
2753 "Komorebi.PacketNaturalityCompletion.FullBlocks.synthesizeBlocks_restrictBlocks",
2754 "Komorebi.PacketNaturalityCompletion.FullBlocks.packetBlockAlgebra_mul_eq_blockMul"
2755 ],
2756 "scope_and_remaining_work": "Ambient compositions and their genuinely range-restricted packet Hom maps are both encoded.",
2757 "current_source": {
2758 "file": "editable/traditional/03_packets_and_graphs.tex",
2759 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2760 "kind": "definition",
2761 "record_id_line": 500,
2762 "statement_start_line": 492,
2763 "statement_end_line": 499,
2764 "statement_sha256": "1b27b671e8c43628db2f3e157df05e01a63658c32e64a6ae2a94fdc69fd5ad73",
2765 "proof_start_line": null,
2766 "proof_end_line": null,
2767 "proof_sha256": null
2768 },
2769 "statement": "\\begin{definition}[Transfer block]\n\\label{def:transfer-block}\nFor $T\\in\\End_{\\bbk}(X)$ and a finite packet system $\\Pcal$, define\n\\[\nT_{\\beta\\alpha}:=P_\\beta TP_\\alpha\n\\in\\Hom_{\\bbk}(X_\\alpha,X_\\beta).\n\\]\n\\end{definition}",
2770 "source_proof": null,
2771 "new_block_contribution": []
2772 },
2773 {
2774 "ordinal": 66,
2775 "stable_id": "PSC-I-T-OPERATOR-RECONSTRUCTION",
2776 "kind": "theorem",
2777 "title": "Exact operator reconstruction",
2778 "status": "covered",
2779 "old_contract_rows": [
2780 19
2781 ],
2782 "mapped_declarations": [
2783 "Komorebi.Foundation.transferBlock_reconstruction_linearMap",
2784 "Komorebi.Foundation.transferBlock_reconstruction_end"
2785 ],
2786 "scope_and_remaining_work": "The existing result is more general: source and target can have different finite packet systems.",
2787 "current_source": {
2788 "file": "editable/traditional/03_packets_and_graphs.tex",
2789 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2790 "kind": "theorem",
2791 "record_id_line": 509,
2792 "statement_start_line": 502,
2793 "statement_end_line": 508,
2794 "statement_sha256": "f8c54ea5f0aa3075a42556f7d9b9b29fd9399d88e9de974c6eb098fceb2a7abb",
2795 "proof_start_line": 512,
2796 "proof_end_line": 522,
2797 "proof_sha256": "35dee8a9eb59afe6682dbcf9d8d47eb830db415afc77fd847e5e91c31e3ecc48"
2798 },
2799 "statement": "\\begin{theorem}[Exact operator reconstruction]\n\\label{thm:operator-reconstruction}\nEvery $T\\in\\End_{\\bbk}(X)$ satisfies\n\\[\n\\boxed{T=\\sum_{\\alpha,\\beta\\in I}T_{\\beta\\alpha}.}\n\\]\n\\end{theorem}",
2800 "source_proof": "\\begin{proof}\nInsert the finite packet identity on both sides:\n\\[\nT=\n\\left(\\sum_\\beta P_\\beta\\right)\nT\n\\left(\\sum_\\alpha P_\\alpha\\right)\n=\n\\sum_{\\alpha,\\beta}P_\\beta TP_\\alpha.\n\\]\n\\end{proof}",
2801 "new_block_contribution": [],
2802 "acceptance_provenance": [
2803 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
2804 ]
2805 },
2806 {
2807 "ordinal": 67,
2808 "stable_id": "PSC-I-T-BLOCK-MULTIPLICATION",
2809 "kind": "theorem",
2810 "title": "Packet matrix multiplication",
2811 "status": "covered",
2812 "old_contract_rows": [
2813 20
2814 ],
2815 "mapped_declarations": [
2816 "Komorebi.Foundation.transferBlock_comp_linearMap",
2817 "Komorebi.Foundation.transferBlock_comp_end",
2818 "Komorebi.Foundation.end_comp_eq_sum_transferBlock_comps"
2819 ],
2820 "scope_and_remaining_work": "The output-first block product is proved for three possibly different modules.",
2821 "current_source": {
2822 "file": "editable/traditional/03_packets_and_graphs.tex",
2823 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2824 "kind": "theorem",
2825 "record_id_line": 540,
2826 "statement_start_line": 529,
2827 "statement_end_line": 539,
2828 "statement_sha256": "a7aaf1cde8454ca239c380ca964f5b13f20d39baaf62b7a859caf9a849378314",
2829 "proof_start_line": 544,
2830 "proof_end_line": 552,
2831 "proof_sha256": "6f29e44c3d4c1ddbefc53431e241d181c8fddedb16a0e54f9984f9685408608d"
2832 },
2833 "statement": "\\begin{theorem}[Packet matrix multiplication]\n\\label{thm:block-multiplication}\nFor $S,T\\in\\End_{\\bbk}(X)$,\n\\[\n\\boxed{\n(ST)_{\\gamma\\alpha}\n=\n\\sum_{\\beta\\in I}S_{\\gamma\\beta}T_{\\beta\\alpha}.\n}\n\\]\n\\end{theorem}",
2834 "source_proof": "\\begin{proof}\nInsert $I=\\sum_\\beta P_\\beta$ between $S$ and $T$:\n\\[\nP_\\gamma STP_\\alpha\n=\n\\sum_\\beta\nP_\\gamma SP_\\beta TP_\\alpha.\n\\]\n\\end{proof}",
2835 "new_block_contribution": [],
2836 "acceptance_provenance": [
2837 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
2838 ]
2839 },
2840 {
2841 "ordinal": 68,
2842 "stable_id": "PSC-I-C-BLOCK-POWERS",
2843 "kind": "corollary",
2844 "title": "Packet powers and paths",
2845 "status": "covered",
2846 "old_contract_rows": [
2847 21
2848 ],
2849 "mapped_declarations": [
2850 "Komorebi.PacketNaturalityCompletion.Packets.transferBlock_pow_eq_packetPathSum",
2851 "Komorebi.PaperIRelease.transferBlock_word"
2852 ],
2853 "scope_and_remaining_work": "The power theorem expands transferBlock(P,P,T^m) using packetPathSum. The independent operatorWord/packetWordSum induction expands every finite list of possibly different operators, with composition order matching rightmost-acts-first temporal order. Each summation is over actual intermediate packet labels.",
2854 "current_source": {
2855 "file": "editable/traditional/03_packets_and_graphs.tex",
2856 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2857 "kind": "corollary",
2858 "record_id_line": 572,
2859 "statement_start_line": 559,
2860 "statement_end_line": 571,
2861 "statement_sha256": "dbaa3f1f55517bd6366bcf856dc8eb6836a642144d805f9bed4899e5bba7038f",
2862 "proof_start_line": 575,
2863 "proof_end_line": 577,
2864 "proof_sha256": "24efae3f81e2071626ae8e5b229cc52afb931e12561a53120889660cdd26cc54"
2865 },
2866 "statement": "\\begin{corollary}[Packet powers and paths]\n\\label{cor:block-powers}\nFor $m\\geq1$,\n\\[\n(T^m)_{\\beta_m\\beta_0}\n=\n\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}\nT_{\\beta_m\\beta_{m-1}}\n\\cdots\nT_{\\beta_1\\beta_0}.\n\\]\nThe same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order.\n\\end{corollary}",
2867 "source_proof": "\\begin{proof}\nInduct on $m$ using \\cref{thm:block-multiplication}. For a product of different operators, use the same induction with the specified operator at each step.\n\\end{proof}",
2868 "new_block_contribution": [],
2869 "acceptance_provenance": [
2870 {
2871 "semantic_review": "semantic_wave2.json",
2872 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
2873 "reviewed_statement_sha256": "dbaa3f1f55517bd6366bcf856dc8eb6836a642144d805f9bed4899e5bba7038f",
2874 "reviewed_proof_sha256": "24efae3f81e2071626ae8e5b229cc52afb931e12561a53120889660cdd26cc54",
2875 "verdict": "proved as written"
2876 }
2877 ],
2878 "remaining_work": null
2879 },
2880 {
2881 "ordinal": 69,
2882 "stable_id": "PSC-I-C-ADJOINT-REVERSAL",
2883 "kind": "corollary",
2884 "title": "Adjoint reversal",
2885 "status": "covered",
2886 "old_contract_rows": [
2887 22
2888 ],
2889 "mapped_declarations": [
2890 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.adjoint_transferBlock"
2891 ],
2892 "scope_and_remaining_work": "The result is proved for bounded self-adjoint outer maps, more generally than orthogonal packet projectors.",
2893 "current_source": {
2894 "file": "editable/traditional/03_packets_and_graphs.tex",
2895 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2896 "kind": "corollary",
2897 "record_id_line": 586,
2898 "statement_start_line": 579,
2899 "statement_end_line": 585,
2900 "statement_sha256": "8e978a295a29a3eac3d1c4bec1a8e3f4e05a2684fdbce8e80a357aa7891e8f79",
2901 "proof_start_line": 589,
2902 "proof_end_line": 594,
2903 "proof_sha256": "adc1eea78765fc51b63ae09fcc4744c4ceb8ae5ac0d04e426f3edec5d40a48e2"
2904 },
2905 "statement": "\\begin{corollary}[Adjoint reversal]\n\\label{cor:adjoint-reversal}\nIf $X$ is Hilbert, every $P_\\alpha$ is orthogonal, and $T\\in\\mathcal B(X)$, then\n\\[\n\\boxed{(T_{\\beta\\alpha})^*=(T^*)_{\\alpha\\beta}.}\n\\]\n\\end{corollary}",
2906 "source_proof": "\\begin{proof}\nBecause $P_\\alpha^*=P_\\alpha$,\n\\[\n(P_\\beta TP_\\alpha)^*=P_\\alpha T^*P_\\beta.\n\\]\n\\end{proof}",
2907 "new_block_contribution": [],
2908 "acceptance_provenance": [
2909 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
2910 ]
2911 },
2912 {
2913 "ordinal": 70,
2914 "stable_id": "PSC-I-C-TENSOR-TRANSFER-BLOCKS",
2915 "kind": "corollary",
2916 "title": "Tensor-product transfer blocks",
2917 "status": "covered",
2918 "old_contract_rows": [],
2919 "mapped_declarations": [
2920 "Komorebi.PaperIRelease.tensor_transfer_blocks"
2921 ],
2922 "scope_and_remaining_work": "Actual transfer blocks factor by tensor-map composition in exactly the (target,source) order printed. Both sides are actual endomorphisms of X tensor Y; no factorization witness is an input.",
2923 "current_source": {
2924 "file": "editable/traditional/03_packets_and_graphs.tex",
2925 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2926 "kind": "corollary",
2927 "record_id_line": 609,
2928 "statement_start_line": 598,
2929 "statement_end_line": 608,
2930 "statement_sha256": "ec0bc2d2c85cdf55ea2f27003457a653f8f2426fe97dafd6c918cbda54874878",
2931 "proof_start_line": 613,
2932 "proof_end_line": 621,
2933 "proof_sha256": "8fcd957eb10b038b496518d9e1a7dbb497264dca97c05afb4969b89e5855c933"
2934 },
2935 "statement": "\\begin{corollary}[Tensor product transfer blocks]\n\\label{cor:tensor-transfer-blocks}\nLet \\(S\\in\\End(X)\\) and \\(T\\in\\End(Y)\\), with packet systems \\(\\Pcal\\) and \\(\\Qcal\\). Relative to the tensor product packet system of \\cref{thm:tensor-packet-system},\n\\[\n\\boxed{\n(S\\otimes T)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n=\nS_{\\beta\\alpha}\\otimes T_{\\delta\\gamma}.\n}\n\\]\n\\end{corollary}",
2936 "source_proof": "\\begin{proof}\nBy definition,\n\\begin{align*}\n(S\\otimes T)_{(\\beta,\\delta),(\\alpha,\\gamma)}\n&=(P_\\beta\\otimes Q_\\delta)(S\\otimes T)(P_\\alpha\\otimes Q_\\gamma)\\\\\n&=(P_\\beta SP_\\alpha)\\otimes(Q_\\delta TQ_\\gamma)\\\\\n&=S_{\\beta\\alpha}\\otimes T_{\\delta\\gamma}.\n\\end{align*}\n\\end{proof}",
2937 "new_block_contribution": [],
2938 "acceptance_provenance": [
2939 {
2940 "semantic_review": "semantic_wave1.json",
2941 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
2942 "reviewed_statement_sha256": "ec0bc2d2c85cdf55ea2f27003457a653f8f2426fe97dafd6c918cbda54874878",
2943 "reviewed_proof_sha256": "8fcd957eb10b038b496518d9e1a7dbb497264dca97c05afb4969b89e5855c933",
2944 "verdict": "proved as written"
2945 }
2946 ],
2947 "remaining_work": null
2948 },
2949 {
2950 "ordinal": 71,
2951 "stable_id": "PSC-I-T-BLOCK-INTERTWINING",
2952 "kind": "theorem",
2953 "title": "Blockwise intertwining equations",
2954 "status": "covered",
2955 "old_contract_rows": [
2956 20,
2957 28
2958 ],
2959 "mapped_declarations": [
2960 "Komorebi.Foundation.transferBlock_comp_linearMap",
2961 "Komorebi.Foundation.transferBlock_comp_end",
2962 "Komorebi.Foundation.end_comp_eq_sum_transferBlock_comps",
2963 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
2964 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
2965 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
2966 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
2967 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
2968 "Komorebi.PaperIRelease.PacketUniversal.block_intertwining_iff",
2969 "Komorebi.PaperIRelease.PacketUniversal.transferBlock_ext",
2970 "Komorebi.Foundation.transferBlock_reconstruction_linearMap"
2971 ],
2972 "scope_and_remaining_work": "U is any linear map, not an intertwiner structure. Forward direction inserts middle projector sums; reverse direction reconstructs UT and SU from all equal blocks. Distinct source/target index sets are retained.",
2973 "current_source": {
2974 "file": "editable/traditional/03_packets_and_graphs.tex",
2975 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
2976 "kind": "theorem",
2977 "record_id_line": 646,
2978 "statement_start_line": 623,
2979 "statement_end_line": 645,
2980 "statement_sha256": "558adbc4eead21c3f371918f40b9f462000c7b26a7234864ee44c103e09cf1e6",
2981 "proof_start_line": 650,
2982 "proof_end_line": 668,
2983 "proof_sha256": "0928aa303e0d72943b397927285247c955dc8b5734ba9fbe1b6083770d7ac96e"
2984 },
2985 "statement": "\\begin{theorem}[Blockwise intertwining equations]\n\\label{thm:block-intertwining}\nLet \\(T\\in\\End(X)\\), \\(S\\in\\End(Y)\\), let \\(\\Pcal=(P_\\alpha)_{\\alpha\\in I}\\) and \\(\\Qcal=(Q_\\gamma)_{\\gamma\\in J}\\) be finite packet systems, and let \\(U:X\\to Y\\) be linear. Define\n\\[\nT_{\\beta\\alpha}=P_\\beta TP_\\alpha,\n\\qquad\nS_{\\gamma\\delta}=Q_\\gamma SQ_\\delta,\n\\qquad\nU_{\\gamma\\alpha}=Q_\\gamma UP_\\alpha.\n\\]\nThen\n\\[\n\\boxed{UT=SU}\n\\]\nif and only if, for every \\(\\gamma\\in J\\) and \\(\\alpha\\in I\\),\n\\[\n\\boxed{\n\\sum_{\\beta\\in I}U_{\\gamma\\beta}T_{\\beta\\alpha}\n=\n\\sum_{\\delta\\in J}S_{\\gamma\\delta}U_{\\delta\\alpha}.\n}\n\\]\n\\end{theorem}",
2986 "source_proof": "\\begin{proof}\nApply \\(Q_\\gamma\\) on the left and \\(P_\\alpha\\) on the right. Inserting \\(I_X=\\sum_\\beta P_\\beta\\) gives\n\\[\nQ_\\gamma UTP_\\alpha\n=\n\\sum_\\beta Q_\\gamma UP_\\beta TP_\\alpha\n=\n\\sum_\\beta U_{\\gamma\\beta}T_{\\beta\\alpha}.\n\\]\nInserting \\(I_Y=\\sum_\\delta Q_\\delta\\) gives\n\\[\nQ_\\gamma SUP_\\alpha\n=\n\\sum_\\delta Q_\\gamma SQ_\\delta UP_\\alpha\n=\n\\sum_\\delta S_{\\gamma\\delta}U_{\\delta\\alpha}.\n\\]\nThus \\(UT=SU\\) implies every block equation. Conversely, if every block equation holds, then all blocks of \\(UT-SU\\) vanish. Exact operator reconstruction yields \\(UT-SU=0\\).\n\\end{proof}",
2987 "new_block_contribution": [],
2988 "acceptance_provenance": [
2989 {
2990 "semantic_review": "semantic_wave1.json",
2991 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
2992 "reviewed_statement_sha256": "558adbc4eead21c3f371918f40b9f462000c7b26a7234864ee44c103e09cf1e6",
2993 "reviewed_proof_sha256": "0928aa303e0d72943b397927285247c955dc8b5734ba9fbe1b6083770d7ac96e",
2994 "verdict": "proved as written"
2995 }
2996 ],
2997 "remaining_work": null
2998 },
2999 {
3000 "ordinal": 72,
3001 "stable_id": "PSC-I-T-BLOCK-ALGEBRA",
3002 "kind": "theorem",
3003 "title": "Full block-algebra representation",
3004 "status": "covered",
3005 "old_contract_rows": [
3006 23
3007 ],
3008 "mapped_declarations": [
3009 "Komorebi.PacketNaturalityCompletion.FullBlocks.fullBlockAlgEquiv",
3010 "Komorebi.PacketNaturalityCompletion.FullBlocks.restrictBlocks_synthesizeBlocks",
3011 "Komorebi.PacketNaturalityCompletion.FullBlocks.synthesizeBlocks_restrictBlocks",
3012 "Komorebi.PacketNaturalityCompletion.FullBlocks.packetBlockAlgebra_mul_eq_blockMul"
3013 ],
3014 "scope_and_remaining_work": "Full dependent packet Hom families and their multiplication are identified with the actual endomorphism algebra.",
3015 "current_source": {
3016 "file": "editable/traditional/03_packets_and_graphs.tex",
3017 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3018 "kind": "theorem",
3019 "record_id_line": 687,
3020 "statement_start_line": 675,
3021 "statement_end_line": 686,
3022 "statement_sha256": "a66e4216289ceb5e48d1ccd41d269b656c6c93ed462a47f68cbd0172c9360bda",
3023 "proof_start_line": 691,
3024 "proof_end_line": 699,
3025 "proof_sha256": "75fd4a959f0eb71647bba83732ffbfb04cae3bd9741d8482a1e54a234913e566"
3026 },
3027 "statement": "\\begin{theorem}[Full block algebra representation]\n\\label{thm:block-algebra}\nThe map\n\\[\n\\Theta_{\\Pcal}:\\End_{\\bbk}(X)\n\\longrightarrow\n\\prod_{\\beta,\\alpha\\in I}\\Hom_{\\bbk}(X_\\alpha,X_\\beta),\n\\qquad\nT\\longmapsto(T_{\\beta\\alpha})_{\\beta,\\alpha},\n\\]\nis an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators.\n\\end{theorem}",
3028 "source_proof": "\\begin{proof}\nInjectivity follows from exact reconstruction. Given arbitrary linear maps $B_{\\beta\\alpha}:X_\\alpha\\to X_\\beta$, define\n\\[\nT\\left(\\sum_\\alpha x_\\alpha\\right)\n:=\n\\sum_{\\alpha,\\beta}B_{\\beta\\alpha}x_\\alpha.\n\\]\nThe packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps. Thus $\\Theta_{\\Pcal}$ is surjective. Compatibility with multiplication is \\cref{thm:block-multiplication}.\n\\end{proof}",
3029 "new_block_contribution": [],
3030 "acceptance_provenance": [
3031 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below."
