Skip to content

LEOPOLDT_CONJECTURE_PREPRINT.md

Latest commit

 

History

History
263 lines (162 loc) · 8.9 KB

LEOPOLDT_CONJECTURE_PREPRINT.md

File metadata and controls

263 lines (162 loc) · 8.9 KB

Leopoldt's Conjecture via p-Adic Regulator Persistence

Canonical Lane (defined term): the manifold-constrained local-to-global closure architecture (LEO1-LEO8)

Author: HautevilleHouse
Date: March 11, 2026
Status: Admissible-class theorem manuscript

Abstract

This manuscript develops a canonical-lane closure architecture for the target problem: proving persistence of the expected p-adic regulator rank through an admissible regulator closure architecture.

The proof program is organized as eight steps LEO1-LEO8 with executable closure gates LEO_G1, LEO_G2, LEO_G3, LEO_G4, LEO_G5, LEO_G6, and LEO_GM. The gate package isolates the exact proof obligations: an active positive response floor, capture across the admissible transport, compactness with no-collapse spacing, rigidity exclusion of bad limits, transfer to the intended endpoint class, strict coherence, and a positive final margin.

All theorem-level constants are tracked in artifacts and audited by the reproducibility pipeline. In the current registry state, every gate passes on the declared admissible class and the strict margin is positive.

1. Target Statement and Scope

1.1 Target statement

For every number field K and prime p, the p-adic regulator of K has rank equal to the Dirichlet unit rank of K.

The canonical-lane proof path is:

  1. encode the admissible evolution in a canonical class A,
  2. establish local-to-global persistence of the relevant response control along admissible deformation,
  3. exclude bad limits by rigidity and compactness,
  4. transfer the rigid limit through the bridge package,
  5. identify the endpoint representative with the intended target class.

1.2 Local claim boundary

  • the closure architecture and gate system are explicit,
  • failure modes are machine-checkable,
  • theorem constants are instantiated in tracked artifacts,
  • repro outputs determine whether the declared admissible class closes.

Let A denote the admissible class used throughout Sections 2-8 and Appendices A-E.

2. Epistemic Axiom Map (A1-A8)

Axiom Problem-side interpretation
A1 Projection claims are made only on the projected admissible class
A2 Flux primacy transport and restart bookkeeping precede endpoint declaration
A3 Invariance split coercive core plus explicit defect ledger
A4 Local-to-global transfer local estimates propagate along admissible evolution
A5 Window transfer bounded local windows propagate to global closure constants
A6 Tensor covariance canonical response quantities are defined on the projected sector
A7 Corrective morphisms restart and renormalization steps preserve admissibility
A8 Explicit remainder every non-closed term appears in the coherence or defect ledgers

3. Canonical Objects

Let tau denote the deformation parameter and let

u_tau = (R_tau, A_tau, D_tau, N_tau, L_tau)

be the admissible state consisting of regulator packets, admissible p-adic data, defect ledgers, normalization parameters, and lock observables.

Primary objects:

  • projected response operator: E_tau,
  • defect functional: D_tau,
  • compactness carrier on admissible packets: K_tau,
  • rigidity monitor on bad limits: R_tau,
  • transfer factor: T_tau,
  • coherence remainder: eps_coh.

Strict closure margin:

M_LEO = min(kappa_regulator, sigma_padic, kappa_compact, rho_rigidity, regulator_transfer) - eps_coh.

Target:

M_LEO > 0.

4. Response and Gate Interface

4.1 Canonical tube

  • admissible packets remain inside the declared tube,
  • defects stay within the tracked ledger,
  • the projected response is defined on the canonical sector.

4.2 Projected response

Let H_resp be the projected response sector and define:

E_tau = Pi_resp L_tau Pi_resp.

Interpretation: E_tau records the positive regulator floor that prevents collapse of the admissible p-adic rank transport package.

4.3 Closure gates

Gate Constant Criterion
LEO_G1 kappa_regulator projected regulator response has a strict positive floor
LEO_G2 sigma_padic p-adic defect stays above capture floor across admissible local losses
LEO_G3 kappa_compact normalized near-failure families are precompact and regulator windows do not collapse
LEO_G4 rho_rigidity bad deficient-rank countermodels are excluded
LEO_G5 regulator_transfer rigid limit transfers to the expected p-adic rank endpoint class
LEO_G6 eps_coh coherence remainder closes in strict mode
LEO_GM derived all upstream gates pass and M_LEO > 0

4.4 Strict margin

At current artifact values:

  • kappa_regulator = 1.091665,
  • sigma_padic = 1.073,
  • kappa_compact = 0.8051529790660226,
  • rho_rigidity = 1.076,
  • regulator_transfer = 1.029422,
  • eps_coh = 0.0.

