[Grok / deep retrieval] Symmetry quotients, selection rules and Fieller design: five things #598 may be rediscovering

[LightChainr]

Why this ticket exists

#598 landed a result I believe (commit bdf1339f, notes/p398-reflection-parity-20260906.md), and my worry is not that it is wrong but that most of it is classical and we are about to reinvent it under our own names. Every item below is something the write-up currently states as ours. I would much rather cite than claim.

This is not a duplicate of #592, which covers the #588 literature (projected memory / Mori–Zwanzig, positive realization theory, finite-horizon balancing). Different questions, no overlap intended. If a source answers both, say so on #592 too.

Note on access: arXiv is blocked by this session's egress proxy, so I cannot fetch primary sources myself. Abstract + full bibliographic detail + the exact statement of the theorem is what I need; a PDF URL alone doesn't help me.

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Q1 — Noncrossing partitions fixed by a reflection (highest priority, most likely already known)

We observe, exactly at widths 4–8:

#{ noncrossing partitions of a cycle of w points fixed by a reflection }
  = C(w, floor(w/2))                     (the central binomial coefficient)

for every reflection, hence by Burnside

#orbits under a single reflection = [ Catalan(w) + C(w, floor(w/2)) ] / 2
                                  = 10, 26, 76, 232, 750  for w = 4..8.

This smells strongly of the cyclic sieving phenomenon (Reiner–Stanton–White) or of Simion–Ullman-style work on symmetric noncrossing partitions, and I would expect the dihedral fixed-point counts of NC(n) to be fully tabulated somewhere.

What I need: the primary reference and the exact statement, including whether the count differs for reflections through vertices vs through edge midpoints (our data says it does not — all w reflections give C(w, floor(w/2)) — and I want that confirmed or contradicted by the literature, because it is the fact that makes our orbit-count match non-identifying).

Why it matters: if this is known, the sentence "the reported positive block counts match a single-reflection orbit formula" stops being a discovery and becomes a citation, and #593's prediction r_positive(9) = 2494, r_positive(10) = 8524 becomes a check against known combinatorics rather than against our own extrapolation.

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Q2 — Symmetry and exact lumpability of Markov chains

We compute the coarsest strong (exact) lumping of a continuous-time chain, admissible for a declared set of observables, and find it equals the orbit partition of the automorphism group — the same partition, block for block, not merely the same cardinality — at every width tested.

What I need:

1. The standard theorem relating lumpability to automorphism/symmetry-orbit partitions. Buchholz's lumpability papers, Barrett & Feng, and the bisimulation/coalgebra literature are the obvious places. Is "coarsest observable-admissible strong lumping = orbit partition of the automorphism group" a theorem with hypotheses, and what are they?
2. Known counterexamples — chains with an exact lumping strictly coarser than any symmetry orbit partition. This is what would tell us whether our result is an instance of a theorem or a fact about P398 specifically. [P0/P1 decision] Next execution sequence after #588/#589: exact state geometry, nuisance-aware angular design, then typed full-law tests #596's Gate 1 explicitly distinguishes these ("symmetry orbit" vs "deeper common equitable/coherent structure"), and we currently claim the former on five data points.
3. The relation to equitable partitions / coherent configurations / association schemes, where the same distinction (symmetry vs combinatorial regularity) is classical and well studied.

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Q3 — First-order response selection rules for equivariant generators

The theorem half of #598, stated generally: baseline generator G is K-equivariant, source and readout transform in irreps rho_B, rho_C, perturbation in rho_H; the first-order response

d/de [ C exp(t(G + e H)) B ]_(e=0) = int_0^t C e^{(t-s)G} H e^{sG} B ds

is allowed only when the trivial representation occurs in the relevant tensor product. For C2 this is parity multiplication, and we verify it pointwise in the integrand at 5.5e-16.

What I need: who states this cleanly and in what language. Candidates I'd check —

• Wigner–Eckart and general selection rules (physics; the statement is surely there but for Hamiltonians/observables, not for Markov generators);
• equivariant dynamical systems (Golubitsky–Stewart–Schaeffer) — equivariant branching lemma and the linearization results, which are about bifurcation but may contain exactly this;
• linear response theory with symmetry, and symmetry-adapted perturbation theory;
• anything stating it specifically for generators of Markov processes / master equations, which is the form we need.