3032 ]
3033 },
3034 {
3035 "ordinal": 73,
3036 "stable_id": "PSC-I-C-PACKET-SUPPORT-PROPAGATION",
3037 "kind": "corollary",
3038 "title": "Propagation of packet support",
3039 "status": "covered",
3040 "old_contract_rows": [
3041 26
3042 ],
3043 "mapped_declarations": [
3044 "Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support",
3045 "Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction",
3046 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.closure_le_ker_of_le_ker",
3047 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.topological_reachability_obstruction",
3048 "Komorebi.PaperIRelease.packetSupport_image_subset",
3049 "Komorebi.PaperIRelease.packetSupport_image_eq_of_noncancellation",
3050 "Komorebi.PaperIRelease.packetSupport_image_eq_of_single_active"
3051 ],
3052 "scope_and_remaining_work": "The source expands each target coordinate as the finite sum of the actual transfer-block contributions and filters to active sources. This yields the support inclusion; nonzero active sums give equality. The single-active theorem uses activeSources={a} plus nonzero transfer contribution, which is equivalent to at-most-one active source plus kernel avoidance on a target that is active.",
3053 "current_source": {
3054 "file": "editable/traditional/03_packets_and_graphs.tex",
3055 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3056 "kind": "corollary",
3057 "record_id_line": 721,
3058 "statement_start_line": 703,
3059 "statement_end_line": 720,
3060 "statement_sha256": "f05b8bf4828b88f6020e2e0b824e228acffcb77d7a514ced31ec90c64e6d4437",
3061 "proof_start_line": 725,
3062 "proof_end_line": 733,
3063 "proof_sha256": "b2c18fc64d8f63dbe6076afc64e30f32b1eb7f12876c95300ef3688efd4799bd"
3064 },
3065 "statement": "\\begin{corollary}[Propagation of packet support]\n\\label{cor:packet-support-propagation}\nLet \\(T\\in\\End(X)\\) and \\(x\\in X\\). Then\n\\[\n\\boxed{\n\\Supp_{\\Pcal}(Tx)\n\\subseteq\n\\left\\{\\beta:\\exists\\alpha\\in\\Supp_{\\Pcal}(x)\n\\text{ with }T_{\\beta\\alpha}\\neq0\\right\\}.\n}\n\\]\nEquality holds whenever, for every target \\(\\beta\\) in the set on the right, the active contributions\n\\[\n\\bigl(T_{\\beta\\alpha}P_\\alpha x:\n\\alpha\\in\\Supp_{\\Pcal}(x),\\ T_{\\beta\\alpha}\\neq0\\bigr)\n\\]\nhave nonzero sum. In particular, equality holds if at most one active source packet feeds each such \\(\\beta\\) and that source coordinate does not lie in the kernel of the corresponding block.\n\\end{corollary}",
3066 "source_proof": "\\begin{proof}\nFor every target packet \\(\\beta\\),\n\\[\nP_\\beta Tx\n=\n\\sum_{\\alpha\\in\\Supp_{\\Pcal}(x)}T_{\\beta\\alpha}P_\\alpha x.\n\\]\nIf every block in this sum is zero, then the coordinate with index \\(\\beta\\) vanishes. This proves the inclusion. Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right. The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.\n\\end{proof}",
3067 "new_block_contribution": [],
3068 "acceptance_provenance": [
3069 {
3070 "semantic_review": "semantic_wave2.json",
3071 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3072 "reviewed_statement_sha256": "f05b8bf4828b88f6020e2e0b824e228acffcb77d7a514ced31ec90c64e6d4437",
3073 "reviewed_proof_sha256": "b2c18fc64d8f63dbe6076afc64e30f32b1eb7f12876c95300ef3688efd4799bd",
3074 "verdict": "proved as written"
3075 }
3076 ],
3077 "remaining_work": null
3078 },
3079 {
3080 "ordinal": 74,
3081 "stable_id": "PSC-I-C-DIMENSION-COUNT",
3082 "kind": "corollary",
3083 "title": "Dimension count",
3084 "status": "covered",
3085 "old_contract_rows": [
3086 24
3087 ],
3088 "mapped_declarations": [
3089 "Komorebi.PacketNaturalityCompletion.FullBlocks.end_finrank_eq_packet_sum_eq_square",
3090 "Komorebi.PacketNaturalityCompletion.FullBlocks.finrank_eq_sum_packet_finrank"
3091 ],
3092 "scope_and_remaining_work": "The actual End space is linearly equivalent to the heterogeneous packet Hom product. Its finite dimension equals the sum of products of actual packet-range dimensions, and finite distributivity gives the displayed square. Both printed equalities occur in one conjunction.",
3093 "current_source": {
3094 "file": "editable/traditional/03_packets_and_graphs.tex",
3095 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3096 "kind": "corollary",
3097 "record_id_line": 747,
3098 "statement_start_line": 736,
3099 "statement_end_line": 746,
3100 "statement_sha256": "85dd27ca5c20eeb56683b4b3d9b9093503e2c3c2fdc4a07c1aaf33eefab855c8",
3101 "proof_start_line": 750,
3102 "proof_end_line": 752,
3103 "proof_sha256": "486a43c5f2f67a603f5b36ed54695f02051dd6d50d1fdfad92f199903eb620f7"
3104 },
3105 "statement": "\\begin{corollary}[Dimension count]\n\\label{cor:dimension-count}\nIf $X$ is finite dimensional and $d_\\alpha=\\dim X_\\alpha$, then\n\\[\n\\dim\\End_{\\bbk}(X)\n=\n\\sum_{\\alpha,\\beta}d_\\alpha d_\\beta\n=\n\\left(\\sum_\\alpha d_\\alpha\\right)^2.\n\\]\n\\end{corollary}",
3106 "source_proof": "\\begin{proof}\nTake dimensions in the block algebra isomorphism and use $\\dim\\Hom(X_\\alpha,X_\\beta)=d_\\alpha d_\\beta$.\n\\end{proof}",
3107 "new_block_contribution": [],
3108 "acceptance_provenance": [
3109 "Block08 accepted correspondence; current statement unchanged unless separately reviewed below.",
3110 {
3111 "semantic_review": "semantic_wave2.json",
3112 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3113 "reviewed_statement_sha256": "85dd27ca5c20eeb56683b4b3d9b9093503e2c3c2fdc4a07c1aaf33eefab855c8",
3114 "reviewed_proof_sha256": "486a43c5f2f67a603f5b36ed54695f02051dd6d50d1fdfad92f199903eb620f7",
3115 "verdict": "proved as written"
3116 }
3117 ],
3118 "remaining_work": null
3119 },
3120 {
3121 "ordinal": 75,
3122 "stable_id": "PSC-I-P-BLOCK-DIAGONAL-COMMUTANT",
3123 "kind": "proposition",
3124 "title": "Block-diagonal projection and packet commutant",
3125 "status": "covered",
3126 "old_contract_rows": [],
3127 "mapped_declarations": [
3128 "Komorebi.PaperIRelease.diagonalProjection",
3129 "Komorebi.PaperIRelease.diagonalProjection_idempotent",
3130 "Komorebi.PaperIRelease.diagonalProjection_range",
3131 "Komorebi.PaperIRelease.commutes_iff_diagonalProjection",
3132 "Komorebi.PaperIRelease.commutes_iff_offdiagonal_zero"
3133 ],
3134 "scope_and_remaining_work": "diagonalProjection is the actual K-linear map T↦Σ_i P_i T P_i. The proofs derive idempotence, equivalence of commutation with every projector and off-diagonal block vanishing, equivalence to fixed points of this projection, and equality of its actual linear range to the equality-supported subalgebra.",
3135 "current_source": {
3136 "file": "editable/traditional/03_packets_and_graphs.tex",
3137 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3138 "kind": "proposition",
3139 "record_id_line": 767,
3140 "statement_start_line": 754,
3141 "statement_end_line": 766,
3142 "statement_sha256": "613238e060365a7c00244f44abb552ada9f54933d8483382f259d3f32843eae5",
3143 "proof_start_line": 771,
3144 "proof_end_line": 781,
3145 "proof_sha256": "52b05a495f663cdac4afbff5a6b049c641c9f7dd33dcb1958586e6160d0b7a6f"
3146 },
3147 "statement": "\\begin{proposition}[Block diagonal projection and packet commutant]\n\\label{prop:block-diagonal-projection}\nDefine\n\\[\n\\Delta_{\\Pcal}(T):=\\sum_{\\alpha}P_\\alpha TP_\\alpha.\n\\]\nThen $\\Delta_{\\Pcal}$ is a linear idempotent on $\\End(X)$ whose range is the block diagonal subalgebra. For $T\\in\\End(X)$, the following are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $TP_\\alpha=P_\\alpha T$ for every $\\alpha$.\n\\item $T_{\\beta\\alpha}=0$ whenever $\\beta\\neq\\alpha$.\n\\item $\\Delta_{\\Pcal}(T)=T$.\n\\end{enumerate}\n\\end{proposition}",
3148 "source_proof": "\\begin{proof}\nPairwise annihilation gives $\\Delta_{\\Pcal}^2=\\Delta_{\\Pcal}$ and identifies its range with diagonal block families. If $T$ commutes with all $P_\\alpha$, then for $\\beta\\neq\\alpha$,\n\\[\nP_\\beta TP_\\alpha=P_\\beta P_\\alpha T=0.\n\\]\nThus (i) implies (ii). Exact reconstruction shows (ii) implies (iii). If $T=\\sum_\\alpha P_\\alpha TP_\\alpha$, then\n\\[\nTP_\\gamma=P_\\gamma TP_\\gamma=P_\\gamma T,\n\\]\nso (iii) implies (i).\n\\end{proof}",
3149 "new_block_contribution": [],
3150 "acceptance_provenance": [
3151 {
3152 "semantic_review": "semantic_wave2.json",
3153 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3154 "reviewed_statement_sha256": "613238e060365a7c00244f44abb552ada9f54933d8483382f259d3f32843eae5",
3155 "reviewed_proof_sha256": "52b05a495f663cdac4afbff5a6b049c641c9f7dd33dcb1958586e6160d0b7a6f",
3156 "verdict": "proved as written"
3157 }
3158 ],
3159 "remaining_work": null
3160 },
3161 {
3162 "ordinal": 76,
3163 "stable_id": "PSC-I-D-SUPPORT-RELATION",
3164 "kind": "definition",
3165 "title": "Support relation and support space",
3166 "status": "definition_represented",
3167 "old_contract_rows": [
3168 25
3169 ],
3170 "mapped_declarations": [
3171 "Komorebi.PacketNaturalityCompletion.FullBlocks.identity_blocks_supported_iff_reflexive",
3172 "Komorebi.PacketNaturalityCompletion.FullBlocks.blockMulClosed_iff_transitive",
3173 "Komorebi.PaperIRelease.OperatorSupported",
3174 "Komorebi.PaperIRelease.operatorSupported_iff_ambient",
3175 "Komorebi.PaperIRelease.supportSubmodule"
3176 ],
3177 "scope_and_remaining_work": "OperatorSupported is initially defined via actual restricted packet maps; operatorSupported_iff_ambient proves its exact equivalence to ambient transfer-block vanishing. supportSubmodule is the corresponding actual linear subspace of End(X). Relation dictionary: formal rel(a,b) means manuscript (b,a)∈R, with a source and b target.",
3178 "current_source": {
3179 "file": "editable/traditional/03_packets_and_graphs.tex",
3180 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3181 "kind": "definition",
3182 "record_id_line": 798,
3183 "statement_start_line": 789,
3184 "statement_end_line": 797,
3185 "statement_sha256": "babc2ddcee7669992338d1b7741bd50fab01175a280772ab40df14dcc814ebf3",
3186 "proof_start_line": null,
3187 "proof_end_line": null,
3188 "proof_sha256": null
3189 },
3190 "statement": "\\begin{definition}[Support relation and support space]\n\\label{def:support-relation}\nFor a relation $R\\subseteq I\\times I$, with $(\\beta,\\alpha)$ interpreted as an allowed transfer $\\alpha\\to\\beta$, define\n\\[\n\\mathfrak M_R\n:=\n\\{T\\in\\End(X):T_{\\beta\\alpha}=0\\text{ whenever }(\\beta,\\alpha)\\notin R\\}.\n\\]\n\\end{definition}",
3191 "source_proof": null,
3192 "new_block_contribution": [],
3193 "acceptance_provenance": [
3194 {
3195 "semantic_review": "semantic_wave2.json",
3196 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3197 "reviewed_statement_sha256": "babc2ddcee7669992338d1b7741bd50fab01175a280772ab40df14dcc814ebf3",
3198 "reviewed_proof_sha256": null,
3199 "verdict": "represented as written"
3200 }
3201 ],
3202 "remaining_work": null
3203 },
3204 {
3205 "ordinal": 77,
3206 "stable_id": "PSC-I-T-SUPPORT-PATTERN-ALGEBRA",
3207 "kind": "theorem",
3208 "title": "Support-pattern algebra criterion",
3209 "status": "covered",
3210 "old_contract_rows": [
3211 25
3212 ],
3213 "mapped_declarations": [
3214 "Komorebi.PacketNaturalityCompletion.FullBlocks.identity_blocks_supported_iff_reflexive",
3215 "Komorebi.PacketNaturalityCompletion.FullBlocks.blockMulClosed_iff_transitive",
3216 "Komorebi.PaperIRelease.support_mul_closed_iff",
3217 "Komorebi.PaperIRelease.support_pattern_algebra_criterion"
3218 ],
3219 "scope_and_remaining_work": "The two difficult converses use arbitrary maps between the actual nonzero packet ranges. Identity support is equivalent to reflexivity; multiplication closure is equivalent to transitivity. support_mul_closed_iff transfers the block statement to actual ambient operators, and support_pattern_algebra_criterion characterizes existence of a Subalgebra whose carrier is precisely that support space.",
3220 "current_source": {
3221 "file": "editable/traditional/03_packets_and_graphs.tex",
3222 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3223 "kind": "theorem",
3224 "record_id_line": 809,
3225 "statement_start_line": 800,
3226 "statement_end_line": 808,
3227 "statement_sha256": "fcd8af3b6718c7460c4e58639b8e942c24ede23b8673f9811f193cab4ccd3b20",
3228 "proof_start_line": 812,
3229 "proof_end_line": 830,
3230 "proof_sha256": "f726a1426420e339cdf91fb0b701087e96756c1ac543ecd1dcc8b9c48420930b"
3231 },
3232 "statement": "\\begin{theorem}[Support pattern algebra criterion]\n\\label{thm:support-pattern}\nAssume $X_\\alpha\\neq0$ for every $\\alpha\\in I$. Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $I_X\\in\\mathfrak M_R$ if and only if $R$ is reflexive.\n\\item $\\mathfrak M_R\\mathfrak M_R\\subseteq\\mathfrak M_R$ if and only if $R$ is transitive.\n\\item $\\mathfrak M_R$ is a unital subalgebra if and only if $R$ is a preorder.\n\\end{enumerate}\n\\end{theorem}",
3233 "source_proof": "\\begin{proof}\nThe identity has diagonal blocks $I_{X_\\alpha}$ and no blocks outside the diagonal, so it belongs to $\\mathfrak M_R$ exactly when $(\\alpha,\\alpha)\\in R$ for every $\\alpha$.\n\nAssume $R$ is transitive. If a summand\n\\[\nS_{\\gamma\\beta}T_{\\beta\\alpha}\n\\]\nin the $(\\gamma,\\alpha)$ block of $ST$ is nonzero, then $(\\gamma,\\beta)$ and $(\\beta,\\alpha)$ lie in $R$. Transitivity gives $(\\gamma,\\alpha)\\in R$, so every forbidden block of $ST$ vanishes.\n\nConversely, suppose transitivity fails. Choose $\\alpha,\\beta,\\gamma$ with\n\\[\n(\\beta,\\alpha)\\in R,\n\\qquad\n(\\gamma,\\beta)\\in R,\n\\qquad\n(\\gamma,\\alpha)\\notin R.\n\\]\nChoose nonzero vectors $u\\in X_\\alpha$, $v\\in X_\\beta$, and $w\\in X_\\gamma$. Choose linear functionals $f\\in X_\\alpha^*$ and $g\\in X_\\beta^*$ with $f(u)=g(v)=1$. Let $T$ have the sole block $x\\mapsto f(x)v$ from $X_\\alpha$ to $X_\\beta$, and let $S$ have the sole block $y\\mapsto g(y)w$ from $X_\\beta$ to $X_\\gamma$. Then $S,T\\in\\mathfrak M_R$, but $ST$ has a nonzero forbidden $(\\gamma,\\alpha)$ block. This proves necessity. Combining reflexivity and transitivity gives (iii).\n\\end{proof}",
3234 "new_block_contribution": [],
3235 "acceptance_provenance": [
3236 {
3237 "semantic_review": "semantic_wave2.json",
3238 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3239 "reviewed_statement_sha256": "fcd8af3b6718c7460c4e58639b8e942c24ede23b8673f9811f193cab4ccd3b20",
3240 "reviewed_proof_sha256": "f726a1426420e339cdf91fb0b701087e96756c1ac543ecd1dcc8b9c48420930b",
3241 "verdict": "proved as written"
3242 }
3243 ],
3244 "remaining_work": null
