An order-field invariant that separates a pair the mod-p reduction cannot

An order-field invariant that separates a pair the mod-p reduction cannot. This deposit reports a result about an order-field invariant, F_ord, computedfor triangle-group families. F_ord distinguishes a pair of families that themod-p conjugacy type cannot separate across 726 primes, and the published tierhierarchy is shown to be a quotient of F_ord on the rows where F_ord isdefined -- strictly finer, with the converse failing at five fields. The deposit contains no new mathematics. Every claim in the paper traces to anappend-only ledger entry from one of two earlier runs, by entry id and by thesha256 of the ledger line carrying it. The machine-checkable claim inventory,the ledgers, the pre-registrations, the run reports and the code all ship asfiles, and a verification script re-checks every trace against the shippedledgers offline. Stated before the result rather than after it, what is NOT claimed: - not that F_ord is a complete invariant. Nothing is claimed on the rows where it abstains, and the abstention is forced by the object rather than chosen by the authors; - not that F_ord, F_tr and F_tri are the same quantity or may be substituted for one another. They are three distinct quantities with three distinct defining expressions and are never merged; - not that the tier oracle's tiers are the same predicate as the paper table's O0-O3. The two were measured against each other; agreement is evidence about the pair, not a definition; - not that the log-k=4 tier is resolved. It is reported open across three disagreeing sources, adjacent to the claim it bears on rather than in a footnote; - not that F_ord extends to the 15 families newly located here. It was deliberately not computed for them; that needs its own hold-out design. Reach is stated rather than implied: 30 of 37 known families are located, andthe seven that are not are named individually with the resolver that failed. Epistemic classes are carried, not flattened. A finite computation over themembers in hand is VERIFIED; only a closed argument is PROVEN; a readingdepending on a normalization this work chose is STRUCTURAL. Each class wasread back from the ledger entry rather than asserted in the prose. One external reference is named but not shipped: the PSL2Z-ARCH run, whoseorthogonality result is the negative complement to this one. It is declared asnot-included, with its locator and the reason, and is intended for its ownrecord cross-referenced from a future version. This disclosure is mechanicalrather than editorial: a check in the verification script requires everyexternally named run to resolve to a DOI, a shipped file, or an explicitnot-included declaration. Related deposits, cross-referenced in CITATION.cff: 10.5281/zenodo.22559256 10.5281/zenodo.22568604 10.5281/zenodo.22634591