A variational doublet structure for a marking chain on the path — exact Rayleigh–Ritz spectrum on a family of fronts, a sign rule for the variational modes, strictness of the variational bounds for k ≥ 4, and open questions on the true modes (KAC-T1, KAC-T2, KAC-T3)

Let Q be the generator of the chain on {0,1}^k in which every occupied vertex of the path P_k flips the state of each of its neighbours at rate one; it arose as the frozen-sign cluster dynamics in a series of notes (Phase R, doi:10.5281/zenodo.21979354, Project LOSCHMIDT, doi:10.5281/zenodo.22233373, Project ZERMELO, doi:10.5281/zenodo.22313877, Project STOSSZAHL, doi:10.5281/zenodo.22437728, Project EHRENFEST, doi:10.5281/zenodo.22734680). (i) For indicators of empty regions ("fronts") on an arbitrary graph we give closed formulas for the Gram matrix, the Dirichlet form and the Walsh–Hadamard transform (Lemma KAC-F1). (ii) On the path, the compression of −Q to the span of the left fronts has the exact spectrum E_n(k) = 3/2 − √2 cos(πn/k); the Walsh transform pairs the left and the right front families and splits every level into a variational doublet of ratio exactly (1 − ε_k)/(1 + ε_k), ε_k = 2^(−k/2), with alternating symmetry labels; min–max gives upper bounds for the true rates in each symmetry sector; at k = 2 both bounds, and at k = 3 the two R-odd ones, are equalities (Theorem KAC-T1 and its notes). (iii) The level polynomials and the mode coefficients of the variational modes have closed forms; on (0,1) the coefficient of the effective member of a doublet changes sign exactly at θ* = √2 − 1 for even levels and nowhere for odd levels (Theorem KAC-T2). (iv) For k ≥ 4 the span of the two front families contains no eigenvector of Q outside ker Q; hence all the upper bounds of (ii) are strict (Theorem KAC-T3). Statements (i)–(iii) are at the variational level; (iv) is a negative statement on the true eigen-equation. Whether the true spectrum has doublets with a splitting of order 2^(−k/2), and whether the true modes obey the sign rule, is open; both are stated as questions (Section 4), the first one in the project's open-problem format with falsification conditions (Appendix C). The form of the trial family is known from kinetically constrained spin models (Cancrini, Martinelli, Roberto and Toninelli, doi:10.1007/s00440-007-0072-3; Shapira, arXiv:2005.13327). What we did not see in the texts we compared is the application to this model and the closed formulas; not having seen it there does not mean that it is absent from the literature, and no novelty is claimed (Section 5). The project had 4 independent proof-review rounds, each by a fresh context-free instance of the AI system that also executed the work, with no mandatory fix in any round; the four statements of this note were confirmed in rounds r2–r4 (round r1 confirmed two geometric lemmas that are only mentioned here, Appendix C). No human refereeing or external peer review has taken place (Section 6, Appendix E). The proofs are reproduced in Japanese, byte for byte from the project's normative document (Appendix B); Appendix A translates the four statements. Version note. v0.2.1 (2026-09-21): deposit version of the note. The project had 4 proof-review rounds (6 confirmed claims, of which the 4 statements of this note; no mandatory fix in any proof round, no statement sent back), 1 manuscript review of the note (accepted conditionally, six mandatory fixes in wording) and 1 narrow difference review of the revised note (accepted); every reviewer was a fresh context-free instance of the same AI system. The record holds the note (note6.pdf), the two-page statement (kac-t1-onepage.pdf), a README, and the archive kac-note6-bundle-v0.2.1.tar.gz: a selection of the project repository at the deposit commit — the TeX sources of the note, the Japanese normative documents with the proofs, the claim ledger and the errata, the two open-problem documents, the machine records with their generators, the findings and scripts of all six review rounds, the progress, intervention and error ledgers, the project charter and the constitution of the executing agent — with a MANIFEST.sha256. The proofs are in Japanese (Appendix B of the note is a byte-checked transcription of the confirmed blocks of the normative document). Sixth work of a series after Phase R (10.5281/zenodo.21979354), Project LOSCHMIDT (10.5281/zenodo.22233373), Project ZERMELO (10.5281/zenodo.22313877), Project STOSSZAHL (10.5281/zenodo.22437728), Project EHRENFEST (10.5281/zenodo.22734680).