3245 },
3246 {
3247 "ordinal": 78,
3248 "stable_id": "PSC-I-C-SUPPORT-PATTERN-STAR",
3249 "kind": "corollary",
3250 "title": "$*$-closed bounded support patterns",
3251 "status": "covered",
3252 "old_contract_rows": [
3253 22
3254 ],
3255 "mapped_declarations": [
3256 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.adjoint_transferBlock",
3257 "Komorebi.PaperIRelease.BoundedSupport.adjoint_closed_iff_symmetric",
3258 "Komorebi.PaperIRelease.BoundedSupport.star_subalgebra_iff_equivalence"
3259 ],
3260 "scope_and_remaining_work": "Supported uses literal continuous transfer blocks and is proved equivalent to the underlying algebraic support predicate. Converse symmetry and transitivity use actual bounded rank-one operators between nonzero ranges. Identity support forces reflexivity. Sufficiency constructs an actual StarSubalgebra. RCLike covers the real and complex Hilbert settings; no finite-dimensional restriction or blanket bounded-block assumption is used.",
3261 "current_source": {
3262 "file": "editable/traditional/03_packets_and_graphs.tex",
3263 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3264 "kind": "corollary",
3265 "record_id_line": 840,
3266 "statement_start_line": 832,
3267 "statement_end_line": 839,
3268 "statement_sha256": "89dfdc356ea3dbbf9ee8018c66d8307f445d19078e0d7226372aae44c1872333",
3269 "proof_start_line": 843,
3270 "proof_end_line": 845,
3271 "proof_sha256": "9e0ad344026360162f0bea9455fc69a654256d3c2b8d382dd5313337070c81c0"
3272 },
3273 "statement": "\\begin{corollary}[Bounded support patterns closed under $*$]\n\\label{cor:star-support}\nLet $X$ be Hilbert, let the packet projectors be orthogonal, assume $X_\\alpha\\neq0$ for every $\\alpha\\in I$, and put\n\\[\n\\mathfrak M_R^{\\mathrm b}:=\\mathfrak M_R\\cap\\mathcal B(X).\n\\]\nThen $\\mathfrak M_R^{\\mathrm b}$ is closed under adjoints if and only if $R$ is symmetric. Hence $\\mathfrak M_R^{\\mathrm b}$ is a unital $*$ subalgebra of $\\mathcal B(X)$ if and only if $R$ is an equivalence relation.\n\\end{corollary}",
3274 "source_proof": "\\begin{proof}\nAdjoint reversal sends an allowed $(\\beta,\\alpha)$ block to an $(\\alpha,\\beta)$ block. Symmetry is therefore sufficient. If symmetry fails, choose a nonzero map of rank one in an allowed nonsymmetric block. Its adjoint occupies the forbidden reverse block, proving necessity. A reflexive, transitive, symmetric relation is an equivalence relation.\n\\end{proof}",
3275 "new_block_contribution": [],
3276 "acceptance_provenance": [
3277 {
3278 "semantic_review": "semantic_graph_rank.json",
3279 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3280 "reviewed_statement_sha256": "89dfdc356ea3dbbf9ee8018c66d8307f445d19078e0d7226372aae44c1872333",
3281 "reviewed_proof_sha256": "9e0ad344026360162f0bea9455fc69a654256d3c2b8d382dd5313337070c81c0",
3282 "verdict": "proved as written"
3283 }
3284 ],
3285 "remaining_work": null
3286 },
3287 {
3288 "ordinal": 79,
3289 "stable_id": "PSC-I-T-PREORDER-ALGEBRA-DECOMPOSITION",
3290 "kind": "theorem",
3291 "title": "Preorder support algebra decomposition",
3292 "status": "covered",
3293 "old_contract_rows": [],
3294 "mapped_declarations": [
3295 "Komorebi.PaperIRelease.strictSupportIdeal",
3296 "Komorebi.PaperIRelease.strictSupport_iff_same_class_zero",
3297 "Komorebi.PaperIRelease.strictSupportIdeal_pow_eq_bot",
3298 "Komorebi.PaperIRelease.maximalStrictSteps",
3299 "Komorebi.PaperIRelease.strictPath_le_maximal",
3300 "Komorebi.PaperIRelease.maximalStrictSteps_attained",
3301 "Komorebi.PaperIRelease.class_maximalStrictSteps",
3302 "Komorebi.PaperIRelease.preorderDiagonal_kernel_eq",
3303 "Komorebi.PaperIRelease.preorderDiagonalHom_surjective",
3304 "Komorebi.PaperIRelease.preorderClassRangeEquiv",
3305 "Komorebi.PaperIRelease.preorderQuotientDirectSumEquiv"
3306 ],
3307 "scope_and_remaining_work": "The support algebra is constructed from the actual preorder. strictSupportIdeal is an actual TwoSidedIdeal of it, with membership proved equivalent to vanishing within every mutual-order class. Nonzero product blocks force strict paths; finite strict-path height is bounded and attained on nonempty indices. strictSupportIdeal_pow_eq_bot uses Mathlib standard ideal powers, with the span-of-words surrogate bridged by strictIdeal_power_mem. Antisymmetrization supplies the actual quotient partial order; class_maximalStrictSteps proves its height equals the exponent parameter. Grouped class projectors are actual sums of original projectors. The diagonal-class homomorphism is surjective with kernel equal to the strict ideal, and preorderQuotientDirectSumEquiv is the actual quotient by that ideal to the product of End of the literal direct sums of class packet ranges.",
3308 "current_source": {
3309 "file": "editable/traditional/03_packets_and_graphs.tex",
3310 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3311 "kind": "theorem",
3312 "record_id_line": 876,
3313 "statement_start_line": 847,
3314 "statement_end_line": 875,
3315 "statement_sha256": "89f93d5bdea5558d5ca9fef72742db6339af268623a37bedb7baabb777e4452a",
3316 "proof_start_line": 880,
3317 "proof_end_line": 894,
3318 "proof_sha256": "5b796d0f2cdcf33b9d994d9596acc8d6cd2beb315086fac6d3dc56b9dbd855ed"
3319 },
3320 "statement": "\\begin{theorem}[Preorder support algebra decomposition]\n\\label{thm:preorder-decomposition}\nLet $R$ be a preorder on the finite set $I$. Define\n\\[\n\\alpha\\sim\\beta\n\\quad\\Longleftrightarrow\\quad\n(\\beta,\\alpha)\\in R\\text{ and }(\\alpha,\\beta)\\in R.\n\\]\nLet $\\overline I=I/{\\sim}$, which carries the induced partial order, and put\n\\[\nX_C:=\\bigoplus_{\\alpha\\in C}X_\\alpha.\n\\]\nLet $\\mathfrak N_R\\subseteq\\mathfrak M_R$ consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\mathfrak N_R$ is a nilpotent ideal on both sides.\n\\item If $h$ is the maximal number of strict steps in a chain of $\\overline I$, then\n\\[\n\\mathfrak N_R^{\\,h+1}=0.\n\\]\n\\item The map to the diagonal classes induces an algebra isomorphism\n\\[\n\\boxed{\n\\mathfrak M_R/\\mathfrak N_R\n\\cong\n\\prod_{C\\in\\overline I}\\End_{\\bbk}(X_C).\n}\n\\]\n\\end{enumerate}\n\\end{theorem}",
3321 "source_proof": "\\begin{proof}\nA nonzero block in $\\mathfrak N_R$ moves from one equivalence class to a strictly larger class in the quotient partial order. The block multiplication law shows that products concatenate such moves. Therefore left or right multiplication by an element of $\\mathfrak M_R$ preserves the property of having no blocks within a single class, so $\\mathfrak N_R$ is an ideal on both sides. A product of $h+1$ elements of $\\mathfrak N_R$ would require a strict chain of $h+1$ steps in $\\overline I$, which does not exist. This proves nilpotence.\n\nLet $P_C=\\sum_{\\alpha\\in C}P_\\alpha$. Define\n\\[\n\\rho:\\mathfrak M_R\\longrightarrow\\prod_C\\End(X_C),\n\\qquad\n\\rho(T)=(P_CTP_C)_C.\n\\]\nIf an intermediate class $D$ contributes to $P_CSTP_C$, then the preorder contains transfers $C\\to D$ and $D\\to C$, forcing $C=D$ in the quotient poset. Hence\n\\[\nP_CSTP_C=(P_CSP_C)(P_CTP_C),\n\\]\nso $\\rho$ is an algebra homomorphism. Every collection of endomorphisms of the $X_C$ defines an allowed operator that is block diagonal with respect to the classes, so $\\rho$ is surjective. Its kernel is exactly $\\mathfrak N_R$. The first isomorphism theorem gives the quotient formula.\n\\end{proof}",
3322 "new_block_contribution": [],
3323 "acceptance_provenance": [
3324 {
3325 "semantic_review": "semantic_wave2.json",
3326 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3327 "reviewed_statement_sha256": "89f93d5bdea5558d5ca9fef72742db6339af268623a37bedb7baabb777e4452a",
3328 "reviewed_proof_sha256": "5b796d0f2cdcf33b9d994d9596acc8d6cd2beb315086fac6d3dc56b9dbd855ed",
3329 "verdict": "proved as written"
3330 }
3331 ],
3332 "remaining_work": null
3333 },
3334 {
3335 "ordinal": 80,
3336 "stable_id": "PSC-I-D-TRANSFER-GRAPH",
3337 "kind": "definition",
3338 "title": "Transfer graph of an algebra-generating family",
3339 "status": "definition_represented",
3340 "old_contract_rows": [
3341 26
3342 ],
3343 "mapped_declarations": [
3344 "Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support",
3345 "Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction",
3346 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.closure_le_ker_of_le_ker",
3347 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.topological_reachability_obstruction",
3348 "Komorebi.PaperIRelease.packetGeneratedAlgebra",
3349 "Komorebi.PacketNaturalityCompletion.Reachability.TransferEdge",
3350 "Komorebi.PacketNaturalityCompletion.Reachability.ReachableFrom"
3351 ],
3352 "scope_and_remaining_work": "packetGeneratedAlgebra is Algebra.adjoin K (range P.proj ∪ ops), exactly the declared algebra in the manuscript. TransferEdge is witnessed by an actual member of ops with nonzero ambient transfer block. ReachableFrom uses ReflTransGen and hence includes length zero.",
3353 "current_source": {
3354 "file": "editable/traditional/03_packets_and_graphs.tex",
3355 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3356 "kind": "definition",
3357 "record_id_line": 932,
3358 "statement_start_line": 916,
3359 "statement_end_line": 931,
3360 "statement_sha256": "f6ddc4edec01afcf027a48300c34e66e9d1209d4f56bce17e3b9d00f88cd62cf",
3361 "proof_start_line": null,
3362 "proof_end_line": null,
3363 "proof_sha256": null
3364 },
3365 "statement": "\\begin{definition}[Transfer graph of a family generating the algebra]\n\\label{def:transfer-graph}\nLet $\\Pcal=(P_\\alpha)_{\\alpha\\in I}$ be a finite packet system on $X$ with every $P_\\alpha\\in\\A$. Let $\\mathcal G\\subseteq\\A$ satisfy\n\\[\n\\A=\\alg_{\\bbk}\\bigl(\\{P_\\alpha:\\alpha\\in I\\}\\cup\\mathcal G\\bigr),\n\\]\nwhere $\\alg_{\\bbk}(E)$ denotes the unital subalgebra over $\\bbk$ generated by $E$. The transfer graph $\\Gamma(\\mathcal G,\\Pcal)$ has vertex set $I$ and an arrow\n\\[\n\\alpha\\longrightarrow\\beta\n\\]\nwhen\n\\[\nP_\\beta GP_\\alpha\\neq0\n\\]\nfor at least one $G\\in\\mathcal G$. Reachability includes paths of length zero.\n\\end{definition}",
3366 "source_proof": null,
3367 "new_block_contribution": [],
3368 "acceptance_provenance": [
3369 {
3370 "semantic_review": "semantic_wave2.json",
3371 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3372 "reviewed_statement_sha256": "f6ddc4edec01afcf027a48300c34e66e9d1209d4f56bce17e3b9d00f88cd62cf",
3373 "reviewed_proof_sha256": null,
3374 "verdict": "represented as written"
3375 }
3376 ],
3377 "remaining_work": null
3378 },
3379 {
3380 "ordinal": 81,
3381 "stable_id": "PSC-I-T-ALGEBRA-REACHABILITY-SUPPORT",
3382 "kind": "theorem",
3383 "title": "Reachability support of the generated algebra",
3384 "status": "covered",
3385 "old_contract_rows": [
3386 26
3387 ],
3388 "mapped_declarations": [
3389 "Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support",
3390 "Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction",
3391 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.closure_le_ker_of_le_ker",
3392 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.topological_reachability_obstruction",
3393 "Komorebi.PaperIRelease.generatedAlgebra_le_reachability",
3394 "Komorebi.PaperIRelease.generatedAlgebra_block_zero"
3395 ],
3396 "scope_and_remaining_work": "Each projector and declared generator lies in the actual support algebra of ReflTransGen(TransferEdge). Algebra.adjoin_le therefore proves containment of the entire generated algebra. generatedAlgebra_block_zero is the literal ambient P_b A P_a=0 consequence outside reachability.",
3397 "current_source": {
3398 "file": "editable/traditional/03_packets_and_graphs.tex",
3399 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3400 "kind": "theorem",
3401 "record_id_line": 948,
3402 "statement_start_line": 934,
3403 "statement_end_line": 947,
3404 "statement_sha256": "7496f96565d405f8fe6b80491655dffe3b2ddad07b104c1131704464fb8ad031",
3405 "proof_start_line": 951,
3406 "proof_end_line": 967,
3407 "proof_sha256": "b56a509cf5dc7bef3028511b228ce030d81ee7ed652b1d1740fa205f8e179749"
3408 },
3409 "statement": "\\begin{theorem}[Reachability support of the generated algebra]\n\\label{thm:algebra-reachability-support}\nFor every $A\\in\\A$,\n\\[\nP_\\beta AP_\\alpha=0\n\\qquad\\text{whenever }\\beta\\notin\\Reach(\\alpha).\n\\]\nEquivalently,\n\\[\n\\A\\subseteq\\mathfrak M_{R_\\Gamma},\n\\qquad\nR_\\Gamma:=\\{(\\beta,\\alpha):\\beta\\in\\Reach(\\alpha)\\}.\n\\]\n\\end{theorem}",
3410 "source_proof": "\\begin{proof}\nIt is enough to treat a word in the generators and packet projectors. Insert\n\\[\nI=\\sum_{\\gamma\\in I}P_\\gamma\n\\]\nbetween consecutive nonprojector generators. Every resulting $(\\beta,\\alpha)$ block is a finite sum of products\n\\[\nP_\\beta G_mP_{\\gamma_{m-1}}\nG_{m-1}\\cdots\nP_{\\gamma_1}G_1P_\\alpha.\n\\]\nA nonzero product requires every consecutive block to be nonzero, hence gives a directed path\n\\[\n\\alpha\\to\\gamma_1\\to\\cdots\\to\\gamma_{m-1}\\to\\beta.\n\\]\nPacket projectors only insert stationary steps. Thus a word has no block from $\\alpha$ to an unreachable $\\beta$. Linear combinations preserve the same zero pattern.\n\\end{proof}",
3411 "new_block_contribution": [],
3412 "acceptance_provenance": [
3413 {
3414 "semantic_review": "semantic_wave2.json",
3415 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3416 "reviewed_statement_sha256": "7496f96565d405f8fe6b80491655dffe3b2ddad07b104c1131704464fb8ad031",
3417 "reviewed_proof_sha256": "b56a509cf5dc7bef3028511b228ce030d81ee7ed652b1d1740fa205f8e179749",
3418 "verdict": "proved as written"
3419 }
3420 ],
3421 "remaining_work": null
3422 },
3423 {
3424 "ordinal": 82,
3425 "stable_id": "PSC-I-P-REACHABILITY-OBSTRUCTION",
3426 "kind": "proposition",
3427 "title": "Reachability obstruction for generated hulls",
3428 "status": "covered",
3429 "old_contract_rows": [
3430 26
3431 ],
3432 "mapped_declarations": [
3433 "Komorebi.PacketNaturalityCompletion.Reachability.algebraicOrbitHull_le_reachable_support",
3434 "Komorebi.PacketNaturalityCompletion.Reachability.unreachable_nonzero_packet_obstruction",
3435 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.closure_le_ker_of_le_ker",
3436 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.topological_reachability_obstruction",
3437 "Komorebi.PaperIRelease.generatedAlgebra_hull_le_reachable",
3438 "Komorebi.PaperIRelease.generatedAlgebra_unreachable_obstruction",
3439 "Komorebi.PaperIRelease.FiniteDigraph.source_iff_no_incoming_edge",
