PTO Window Calculus I: Stage--B--H- A Recorded--Ledger Window Calculus for Knot and Link Diagrams with Coherence Certificates and Evidence-on-Loops (FAO/10TWF)}

This Zenodo upload (embargoed) contains PTO Window Calculus I: Stage–B–H, a fixed-container recorded-ledger framework for knot and link diagrams on a fixed projection diagram (PD) with a fixed PD/lex recording basis. The core mathematics is single-state, disk-local, and finitely auditable: per-disk observables are recorded in a fixed bin order, stabilized by three commuting closures (with a harmony join), and compared only after recording inside the fixed container. The manuscript is explicitly stage-typed. Part I develops the Stage-B calculus (deterministic gauge, local audits on disks with k\le 3, typed value dualities vs. index reparametrizations, defects as checkable certificates, and selected diagnostics including alternating/adequacy/flype-typed interfaces under stated hypotheses). It also includes explicit Stage–C / proto–D / stage-D/proto–H / Stage–H gates as certificate/evidence clauses (never as hidden assumptions): cover-coherence is expressed via a witnessable obstruction equation on an overlap graph, and evidence-on-loops is expressed via holonomy checks on declared protocol cycles. Part II is a packaging/application layer for humane society and education: it applies the same typed discipline to FAO / 10-Tier World Family (10TWF)—i.e., the family-to-world-family social system—by modeling education, verification, and transmissible evidence as protocol graphs (relationships + episodes) with certificate gates and evidence-on-loops (holonomy) closure. This layer treats “constitution / undividedness / mutual satisfaction” as typed, checkable coherence conditions on declared social/evidence networks, without changing any Stage-B knot-theoretic proofs in Part I. Madhyasth Darshan (A. Nagraj, 1920–2016) is used only as an interpretive typing lens (notation/organization and stage semantics), not as a premise for the mathematical arguments.