Weil Defect

A public research repository for a defect-theoretic analysis of Weil's quadratic form, its finite negative-index structure, spectral screening, persistent negative directions, and compact-window neutral modes.

Status: Horizon 1 complete; downstream actual-zeta interfaces remain open.

Scope: This repository packages an independent Weil-defect theory to its stated stop boundary. It does not claim a proof of the Riemann Hypothesis.

Scope and separation

The project separates two mathematical layers that must not be conflated:

• Weil-defect theory: finite negative index, screening, persistence, compact-window neutrality, and their operator-theoretic structure.
• Actual-zeta exclusion: the additional arithmetic input needed to rule out the remaining defect geometries for the actual zeta divisor.

The first layer can be developed and audited independently of the second. The working architecture is

$$\boxed{ \text{finite negative index} \longrightarrow \text{spectral screening / support filtration} \longrightarrow \begin{cases} \text{fixed-packet negative persistence},\\\ \text{fixed-packet attained-neutral persistence},\\\ \text{moving/infinite-sector or background noncompactness}. \end{cases} }$$

Only after this defect geometry is isolated does the program ask whether the actual zeta divisor can realize the remaining branches.

Horizon 1

The repository is currently organized under Horizon 1 — Independent Weil-Defect Theory. Its stop boundary is deliberately before RH closure.

Phase	Scope	Status
H1-P0	Consolidation and custody	Complete
H1-P1	Abstract defect calculus	Complete
H1-P2	Zeta-Weil specialization	Complete
H1-P3	Defect morphology theorem	Complete
H1-P4	Proof audit and theorem normalization	Complete
LEAN-H1	Lean certification track	Exhausted
H1-P5	Public mathematical package	Complete
Horizon 1	Independent Weil-defect theory	Complete

Source pinning, internal proof audit, composite morphology audit, examples sharpness audit, Lean certification, and public packaging are complete for the Horizon-1 inventory. No post-Horizon research cursor is currently selected. Stable theorem IDs and the canonical audit surfaces are recorded in the Theorem Ledger, Dependency Audit, Imported Source Pins, Internal Proof Audit, Composite Morphology Audit, and Examples and Sharpness Audit.

Project terms such as horizon, phase, standing, interface, custody, and screening are fixed in the Terminology Registry.

Formal verification has reached its exhaustion condition. See the Lean Formalization Track and Lean Status Ledger. The public package has passed its final adversarial audit. See Public Package Architecture, the Weil-Defect Manuscript, the Public Theorem Index, the Public Verification Matrix, the Public Dependency Map, the Public Examples and Sharpness, the RH-Facing Interface Appendix, and the Public Package Audit.

Core mathematical picture

The abstract defect calculus is organized around

$$D=S_{+}S_{+}^{*}-S_{-}S_{-}^{*},$$

with the sign problem reduced to contractive Douglas screening. In the monotone support-filtration setting of WD-T16/WD-T17, a normalized critical or negative right-approaching sequence whose negative coordinate remains in one fixed finite selected sector has a nonzero nonpositive right-limit subsequence. Accordingly, loss of a persistent selected ray while selected negative mass remains anchored requires leaving the fixed finite-sector regime. Unselected- background noncompactness is a separate mechanism: after a selected ray is anchored it may obstruct strong full-coefficient compactness, but it does not erase that selected ray.

Under the zeta-Weil specialization, the selected raw residue vector satisfies

$$\mathbf{1}^{T}v=0.$$

For the associated rational response, that zero-moment identity removes the first Laurent term and yields

$$R_{v}(z)=O(|z|^{-2}).$$

The explicit-formula attachment sharpens the far-field contribution to

$$\mathcal{F}_{v,R} = O\!\left(\frac{\log R}{R}\right),$$

leaving a weighted completed $\Xi$ next-jet field as the negative-branch arithmetic obstruction.

On the neutral branch, fixed compact support produces only finitely many prime-power translations, with principal order

$$\Psi_{c}(t)=\log|t|+O_{c}(1).$$

These statements are packaged in the negative, neutral, and noncompact morphology documents linked below.

Current theorem picture

Component	Current standing
Finite Weil negative index	Imported theorem + exact specialization
Rank-one defect specialization	Internal proof; one-channel specialization of the abstract calculus
Support-filtration right-limit geometry	Internal proof with stated inputs
Screening taxonomy and custody distinctions	Internal/structural package
Persistent normalized Weil negativity	Conditional theorem
Quartet zero-moment law	Internal proof in the selected quartet model
$O(\lvert z\rvert^{-2})$ far-field decay	Internal proof
Reduction to weighted near next-jet representation	Internal deduction under stated source/multiplier hypotheses
Compact-window neutral equation $W_{c}k=0$	Conditional theorem
Fixed-window log-order operator + finite prime shifts	Derived
Actual-zeta next-jet exclusion	Open
Neutral null-extension rigidity	Open
RH	Open

See Proof Status for precise hypotheses, mathematical standing, and verification state.

