Statistical Structure of the Historical Orderings of the I Ching Hexagrams: Pair Rule, Family Gradient, and the Limits of Demonstrability

Description

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Version 3 (2026-08-03). The first version in which the text of the paper changes. Two sentences are corrected, one printed approximation is replaced by its exact value, and page pointers are added for the two orderings the paper reconstructs. No result, figure, table or conclusion is affected.

Corrections to the text. The description of the block-permuting null said that pairs are permuted within and across the sixteen blocks. The code permutes the blocks among themselves and a pair never leaves its block; under the literal reading of the old sentence, the theorem printed two paragraphs below was false. Separately, the family of six nulls was described as one of increasing structure, which describes a chain: the six are ordered by containment, and that order is partial, with two of the five consecutive steps incomparable. Finally, the Bonferroni product 0.038 × 9 was printed as approximately 0.35; it is exactly 0.342, and printing the exact value turns a tolerance in the verification suite into an equality. The appendix now begins on its own page, so its table no longer prints among the references, and the paper's main tables and figure are now pointed to from the prose.

What this version adds. Page pointers for both reconstructed orderings, and the statement that each was checked position by position against its printed source. Mawangdui against Shaughnessy (1996): the two trigram orders on p. 17, the sequence on pp. 28-29. Jing Fang against Nielsen (2003): p. 3, Table 2, entry BA GONG GUA. Each matched in all 64 positions, with no discrepancy. Those sources are printed, third-party and independent of this work; the verification suite of the replication package is not, and until this version it had never compared either ordering against any source.

Replication package. The suite grows from 202 checks to 270. Two figures printed in the paper had no assertion behind them and now have one, so the abstract's closing sentence, that every numerical claim in this paper is asserted by an automated test suite, is true as written for the first time, without a word of it changing. The suite also gains a correction to one check that declared a value the paper does not print, a check that every published count matches what the script actually runs, a check that the distributed PDF prints the text of the version it ships with, a check that no table prints on a page of the reference list, and a check that every bibliography entry is named in the body and every citation has an entry. ERRATA.md is new and carries the full log: twenty-two items, twelve findings and ten examined and found sound, each with its evidence, its category and where it was corrected. V3-PLAN.md records the sequence by which this version was built.

Provenance. The two corrected sentences were found in the course of the work reported in Uninformative rungs: an order-theoretic stopping criterion for nested reference sets (10.5281/zenodo.21750029), which this version now cites. Neither correction depends on that work being correct: each is established from this package alone.

Identifiers. The archive sentence in this version prints two. The concept DOI 10.5281/zenodo.21609653 always resolves to the latest version; 10.5281/zenodo.21776041 is the DOI of this version, and it is the one against which the figures of this version were checked.