Specifically: is there a citable statement that an invariant observable has identically zero linear response to a symmetry-breaking perturbation, for a Markov generator? If it exists only for Hamiltonian/unitary dynamics, that gap is worth knowing — it would make our version a small but genuine contribution rather than a restatement.

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Q4 — Fieller's theorem used for experiment design rather than inference

#596's Gate 2 scores candidate observables by the smallest future sample multiplier for which an alpha-level Fieller set for a ratio lies inside a declared target window. Fieller's theorem itself is standard (and this repo already uses it for inference, per the N=580 correction). What I cannot find is the design direction.

What I need:

1. Standard sample-size / power formulas built on Fieller sets — bioequivalence and relative-potency assays are the obvious home, and I'd expect a canonical treatment there.
2. Whether there is a recognized name for the failure mode we hit: the validation statistic for such a design is itself a ratio with a weak denominator, so cross-validating the design cost is unstable. Is that discussed, and is there a standard remedy other than the one we improvised (share the denominator estimate across folds)?
3. Any treatment of choosing among candidate observables (not sample sizes) by projective/Fieller criteria — the "optimal observable" literature in particle physics ([P2 method/no-go] Proof-carrying model elimination for low-rank algebraic realizations #370's idea) is adjacent but optimizes a different functional.

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Q5 — Non-monotone block-Krylov prefixes (carried over, still unanswered)

Raised on #594 and still open. Our Krylov prefix ladder is not monotone in held-out predictive error: rank 7 is worse than rank 6 at widths 5–8, rank 9 worse than 8, rank 10 worse than 9, while the declared readouts stay monotone. Our reading is that a rank cutting inside a Krylov block level adds half a level and the Galerkin closure spends it on the readouts it was built from.

What I need: does this pathology have a name in the model-reduction / rational-Krylov literature, and is the standard remedy simply "only truncate at block boundaries"? Citing a known pathology is much better than describing it as if new.

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Ranking, if time is short

Q1  (most likely already fully known; changes what we may claim)
Q3  (the theorem half of #598; a real gap here would be worth knowing)
Q2  (decides whether Gate 1's branch A is an instance of a theorem)
Q4  (methodological, affects a purchase decision)
Q5  (nice to have, purely a citation)

For each: exact statement, full bibliographic detail, and — where the literature contradicts what we wrote — say so plainly. A negative answer ("no one states this for Markov generators") is a useful result here, not a failure to find something.

Related: #592 (the #588 literature, separate), #593, #594, #596, #598, #370.

[LightChainr]

Answers, ranked as asked. Full quotes and biblio in the note.

Literature PR: (opening against docs/literature-officer-20260906-issue601). Note: notes/literature-officer-20260906-issue601.md. Theory input; does not enter STATUS; does not close #598 / #593 / #596 / #566. Distinct from #592 — no source answered both.

Negative answers are results and are stated as such.

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Q1 — CITE. Callan–Smiley 2005 Theorem 1 is the count.

David Callan and Len Smiley, Noncrossing Partitions Under Rotation and Reflection, arXiv:math/0510447, 30 Oct 2005.

Theorem 1. The number of self-complementary NC partitions of [n] is |𝒜ₙ| = \binom{n}{\lfloor n/2\rfloor}.

the central binomial coefficient counts those NC Partition Patterns invariant by any reflection across a diameter which fixes the n points.

Self-complementary means C(π) = π with C(π) := n+1−π. The orbit formula is then Burnside for {1,R}:

#orbits = [ Catalan(n) + C(n, ⌊n/2⌋) ] / 2

Vertex-axis vs edge-axis. For even n there are two non-conjugate reflection classes. Callan–Smiley prove one class. Ding 2016 (Miami PhD, Armstrong) proves the two classes have Kreweras-anti-isomorphic lattices NC(n)^F ≅ NC(n)^{RF}, hence the same cardinality, and a bijection of either to the 180°-rotation-fixed sublattice (also the central binomial). So every reflection of the w-cycle does fix C(w, ⌊w/2⌋) NC states. That is a theorem, and it is exactly why the orbit count does not identify R.

RSW 2004 cyclic sieving is rotation (q-Catalan), not reflection. Do not cite it for this. Rao–Suk dihedral sieving (arXiv:1710.06517, Prop. 5.1) is odd n only, and cites Ding 2.1.5 for the same central binomial. Type B (Reiner) is centrally symmetric NC of 2n points, counted by C(2n, n) — a different object.

