Reading orientation
This guide follows the paper's mathematical route from finite Frobenius packets to finite wild comparison, then explains the actual arithmetic constructions developed in Blocks 166–167.
The general arithmetic realization endpoint is stated as Conjecture 66. The comparison with the established local Langlands correspondence remains a theorem conditional on an actual realization and H5 for that same candidate.
All mathematical artwork is deterministic Python/Matplotlib vector output. The complete editable source and its figure generator accompany this guide.
How to use this companion
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Read the first two parts as an atlas of finite mathematical objects. The third part explains where finite objects acquire an actual arithmetic carrier. The final part separates the remaining realization conjecture from the conditional comparison theorem.
Each spread develops one construction or one distinction. Displays preserve the normalizations used in the accompanying paper. The source line identifies an original manuscript section or a retained proof note. These are retained source anchors, rather than a claim that every argument has been newly formalized. The separate research atlas provides the later conductor, occurrence and native-marking reconciliation.
The symbol \(K\) denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, \(F={\mathbb F}_p((t))\) is the base field and \(K=F[\beta]\) is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.
Theorem numbering. The historical proof obligation called “66” corresponds to the endpoint printed as Theorem 16.2 in the original manuscript. Conjecture 66 now isolates its general arithmetic realization requirement. It is not a claim that row 66 of the historical theorem register is this endpoint.
Supplementary materials. The accompanying release includes Markdown (.md) files containing detailed proofs, construction notes, proof audits, and research records. These documents distinguish established results, conditional results, and the remaining obligations associated with Conjecture 66.
Evidence. Blocks 166–167 contain ordinary mathematical arguments using the stated classical type-theory results. Their bounded exact computations support specified identities. They are not fresh Lean certificates for the arbitrary-length arithmetic theorems. The separate Lean guide states the actual formalization scope.
Two visual routes, one claim map
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This is the core visual guide. It explains the mathematical objects and the restricted arithmetic constructions in the monograph. The separate 36-page research atlas explains the subsequent comparison tests, failed routes and remaining certificate. The two guides are complementary, not competing versions of the same document.
| Start with the question | Continue to the linked reader |
|---|---|
| What is being realized? | Monograph realization contract and research plate V01. Finite comparison and arithmetic existence are different assertions. |
| How does recovery preserve the candidate? | Monograph restricted constructions and research plate V10. A changed assembly carrier does not mean a changed inducing representation. |
| Why do numerical tests not settle attachment? | Progress Sections 22–25 and research plates V11–V19. The full and skipped rank-eight profiles have capacities six and four despite the same three numerical invariants. |
| What does a dyadic marking retain? | Progress Sections 26–29 and research plates V20–V29. Squares, section products, native choices and phase data must not be collapsed into one commutator form. |
| What is formally checked? | Lean guide: the existence input and the conditional signature. A supplied field in a structure is not constructed by typesetting its declaration. |
Keep the five reader PDFs together. These links use the actual companion pages, while the original lessons keep their source-note and section anchors. No diagram in either guide certifies arithmetic inputs that its source argument leaves open.
Reading corrections without changing the contract
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One candidate throughout. A source-defined induction candidate and a directly constructed type candidate are not identified merely by notation. Occurrence, full-character normalization and the noncompact extension must concern the same representation. The Part II occurrence analysis retains the excluded original route and the corrected route's remaining obligation.
A transported mark is not a prescribed native mark. The positive scalar-pushout attachment carries the supplied finite marking through a center-fixing homomorphism. It does not identify that output with an independently fixed host subgroup. Conversely, the stronger native problem must not be inserted as a new premise into the different transported-marking result.
The equal-capacity example is conditional. At rank eight the full profile has exactly six quotient dimensions. The finite Arf test applies when the actual native refinement is independently prescribed. The abstract six-dimensional forms in the research atlas are examples, not a computation of that candidate's native square bits.
A successful finite repair has inputs. The transvection starts with an allowed symplectic map and an isotropic discrepancy vector. The odd-action correction starts with genuine supplied finite actions. Neither constructs the arithmetic normalizer or chooses every honest phase. See plates V25–V29.
Evidence and publication status
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This revision rebuilds the core guide from its editable source and preserves its Python-generated vector figures. The historical matrix calculations and ordinary construction arguments keep their dates and scopes. The Lean and evidence guide records the separate current compiler replay; the PDF build itself does not certify arithmetic constructors.
The versioned ledger distinguishes publication and formal verification from the open arithmetic endpoints. The selected Henniart–Lemaire source interface has now been checked against the full official text, including its normalization conventions. The complete Block 2 editable archive remains unavailable. Its research-reader pages are preserved. No external peer review is claimed.
For the final same-candidate comparison, read the monograph comparison proposition alongside research plate V32. The conjecture itself does not assert the independent H5 step.