RH-facing interfaces

Horizon 1 deliberately stops before two primary actual-zeta interfaces, with one stronger special-packet refinement on the negative side:

• AZ-NEXTJET-LOC — control or exclusion of the weighted near next-jet field.
• C-ACTUAL-KPH-FLOOR — stronger special-packet KPH/transversality floor that can serve as a sufficient refinement of AZ-NEXTJET-LOC where its packet hypotheses apply.
• AZ-FIN-WEIL-NULL-EXTENSION — exterior support/null-extension rigidity for an actual compact-window neutral mode.

The negative morphology leaves AZ-NEXTJET-LOC as the next unresolved actual-zeta obligation after

$$\mathbf{1}^{T}v=0 \quad\Longrightarrow\quad R_{v}(z)=O(|z|^{-2}) \quad\Longrightarrow\quad \mathcal{F}_{v,R}=O\!\left(\frac{\log R}{R}\right).$$

The neutral morphology leaves AZ-FIN-WEIL-NULL-EXTENSION as its unresolved support/right-limit obligation after deriving the compact-window equation

$$W_{c}k=0.$$

These interfaces are downstream obligations; none is a premise of the Horizon-1 morphology theorem from which it emerges.

Repository map

Public Horizon-1 package

• Public Package Architecture — reader-facing package structure and custody rules.
• Weil-Defect Manuscript — assembled mathematical narrative.
• Public Theorem Index — stable-ID lookup with hypotheses, outputs, dependencies, standing, and Lean status.
• Public Verification Matrix — mathematical standing, source ancestry, formal status, and certificate evidence.
• Public Dependency Map — compressed theorem/import/interface DAG.
• Public Examples and Sharpness — canonical failure modes attached to theorem boundaries.
• RH-Facing Interface Appendix — exact downstream obligations left open by Horizon 1.
• Public Package Audit — final package and reader-surface consistency checks.

Horizon and audit control

• Horizon 1 — phase gates and the Horizon-1 stop boundary.
• Theorem Ledger — stable theorem IDs, mathematical standing, H1-P4 audit/source status, and historical aliases.
• Dependency Audit — normalized theorem DAG, imported-source boundaries, scope guards, and audit queue.
• Imported Source Pins — exact load-bearing external statements, equation locations, and convention map.
• Proof Status — compact theorem and phase status.
• Terminology Registry — canonical project vocabulary.
• Research Map — dependency graph and current frontier.
• References — background literature used by the program.

H1-P1 — Abstract defect calculus

• Abstract Defect Calculus — zeta-independent operator theory.
• Restricted-Channel Transfer — selected finite sectors, background budget, and shorting.
• Support Filtration and Persistence — right limits, endpoint jumps, representative blow-up, and moving-sector escape.

H1-P2 — Zeta-Weil specialization

• Zeta-Weil Specialization Map — mapping from abstract carriers to Weil/divisor/arithmetic structure.
• Quartet Channel and Residue Structure — pair geometry, finite inertia, zero moments, and compact synthesis.
• Explicit-Formula Arithmetic Attachment — far-tail localization, completed $\Xi$ next jets, finite prime shifts, and logarithmic form order.

H1-P3 — Defect morphology

• Negative Defect Morphology Theorem — fixed-packet negative endpoint morphology and stop line.
• Neutral Defect Morphology Theorem — attained-neutral null mode, logarithmic operator, and null-extension stop line.
• Noncompact Background Morphology Theorem — moving selected escape, background escape, and fixed full-divisor negative weak limits.

Consolidation record

• Initial Consolidation — first full public consolidation checkpoint.

Background

The project sits in the lineage of work on Weil's criterion, finite truncations of the Weil quadratic form, compact-window positivity, and operator realizations of the explicit formula.

Particularly relevant references include:

• Enrico Bombieri, Remarks on Weil's quadratic functional in the theory of prime numbers, I.
• Masatoshi Suzuki, Weil's quadratic form via the screw function.
• Recent work on certified compact-window Weil positivity and Landau–Widom-type spectral behavior.

See References for the repository's source list.

Verification convention

The project separates mathematical standing from verification status. An internal proof under stated hypotheses is not, by that fact alone, independently certified or formally verified.

A stable theorem ID is a name, not a certification mark. Verification claims are promoted only by an explicit later audit or certificate artifact.