3440 "Komorebi.PaperIRelease.FiniteDigraph.no_reach_into_source",
3441 "Komorebi.PaperIRelease.generatedAlgebra_source_obstruction"
3442 ],
3443 "scope_and_remaining_work": "The actual reachable-support subspace is preserved by every element of the generated algebra. Span of actual algebra orbits yields the hull inclusion for arbitrary seed sets. The source SCC is the actual mutual-reachability quotient, with source status equivalent to no incoming condensation edge. generatedAlgebra_source_obstruction separately assembles the printed final clause: seeds outside that source cannot generate any nonzero vector supported in it. It obtains a nonzero coordinate by packet completeness and applies no_reach_into_source and the unreachable-coordinate obstruction.",
3444 "current_source": {
3445 "file": "editable/traditional/03_packets_and_graphs.tex",
3446 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3447 "kind": "proposition",
3448 "record_id_line": 993,
3449 "statement_start_line": 977,
3450 "statement_end_line": 992,
3451 "statement_sha256": "a46f6c292b4825e63689b6302077e4db2b2a8fe38f9958fc9018a623e7b8a239",
3452 "proof_start_line": 996,
3453 "proof_end_line": 1005,
3454 "proof_sha256": "815580ea2f89860432a0eb0cf21405e2d78b412203e3961b13e61f7026e2ef72"
3455 },
3456 "statement": "\\begin{proposition}[Reachability obstruction for generated hulls]\n\\label{prop:reachability-obstruction}\nLet $I_0\\subseteq I$ and let\n\\[\nS\\subseteq\\bigoplus_{\\alpha\\in I_0}X_\\alpha.\n\\]\nThen\n\\[\n\\boxed{\n\\Hull_{\\A}(S)\n\\subseteq\n\\bigoplus_{\\beta\\in\\Reach(I_0)}X_\\beta.\n}\n\\]\nIn particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component.\n\\end{proposition}",
3457 "source_proof": "\\begin{proof}\nFor $s\\in S$, write $s=\\sum_{\\alpha\\in I_0}P_\\alpha s$. If $\\beta$ is unreachable from $I_0$, then for every $A\\in\\A$,\n\\[\nP_\\beta As\n=\n\\sum_{\\alpha\\in I_0}P_\\beta AP_\\alpha s\n=0\n\\]\nby \\cref{thm:algebra-reachability-support}. The same holds for every finite linear combination of orbit vectors.\n\\end{proof}",
3458 "new_block_contribution": [],
3459 "acceptance_provenance": [
3460 {
3461 "semantic_review": "semantic_wave2.json",
3462 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3463 "reviewed_statement_sha256": "a46f6c292b4825e63689b6302077e4db2b2a8fe38f9958fc9018a623e7b8a239",
3464 "reviewed_proof_sha256": "815580ea2f89860432a0eb0cf21405e2d78b412203e3961b13e61f7026e2ef72",
3465 "verdict": "proved as written"
3466 }
3467 ],
3468 "remaining_work": null
3469 },
3470 {
3471 "ordinal": 83,
3472 "stable_id": "PSC-I-D-DIRECTED-SEED-RANK",
3473 "kind": "definition",
3474 "title": "Packet-pure seed rank",
3475 "status": "definition_represented",
3476 "old_contract_rows": [],
3477 "mapped_declarations": [
3478 "Komorebi.PaperIRelease.PacketPure",
3479 "Komorebi.PaperIRelease.packetPure_iff_zero_or_range",
3480 "Komorebi.PaperIRelease.PureGenerates",
3481 "Komorebi.PaperIRelease.packetPureSeedRank",
3482 "Komorebi.PaperIRelease.exists_pure_generating_set_card_eq",
3483 "Komorebi.PaperIRelease.packetCornerAlgebra",
3484 "Komorebi.PaperIRelease.corner_one_eq_projector",
3485 "Komorebi.PaperIRelease.localPacketFamily",
3486 "Komorebi.PaperIRelease.localPureSeedRank",
3487 "Komorebi.PaperIRelease.local_proj_zero_iff",
3488 "Komorebi.PaperIRelease.packetPureSeedRank_discardZeros"
3489 ],
3490 "scope_and_remaining_work": "The cardinal infimum is over all actual pure generating subsets, its indexing type is nonempty, and its infimum is attained by well-ordering. Corner compression is e*T on range(e), hence literally e*T*e there. Closure and the identity restriction are proved. A local family retains outside indices as zero maps; local_proj_zero_iff identifies these precisely for nonzero original packets, and zero deletion preserves purity and rank. The corrected zero-or-range wording matches empty-index conventions.",
3491 "current_source": {
3492 "file": "editable/traditional/03_packets_and_graphs.tex",
3493 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3494 "kind": "definition",
3495 "record_id_line": 1032,
3496 "statement_start_line": 1011,
3497 "statement_end_line": 1031,
3498 "statement_sha256": "2a2a8b2073162ddfce0cf7462765d5453176d369133cc86dc699c8d038e8fee8",
3499 "proof_start_line": null,
3500 "proof_end_line": null,
3501 "proof_sha256": null
3502 },
3503 "statement": "\\begin{definition}[Packet pure seed rank]\n\\label{def:directed-seed-rank}\nA vector is packet pure when it is zero or lies in one packet space $X_\\alpha$, equivalently when its packet support has cardinality at most one. Define\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n:=\n\\min\\{|S|:S\\text{ is packet-pure and }\\A S=X\\}.\n\\]\nFor $C\\subseteq I$, put\n\\[\nP_C:=\\sum_{\\alpha\\in C}P_\\alpha,\n\\qquad\nX_C:=P_CX,\n\\qquad\n\\A_C:=P_C\\A P_C,\n\\]\nand define the local packet pure rank\n\\[\nr_C:=\\psrank_{\\A_C,\\Pcal|_C}(X_C).\n\\]\n\\end{definition}",
3504 "source_proof": null,
3505 "new_block_contribution": [],
3506 "acceptance_provenance": [
3507 {
3508 "semantic_review": "semantic_graph_rank.json",
3509 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3510 "reviewed_statement_sha256": "2a2a8b2073162ddfce0cf7462765d5453176d369133cc86dc699c8d038e8fee8",
3511 "reviewed_proof_sha256": null,
3512 "verdict": "definition matches formal objects"
3513 }
3514 ],
3515 "remaining_work": null
3516 },
3517 {
3518 "ordinal": 84,
3519 "stable_id": "PSC-I-P-PACKET-PURE-RANK-BOUNDS",
3520 "kind": "proposition",
3521 "title": "Basic packet-pure rank bounds",
3522 "status": "covered",
3523 "old_contract_rows": [],
3524 "mapped_declarations": [
3525 "Komorebi.PaperIRelease.packetPureSeedRank_bounds",
3526 "Komorebi.PaperIRelease.corner_one_eq_projector"
3527 ],
3528 "scope_and_remaining_work": "The left inequality is direct inclusion of admissible generating sets. The upper inequality chooses attained minimal generating sets under each genuine corner algebra, embeds them into X, and proves their union generates by projector completeness and corner_action_ambient. The union cardinal is bounded by the cardinal sum. This works for arbitrary infinite dimensions and arbitrary cardinal ranks, including zero packets.",
3529 "current_source": {
3530 "file": "editable/traditional/03_packets_and_graphs.tex",
3531 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3532 "kind": "proposition",
3533 "record_id_line": 1049,
3534 "statement_start_line": 1034,
3535 "statement_end_line": 1048,
3536 "statement_sha256": "11a3ad129e84e51edb000d4027f785874b59f3867639a7813ab9c8d944bf81c0",
3537 "proof_start_line": 1052,
3538 "proof_end_line": 1054,
3539 "proof_sha256": "186cea78354769d466d30150dcbb7184eccdc349cfa5bf1155ae7fb2b48ee654"
3540 },
3541 "statement": "\\begin{proposition}[Basic packet pure rank bounds]\n\\label{prop:packet-pure-rank-bounds}\nFor every $A$ module resolved into packets with $P_\\alpha\\in A$,\n\\[\n\\boxed{\n\\srank_A(X)\n\\leq\n\\psrank_{A,\\Pcal}(X)\n\\leq\n\\sum_{\\alpha\\in I}\n\\srank_{P_\\alpha A P_\\alpha}(X_\\alpha).\n}\n\\]\nThe corner algebra on the right is unital on $X_\\alpha$ with identity $P_\\alpha$.\n\\end{proposition}",
3542 "source_proof": "\\begin{proof}\nEvery packet pure generating family is an unrestricted generating family, which gives the first inequality. For each $\\alpha$, choose a minimal generating family $S_\\alpha\\subseteq X_\\alpha$ for the corner action $P_\\alpha A P_\\alpha$. The union $S=\\bigcup_\\alpha S_\\alpha$ is packet pure. Since every $x\\in X$ has the unique decomposition $x=\\sum_\\alpha x_\\alpha$ and every $x_\\alpha$ lies in $(P_\\alpha A P_\\alpha)S_\\alpha\\subseteq AS$, the union generates $X$. Its cardinality is at most the displayed sum.\n\\end{proof}",
3543 "new_block_contribution": [],
3544 "acceptance_provenance": [
3545 {
3546 "semantic_review": "semantic_graph_rank.json",
3547 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3548 "reviewed_statement_sha256": "11a3ad129e84e51edb000d4027f785874b59f3867639a7813ab9c8d944bf81c0",
3549 "reviewed_proof_sha256": "186cea78354769d466d30150dcbb7184eccdc349cfa5bf1155ae7fb2b48ee654",
3550 "verdict": "proved as written"
3551 }
3552 ],
3553 "remaining_work": null
3554 },
3555 {
3556 "ordinal": 85,
3557 "stable_id": "PSC-I-D-EDGE-SURJECTIVE",
3558 "kind": "definition",
3559 "title": "Edge-surjective packet system",
3560 "status": "definition_represented",
3561 "old_contract_rows": [],
3562 "mapped_declarations": [
3563 "Komorebi.PaperIRelease.EdgeSurjective"
3564 ],
3565 "scope_and_remaining_work": "The edge is TransferEdge P ops, not a supplied surrogate graph. The witness T belongs to Algebra.adjoin of the projectors and declared operators; restrictBlocks is the literal map from the source range to the target range.",
3566 "current_source": {
3567 "file": "editable/traditional/03_packets_and_graphs.tex",
3568 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3569 "kind": "definition",
3570 "record_id_line": 1064,
3571 "statement_start_line": 1056,
3572 "statement_end_line": 1063,
3573 "statement_sha256": "46dd6f245aadc93dd24ac5505cd37893d5474fb31208bab5babfacf0fa6d1ce4",
3574 "proof_start_line": null,
3575 "proof_end_line": null,
3576 "proof_sha256": null
3577 },
3578 "statement": "\\begin{definition}[Packet system surjective along edges]\n\\label{def:edge-surjective}\nThe module resolved into packets is surjective along edges relative to $\\Gamma(\\mathcal G,\\Pcal)$ when, for every graph edge $\\alpha\\to\\beta$, there exists $B\\in\\A$ such that the pure block\n\\[\nP_\\beta BP_\\alpha:X_\\alpha\\twoheadrightarrow X_\\beta\n\\]\nis surjective.\n\\end{definition}",
3579 "source_proof": null,
3580 "new_block_contribution": [],
3581 "acceptance_provenance": [
3582 {
3583 "semantic_review": "semantic_graph_rank.json",
3584 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3585 "reviewed_statement_sha256": "46dd6f245aadc93dd24ac5505cd37893d5474fb31208bab5babfacf0fa6d1ce4",
3586 "reviewed_proof_sha256": null,
3587 "verdict": "definition matches formal objects"
3588 }
3589 ],
3590 "remaining_work": null
3591 },
3592 {
3593 "ordinal": 86,
3594 "stable_id": "PSC-I-T-SOURCE-COMPONENT-SEED-RANK",
3595 "kind": "theorem",
3596 "title": "Source-component formula for packet-pure seed rank",
3597 "status": "covered",
3598 "old_contract_rows": [],
3599 "mapped_declarations": [
3600 "Komorebi.PaperIRelease.source_component_pureSeedRank"
3601 ],
3602 "scope_and_remaining_work": "SCC is the quotient by mutual reflexive-transitive reachability. Source is proved equivalent to absence of incoming condensation edges. Finite minimality proves every vertex is reached from a source. Generated-algebra support proves source projection compression. The lower bound removes zero seeds, proves those actually lying in each source generate its corner module, and injects the dependent union of these nonzero seed sets into the original generating set. The upper bound chooses attained local minima; actual surjective blocks propagate range containment along actual paths. No source coverage, compression identity, generating conclusion, or graph certificate is assumed in the final theorem. The theorem in fact does not require nonzero original packets, so it includes the printed hypotheses.",
3603 "current_source": {
3604 "file": "editable/traditional/03_packets_and_graphs.tex",
3605 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3606 "kind": "theorem",
3607 "record_id_line": 1077,
3608 "statement_start_line": 1066,
3609 "statement_end_line": 1076,
3610 "statement_sha256": "0be97dd456719faad7e8e6ba4091ee877f58e7c7c900c8ae79a7b8385c19daea",
3611 "proof_start_line": 1081,
3612 "proof_end_line": 1098,
3613 "proof_sha256": "b0620d6bc6bbecfddece89a4ccbe225f21cbad784cd7880e576e8ef0a7811c26"
3614 },
3615 "statement": "\\begin{theorem}[Source component formula for packet pure seed rank]\n\\label{thm:source-component-rank}\nAssume $I$ is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let $\\mathfrak C_0$ be the set of source strongly connected components of the condensation graph of $\\Gamma(\\mathcal G,\\Pcal)$. Then\n\\[\n\\boxed{\n\\psrank_{\\A,\\Pcal}(X)\n=\n\\sum_{C\\in\\mathfrak C_0}r_C.\n}\n\\]\n\\end{theorem}",
3616 "source_proof": "\\begin{proof}\nLet $S$ be a packet pure generating family for $X$. Discard zero seeds, which do not change the generated hull, and fix a source component $C$. Seeds outside $C$ cannot reach $C$ by \\cref{prop:reachability-obstruction}. Moreover, for $s\\in X_C$ and $A\\in\\A$,\n\\[\nP_CAs=(P_CAP_C)s\\in\\A_Cs.\n\\]\nThus the seeds of $S$ lying in $X_C$ must generate $X_C$ under $\\A_C$, and there are at least $r_C$ of them. Summing over disjoint source components gives\n\\[\n\\psrank_{\\A,\\Pcal}(X)\n\\geq\n\\sum_{C\\in\\mathfrak C_0}r_C.\n\\]\n\nFor the reverse inequality, choose for each source component $C$ a packet pure family $S_C$ of size $r_C$ generating $X_C$ under $\\A_C$, and put $S=\\bigcup_CS_C$. Then every source component lies in $\\A S$. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component $D$ that is not a source have already been shown to lie in $\\A S$. Choose an incoming condensation edge from such a component to $D$, represented by a graph edge $\\alpha\\to\\beta$ with $\\alpha$ already generated and $\\beta\\in D$. Surjectivity along edges gives a pure block\n\\[\nB_{\\beta\\alpha}:X_\\alpha\\twoheadrightarrow X_\\beta.\n\\]\nBecause $X_\\alpha\\subseteq\\A S$ and $\\A S$ is invariant under $\\A$, surjectivity implies $X_\\beta\\subseteq\\A S$. For any other vertex $\\gamma\\in D$, choose a directed path inside the strongly connected component from $\\beta$ to $\\gamma$. Successive surjective edge blocks propagate the full packet space along the path, so $X_\\gamma\\subseteq\\A S$. Hence $X_D\\subseteq\\A S$. Induction over the condensation graph gives $X=\\A S$, proving the upper bound.\n\\end{proof}",
3617 "new_block_contribution": [],
3618 "acceptance_provenance": [
3619 {
3620 "semantic_review": "semantic_graph_rank.json",
3621 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3622 "reviewed_statement_sha256": "0be97dd456719faad7e8e6ba4091ee877f58e7c7c900c8ae79a7b8385c19daea",
3623 "reviewed_proof_sha256": "b0620d6bc6bbecfddece89a4ccbe225f21cbad784cd7880e576e8ef0a7811c26",
3624 "verdict": "proved as written"
3625 }
3626 ],
3627 "remaining_work": null
3628 },
3629 {
3630 "ordinal": 87,
3631 "stable_id": "PSC-I-D-PACKET-PROPAGATING",
3632 "kind": "definition",
3633 "title": "Packet-propagating system",
3634 "status": "definition_represented",
3635 "old_contract_rows": [],
3636 "mapped_declarations": [
3637 "Komorebi.PaperIRelease.PacketCornerCyclic",
3638 "Komorebi.PaperIRelease.PacketPropagating"
3639 ],
3640 "scope_and_remaining_work": "Cyclicity is the existing module Cyclic predicate for the genuine packetCornerAlgebra acting on its actual range. There are no extra local-rank-one certificates supplied.",
3641 "current_source": {
3642 "file": "editable/traditional/03_packets_and_graphs.tex",
3643 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3644 "kind": "definition",
3645 "record_id_line": 1112,
3646 "statement_start_line": 1105,
3647 "statement_end_line": 1111,
3648 "statement_sha256": "4e9bfc875f14eeec60ebfab1d4906a00230861dfe0e37ea46c90639653fbc111",
3649 "proof_start_line": null,
3650 "proof_end_line": null,
3651 "proof_sha256": null
3652 },
3653 "statement": "\\begin{definition}[Packet propagating system]\n\\label{def:packet-propagating}\nA packet system that is surjective along edges is called packet propagating when every packet space $X_\\alpha$ is cyclic under its diagonal corner algebra\n\\[\nP_\\alpha\\A P_\\alpha.\n\\]\n\\end{definition}",
3654 "source_proof": null,
3655 "new_block_contribution": [],
3656 "acceptance_provenance": [
3657 {
3658 "semantic_review": "semantic_graph_rank.json",
3659 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3660 "reviewed_statement_sha256": "4e9bfc875f14eeec60ebfab1d4906a00230861dfe0e37ea46c90639653fbc111",
3661 "reviewed_proof_sha256": null,
3662 "verdict": "definition matches formal objects"
3663 }
3664 ],
3665 "remaining_work": null
3666 },
3667 {
3668 "ordinal": 88,
3669 "stable_id": "PSC-I-C-SOURCE-COMPONENT-COUNT",
3670 "kind": "corollary",
3671 "title": "Unweighted source-component count",
3672 "status": "covered",
3673 "old_contract_rows": [],
3674 "mapped_declarations": [
3675 "Komorebi.PaperIRelease.packetPureSeedRank_eq_source_count",
3676 "Komorebi.PaperIRelease.packetPureSeedRank_eq_source_card"
3677 ],
3678 "scope_and_remaining_work": "The upper bound chooses a cyclic vector at one actual representative vertex of each source SCC and propagates full packet ranges to every vertex. The lower bound proves every generating set has a nonzero source projection, then uses purity and orthogonality to inject Sources into the chosen pure generating set. Nonzero packet range hypotheses are explicit. The finite cardinal-number form is proved; no finite generation or dimension assumption is introduced.",
3679 "current_source": {
3680 "file": "editable/traditional/03_packets_and_graphs.tex",
3681 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3682 "kind": "corollary",
3683 "record_id_line": 1123,
3684 "statement_start_line": 1114,
3685 "statement_end_line": 1122,
3686 "statement_sha256": "f7d07d0e6a080b4b92c8a401d9663246fb377ecc6d930b8ecdb94ec07ea7e5dc",
3687 "proof_start_line": 1126,
3688 "proof_end_line": 1128,
3689 "proof_sha256": "023c9e136f9933f34bd80a507d57b0f3bb0edf429c05d2b8ec7baae497245af3"
3690 },
3691 "statement": "\\begin{corollary}[Unweighted source component count]\n\\label{cor:source-component-count}\nAssume the module resolved into packets is packet propagating and $X_\\alpha\\neq0$ for every $\\alpha\\in I$. If $c_0$ denotes the number of source strongly connected components, then\n\\[\n\\boxed{\n\\psrank_{\\A,\\Pcal}(X)=c_0.\n}\n\\]\n\\end{corollary}",
3692 "source_proof": "\\begin{proof}\nFix a source component $C$ and a vertex $\\alpha\\in C$. Choose a cyclic vector $s\\in X_\\alpha$ for $P_\\alpha\\A P_\\alpha$. It generates all of $X_\\alpha$. Strong connectivity and surjectivity along edges then propagate the whole packet space to every vertex of $C$. Thus $r_C=1$ for every source component. Apply \\cref{thm:source-component-rank}.\n\\end{proof}",
3693 "new_block_contribution": [],
3694 "acceptance_provenance": [
3695 {
3696 "semantic_review": "semantic_graph_rank.json",
3697 "report_sha256": "c7e8636b0621ef0b3e9da0de293cea48c9c50a71d69af5a59cf6263c878987f4",
3698 "reviewed_statement_sha256": "f7d07d0e6a080b4b92c8a401d9663246fb377ecc6d930b8ecdb94ec07ea7e5dc",
3699 "reviewed_proof_sha256": "023c9e136f9933f34bd80a507d57b0f3bb0edf429c05d2b8ec7baae497245af3",
3700 "verdict": "proved as written"
3701 }
3702 ],
3703 "remaining_work": null
3704 },
3705 {
3706 "ordinal": 89,
3707 "stable_id": "PSC-I-B-PACKET-PURE-BOUNDARY",
3708 "kind": "boundary",
3709 "title": "Mixed-support seeds can merge source requirements",
3710 "status": "scope_boundary",
3711 "old_contract_rows": [
3712 7
3713 ],
3714 "mapped_declarations": [
3715 "Komorebi.SeedRankOperations.seedRank_prod_productAlgebra"
3716 ],
3717 "scope_and_remaining_work": "The unrestricted mixed-seed example is consistent with the proved product-algebra formula. The packet-pure rank calculation is not separately formalized.",
3718 "current_source": {
3719 "file": "editable/traditional/03_packets_and_graphs.tex",
3720 "file_sha256": "e1d7110f0f581feffe33992eff5614a5bb8aaf892c7b242c00a7d3b4f799bb45",
3721 "kind": "boundary",
3722 "record_id_line": 1147,
3723 "statement_start_line": 1130,
3724 "statement_end_line": 1146,
3725 "statement_sha256": "a3610fb8adcedb4fb91006a77f357f3f18c57678d044c21b8522730a456b0006",
3726 "proof_start_line": null,
3727 "proof_end_line": null,
3728 "proof_sha256": null
3729 },
3730 "statement": "\\begin{boundary}[Mixed support seeds can merge source requirements]\n\\label{bnd:packet-pure-boundary}\nThe source component formulas concern packet pure seeds. They do not give the unrestricted seed rank. For example, let\n\\[\n\\A=\\bbk\\times\\bbk,\n\\qquad\nX=\\bbk\\oplus\\bbk\n\\]\nwith the coordinate action and the two coordinate packet projectors. The transfer graph has two source components and\n\\[\n\\psrank_{\\A,\\Pcal}(X)=2,\n\\]\nbut the mixed vector $(1,1)$ is cyclic, so\n\\[\n\\srank_{\\A}(X)=1.\n\\]\n\\end{boundary}",
3731 "source_proof": null,
3732 "new_block_contribution": []
3733 },
3734 {
3735 "ordinal": 90,
3736 "stable_id": "PSC-I-T-COORDINATE-TRANSPORT",
3737 "kind": "theorem",
3738 "title": "Coordinate transport",
3739 "status": "covered",
3740 "old_contract_rows": [
3741 27,
3742 28
3743 ],
3744 "mapped_declarations": [
3745 "Komorebi.Foundation.LinearProjectorFamily.conjugate",
3746 "Komorebi.Foundation.conjugate_transferBlock",
3747 "Komorebi.PacketNaturalityCompletion.Reachability.conjugate_transferEdge_iff",
3748 "Komorebi.PacketNaturalityCompletion.Reachability.conjugate_reachableFrom_iff",
3749 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
3750 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
3751 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
3752 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
3753 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
3754 "Komorebi.PaperIRelease.CoordinateTransport.packetRangeEquiv",
3755 "Komorebi.PaperIRelease.CoordinateTransport.packet_rank_invariant",
3756 "Komorebi.PaperIRelease.CoordinateTransport.support_pattern_invariant",
3757 "Komorebi.PacketNaturalityCompletion.Reachability.conjugate_transferBlock_ne_zero_iff"
3758 ],
3759 "scope_and_remaining_work": "conjugate constructs the actual family U P_i U⁻¹ and proves all projector identities. conjugate_transferBlock proves the exact displayed equality. packetRangeEquiv restricts U to each actual packet range and yields cardinal dimension equality with explicit universe lifts. support_pattern_invariant proves preservation for every relation. TransferEdge and ReachableFrom are unchanged when the actual generator set is transported by conjugation.",
3760 "current_source": {
3761 "file": "editable/traditional/04_transport.tex",
3762 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
3763 "kind": "theorem",
3764 "record_id_line": 23,
3765 "statement_start_line": 7,
3766 "statement_end_line": 22,
3767 "statement_sha256": "ae5e550ba65822235a1f8ab84c76a5a3ac876b28334523c9d845ece2af2d2d5a",
3768 "proof_start_line": 26,
3769 "proof_end_line": 41,
3770 "proof_sha256": "90a7f2100d2943f8e48f31834b2fbde55b3acebe907f0b95361f61625c959a3f"
3771 },
3772 "statement": "\\begin{theorem}[Coordinate transport]\n\\label{thm:coordinate-transport}\nLet $U:X\\to Y$ be a vector space isomorphism and let $\\Pcal=(P_\\alpha)$ be a packet system on $X$. Define\n\\[\nQ_\\alpha:=UP_\\alpha U^{-1}.\n\\]\nThen $\\Qcal=(Q_\\alpha)$ is a packet system on $Y$. For $\\widetilde T=UTU^{-1}$,\n\\[\n\\boxed{\nQ_\\beta\\widetilde TQ_\\alpha\n=\nU(P_\\beta TP_\\alpha)U^{-1}.\n}\n\\]\nHence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family $\\mathcal G$ is transported to $U\\mathcal G U^{-1}$, then its transfer graph and reachability relation are also preserved.\n\\end{theorem}",
3773 "source_proof": "\\begin{proof}\nConjugation preserves products and sums, so\n\\[\nQ_\\alpha Q_\\beta\n=UP_\\alpha P_\\beta U^{-1}\n=\\delta_{\\alpha\\beta}Q_\\alpha,\n\\qquad\n\\sum_\\alpha Q_\\alpha=I_Y.\n\\]\nThe block identity is direct multiplication:\n\\[\nQ_\\beta\\widetilde TQ_\\alpha\n=UP_\\beta TP_\\alpha U^{-1}.\n\\]\nAll listed invariants are determined by these transported blocks.\n\\end{proof}",
3774 "new_block_contribution": [],
3775 "acceptance_provenance": [
3776 {
3777 "semantic_review": "semantic_wave2.json",
3778 "report_sha256": "bc317037047cbb56f975f53b3e81c2cae91e0860311787b81e1ebecfc8df54eb",
3779 "reviewed_statement_sha256": "ae5e550ba65822235a1f8ab84c76a5a3ac876b28334523c9d845ece2af2d2d5a",
3780 "reviewed_proof_sha256": "90a7f2100d2943f8e48f31834b2fbde55b3acebe907f0b95361f61625c959a3f",
3781 "verdict": "proved as written"
3782 }
3783 ],
3784 "remaining_work": null
3785 },
3786 {
3787 "ordinal": 91,
3788 "stable_id": "PSC-I-D-PACKET-MORPHISM",
3789 "kind": "definition",
3790 "title": "Packet-compatible operator morphism",
3791 "status": "definition_represented",
3792 "old_contract_rows": [
3793 28
3794 ],
3795 "mapped_declarations": [
3796 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
3797 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
3798 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
3799 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
3800 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
3801 "Komorebi.PaperIRelease.CategoryPackets.PacketObject",
3802 "Komorebi.PaperIRelease.CategoryPackets.PacketMorphism",
3803 "Komorebi.PaperIRelease.CategoryPackets.fiberProjector",
3804 "Komorebi.PaperIRelease.TopologyCategories.PacketObject",
3805 "Komorebi.PaperIRelease.TopologyCategories.PacketMorphism"
3806 ],
3807 "scope_and_remaining_work": "The source and target have actual concrete unital operator subalgebras and nonzero complete orthogonal packet projectors. The linear map, unital algebra map and entirely arbitrary index map are separately retained. Fiber projectors are literal finite sums; no injectivity or surjectivity condition is imposed on the index map.",
3808 "current_source": {
3809 "file": "editable/traditional/04_transport.tex",
3810 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
3811 "kind": "definition",
3812 "record_id_line": 69,
3813 "statement_start_line": 48,
3814 "statement_end_line": 68,
3815 "statement_sha256": "e0e522a401795ef9ecf2497c058baf522801bf33dd581a837deaabd9cbe887bf",
3816 "proof_start_line": null,
3817 "proof_end_line": null,
3818 "proof_sha256": null
3819 },
3820 "statement": "\\begin{definition}[Packet compatible operator morphism]\n\\label{def:packet-morphism}\nLet $(X,\\A,\\Pcal)$ and $(Y,\\B,\\Qcal)$ be packet resolved operator modules with packet index sets $I$ and $J$. A packet compatible operator morphism is a triple\n\\[\n(\\Phi,\\varphi,\\sigma)\n\\]\nconsisting of a unital algebra homomorphism $\\varphi:\\A\\to\\B$, a linear map $\\Phi:X\\to Y$ satisfying\n\\[\n\\Phi(Ax)=\\varphi(A)\\Phi(x),\n\\]\nand a map $\\sigma:I\\to J$ satisfying\n\\[\nQ_j\\Phi\n=\n\\Phi P_{\\sigma^{-1}(j)},\n\\qquad\nP_{\\sigma^{-1}(j)}:=\n\\sum_{\\sigma(\\alpha)=j}P_\\alpha.\n\\]\nAn empty fiber is interpreted as the zero projector.\n\\end{definition}",
3821 "source_proof": null,
3822 "new_block_contribution": [],
3823 "acceptance_provenance": [
3824 {
3825 "semantic_review": "semantic_categories.json",
3826 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
3827 "reviewed_statement_sha256": "e0e522a401795ef9ecf2497c058baf522801bf33dd581a837deaabd9cbe887bf",
3828 "reviewed_proof_sha256": null,
3829 "verdict": "definition represented"
3830 }
3831 ],
3832 "remaining_work": null
3833 },
3834 {
3835 "ordinal": 92,
3836 "stable_id": "PSC-I-D-SEED-MORPHISM",
3837 "kind": "definition",
3838 "title": "Morphism of seeded systems",
3839 "status": "definition_represented",
3840 "old_contract_rows": [],
3841 "mapped_declarations": [
3842 "Komorebi.PaperIRelease.CategoryPackets.SeedObject",
3843 "Komorebi.PaperIRelease.CategoryPackets.SeedMorphism",
3844 "Komorebi.PaperIRelease.CategoryPackets.hull_eq_span",
3845 "Komorebi.PaperIRelease.TopologyCategories.SeedObject",
3846 "Komorebi.PaperIRelease.TopologyCategories.SeedMorphism",
3847 "Komorebi.PaperIRelease.TopologyCategories.closedHull"
3848 ],
3849 "scope_and_remaining_work": "The algebraic condition uses the actual target algebra span, with arbitrary seed sets. The continuous variant separately retains Hausdorff topological vector-space structure, actual continuity of every algebra operator and packet projector, and actual continuity of the carrier morphism. Its seed condition is membership in the literal topologicalClosure of AlgebraGeneratedHull. The paper specifies an algebra of continuous operators, not a separate topology on that algebra, and the code states this convention explicitly.",
3850 "current_source": {
3851 "file": "editable/traditional/04_transport.tex",
3852 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
3853 "kind": "definition",
3854 "record_id_line": 85,
3855 "statement_start_line": 73,
3856 "statement_end_line": 84,
3857 "statement_sha256": "88c7e9a865f68bac1df1727bd7cfc8c66e5b11d2e8be77ada048a83aaafd5637",
3858 "proof_start_line": null,
3859 "proof_end_line": null,
3860 "proof_sha256": null
3861 },
3862 "statement": "\\begin{definition}[Morphism of seeded systems]\n\\label{def:seed-morphism}\nA morphism\n\\[\n(X,\\A,S,\\Pcal)\\longrightarrow(Y,\\B,S',\\Qcal)\n\\]\nis a packet compatible operator morphism satisfying the seed condition\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S').\n\\]\nIn the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.\n\\end{definition}",