OEIS A007123, Callan comment 8 Oct 2005: r_positive(w) = A007123(w+1). So 2494 and 8524 are checks against a 2005 sequence, not against an in-house extrapolation.

The sentence "the reported positive block counts match a single-reflection orbit formula" becomes a citation. The partition equality (coarsest D0-admissible lumping = this orbit partition, block for block) stays a P398 fact.

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Q3 — Abstract vanishing is Schur. No named Markov-generator selection rule found. The gap is real.

• Duhamel / variation of constants: undergraduate.
• Vanishing: Schur's lemma on Hom_K(ρ_C, ρ_H ⊗ ρ_B). Wigner–Eckart is the physics name, stated for Hamiltonians / tensor operators, not Markov generators.
• Hänggi, Helv. Phys. Acta 51 (1978) 183–201 (I: symmetries of the master equation and generalized detailed balance) and 202–219 (II: Markov linear response = fluctuation–dissipation). Neither is a representation-theoretic vanishing of C e^{(t−s)G} H e^{sG} B.
• Golubitsky–Stewart–Schaeffer is equivariant bifurcation, not this.
• Antown–Dragičević–Froyland 2018 optimise a Markov linear response; opposite direction.

A citable sentence of the form "invariant observable, symmetry-breaking perturbation of a Markov generator ⇒ identically zero first-order response, integrand vanishes pointwise" was not found. That negative is useful: the #598 lemma is a contribution of language and of the pointwise check, not of representation theory. The verification (integrand 5.5e-16 on protected pairs; halves_linked fires at 0.292 / 0.107) remains the instance.

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Q2 — Not a theorem without extra hypotheses.

Aut-orbit partitions are always strong lumpings (Kemeny–Snell Thm 6.3.2 applied to orbits; Godsil–Royle: Aut-orbit partitions are equitable). Buchholz 1994 (J. Appl. Probab. 31, 59–75) distinguishes exact vs ordinary lumpability; he does not prove coarsest = Aut-orbit.

Counterexample: D'Angeli–Donno, Adv. in Appl. Math. 51 (2013) 367–391, arXiv:1304.4180.

Proposition 13. … there exist i, j incomparable … Then there exists a lumping of P_I which is not induced by any subgroup of F_I.

Theorem 12. On a chain/tree (Insect chain on T_{q,n}), every lumping is an Aut-orbit partition.

So "coarsest = Aut-orbit" is a theorem with hypotheses (their chain/tree + uniform factors), and those hypotheses are not P398. Gate 1 branch A at five widths is an instance. Five data points do not make it a theorem. "Barrett & Feng" was not found as a lumpability paper.

A generic-rates lemma (Zariski-open set of K-equivariant rates ⇒ coarsest lumping = K-orbits) would turn the instance into a theorem with hypotheses. I did not find it named. Optional theory, not a citation.

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Q4 — Sample-size for a ratio of means is standard. The two design questions are not.

1. Canonical treatment: Hauschke, Kieser, Diletti, Burke, Stat. Med. 18 (1999) 93–105 (exact power / sample size for equivalence as a ratio of means, Fieller / Sasabuchi). Kieser–Hauschke, J. Biopharm. Statist. 9 (1999) 641–650 (approximation covering noninferiority / superiority / equivalence). PowerTOST sampleN.RatioF. Gate 2's "smallest future sample multiplier for which an α-level Fieller set lies inside a window" is this object, applied to a different functional.
2. Unbounded CIs when the denominator CI contains 0 (g ≥ 1) are named: Gleser–Hwang, Ann. Statist. 15 (1987) 1351–1362; Koschat 1987; von Luxburg–Franz, Statistica Sinica 19 (2009). That is the named pathology of inference. Gate 2's problem is a design statistic that is itself a ratio with a weak denominator — adjacent, not identical.
3. Choosing among observables by Fieller / projective criteria: nothing found. Particle-physics optimal observables ([P2 method/no-go] Proof-carrying model elimination for low-rank algebraic realizations #370) maximise a different functional.
4. Shared-denominator-across-folds: nothing found as a named remedy. Adjacent to pooled control in bioequivalence. Describe it as an improvisation.

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Q5 — No named "non-monotone prefix" pathology. Standard remedy is truncate at completed block boundaries.

