THE METRIC COST OF SELECTION: Proper Length Invariance, Smooth Horizons, and Projection-Constrained General Relativity (Document 74)

Abstract

Document 74 // Monograph Series on Causal Inertia & Epistemic Governance Author: Marius Egerhei Torjusen (ORCID: 0009-0006-0431-6637) Affiliation: Chief Architect, ReismannPoint Systems // Brahman Ventures (Locus Zero Research Division, Risør, Norway) Series Lineage: Direct operational, mathematical, and physical successor to Document 70 (10.5281/zenodo.22796381), Document 71 (10.5281/zenodo.22801868), Document 72 (10.5281/zenodo.22802491), and Document 73 (10.5281/zenodo.22802706). Status: Formally Prepared Research Monograph // CERN/Zenodo Candidate Executive Summary & Abstract: We formulate the rigorous relativistic, operator-theoretic, and thermodynamic foundation for Projection-Constrained General Relativity (PC-GR), proving that spacetime curvature is the metric response to ontological selection. Resolving the long-standing pathology of infinite boundary stresses arising from mathematically idealized, infinitely thin interfaces ("hard gates", P_K^2 = P_K), we establish that physical realization requires a finite, Lorentz-invariant proper length scale ε > 0 ("smooth gates", P_K^(ε)). By defining the boundary distance d(x, ∂A) covariantly via the proper distance eikonal equation g^μν ∇_μ d ∇_ν d = 1, we prove that the regularizing scale ε is a frame-independent Lorentz scalar, resolving all frame-dependent Lorentz-contraction ambiguities. We derive the irreversible stress-energy tensor Σ^irr_μν directly from the metric variation of a kinetic selection action: S_Π[g, ψ, P_K] = ∫_M [ λ_1/2 ||Δ_irr||^2 + λ_2/2 g^αβ ∇_α P_K^(ε) ∇_β P_K^(ε) + B_A(x) ] √(-g) d^4x where the rejected dynamic residual is Δ_irr = (I - P_K^(ε)) U(t) ψ_0. We prove that the resulting boundary stress-energy tensor is strictly proportional to ε^-2 sech^4(d(x) / 2ε), proving that the sech^4 exponent is the exact mathematical consequence of field gradient squaring. The modified Einstein field equations: G_μν + Λ g_μν = 8π G ( T_μν + Σ^irr_μν ) are proved to possess smooth, regular, C^∞ solutions throughout spacetime, converging to classical General Relativity (G_μν = 8π G T_μν) if and only if all generated dynamics are unconditionally admissible (Σ^irr_μν ≡ 0). We establish a two-level physical demarcation for ε: Level A (phenomenological / de Broglie wavelength ε ~ λ_dB) and Level B (fundamental Planck scale ε ~ ℓ_P = √(ℏG/c^3) enforced by the Generalized Uncertainty Principle). The entire theoretical framework is validated against the deterministic L1 software engine unitary_projection_stress_engine.py under the monorepo governance axiom INSPIRED_BY != IMPLEMENTS != PROVES. Core Architectural Pillars: 1. The Proper Distance Eikonal: Invariant formulation g^μν ∇_μ d ∇_ν d = 1 guarantees that ε is a Lorentz scalar invariant, eliminating observer-dependent frame contraction. 2. Analytical Proof of the sech^4 Profile: Kinetic field gradient squaring in the action principle generates Σ^irr_μν ∝ ε^-2 sech^4(d/2ε), explaining the exact physical profile of boundary tension. 3. Lemma 1 (Zero-Selection Equilibrium): Classical Einstein GR is the exact equilibrium limit where dynamic generation requires zero projection rejection (Σ^irr_μν = 0). 4. Total Noether Conservation: Boundary filtering obeys Δ J^μ_field + Δ J^μ_barrier + Δ J^μ_environment ≡ 0, converting rejected field momentum into horizon heat. 5. GUP and the Planck Bound Poka-Yoke: Compressing a boundary below ℓ_P creates a micro-black hole, shielding the boundary behind an event horizon and proving that hard gates (ε = 0) are physically impossible in our universe. 6. Deterministic L1 Verification: Validated by 7 unit tests in test_unitary_projection_stress_engine.py (100% pass rate). 7. The Monograph Quintych Closure: Synthesizes Documents 70, 71, 72, 73, and 74 into a unified epistemic and physical paradigm. The Final Quintych Canon: "Unitary flow generates candidates. Admissibility projects and filters. Spacetime absorbs the metric shear cost. The immutable ledger witnesses."