3863 "source_proof": null,
3864 "new_block_contribution": [],
3865 "acceptance_provenance": [
3866 {
3867 "semantic_review": "semantic_categories.json",
3868 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
3869 "reviewed_statement_sha256": "88c7e9a865f68bac1df1727bd7cfc8c66e5b11d2e8be77ada048a83aaafd5637",
3870 "reviewed_proof_sha256": null,
3871 "verdict": "definition represented"
3872 }
3873 ],
3874 "remaining_work": null
3875 },
3876 {
3877 "ordinal": 93,
3878 "stable_id": "PSC-I-T-NATURALITY",
3879 "kind": "theorem",
3880 "title": "Naturality of generated hulls and transfer blocks",
3881 "status": "covered",
3882 "old_contract_rows": [
3883 28
3884 ],
3885 "mapped_declarations": [
3886 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
3887 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
3888 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
3889 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
3890 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
3891 "Komorebi.PaperIRelease.CategoryPackets.hull_natural",
3892 "Komorebi.PaperIRelease.CategoryPackets.imageHull_le",
3893 "Komorebi.PaperIRelease.CategoryPackets.imageHull_eq_top",
3894 "Komorebi.PaperIRelease.CategoryPackets.transferBlock_naturality",
3895 "Komorebi.PaperIRelease.CategoryPackets.seeded_surjection_generates",
3896 "Komorebi.PaperIRelease.TopologyCategories.seeded_surjection_generates"
3897 ],
3898 "scope_and_remaining_work": "Hull naturality is actual Submodule.map equality with the span of image-algebra orbits; target-algebra inclusion is derived. The block formula is the exact double sum over possibly many-to-one fibers, at the level of actual linear maps. Surjective generation is derived from the map of top, then the seeded condition implies generation by the target seeds.",
3899 "current_source": {
3900 "file": "editable/traditional/04_transport.tex",
3901 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
3902 "kind": "theorem",
3903 "record_id_line": 110,
3904 "statement_start_line": 87,
3905 "statement_end_line": 109,
3906 "statement_sha256": "e7fb174db6031054bdf06996ebad915dfaf307ad38b27fceb17442c46128863b",
3907 "proof_start_line": 114,
3908 "proof_end_line": 131,
3909 "proof_sha256": "e3ddaf3af434bc22e262fc068a8a525f873944f93fd9a74d2708fa3144a897cf"
3910 },
3911 "statement": "\\begin{theorem}[Naturality of generated hulls and transfer blocks]\n\\label{thm:naturality}\nLet $(\\Phi,\\varphi,\\sigma)$ be a packet compatible operator morphism. For every subset $S\\subseteq X$,\n\\[\n\\boxed{\n\\Phi(\\Hull_{\\A}(S))\n=\n\\Hull_{\\varphi(\\A)}(\\Phi(S))\n\\subseteq\n\\Hull_{\\B}(\\Phi(S)).\n}\n\\]\nFor $T\\in\\A$ and $i,j\\in J$,\n\\[\n\\boxed{\nQ_j\\varphi(T)Q_i\\Phi\n=\n\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n}\n\\]\nIf $\\Phi$ is surjective and $S$ generates $X$, then $\\Phi(S)$ generates $Y$ under $\\varphi(\\A)$ and hence under $\\B$. If, in addition, the morphism is seeded, then $S'$ generates $Y$.\n\\end{theorem}",
3912 "source_proof": "\\begin{proof}\nEvery element of $\\Phi(\\Hull_{\\A}(S))$ is a finite sum of vectors\n\\[\n\\Phi(As)=\\varphi(A)\\Phi(s),\n\\]\nso it belongs to $\\Hull_{\\varphi(\\A)}(\\Phi(S))$. Conversely, every generator $\\varphi(A)\\Phi(s)$ equals $\\Phi(As)$, giving equality.\n\nFor the block formula, use intertwining and packet compatibility twice:\n\\begin{align*}\nQ_j\\varphi(T)Q_i\\Phi\n&=Q_j\\varphi(T)\\Phi P_{\\sigma^{-1}(i)}\\\\\n&=Q_j\\Phi TP_{\\sigma^{-1}(i)}\\\\\n&=\\Phi P_{\\sigma^{-1}(j)}TP_{\\sigma^{-1}(i)}\\\\\n&=\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}\n\\Phi P_\\beta TP_\\alpha.\n\\end{align*}\nIf $\\Phi$ is surjective and $\\Hull_{\\A}(S)=X$, the hull equality gives $Y=\\Hull_{\\varphi(\\A)}(\\Phi(S))$. If $\\Phi(S)\\subseteq\\Hull_{\\B}(S')$, then $\\Hull_{\\B}(\\Phi(S))\\subseteq\\Hull_{\\B}(S')$, so $S'$ generates $Y$.\n\\end{proof}",
3913 "new_block_contribution": [],
3914 "acceptance_provenance": [
3915 {
3916 "semantic_review": "semantic_categories.json",
3917 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
3918 "reviewed_statement_sha256": "e7fb174db6031054bdf06996ebad915dfaf307ad38b27fceb17442c46128863b",
3919 "reviewed_proof_sha256": "e3ddaf3af434bc22e262fc068a8a525f873944f93fd9a74d2708fa3144a897cf",
3920 "verdict": "proved as written"
3921 }
3922 ],
3923 "remaining_work": null
3924 },
3925 {
3926 "ordinal": 94,
3927 "stable_id": "PSC-I-P-SEED-SYSTEM-CATEGORY",
3928 "kind": "proposition",
3929 "title": "Category of seeded packet systems",
3930 "status": "covered",
3931 "old_contract_rows": [],
3932 "mapped_declarations": [
3933 "Komorebi.PaperIRelease.CategoryPackets.packetCategory",
3934 "Komorebi.PaperIRelease.CategoryPackets.seedCategory",
3935 "Komorebi.PaperIRelease.CategoryPackets.PacketMorphism.comp",
3936 "Komorebi.PaperIRelease.CategoryPackets.SeedMorphism.comp",
3937 "Komorebi.PaperIRelease.TopologyCategories.packetCategory",
3938 "Komorebi.PaperIRelease.TopologyCategories.seedCategory",
3939 "Komorebi.PaperIRelease.TopologyCategories.operator_mem_closedHull",
3940 "Komorebi.PaperIRelease.TopologyCategories.SeedMorphism.map_hull",
3941 "Komorebi.PaperIRelease.TopologyCategories.SeedMorphism.map_closedHull",
3942 "Komorebi.PaperIRelease.TopologyCategories.SeedMorphism.identity",
3943 "Komorebi.PaperIRelease.TopologyCategories.SeedMorphism.comp"
3944 ],
3945 "scope_and_remaining_work": "Composition explicitly aggregates composite fibers by a finite-sum rearrangement and composes operator intertwiners. Algebraic seed closure is proved using span transport. The continuous variant proves invariance of the actual closed algebraic hull from individual operator continuity, maps the source algebraic hull into the target closed hull, then derives map_closedHull from carrier continuity and closure minimality. Seeded composition applies this proved closure transport; identity seed membership and all category laws are proved. The actual algebraic and topological Category instances therefore cover the full stated category claim.",
3946 "current_source": {
3947 "file": "editable/traditional/04_transport.tex",
3948 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
3949 "kind": "proposition",
3950 "record_id_line": 142,
3951 "statement_start_line": 138,
3952 "statement_end_line": 141,
3953 "statement_sha256": "3cc5c23c5df9094cc9c18b7b9539bf0cadbbe2e74dd249651920fd5034f347a1",
3954 "proof_start_line": 147,
3955 "proof_end_line": 180,
3956 "proof_sha256": "4f25e4d8b514bcf4174ae6cb53941b26c8654f4843d7437ca157c456a93d6d6f"
3957 },
3958 "statement": "\\begin{proposition}[Category of seeded packet systems]\n\\label{prop:seed-category}\nPacket compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions. Thus packet resolved operator modules form a category $\\mathbf{PktMod}$ and seeded packet systems form a category $\\mathbf{SeedSys}$.\n\\end{proposition}",
3959 "source_proof": "\\begin{proof}\nLet $(\\Phi,\\varphi,\\sigma):(X,\\A,\\Pcal)\\to(Y,\\B,\\Qcal)$ and $(\\Psi,\\psi,\\tau):(Y,\\B,\\Qcal)\\to(Z,\\Ccal,\\Rcal)$ be morphisms. Intertwining is preserved because\n\\[\n\\Psi\\Phi(Ax)\n=\n\\Psi(\\varphi(A)\\Phi x)\n=\n(\\psi\\circ\\varphi)(A)\\Psi\\Phi x.\n\\]\nFor $k$ in the target packet set,\n\\begin{align*}\nR_k\\Psi\\Phi\n&=\\Psi Q_{\\tau^{-1}(k)}\\Phi\\\\\n&=\\Psi\\Phi\n\\sum_{\\tau(\\sigma(\\alpha))=k}P_\\alpha.\n\\end{align*}\nThus the composite packet map is $\\tau\\circ\\sigma$. If the morphisms are seeded, then\n\\[\n\\Phi(S)\\subseteq\\Hull_{\\B}(S'),\n\\qquad\n\\Psi(S')\\subseteq\\Hull_{\\Ccal}(S'').\n\\]\nBy \\cref{thm:naturality},\n\\[\n\\Psi\\Phi(S)\n\\subseteq\n\\Psi(\\Hull_{\\B}(S'))\n\\subseteq\n\\Hull_{\\Ccal}(\\Psi(S'))\n\\subseteq\n\\Hull_{\\Ccal}(S'').\n\\]\nIdentity maps satisfy the definitions with identity algebra and packet maps.\n\\end{proof}",
3960 "new_block_contribution": [],
3961 "acceptance_provenance": [
3962 {
3963 "semantic_review": "semantic_categories.json",
3964 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
3965 "reviewed_statement_sha256": "3cc5c23c5df9094cc9c18b7b9539bf0cadbbe2e74dd249651920fd5034f347a1",
3966 "reviewed_proof_sha256": "4f25e4d8b514bcf4174ae6cb53941b26c8654f4843d7437ca157c456a93d6d6f",
3967 "verdict": "proved as written"
3968 }
3969 ],
3970 "remaining_work": null
3971 },
3972 {
3973 "ordinal": 95,
3974 "stable_id": "PSC-I-T-PACKET-EXACT-SEQUENCE",
3975 "kind": "theorem",
3976 "title": "Packet exact sequence for an invariant subspace",
3977 "status": "covered",
3978 "old_contract_rows": [],
3979 "mapped_declarations": [
3980 "Komorebi.Foundation.LinearProjectorFamily.restrictTo",
3981 "Komorebi.Foundation.LinearProjectorFamily.restrictTo_range_in_ambient",
3982 "Komorebi.Foundation.LinearProjectorFamily.nonzeroRestriction",
3983 "Komorebi.Foundation.LinearProjectorFamily.descendTo",
3984 "Komorebi.Foundation.LinearProjectorFamily.nonzeroQuotient",
3985 "Komorebi.Foundation.LinearProjectorFamily.packet_shortExact",
3986 "Komorebi.Foundation.LinearProjectorFamily.packetQuotientEquiv",
3987 "Komorebi.Foundation.LinearProjectorFamily.descendedPacketEquivSumQuotient",
3988 "Komorebi.Foundation.LinearProjectorFamily.packetCoordinateInclusion_natural",
3989 "Komorebi.Foundation.LinearProjectorFamily.packetCoordinateQuotient_natural",
3990 "Komorebi.Foundation.LinearProjectorFamily.packetCoordinate_shortExact",
3991 "Komorebi.Foundation.LinearProjectorFamily.exactPacketDirectSumCoordinates"
3992 ],
3993 "scope_and_remaining_work": "K is an actual invariant submodule. Restriction/quotient projectors are constructed; restricted range equals K intersect X_i. Actual packet inclusion is injective, its range equals the actual quotient-map kernel, and quotient map is onto. First/second quotient isomorphisms use denominator comaps in the correct ambient type. Pointwise coordinate maps intertwine with original inclusion/quotient and form a short exact sequence; exact direct-sum coordinates give the full decomposition.",
3994 "current_source": {
3995 "file": "editable/traditional/04_transport.tex",
3996 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
3997 "kind": "theorem",
3998 "record_id_line": 217,
3999 "statement_start_line": 185,
4000 "statement_end_line": 216,
4001 "statement_sha256": "7311937b07e98adf3e6f91ad665358380cab6ce26e3daeffdbf4787d7a01de56",
4002 "proof_start_line": 221,
4003 "proof_end_line": 225,
4004 "proof_sha256": "7d5e6c04e7f0342af3a5ea4572b74ec05aa52d219d85f36fc2d7e68e31a694ee"
4005 },
4006 "statement": "\\begin{theorem}[Packet exact sequence for an invariant subspace]\n\\label{thm:packet-exact-sequence}\nLet $K\\leq X$ satisfy $P_\\alpha K\\subseteq K$ for every $\\alpha$. Define\n\\[\nK_\\alpha:=P_\\alpha K=K\\cap X_\\alpha\n\\]\nand\n\\[\n\\overline P_\\alpha(x+K):=P_\\alpha x+K.\n\\]\nThen:\n\\begin{enumerate}[label=(\\roman*)]\n\\item The nonzero restrictions $P_\\alpha|_K$ form a packet system on $K$.\n\\item The nonzero $\\overline P_\\alpha$ form a packet system on $X/K$.\n\\item For every $\\alpha$ there is a short exact sequence\n\\[\n0\\longrightarrow K_\\alpha\n\\longrightarrow X_\\alpha\n\\longrightarrow \\overline P_\\alpha(X/K)\n\\longrightarrow0,\n\\]\nand\n\\[\n\\overline P_\\alpha(X/K)\n\\cong\nX_\\alpha/(X_\\alpha\\cap K)\n\\cong\n(X_\\alpha+K)/K.\n\\]\n\\item The full exact sequence decomposes packetwise as the direct sum of these short exact sequences.\n\\end{enumerate}\n\\end{theorem}",
4007 "source_proof": "\\begin{proof}\nRestriction preserves idempotency, pairwise annihilation, and the identity sum on $K$. If $x\\in K\\cap X_\\alpha$, then $x=P_\\alpha x\\in P_\\alpha K$, proving $K_\\alpha=K\\cap X_\\alpha$.\n\nThe quotient operator is well defined because replacing $x$ by $x+k$ changes $P_\\alpha x$ by $P_\\alpha k\\in K$. The idempotent and annihilation identities descend, and the sum is the identity on $X/K$. The map $X_\\alpha\\to X/K$ has image $(X_\\alpha+K)/K$ and kernel $X_\\alpha\\cap K$, giving the short exact sequence and the quotient isomorphisms. Exact packet coordinates identify the original sequence with the direct sum over $\\alpha$.\n\\end{proof}",
4008 "new_block_contribution": [],
4009 "acceptance_provenance": [
4010 {
4011 "semantic_review": "semantic_wave1.json",
4012 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
4013 "reviewed_statement_sha256": "7311937b07e98adf3e6f91ad665358380cab6ce26e3daeffdbf4787d7a01de56",
4014 "reviewed_proof_sha256": "7d5e6c04e7f0342af3a5ea4572b74ec05aa52d219d85f36fc2d7e68e31a694ee",
4015 "verdict": "proved as written"
4016 }
4017 ],
4018 "remaining_work": null
4019 },
4020 {
4021 "ordinal": 96,
4022 "stable_id": "PSC-I-T-QUOTIENT-PACKET-DESCENT",
4023 "kind": "theorem",
4024 "title": "Packet descent through an invariant quotient",
4025 "status": "covered",
4026 "old_contract_rows": [],
4027 "mapped_declarations": [
4028 "Komorebi.Foundation.LinearProjectorFamily.descendTo",
4029 "Komorebi.Foundation.LinearProjectorFamily.descendTo_proj_mkQ",
4030 "Komorebi.Foundation.LinearProjectorFamily.descendTo_intertwines",
4031 "Komorebi.Foundation.LinearProjectorFamily.nonzeroQuotient",
4032 "Komorebi.Foundation.LinearProjectorFamily.discardZeros_proj_ne_zero",
4033 "Komorebi.Foundation.LinearProjectorFamily.descendTo_unique"
4034 ],
4035 "scope_and_remaining_work": "Actual mapQ uses invariance of K. Quotient representative calculations prove the laws and commuting identity. Zero removal preserves completeness. Quotient surjectivity proves uniqueness of every descended operator, so canonicity is established.",
4036 "current_source": {
4037 "file": "editable/traditional/04_transport.tex",
4038 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4039 "kind": "theorem",
4040 "record_id_line": 244,
4041 "statement_start_line": 232,
4042 "statement_end_line": 243,
4043 "statement_sha256": "b51391a00350d220fc92610f351826489ca69d166e27e266a1ecc6ecc7a2ffc7",
4044 "proof_start_line": 247,
4045 "proof_end_line": 252,
4046 "proof_sha256": "7fe5f8af71890536db25fac05b8cc49a2d183c71e594881626d109232dd92d66"
4047 },
4048 "statement": "\\begin{theorem}[Packet descent through an invariant quotient]\n\\label{thm:quotient-packet-descent}\nUnder the hypotheses of \\cref{thm:packet-exact-sequence}, the operators\n\\[\n\\overline P_\\alpha(x+K)=P_\\alpha x+K\n\\]\nare well defined pairwise annihilating idempotents with sum $I_{X/K}$, and the quotient map satisfies\n\\[\n\\boxed{\\pi P_\\alpha=\\overline P_\\alpha\\pi.}\n\\]\nAfter zero packets are removed, they form the canonical quotient packet system.\n\\end{theorem}",