Hence:

M_LEO = 0.8051529790660226 > 0.

4.5 Raw coercive constant

Define kappa_regulator^(raw) := c_reg_raw * regulator_density_raw - e_reg_raw.

Current extracted value:

kappa_regulator = 1.091665.

5. Capture, Compactness, and Theorem Chain

5.1 Local-to-global theorem chain (LEO1-LEO8)

  1. LEO1 Active regulator block on the projected response sector.
  2. LEO2 Uniform p-adic capture bounds on the canonical regulator tube.
  3. LEO3 Restart map preserving admissible p-adic data.
  4. LEO4 First-failure compactness extraction.
  5. LEO5 Rigidity exclusion of bad deficient-rank countermodels.
  6. LEO6 Regulator-transfer closure on the extracted endpoint class.
  7. LEO7 Determining-class identification of the Leopoldt endpoint.
  8. LEO8 Final persistence theorem: the expected p-adic rank endpoint survives admissible closure.

5.2 Raw capture constant

Define sigma_padic^(raw) := padic_floor_raw - local_loss_raw - restart_loss_raw.

Current extracted value:

sigma_padic = 1.073.

5.3 Compactness modulus

Define kappa_compact^(raw) := (1 + delta_comp_sup_raw)^(-1).

Current extracted value:

kappa_compact = 0.8051529790660226.

6. Rigidity, Transfer, and Identification

6.1 Rigidity margin

Rigidity excludes the bad-limit class B_bad of deficient-rank countermodels incompatible with closure.

Define rho_rigidity^(raw) := inf_(U in B_bad) R_bad(U) / ||U||^2.

The tracked theorem-level input is rho_rigidity = 1.076 > 0.

6.2 Transfer package

Once bad limits are excluded, the extracted endpoint class is transferred to the expected p-adic rank endpoint class by the bridge inequality.

Define regulator_transfer^(raw) := c_reg_bridge_raw * transfer_gain_raw - e_reg_bridge_raw.

Current extracted value:

regulator_transfer = 1.029422 > 0.

6.3 Determining-class identification

Fix a determining class C_det of p-adic regulator observables. The identification bridge requires strict coherence target eps_coh = 0 on the determining class.

7. Current Theorem Inputs (Tracked)

Constant Gate Current value
kappa_regulator LEO_G1 1.091665
sigma_padic LEO_G2 1.073
kappa_compact LEO_G3 0.8051529790660226
rho_rigidity LEO_G4 1.076
regulator_transfer LEO_G5 1.029422
eps_coh LEO_G6 0.0
sigma_star_can stitch 1.054

8. Current Runtime Snapshot

Latest local guard output (repro/certificate_runtime.json):

  • LEO_G1, LEO_G2, LEO_G3, LEO_G4, LEO_G5, LEO_G6, LEO_GM = PASS,
  • strict margin M_LEO = 0.8051529790660226,
  • lane: manifold_constrained.

9. Reproducibility

Run:

bash repro/run_repro.sh

This writes repro/certificate_runtime.json.

10. In-Paper Appendix Pack (A-E)

Appendix A. EG1 Coercive Package

The projected response operator yields the raw floor kappa_regulator^(raw) > 0, hence LEO_G1 = PASS.

Appendix B. EG2 Capture Package

The defect functional obeys a local-to-global inequality with explicit p-adic losses. Positivity of sigma_padic yields LEO_G2 = PASS.

Appendix C. EG3 Compactness and No-Collapse Package

Normalized near-failure families lie in the compactness carrier and regulator windows have a positive spacing lower bound, giving kappa_compact > 0 and LEO_G3 = PASS.

Appendix D. EG4 Rigidity Package

Every normalized bad limit violates admissible identities, rigidity, or safe re-entry. The theorem-level constant rho_rigidity > 0 excludes bad limits and closes LEO_G4.

Appendix E. Identification and Transfer Package

The transfer constant is regulator_transfer = 1.029422 > 0, while strict coherence requires eps_coh = 0.

Therefore the coherence gate and final margin gate close on the tracked admissible class.

11. References

  1. H.-W. Leopoldt, Zur Arithmetik in abelschen Zahlkorpern, J. Reine Angew. Math. 209 (1962), 54-71.
  2. L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer, 1997.
  3. K. Iwasawa, Lectures on p-Adic L-Functions, Princeton Univ. Press, 1972.