The δ-calculus and OSMM: a machine-checked foundation for distinction-to-arithmetic, classical-to-δ transport, and an operating system for machine mathematics

This deposit publishes an eleven-paper research stack on the delta-calculus and OSMM (an operating system for machine mathematics). The shared aim is a parameter-free, machine-auditable path from a licensed act of distinction to arithmetic structure, and from that foundation to scale classification, numeric closure, and production verification discipline.

The mathematical core develops the delta-calculus: intuitionistic first-order arithmetic over the licensed signature {0, S, +, .}. The term syntax is the initial algebra of that signature, and an exhaustive primitive audit shows that the object language introduces no set former, membership symbol, type former, universe, comprehension rule, or propositions-as-types constructor. From that initiality the stack constructs the number carriers N_delta, Z_delta, Q_delta with exact free-versus-constructed provenance (N free and initial; Z by group completion; Q by fractions), proves that generation terminates at Q relative to a fixed three-stage grammar, and develops the classical index-period structure theory of the derived monogenic algebra. A companion paper prices the logical cost of passing beyond Q to the continuum: what is refused without additional posits, what decision principles (LPO, WLPO, Markov, and related principles) buy, and which constructive routes remain available.

At scale, a pinned Mathlib corpus of 244,075 propositional theorems is classified fail-closed into delta-native transport versus typed exclusions. Under the successor registry seal, 445 statements transport (against a pre-registered baseline of 151), with zero floor regressions; the remaining mass is accounted for by typed walls, including a certified design boundary for universe polymorphism. A graded StrengthReceipt claim algebra replaces Boolean "proved" with audited strength grades, certificates, registry charters, and pre-registered kill/retain/diagnostic bets. Methods papers isolate two honesty invariants that travel beyond this stack: intrinsic definability of delta faces (classical agreement is a theorem, never a pullback definition) and proof-carrying numeric closure for transport squares (edge-indexed witnesses that close the same-value laundering loophole of a writable Boolean).

The OSMM half of the stack presents verification as an operating system: gates as the only promotion path; controls, decoys, and semantic mutants per gate; append-only signed hash-chained receipts; staged enforcement (ADVISORY -> WARN -> BLOCK) with fail-closed frozen gates; and a pre-registered product transaction as the consumer-facing end. An anti-vacuity battery reports 8 executable gates, 115 semantic mutants, and 0 undetected mutants under a deterministic digest. A matched-compute active-transport experiment is published as an honest negative under its registered criterion. A unification paper shows that OSMM can re-verify the delta strength ledger in pure stdlib Python without invoking the proof assistant, rejecting decoys and mutants against a shared claim language. The series closes with a methodology paper on pre-registered honesty architecture: frozen verdict functions, decoy gating, credit-bearing negatives, and panel review as adversarial pre-registration, evidenced by the stack's own DIAGNOSTIC, retained, and negative terminals.