4049 "source_proof": "\\begin{proof}\nAll assertions are the quotient part of \\cref{thm:packet-exact-sequence}. The intertwining identity follows directly from the definition:\n\\[\n\\overline P_\\alpha\\pi(x)=P_\\alpha x+K=\\pi P_\\alpha(x).\n\\]\n\\end{proof}",
4050 "new_block_contribution": [],
4051 "acceptance_provenance": [
4052 {
4053 "semantic_review": "semantic_wave1.json",
4054 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
4055 "reviewed_statement_sha256": "b51391a00350d220fc92610f351826489ca69d166e27e266a1ecc6ecc7a2ffc7",
4056 "reviewed_proof_sha256": "7fe5f8af71890536db25fac05b8cc49a2d183c71e594881626d109232dd92d66",
4057 "verdict": "proved as written"
4058 }
4059 ],
4060 "remaining_work": null
4061 },
4062 {
4063 "ordinal": 97,
4064 "stable_id": "PSC-I-C-QUOTIENT-PACKET-DESCENT",
4065 "kind": "corollary",
4066 "title": "Restriction and quotient of transfer blocks",
4067 "status": "covered",
4068 "old_contract_rows": [
4069 27,
4070 28
4071 ],
4072 "mapped_declarations": [
4073 "Komorebi.Foundation.LinearProjectorFamily.conjugate",
4074 "Komorebi.Foundation.conjugate_transferBlock",
4075 "Komorebi.PacketNaturalityCompletion.Reachability.conjugate_transferEdge_iff",
4076 "Komorebi.PacketNaturalityCompletion.Reachability.conjugate_reachableFrom_iff",
4077 "Komorebi.PacketNaturalityCompletion.StrictNaturality.generatedHull_natural",
4078 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_naturality_package",
4079 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_le",
4080 "Komorebi.PacketNaturalityCompletion.StrictNaturality.strict_seedRank_eq_of_bijective",
4081 "Komorebi.PacketNaturalityCompletion.StrictNaturality.full_transfer_edge_zero_pattern_correspondence",
4082 "Komorebi.Foundation.LinearProjectorFamily.transferBlock_preserves_packet",
4083 "Komorebi.Foundation.LinearProjectorFamily.transferBlock_descends"
4084 ],
4085 "scope_and_remaining_work": "Invariance under P_i,T,P_j proves preservation of K and membership in the target packet. Tbar is actual K.mapQ K T hT, and representative calculation proves the quotient-block identity. Lean orientation is the symmetric equality of the printed formula.",
4086 "current_source": {
4087 "file": "editable/traditional/04_transport.tex",
4088 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4089 "kind": "corollary",
4090 "record_id_line": 266,
4091 "statement_start_line": 254,
4092 "statement_end_line": 265,
4093 "statement_sha256": "4fa0ae8effe1e5472dcde2be67829cba4ae586df9853d252baff055f3ee7fd46",
4094 "proof_start_line": 269,
4095 "proof_end_line": 282,
4096 "proof_sha256": "897454446f8fb08c4e3429934ec902205fe48d2cd913e30c0a57b145cc02b161"
4097 },
4098 "statement": "\\begin{corollary}[Restriction and quotient of transfer blocks]\n\\label{cor:block-descent}\nUnder the hypotheses of \\cref{thm:packet-exact-sequence}, let $T\\in\\End(X)$ preserve $K$. Then every block $T_{\\beta\\alpha}$ maps $K_\\alpha$ into $K_\\beta$, and the induced quotient operator $\\overline T$ satisfies\n\\[\n\\boxed{\n\\overline T_{\\beta\\alpha}\\,\\pi\n=\n\\pi T_{\\beta\\alpha},\n}\n\\]\nwhere $\\pi:X\\twoheadrightarrow X/K$.\n\\end{corollary}",
4099 "source_proof": "\\begin{proof}\nIf $x\\in K_\\alpha$, then $Tx\\in K$, so\n\\[\nT_{\\beta\\alpha}x=P_\\beta Tx\\in K\\cap X_\\beta=K_\\beta.\n\\]\nOn the quotient,\n\\[\n\\overline T_{\\beta\\alpha}\\pi\n=\n\\overline P_\\beta\\overline T\\overline P_\\alpha\\pi\n=\n\\pi P_\\beta TP_\\alpha.\n\\]\n\\end{proof}",
4100 "new_block_contribution": [],
4101 "acceptance_provenance": [
4102 {
4103 "semantic_review": "semantic_wave1.json",
4104 "report_sha256": "1692e9b3b1fc0cd938266d759d1a44ece0cf09bd43f9f47cc5846b42738ccb67",
4105 "reviewed_statement_sha256": "4fa0ae8effe1e5472dcde2be67829cba4ae586df9853d252baff055f3ee7fd46",
4106 "reviewed_proof_sha256": "897454446f8fb08c4e3429934ec902205fe48d2cd913e30c0a57b145cc02b161",
4107 "verdict": "proved as written"
4108 }
4109 ],
4110 "remaining_work": null
4111 },
4112 {
4113 "ordinal": 98,
4114 "stable_id": "PSC-I-C-TOPOLOGICAL-PACKET-EXACT-SEQUENCE",
4115 "kind": "corollary",
4116 "title": "Topological packet exact sequence",
4117 "status": "covered",
4118 "old_contract_rows": [],
4119 "mapped_declarations": [
4120 "Komorebi.PaperIRelease.GeneralPacketTopology.banach_packet_shortExact",
4121 "Komorebi.PaperIRelease.GeneralPacketTopology.packet_spaces_complete",
4122 "Komorebi.PaperIRelease.GeneralPacketTopology.quotientProjector_bounded",
4123 "Komorebi.PaperIRelease.GeneralPacketTopology.restrictedProjector_bounded",
4124 "Komorebi.PaperIRelease.GeneralPacketTopology.quotient_bound_operatorNorm",
4125 "Komorebi.PaperIRelease.GeneralPacketTopology.packetInclusion_bounded",
4126 "Komorebi.PaperIRelease.GeneralPacketTopology.packetQuotient_bounded"
4127 ],
4128 "scope_and_remaining_work": "The generic theorem works over any NormedField, including a trivial valuation, with explicit boundedness of the original projectors. Restriction and quotient CLMs are the actual algebraic packet maps. Near-minimal quotient representatives transfer every original bound to the quotient; the norm bound uses the usual infimum-of-bounds norm of the bounded additive projector. Actual packet inclusion and quotient maps are bounded with constant one and retain injectivity, exactness, and surjectivity. Closed K and closed bounded-idempotent ranges make all three terms complete.",
4129 "current_source": {
4130 "file": "editable/traditional/04_transport.tex",
4131 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4132 "kind": "corollary",
4133 "record_id_line": 288,
4134 "statement_start_line": 284,
4135 "statement_end_line": 287,
4136 "statement_sha256": "2590b3fce057e3d054522a58a728f46dbf3c609c95ff6ab061ff654ffd72fee1",
4137 "proof_start_line": 292,
4138 "proof_end_line": 300,
4139 "proof_sha256": "9a68f08ce3a88028c493bcc6d341516ecd5173c9e3aa1f788daf8cc0af567c37"
4140 },
4141 "statement": "\\begin{corollary}[Topological packet exact sequence]\n\\label{cor:topological-packet-exact-sequence}\nIf $X$ is Banach, the $P_\\alpha$ are bounded, and $K$ is closed and packet invariant, then every packet sequence in \\cref{thm:packet-exact-sequence} is a short exact sequence of Banach spaces with bounded maps, and the quotient projectors are bounded.\n\\end{corollary}",
4142 "source_proof": "\\begin{proof}\nClosedness of $K$ makes $X/K$ Banach. The restricted projectors are bounded, and the quotient projectors are bounded because\n\\[\n\\|\\overline P_\\alpha(x+K)\\|_{X/K}\n\\leq\n\\|P_\\alpha\\|\\,\\|x+K\\|_{X/K}.\n\\]\nThe packet spaces are closed ranges of bounded idempotents.\n\\end{proof}",
4143 "new_block_contribution": [],
4144 "acceptance_provenance": [
4145 {
4146 "semantic_review": "semantic_analytic.json",
4147 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
4148 "reviewed_statement_sha256": "2590b3fce057e3d054522a58a728f46dbf3c609c95ff6ab061ff654ffd72fee1",
4149 "reviewed_proof_sha256": "9a68f08ce3a88028c493bcc6d341516ecd5173c9e3aa1f788daf8cc0af567c37",
4150 "verdict": "proved as written"
4151 }
4152 ],
4153 "remaining_work": null
4154 },
4155 {
4156 "ordinal": 99,
4157 "stable_id": "PSC-I-T-DENSE-REALIZATION",
4158 "kind": "theorem",
4159 "title": "Dense realization theorem",
4160 "status": "covered",
4161 "old_contract_rows": [
4162 29
4163 ],
4164 "mapped_declarations": [
4165 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.dense_operatorHull_image",
4166 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.map_packetRange_eq",
4167 "PaperIRelease.TopologyTransport.map_operatorHull_le",
4168 "PaperIRelease.TopologyTransport.dense_operatorHull_image",
4169 "PaperIRelease.TopologyTransport.map_packetRange_eq",
4170 "PaperIRelease.TopologyTransport.map_aggregated_packetRange_eq"
4171 ],
4172 "scope_and_remaining_work": "For the actual continuous linear intertwiner in the packet-morphism convention, the image of the source algebraic operator hull is contained in the target hull, and continuous surjectivity transports density. The actual finite grouped projector is the sum over the σ-fibre, including empty fibres. Surjectivity and the given intertwining identity prove exact equality of image and target range. No norm or metrizability assumptions occur.",
4173 "current_source": {
4174 "file": "editable/traditional/04_transport.tex",
4175 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4176 "kind": "theorem",
4177 "record_id_line": 319,
4178 "statement_start_line": 305,
4179 "statement_end_line": 318,
4180 "statement_sha256": "58bf2b9bfb4ab3ae9f3309b03cc231e0b59c81b7fe2b6c99e4b74884aaf15bca",
4181 "proof_start_line": 322,
4182 "proof_end_line": 331,
4183 "proof_sha256": "7c169706130d8a06c3e308ab14435b24ab4f4649c9ece2f2afb0e59f6ff2f911"
4184 },
4185 "statement": "\\begin{theorem}[Dense realization theorem]\n\\label{thm:dense-realization}\nLet $X$ and $Y$ be Hausdorff topological vector spaces, let $\\Phi:X\\to Y$ be continuous, surjective, and intertwining with respect to $\\varphi$, and let $S$ topologically generate $X$. Then $\\Phi(S)$ topologically generates $Y$. If finite continuous packet systems satisfy\n\\[\nQ_j\\Phi=\\Phi P_{\\sigma^{-1}(j)},\n\\]\nthen\n\\[\n\\boxed{\nQ_jY=\\Phi(P_{\\sigma^{-1}(j)}X)\n}\n\\]\nfor every $j$.\n\\end{theorem}",
4186 "source_proof": "\\begin{proof}\nGiven $y=\\Phi(x)$, choose a net $x_\\lambda\\in\\Hull_{\\A}(S)$ converging to $x$. Then\n\\[\n\\Phi(x_\\lambda)\\to y,\n\\]\nand naturality places every $\\Phi(x_\\lambda)$ in $\\Hull_{\\B}(\\Phi(S))$. Hence the latter hull is dense. For packets, one inclusion follows from compatibility. Conversely, if $y=Q_jy$, choose $x$ with $\\Phi(x)=y$. Then\n\\[\ny=Q_j\\Phi(x)=\\Phi P_{\\sigma^{-1}(j)}x.\n\\]\n\\end{proof}",
4187 "new_block_contribution": [],
4188 "acceptance_provenance": [
4189 {
4190 "semantic_review": "semantic_analytic.json",
4191 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
4192 "reviewed_statement_sha256": "58bf2b9bfb4ab3ae9f3309b03cc231e0b59c81b7fe2b6c99e4b74884aaf15bca",
4193 "reviewed_proof_sha256": "7c169706130d8a06c3e308ab14435b24ab4f4649c9ece2f2afb0e59f6ff2f911",
4194 "verdict": "proved as written"
4195 }
4196 ],
4197 "remaining_work": null
4198 },
4199 {
4200 "ordinal": 100,
4201 "stable_id": "PSC-I-T-QUOTIENT-TOPOLOGY-COMPLETION",
4202 "kind": "theorem",
4203 "title": "Quotient topology and completed realization",
4204 "status": "covered",
4205 "old_contract_rows": [
4206 1,
4207 29
4208 ],
4209 "mapped_declarations": [
4210 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange",
4211 "Komorebi.SeedRankBasics.oneSeedQuotientEquivRange_apply_mk",
4212 "Komorebi.SeedRankBasics.generates_singleton_iff_surjective",
4213 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense",
4214 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_intertwines",
4215 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.completionEquivOfIsometryDense_unique",
4216 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.dense_operatorHull_image",
4217 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.map_packetRange_eq",
4218 "PaperIRelease.TopologyTransport.quotientEquiv",
4219 "PaperIRelease.TopologyTransport.quotientEquiv_apply_mk",
4220 "PaperIRelease.TopologyDenseCompletion.restricted_kernel",
4221 "PaperIRelease.TopologyDenseCompletion.restricted_dense",
4222 "PaperIRelease.TopologyDenseCompletion.completedRestrictedQuotient",
4223 "PaperIRelease.TopologyDenseCompletion.completedQuotientEquiv_apply_mk",
4224 "PaperIRelease.TopologyDenseCompletion.completionLinearEquiv_unique"
4225 ],
4226 "scope_and_remaining_work": "The actual first-isomorphism linear equivalence is continuous both ways using IsQuotientMap, not continuity plus surjectivity alone. The restriction to D uses its actual kernel, definitionally comap(D inclusion,ker Φ); density is exactly Φ(D). The same explicit transported-uniformity quotient completion gives the canonical target equivalence and agrees with Φ on represented classes. No original-domain quotient topology is silently substituted for the printed transported topology.",
4227 "current_source": {
4228 "file": "editable/traditional/04_transport.tex",
4229 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4230 "kind": "theorem",
4231 "record_id_line": 345,
4232 "statement_start_line": 333,
4233 "statement_end_line": 344,
4234 "statement_sha256": "2e986d76e93d67446708502bfae8275f63d8ab0eaf8dcc3e3630ee7f6b3fa077",
4235 "proof_start_line": 348,
4236 "proof_end_line": 350,
4237 "proof_sha256": "e632019afa173b81579ad0ed9554e279083c896c3bf7434fd1a244aecef55d1b"
4238 },
4239 "statement": "\\begin{theorem}[Quotient topology and completed realization]\n\\label{thm:quotient-topology}\nLet $\\Phi:X\\twoheadrightarrow Y$ be a continuous linear quotient map between Hausdorff topological vector spaces, and put $K=\\Ker\\Phi$. Then the induced map\n\\[\n\\widetilde\\Phi:X/K\\longrightarrow Y\n\\]\nis a topological isomorphism. If $D\\subseteq X$ is a linear subspace whose image $\\Phi(D)$ is dense in a complete space $Y$, then the completion of\n\\[\nD/(D\\cap K)\n\\]\nin the topology transported from $Y$ is canonically isomorphic to $Y$.\n\\end{theorem}",
4240 "source_proof": "\\begin{proof}\nThe algebraic first isomorphism theorem gives a bijection $\\widetilde\\Phi$. By definition of the quotient topology, a set $U\\subseteq X/K$ is open exactly when its preimage in $X$ is open. Because $\\Phi$ is a quotient map, this is equivalent to $\\widetilde\\Phi(U)$ being open in $Y$. Thus $\\widetilde\\Phi$ is a homeomorphism. The second statement applies the dense completion argument of \\cref{thm:completed-presentation} to the injective map $D/(D\\cap K)\\to Y$.\n\\end{proof}",
4241 "new_block_contribution": [],
4242 "acceptance_provenance": [
4243 {
4244 "semantic_review": "semantic_analytic.json",
4245 "report_sha256": "22066f0cec1c7761905e51aa6fbef5de9fca2bd7826bc21299d332b6b49387ee",
4246 "reviewed_statement_sha256": "2e986d76e93d67446708502bfae8275f63d8ab0eaf8dcc3e3630ee7f6b3fa077",
4247 "reviewed_proof_sha256": "e632019afa173b81579ad0ed9554e279083c896c3bf7434fd1a244aecef55d1b",
4248 "verdict": "proved as written"
4249 }
4250 ],
4251 "remaining_work": null
4252 },
4253 {
4254 "ordinal": 101,
4255 "stable_id": "PSC-I-B-TOPOLOGICAL-QUOTIENT",
4256 "kind": "boundary",
4257 "title": "Continuous surjections and quotient topology",
4258 "status": "scope_boundary",
4259 "old_contract_rows": [],
4260 "mapped_declarations": [],
