PAL - Primitive Axiom Layers

Abstract

Primitive Axiom Layers (PAL) is a mechanical framework for keeping reasoning accountable wherever a distinction is supplied, interpreted, carried, compared, coupled, checked, or provisionally closed. It records what was admitted, which witness licensed it, what trace survives, what authority remains unavailable, and which residuals or reopening conditions must be preserved. Its purpose is not to eliminate uncertainty, but to prevent missing evidence, unresolved possibility, or borrowed authority from disappearing inside a finished-looking result. PAL’s primitive floor keeps Ω as a metalinguistic name for unresolved possibility; no PAL constructor consumes, represents, or exhausts it. A0 is a supplied cut rather than a derivation from Ω. Subsequent layers earn orientation, recoverable trace, relation, transport, feedback, organization, access, response, comparison, coupling, cadence, slack, and scoped closure. Closure remains local and reopenable. Earlier records are retained rather than rewritten, and no proof, successful test, repeated wording, or later outcome can retroactively manufacture a missing witness or enlarge an earlier authority ceiling. The framework is organized across five coordinated surfaces. The Mechanical Structural Spine controls PAL’s identity, order, invariants, and canonical cards. The Mathematical Realization Atlas supplies removable mathematical models and countermodels without allowing mathematics to redefine PAL. The Obligation and Decision Ledger preserves decisions, residuals, open burdens, and reopening addresses. The Conformance Tests provide bounded evidence about declared artifacts and implementations. The Compatibility Note governs migration from earlier PAL releases without silently upgrading their claims or authority. Each surface is authoritative only within its stated role. PAL v2.2 makes the framework’s account and receipt mechanics explicit. It distinguishes five account operations—ROOT, APPEND, CHILD, VERSION, and SUCCESSOR—and clarifies that an AccountHeader is metadata pointing to an opening A0 CutReceipt, not the cut or its witness. It introduces immutable, versioned ReceiptHeaders; separates Cut, Transition, Occurrence, Closure, Test, Transport, Residual, and LegacyReceiptRef payloads; and keeps TestReceipt evidence distinct from the later ClosureReceipt that assigns an A15 claim status. It also adds typed admission obligations, complete claim-status envelopes, stable residual lineage, and conservative migration rules for records created before these schemas existed. The v2.2 Mathematical Atlas adds bounded realizations concerning ordinary 1 and model-level underdetermination, complete generator-run-space coverage, and the difference between finite injectivity and a uniform completion guarantee. The accompanying tests add adversarial checks for account topology, semantic identity, legacy migration, residual lineage, generator coverage, transport, and completion claims. Version 2.2 preserves PAL’s existing primitive floor, A0–A15 spine, four-status closure grammar, and frozen O01–O51 obligation history. First-occurrence/source debt remains open, multi-parent native account lineage remains unresolved, and no fifth status or constructor from Ω is introduced. PAL is not presented as a complete ontology, physical or consciousness theory, ethical authority, or universal proof engine. It is a structured way to retain the distinctions, traces, limits, and open handles that responsible reasoning would otherwise be tempted to erase.