Epistemic Justification Beyond Cognitive Surveyability: A Formal and Historical Account of Deductive Warrant under Trans-Cognitive Verification

When formal proofs or deductive verifications exceed human cognitive surveyability, does epistemic justification require an agent-accessible chain of reasoning, or must epistemology shift from internalist warrant to structural-systemic reliability? Since the seventeenth-century rationalist and empiricist programs of Descartes and Locke, Western epistemology has rested upon the axiom of epistemic transparency: for an epistemic agent S to possess genuine a priori demonstrative knowledge of a theorem phi, S must be capable of cognitively surveying the inferential chain connecting primitive axioms Gamma to phi. Even when Bolzano, Frege, and Russell purged psychologism and spatio-temporal intuition from deductive logic, they treated formalization as a vehicle for rendering every inferential transition perspicuous to rational inspection. However, automated theorem provers, Boolean satisfiability (SAT) solvers, and interactive proof assistants routinely generate formal derivations that are deductively sound yet impossible for any finite biological mind to survey in full—violating the Wittgensteinian criterion of cognitive surveyability (Übersichtlichkeit). This paper provides a systematic formal and historical analysis of this problem across eight sections: (1) Section 1 reconstructs the classical transparency paradigm across Descartes, Leibniz, Locke, and Kant, isolating the working-memory and temporal synthesis bounds built into classical deductio; (2) Section 2 demonstrates how the Bolzanian-Fregean-Hilbertian quest for local step-wise gaplessness (Lückenlosigkeit) triggered an inverse scaling law that dialectically destroyed global proof surveyability; (3) Section 3 analyzes the Wittgensteinian critique of un-surveyable derivations, formalizing the distinction between a normative grammatical criterion and an empirical physical experiment; (4) Section 4 proves via Gödelian speed-up, Gentzen cut-elimination, and propositional proof complexity lower bounds that trans-surveyable derivations (||D|| >> kappa(S)) are an inescapable metamathematical feature of formal systems rather than an artifact of unoptimized proof search; (5) Section 5 introduces Bounded Justification Logic (BJL_kappa), formally deriving the inconsistency of the Classical Surveyability Trilemma; (6) Section 6 evaluates the Tymoczko–Burge–Kripke debate on computer-assisted proofs, refuting both quasi-empiricist hardware inductivism and ungrounded testimonial transmission; (7) Section 7 constructs a positive theory of Structural-Systemic Warrant (SSW), demonstrating how de Bruijn meta-kernel soundness compression and substrate-invariant equivalence preserve derivative a priori warrant while bifurcating demonstrative warrant (Dass-Wissen) from explanatory illumination (Warum-Wissen); (8) Section 8 addresses foundational objections from rule-following skepticism, Gödelian second-incompleteness limits, and zero-knowledge probabilistic verification (ZKPs and PCPs), synthesizing the general epistemological conclusions.