TypedClosure: Lorentz Through the Full Regime Stack NQ → Partial-Q → Q → Potential

This paper re-expresses special relativity's Lorentz transformation as a sequence of typed regime transitions within the TypedClosure grammar. No new physics is claimed. The contribution is a clean regime-typed restatement of standard SR kinematics that reveals two things not visible in the untyped formulation: a finite upper bound on Lorentz factor — the Roof — that precedes the c-ceiling, and the identification of the v > c imaginary domain as the exact semantic counterpart of the TypedClosure Potential layer. All mathematical claims operate in PURE mode. The four-regime stack. The Lorentz factor γ threads through all four regimes as a single continuous parameter. In NQ (Regime B, committed record, v < c), all six standard kinematic quantities — time dilation, length contraction, relativistic mass, density, momentum, energy — are re-expressed as typed claim headers with explicit ledger invariants and falsifiers. The Lorentz scalar m₀²c⁴ = E² − p²c² is identified as the noun-resistance ledger: the invariant record of the noun's core identity under all boosts. In Partial-Q (the transformative zone, γ large but finite), committed NQ records come under internal stress: pair creation thresholds, the Schwinger limit, and radiation-reaction back-reaction all mark physically meaningful Partial-Q entry points. In Q (Regime A, verb-dominant propagation), the photon is the canonical pure-Q token — no rest mass, no committed noun identity, propagation at c. In Potential (v > c, imaginary domain), all Lorentz quantities become imaginary; every URM record-formation primitive — mass commitment, length commitment, energy commitment, compaction — is simultaneously inadmissible. The density two-sided squeeze is the central structural result. Relativistic density grows as γ², the product of mass growing as γ (LR-NQ-3) and volume shrinking as 1/γ (LR-NQ-2). This means an accelerating massive noun is being squeezed simultaneously toward two walls: the Roof (Regime C, Planck compaction) from the noun side, and the c-ceiling from the verb side. The Roof fires at a finite Lorentz factor γ_Roof = √(ρ_P/ρ₀) — large but reachable in principle. For a proton, γ_Roof ≈ 3×10³⁹; the LHC achieves γ ≈ 7.5×10³, twelve orders of magnitude below. The c-ceiling requires γ → ∞ and is never reached. The structural consequence is LR-SQUEEZE: for any massive object, the Roof fires before the c-ceiling. The kinematic limit of acceleration is not c but the Roof, and the Roof produces a black hole — the result of the two-sided squeeze completing. The Potential domain identification is the second structural result. At v > c, γ becomes purely imaginary: γ = iγ_T where γ_T = 1/√(β²−1). Every quantity that inherits γ becomes imaginary. Imaginary mass, imaginary energy, imaginary momentum, imaginary length: none can be committed to a real NQ record. The bookkeeping invariant (Axiom 5) requires a real-valued ledger functional; an imaginary energy violates it. LR-POT-1 establishes that the imaginary Lorentz domain is the exact semantic counterpart of the Potential layer in the TypedClosure/URM grammar: a domain where the grammar runs formally but over imaginary numbers — a grammar that runs on no constructible record. The tachyon is not a particle with imaginary mass; it is a Potential-domain token typed as such, and its admissibility conditions are an open obligation (LR-PO-2). Cross-domain parallels to black holes and Navier–Stokes are drawn throughout. The Roof boundary in SR maps to the horizon boundary in black holes (χ_twist → 1) and the BKM saturation boundary in NS (χ_{m,NS} → 0): in all three domains a typed admissibility boundary is approached, a reflecting condition fires, and committed records cannot persist in-place. Time dilation in SR maps to the external write cadence / proper time dilation in the NS self-similar frame and the near-horizon redshift in black holes. The Partial-Q transformative zone maps to State P in NS and to the near-horizon zone in black holes. The three domains are three instantiations of one typed regime stack. Open obligations. LR-PO-1: derive the Roof condition in the proper frame of the accelerating noun rather than the lab frame (the lab-frame density is frame-dependent; the proper-frame compaction argument must be established independently). LR-PO-2: formally retype the tachyon as a Potential-domain token and establish its admissibility conditions, if any, within the grammar. Bundle DOI: 10.5281/zenodo.18777607 Keywords: typed closure, Lorentz transformation, special relativity, regime stack, NQ regime, Partial-Q, Q regime, Potential domain, Lorentz factor, time dilation, length contraction, relativistic mass, relativistic density, Roof condition, Planck compaction, c-ceiling, two-sided squeeze, imaginary domain, tachyon, Potential layer, URM grammar, black hole analogy, Navier–Stokes analogy, LR-SQUEEZE, LR-POT-1, PURE mode, noun-resistance, regime typing, cross-domain, SR kinematics, falsification, proof obligations