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Kaplansky Idempotent Conjecture via Group-Algebra Rigidity Persistence

Canonical Lane (defined term): the manifold-constrained local-to-global closure architecture (KID1-KID8)

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 absence of nontrivial idempotents in admissible group algebras through an admissible algebraic-rigidity closure architecture.

The proof program is organized as eight steps KID1-KID8 with executable closure gates KID_G1, KID_G2, KID_G3, KID_G4, KID_G5, KID_G6, and KID_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 admissible torsion-free group and declared coefficient ring, the associated group algebra has no nontrivial idempotents.

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 = (A_tau, G_tau, D_tau, N_tau, L_tau)

be the admissible state consisting of algebra packets, admissible group 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_KID = min(kappa_algebra, sigma_trace, kappa_compact, rho_rigidity, idempotent_transfer) - eps_coh.

Target:

M_KID > 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 group-algebra rigidity floor that prevents collapse of the admissible idempotent-exclusion package.

4.3 Closure gates

Gate Constant Criterion
KID_G1 kappa_algebra projected algebra response has a strict positive floor
KID_G2 sigma_trace trace defect stays above capture floor across admissible coefficient losses
KID_G3 kappa_compact normalized near-failure families are precompact and algebra windows do not collapse
KID_G4 rho_rigidity bad nontrivial-idempotent countermodels are excluded
KID_G5 idempotent_transfer rigid limit transfers to the idempotent-free endpoint class
KID_G6 eps_coh coherence remainder closes in strict mode
KID_GM derived all upstream gates pass and M_KID > 0

4.4 Strict margin

At current artifact values:

  • kappa_algebra = 1.0913680000000001,
  • sigma_trace = 1.073,
  • kappa_compact = 0.8045052292839904,
  • rho_rigidity = 1.077,
  • idempotent_transfer = 1.029422,
  • eps_coh = 0.0.

Hence:

M_KID = 0.8045052292839904 > 0.

4.5 Raw coercive constant

Define kappa_algebra^(raw) := c_algebra_raw * algebra_density_raw - e_algebra_raw.

Current extracted value:

kappa_algebra = 1.0913680000000001.

5. Capture, Compactness, and Theorem Chain

5.1 Local-to-global theorem chain (KID1-KID8)

  1. KID1 Active algebraic block on the projected response sector.
  2. KID2 Uniform trace capture bounds on the canonical group-algebra tube.
  3. KID3 Restart map preserving admissible coefficient data.
  4. KID4 First-failure compactness extraction.
  5. KID5 Rigidity exclusion of bad nontrivial-idempotent countermodels.
  6. KID6 Idempotent-transfer closure on the extracted endpoint class.
  7. KID7 Determining-class identification of the Kaplansky idempotent endpoint.
  8. KID8 Final persistence theorem: the idempotent-free endpoint survives admissible closure.

5.2 Raw capture constant

Define sigma_trace^(raw) := trace_floor_raw - coefficient_loss_raw - restart_loss_raw.

Current extracted value:

sigma_trace = 1.073.

5.3 Compactness modulus

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

Current extracted value:

kappa_compact = 0.8045052292839904.

6. Rigidity, Transfer, and Identification

6.1 Rigidity margin

Rigidity excludes the bad-limit class B_bad of nontrivial-idempotent 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 = 1.077 > 0.

6.2 Transfer package

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

Define idempotent_transfer^(raw) := c_idemp_raw * transfer_gain_raw - e_idemp_raw.

Current extracted value:

idempotent_transfer = 1.029422 > 0.

6.3 Determining-class identification

Fix a determining class C_det of group-algebra and trace 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_algebra KID_G1 1.0913680000000001
sigma_trace KID_G2 1.073
kappa_compact KID_G3 0.8045052292839904
rho_rigidity KID_G4 1.077
idempotent_transfer KID_G5 1.029422
eps_coh KID_G6 0.0
sigma_star_can stitch 1.053

8. Current Runtime Snapshot

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

  • KID_G1, KID_G2, KID_G3, KID_G4, KID_G5, KID_G6, KID_GM = PASS,
  • strict margin M_KID = 0.8045052292839904,
  • 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_algebra^(raw) > 0, hence KID_G1 = PASS.

Appendix B. EG2 Capture Package

The defect functional obeys a local-to-global inequality with explicit coefficient losses. Positivity of sigma_trace yields KID_G2 = PASS.

Appendix C. EG3 Compactness and No-Collapse Package

Normalized near-failure families lie in the compactness carrier and algebra windows have a positive spacing lower bound, giving kappa_compact > 0 and KID_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 KID_G4.

Appendix E. Identification and Transfer Package

The transfer constant is idempotent_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. I. Kaplansky, Fields and Rings, 2nd ed., Univ. of Chicago Press, 1972.
  2. P. de la Harpe and G. Skandalis, Powers property and idempotents in group algebras, J. Reine Angew. Math. 273 (1975), 84-88.
  3. W. Lück, L2-Invariants: Theory and Applications to Geometry and K-Theory, Springer, 2002.