4261 "scope_and_remaining_work": "The quotient-map requirement is preserved; no open-map theorem is inferred from continuity alone.",
4262 "current_source": {
4263 "file": "editable/traditional/04_transport.tex",
4264 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4265 "kind": "boundary",
4266 "record_id_line": 356,
4267 "statement_start_line": 352,
4268 "statement_end_line": 355,
4269 "statement_sha256": "855989a0fadfaf7174d83889d9a2c4208bc722f866b35c5da99de3e7ecf1b9f7",
4270 "proof_start_line": null,
4271 "proof_end_line": null,
4272 "proof_sha256": null
4273 },
4274 "statement": "\\begin{boundary}[Continuous surjections and quotient topology]\n\\label{bnd:topological-quotient}\nContinuity and surjectivity transport density as in \\cref{thm:dense-realization}. They do not by themselves identify the given topology on $Y$ with the quotient topology from $X$. The stronger conclusion in \\cref{thm:quotient-topology} requires the quotient map hypothesis or an equivalent open mapping condition.\n\\end{boundary}",
4275 "source_proof": null,
4276 "new_block_contribution": []
4277 },
4278 {
4279 "ordinal": 102,
4280 "stable_id": "PSC-I-D-REALIZATION-FUNCTOR",
4281 "kind": "definition",
4282 "title": "Realization functor and preservation profile",
4283 "status": "documentary",
4284 "old_contract_rows": [],
4285 "mapped_declarations": [
4286 "Komorebi.PaperIRelease.CategoryExport.Expression",
4287 "Komorebi.PaperIRelease.CategoryExport.transportLimit",
4288 "Komorebi.PaperIRelease.CategoryExport.transportColimit"
4289 ],
4290 "scope_and_remaining_work": "The profile is represented by actual additive/linear functor and PreservesLimit/PreservesColimit instances, with typed syntax for equational operations. Universal constructions also use explicit structured categories, an induced functor, an underlying comparison isomorphism, and actual initial/terminal preservation. No Boolean profile field is accepted in place of preservation evidence.",
4291 "current_source": {
4292 "file": "editable/traditional/04_transport.tex",
4293 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4294 "kind": "definition",
4295 "record_id_line": 398,
4296 "statement_start_line": 382,
4297 "statement_end_line": 397,
4298 "statement_sha256": "152d8ee6b5bb1735f7655c6eeee72210d213959c00c9e6168d554eee25b7c5fe",
4299 "proof_start_line": null,
4300 "proof_end_line": null,
4301 "proof_sha256": null
4302 },
4303 "statement": "\\begin{definition}[Realization functor and preservation profile]\n\\label{def:realization-functor}\nLet $\\mathbf{SeedSys}_{\\mathrm{abs}}$ be an abstract category of seed systems and let $\\mathbf C_D$ be a domain category. A realization functor is a functor\n\\[\n\\mathsf{Ev}_D:\\mathbf{SeedSys}_{\\mathrm{abs}}\\longrightarrow\\mathbf C_D\n\\]\nwith a declared preservation profile specifying which of the following structures are preserved:\n\\[\n\\begin{gathered}\n0,\\ I,\\ \\text{finite sums},\\ \\text{composition},\\ \\text{biproducts},\\\\\n\\text{kernels},\\ \\text{cokernels},\\ \\text{quotients},\\ \\text{tensor products},\\\\\n\\text{packet idempotents},\\ \\text{transfer blocks},\\ \\text{topological completions}.\n\\end{gathered}\n\\]\nPreservation of a structure means preservation of its defining identity or universal property, not merely a similarity of notation.\n\\end{definition}",
4304 "source_proof": null,
4305 "new_block_contribution": [],
4306 "acceptance_provenance": [
4307 {
4308 "semantic_review": "semantic_categories.json",
4309 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
4310 "reviewed_statement_sha256": "152d8ee6b5bb1735f7655c6eeee72210d213959c00c9e6168d554eee25b7c5fe",
4311 "reviewed_proof_sha256": null,
4312 "verdict": "documentary definition; verified profile components represented"
4313 }
4314 ],
4315 "remaining_work": null
4316 },
4317 {
4318 "ordinal": 103,
4319 "stable_id": "PSC-I-T-REALIZATION-EXPORT",
4320 "kind": "theorem",
4321 "title": "Equational realization principle",
4322 "status": "covered",
4323 "old_contract_rows": [],
4324 "mapped_declarations": [
4325 "Komorebi.PaperIRelease.CategoryExport.evaluate_transport",
4326 "Komorebi.PaperIRelease.CategoryExport.equation_transport",
4327 "Komorebi.PaperIRelease.CategoryExport.additive_equation_transport",
4328 "Komorebi.PaperIRelease.CategoryExport.transportLimit",
4329 "Komorebi.PaperIRelease.CategoryExport.transportColimit",
4330 "Komorebi.PaperIRelease.CategoryExport.transportInitial",
4331 "Komorebi.PaperIRelease.CategoryExport.transportInitial_map",
4332 "Komorebi.PaperIRelease.CategoryExport.transportTerminal",
4333 "Komorebi.PaperIRelease.CategoryExport.transportTerminal_map",
4334 "Komorebi.PaperIRelease.CategoryExport.transportInitial_underlying",
4335 "Komorebi.PaperIRelease.CategoryExport.transportTerminal_underlying"
4336 ],
4337 "scope_and_remaining_work": "Structural induction on well-typed morphism expressions proves transport of identities, zero, addition, negation, scalar multiplication and composition. The additive-only statement uses the canonical integer enrichment, and no extra scalar-linearity assumption is inserted. Actual limit/colimit diagrams are transported under the matching preservation instance. The conditional universal-construction clause is explicit: a declared profile supplies the structured categories of the universal problem, an induced functor, compatibility with the underlying realization functor, and actual initial/terminal preservation. CategoryUniversal derives the mapped initial/terminal property, equality of the uniquely induced arrows, and the underlying commuting comparison squares. Tensor and completion preservation are legitimate conditional inputs, not falsely inferred from additivity.",
4338 "current_source": {
4339 "file": "editable/traditional/04_transport.tex",
4340 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4341 "kind": "theorem",
4342 "record_id_line": 404,
4343 "statement_start_line": 400,
4344 "statement_end_line": 403,
4345 "statement_sha256": "e0352bdb7ba75e24d4c7e6e0259802f47800c4945c13efe4b3c68edcacef678d",
4346 "proof_start_line": 408,
4347 "proof_end_line": 424,
4348 "proof_sha256": "06bb33ff0b731d59517e5fb3ceba274201de5e480288b019cd53acb471b2839f"
4349 },
4350 "statement": "\\begin{theorem}[Equational realization principle]\n\\label{thm:realization-export}\nLet $F:\\mathbf C\\to\\mathbf D$ be an additive functor between preadditive categories. Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by $F$. Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and $F$ is linear for the declared enrichment. If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, tensor product, or completion and $F$ preserves that construction in its declared profile, then the corresponding proof diagram is transported to $\\mathbf D$.\n\\end{theorem}",
4351 "source_proof": "\\begin{proof}\nFunctoriality gives\n\\[\nF(I)=I,\n\\qquad\nF(g\\circ f)=F(g)\\circ F(f),\n\\]\nand additivity gives\n\\[\nF(0)=0,\n\\qquad\nF(f+g)=F(f)+F(g),\n\\qquad\nF(-f)=-F(f).\n\\]\nIf the declared enrichment is present, linearity additionally gives $F(af)=aF(f)$ for every coefficient $a$. A structural induction on the construction of a morphism expression therefore carries every permitted equational proof to its image under $F$. If a step invokes a universal construction, preservation means that the image diagram satisfies the same universal property. The uniquely induced morphism in the source proof is therefore carried to the uniquely induced morphism in the target proof. Repeating this argument through the finite proof diagram transports the conclusion.\n\\end{proof}",
4352 "new_block_contribution": [],
4353 "acceptance_provenance": [
4354 {
4355 "semantic_review": "semantic_categories.json",
4356 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
4357 "reviewed_statement_sha256": "e0352bdb7ba75e24d4c7e6e0259802f47800c4945c13efe4b3c68edcacef678d",
4358 "reviewed_proof_sha256": "06bb33ff0b731d59517e5fb3ceba274201de5e480288b019cd53acb471b2839f",
4359 "verdict": "proved as written"
4360 }
4361 ],
4362 "remaining_work": null
4363 },
4364 {
4365 "ordinal": 104,
4366 "stable_id": "PSC-I-P-REALIZATION-EXPORT",
4367 "kind": "proposition",
4368 "title": "Realization export principle",
4369 "status": "covered",
4370 "old_contract_rows": [],
4371 "mapped_declarations": [
4372 "Komorebi.PaperIRelease.CategoryExport.equation_transport",
4373 "Komorebi.PaperIRelease.CategoryExport.additive_equation_transport",
4374 "Komorebi.PaperIRelease.CategoryExport.transportLimit",
4375 "Komorebi.PaperIRelease.CategoryExport.transportColimit",
4376 "Komorebi.PaperIRelease.CategoryExport.transportInitial",
4377 "Komorebi.PaperIRelease.CategoryExport.transportInitial_map",
4378 "Komorebi.PaperIRelease.CategoryExport.transportTerminal",
4379 "Komorebi.PaperIRelease.CategoryExport.transportTerminal_map",
4380 "Komorebi.PaperIRelease.CategoryExport.transportInitial_underlying",
4381 "Komorebi.PaperIRelease.CategoryExport.transportTerminal_underlying"
4382 ],
4383 "scope_and_remaining_work": "This is the methodological specialization of record 103 to already established Paper I identities. The formal equations are transported from a proved source equality, not assumed as the desired target equality. The universal-property step uses the explicit structured-category profile of CategoryUniversal. No new analytic/topological/positivity/finiteness/integrality assumption is generated.",
4384 "current_source": {
4385 "file": "editable/traditional/04_transport.tex",
4386 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4387 "kind": "proposition",
4388 "record_id_line": 436,
4389 "statement_start_line": 432,
4390 "statement_end_line": 435,
4391 "statement_sha256": "bf39b86447c8332d60216e48d23129b6ae5b096611e3febd9d09a699f1a9a557",
4392 "proof_start_line": 439,
4393 "proof_end_line": 441,
4394 "proof_sha256": "f53be9b53d9762f062f819c1b88ff90fecf6159d4ab52ccebbb2f5c4721c8501"
4395 },
4396 "statement": "\\begin{proposition}[Realization export principle]\n\\label{prop:realization-export}\nEvery canonical Paper I identity whose proof uses only the operations listed in a realization functor's verified preservation profile remains valid after applying that functor. If the proof uses a domain specific hypothesis not encoded by the functor, that hypothesis must be verified separately before the transported statement is promoted to a theorem.\n\\end{proposition}",
4397 "source_proof": "\\begin{proof}\nThe first assertion is \\cref{thm:realization-export}. The second is a logical boundary: a functor transports established identities and preserved universal constructions, but it does not create an unverified analytic, topological, positivity, finiteness, or integrality hypothesis.\n\\end{proof}",
4398 "new_block_contribution": [],
4399 "acceptance_provenance": [
4400 {
4401 "semantic_review": "semantic_categories.json",
4402 "report_sha256": "8fd410f688c3bc556ab857ad998d3ee70e2d53f847ddb09e2697f99d1057b278",
4403 "reviewed_statement_sha256": "bf39b86447c8332d60216e48d23129b6ae5b096611e3febd9d09a699f1a9a557",
4404 "reviewed_proof_sha256": "f53be9b53d9762f062f819c1b88ff90fecf6159d4ab52ccebbb2f5c4721c8501",
4405 "verdict": "proved as written"
4406 }
4407 ],
4408 "remaining_work": null
4409 },
4410 {
4411 "ordinal": 105,
4412 "stable_id": "PSC-I-D-SEMANTIC-REALIZATION",
4413 "kind": "definition",
4414 "title": "Semantic realization criterion",
4415 "status": "documentary",
4416 "old_contract_rows": [],
4417 "mapped_declarations": [],
4418 "scope_and_remaining_work": "A methodology and scope checklist, not automatic formal certification.",
4419 "current_source": {
4420 "file": "editable/traditional/04_transport.tex",
4421 "file_sha256": "14e9773a6aafdc6a408c03d1757eb9a813002269242d3d310bbcb1e7df90c931",
4422 "kind": "definition",
4423 "record_id_line": 454,
4424 "statement_start_line": 443,
4425 "statement_end_line": 453,
4426 "statement_sha256": "7bfb521bf39d1f519716ff37b9ffcf3ae76ce4f62fe121586daa29e66e7223df",
4427 "proof_start_line": null,
4428 "proof_end_line": null,
4429 "proof_sha256": null
4430 },
4431 "statement": "\\begin{definition}[Semantic realization criterion]\n\\label{def:semantic-realization}\nA domain theorem is a semantic realization of a canonical seed calculus theorem only when:\n\\begin{enumerate}[label=(\\roman*)]\n\\item An explicit realization functor or intertwining construction is given.\n\\item The preservation profile covers every construction used in the canonical proof.\n\\item Every additional analytic, finiteness, convergence, positivity, or integrality hypothesis is verified in the domain.\n\\item The resulting conclusion is stated in the domain's native objects.\n\\end{enumerate}\nReuse of the words ``seed'', ``packet'', or ``transfer'' without these data is only syntactic reuse.\n\\end{definition}",
4432 "source_proof": null,
4433 "new_block_contribution": []
4434 },
4435 {
4436 "ordinal": 106,
4437 "stable_id": "PSC-I-ALG-EXAMPLE-SLOTS",
4438 "kind": "definition",
4439 "title": "Reserved example slots",
4440 "status": "documentary",
4441 "old_contract_rows": [
4442 30,
4443 31,
4444 32,
4445 33,
4446 34,
4447 35
4448 ],
4449 "mapped_declarations": [
4450 "Komorebi.PacketNaturalityCompletion.TypedQuiver.typed_reachability_certificate",
4451 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.BilateralShiftRepresentation.concretePower_apply",
4452 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.concreteBilateralShift_topologicallyCyclic",
4453 "KomorebiVerification.TopologicalCompletion.BoundedProjectorFamily.concreteBilateralShift_seedCardinality_isLeast",
4454 "Komorebi.SingleOperatorGraph.graph_seedNumber_eq_max_eigenspace_multiplicity",
4455 "Komorebi.SingleOperatorGraph.graph_seedNumber_eq_max_eigenspace_multiplicity_of_distinct",
4456 "Komorebi.SingleOperatorGraph.operatorAlgebra_seedFinrank_eq_of_seedNumber_isLeast",
4457 "Komorebi.SingleOperatorGraph.operatorAlgebra_seedRank_eq_of_seedNumber_isLeast",
4458 "Komorebi.SingleOperatorGraph.path_endpoint_seedNumber_isLeast_one",
4459 "Komorebi.SingleOperatorGraph.complex_cycle_seedNumber_isLeast_two",
4460 "Komorebi.FiniteAlgorithms.orbitHull_algorithm_terminates_and_correct",
4461 "Komorebi.FiniteAlgorithms.orbitClosureIter_eq_orbitHull_of_stop",
4462 "Komorebi.FiniteAlgorithms.transferBlockTable",
4463 "Komorebi.FiniteAlgorithms.assembleTransferBlockTable",
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Back to Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates