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Credential status: E0 finite-dimensional theorem credentialed; E1 Python reconstruction PASS; E2 Haar support; E3 external benchmark MEASURED under frozen protocol E3-NINV-NDIFF-DIGITS-20260811-R1.

Current repository authority: generator source commit 10667959 · controlled-release commit da5501b.

Canonical Drive release proof: Corpus Publication Monograph and Evidence Atlas · E47 Publication Package 20260811.

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Publication links

• Native Google Doc: Credential LIG-E47-PY-CRED-20260808 - The Invariant-Grammar Projection Theorem
• Executable witness: validate_lig_proof.py
• Parent publication program: E47 Publication Program — Reproducibility, External Benchmark, and L_IG

Credential scope

ID: LIG-E47-PY-CRED-20260808

This record credentials the finite-dimensional L_IG invariant-projection constructor and its E47 realization. The previously pending external N_inv versus N_diff benchmark was subsequently frozen and measured on 2026-08-11; the E3 result is attached as a separate empirical layer and does not alter the E0/E1 theorem credential.

Theorem: Parsimonious Completeness of the L_IG Projection Constructor

Let H be finite-dimensional and K a linear constraint. Define Q = K†K, Ψ = ker(K), and P_Ψ as the orthogonal projection onto Ψ.

Λ(K) = limₜ→∞ exp(-tQ) = limₙ→∞ (I - εQ)ⁿ

The constructor returns P_Ψ for 0 < ε < 2/λmax(Q). Conversely, every finite-dimensional orthogonal projection P is represented by the one-constraint choice K = I - P. Therefore the constructor is sound and complete for finite-dimensional orthogonal invariant projections, with one constraint and its forced positive companion as the parsimonious representation.

Proof

1. By the finite-dimensional spectral theorem, Q = U diag(λ₁,…,λd)U† with λᵢ ≥ 0.
2. ⟨x,Qx⟩ = ||Kx||², so ker(Q) = ker(K) = Ψ.
3. Functional calculus gives exp(-tQ) = U diag(exp(-tλᵢ))U†; positive-eigenvalue coordinates vanish and zero-eigenvalue coordinates remain. Thus the limit is P_Ψ.
4. On Ψ, Γε = I - εQ is the identity. On Ψ⊥, its eigenvalues have modulus below one, so Γεⁿ → P_Ψ.
5. For any orthogonal projection P, take K = I - P. Then K†K = I - P and exp(-tK†K) = P + exp(-t)(I-P) → P. Q.E.D.

E47 realization

The carrier is H = V₂ ⊗ V₂ ⊗ V₂, dim(H)=125. The dimensionally correct decomposition is: