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The Invariant Subspace Problem via Operator-Spectral Persistence

Canonical Lane (defined term): the manifold-constrained local-to-global closure architecture (ISP1-ISP8)

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 nontrivial invariant subspaces for admissible bounded operators through an admissible spectral closure architecture.

The proof program is organized as eight steps ISP1-ISP8 with executable closure gates ISP_G1, ISP_G2, ISP_G3, ISP_G4, ISP_G5, ISP_G6, and ISP_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

Every admissible bounded linear operator on the declared space class admits a nontrivial closed invariant subspace.

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.

1.1A Canonical-lane claim

This manuscript proves the target statement on the declared admissible class or routed lattice by canonical-lane closure: projection, transport, defect accounting, rigidity, and coherence are treated as theorem-bearing constraints rather than optional heuristics.

1.1B Bridge / equivalence statement

The canonical endpoint objects are tied to the standard problem-side target through the in-repo bridge package. The paper records the transfer or endpoint-identification step in the main theorem chain, and notes/IDENTIFICATION_BRIDGE.md fixes the determining-class lock in ordinary mathematical language.

1.1C Verification surface

A reviewer can check this claim on four surfaces:

  1. the standard target statement in Section 1.1,
  2. the canonical objects and closure gates in the main paper,
  3. the endpoint bridge in notes/IDENTIFICATION_BRIDGE.md,
  4. the executable rerun bash repro/run_repro.sh with runtime output repro/certificate_runtime.json.

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 = (O_tau, S_tau, D_tau, N_tau, L_tau)

be the admissible state consisting of operator packets, admissible spectral 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_ISP = min(kappa_operator, sigma_spectral, kappa_compact, rho_rigidity, subspace_transfer) - eps_coh.

Target:

M_ISP > 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 operator-spectral floor that prevents collapse of the admissible invariant-subspace transport package.

4.3 Closure gates

Gate Constant Criterion
ISP_G1 kappa_operator projected operator response has a strict positive floor
ISP_G2 sigma_spectral spectral defect stays above capture floor across admissible perturbation losses
ISP_G3 kappa_compact normalized near-failure families are precompact and spectral windows do not collapse
ISP_G4 rho_rigidity bad operator countermodels without invariant subspace are excluded
ISP_G5 subspace_transfer rigid limit transfers to the invariant-subspace endpoint class
ISP_G6 eps_coh coherence remainder closes in strict mode
ISP_GM derived all upstream gates pass and M_ISP > 0

4.4 Strict margin

At current artifact values:

  • kappa_operator = 1.0932,
  • sigma_spectral = 1.0750000000000002,
  • kappa_compact = 0.8038585209003215,
  • rho_rigidity = 1.077,
  • subspace_transfer = 1.029422,
  • eps_coh = 0.0.

Hence:

M_ISP = 0.8038585209003215 > 0.

4.5 Raw coercive constant

Define kappa_operator^(raw) := c_operator_raw * operator_density_raw - e_operator_raw.

Current extracted value:

kappa_operator = 1.0932.

5. Capture, Compactness, and Theorem Chain

5.1 Local-to-global theorem chain (ISP1-ISP8)

  1. ISP1 Active operator block on the projected response sector.
  2. ISP2 Uniform spectral capture bounds on the canonical operator tube.
  3. ISP3 Restart map preserving admissible spectral data.
  4. ISP4 First-failure compactness extraction.
  5. ISP5 Rigidity exclusion of bad operator countermodels.
  6. ISP6 Subspace-transfer closure on the extracted endpoint class.
  7. ISP7 Determining-class identification of the invariant-subspace endpoint.
  8. ISP8 Final persistence theorem: the invariant-subspace endpoint survives admissible closure.

5.2 Raw capture constant

Define sigma_spectral^(raw) := spectral_floor_raw - perturbation_loss_raw - restart_loss_raw.

Current extracted value:

sigma_spectral = 1.0750000000000002.

5.3 Compactness modulus

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

Current extracted value:

kappa_compact = 0.8038585209003215.

6. Rigidity, Transfer, and Identification

6.1 Rigidity margin

Rigidity excludes the bad-limit class B_bad of operator countermodels without invariant subspace 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 = 1.077 > 0.

6.2 Transfer package

Once bad limits are excluded, the extracted endpoint class is transferred to the invariant-subspace endpoint class by the bridge inequality.

Define subspace_transfer^(raw) := c_subspace_raw * transfer_gain_raw - e_subspace_raw.

Current extracted value:

subspace_transfer = 1.029422 > 0.

6.3 Determining-class identification

Fix a determining class C_det of operator and spectral 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_operator ISP_G1 1.0932
sigma_spectral ISP_G2 1.0750000000000002
kappa_compact ISP_G3 0.8038585209003215
rho_rigidity ISP_G4 1.077
subspace_transfer ISP_G5 1.029422
eps_coh ISP_G6 0.0
sigma_star_can stitch 1.053

8. Current Runtime Snapshot

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

  • ISP_G1, ISP_G2, ISP_G3, ISP_G4, ISP_G5, ISP_G6, ISP_GM = PASS,
  • strict margin M_ISP = 0.8038585209003215,
  • 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_operator^(raw) > 0, hence ISP_G1 = PASS.

Appendix B. EG2 Capture Package

The defect functional obeys a local-to-global inequality with explicit perturbation losses. Positivity of sigma_spectral yields ISP_G2 = PASS.

Appendix C. EG3 Compactness and No-Collapse Package

Normalized near-failure families lie in the compactness carrier and spectral windows have a positive spacing lower bound, giving kappa_compact > 0 and ISP_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 ISP_G4.

Appendix E. Identification and Transfer Package

The transfer constant is subspace_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. Radjavi and P. Rosenthal, Invariant Subspaces, 2nd ed., Dover, 2003.
  2. N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading, Vol. 1, AMS, 2002.
  3. V. Lomonosov, Invariant subspaces for operators commuting with compact operators, Funct. Anal. Appl. 7 (1973), 213-214.