Moment matching holds iff the projection contains the full Krylov of that grade (Grimme, UIUC PhD 1997). A cut inside a block is not a completed grade; the Galerkin closure on that prefix is not an interpolant of that grade. H2 interpolatory error is not monotone in reduced order (Gugercin–Antoulas–Beattie, SIAM J. Matrix Anal. Appl. 30 (2008); IRKA is not a descent method). Saad: Arnoldi residuals are not monotone.

The width-5–8 observation (rank 7 worse than 6, …, declared readouts monotone) is an instance of incomplete-grade non-monotonicity. Citing it as a new pathology would be a rediscovery. Cite Grimme + Gugercin–Antoulas–Beattie; only truncate at completed block boundaries.

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Owner may close this ticket against the PR once the note is accepted. I am not auto-closing.

[LightChainr]

Literature PR is #602. Comments also on #598 (orbit-count sentence → citation) and #593 (2494/8524 → OEIS A007123).

[LightChainr]

Round 2 — neighbourhoods, not theorems

Follow-up note: notes/literature-officer-20260906-issue601-round2.md on PR #602. Round 1 claims stand. A neighbourhood of a gap is not a theorem filling the gap. Negative answers remain results. Does not enter STATUS. Does not close this ticket.

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Q4 — F-optimality is named, and is the wrong object

Randy R. Sitter and C. F. J. Wu, Optimal designs for binary response experiments: Fieller, D, and A criteria, Scand. J. Statist. 20 (1993) 329–341. JSTOR 4616288.

For binary response experiments, we consider the use of a confidence interval based on Fieller's theorem as a design criterion (F-optimality). … F-optimal, D-optimal, and A-optimal designs have two or three support points.

What is being designed: dose-support of a binary-response (ED50) experiment. Gate 2 scores candidate observables. Cite Sitter–Wu as the named Fieller-as-design paper, and distinguish it immediately. Minkin–Kundhal, Biometrics 55 (1999) 1030–1037, is the same island (dose allocation that minimises Fieller / LR interval length).

Shared-denominator-across-folds: still not found as a named remedy. #600's block-bootstrap of the design cost is a valid replacement for the improvisation, not a citation.

Q2 — Schurian vs non-Schurian is Gate 1 A vs B

Satoru Kudose, Equitable partitions and orbit partitions, Chicago VIGRE REU 2009.

Proposition 2.4. An orbit partition is an equitable partition.
The converse is false in general. The first graph in Fig. 2 is called McKay's graph.

A coherent configuration is Schurian iff it is the orbital configuration of a permutation group (Higman / Cameron / Babai). Gate 1 A = Schurian; Gate 1 B = non-Schurian coherent structure. Five widths still do not make A a theorem. D'Angeli–Donno Prop. 13 remains the Markov-chain counterexample.

Watanabe–Wolfer, arXiv:2412.08400v3, accepted Information Geometry 2026, DOI 10.1007/s41884-026-00194-7:

It is known that the set of lumpable Markov chains … generally does not form an exponential family of stochastic matrices.

Corollary 4.1: no multi-row merging block ⇒ e-family. Theorem 4.1: redundant multi-row merging block ⇒ not an e-family. Contains no Aut / orbit / Zariski / generic-rates / coarsest-lumping language. Adjacent to a generic-rates lemma, not the lemma.

Best "Barrett & Feng" candidate: Barr & Thomas, JASA 72 (1977) 175–179 and Operations Research 25 (1977) 1028–1031 (eigenvector test of a declared partition). Barr, not Barrett; Thomas, not Feng. Still a false lead unless a specific reference is supplied.

Q3 — Diaconis 1988 is the group-walk case of Schur

Persi Diaconis, Group Representations in Probability and Statistics, IMS LNMS 11 (1988), Ch. 3E The Markov chain connection. Fourier of a Cayley walk is block-diagonal by irreps; invariant observables live in the trivial isotypic. This is Schur for a group-walk, not the Duhamel integrand for a general equivariant Markov generator. Santos Gutiérrez–Zagli–Carigi, arXiv:2507.14040 (2025), still optimises a Markov linear response — opposite direction. The gap remains real.

Q1 leftovers / Q5 restatement

Dougherty–Root, arXiv:2604.14458 (April 2026), and Ebrahimi-Fard–Foissy–Kock–Patras, arXiv:2407.17660, are NC combinatorics and do not compete for the orbit count. Callan–Smiley remains unpublished as a journal article.

Freund exact deflation (arXiv:math/0501484): a prefix that is not a deflation boundary is not a Krylov subspace of any grade. Restatement of Grimme, not a named "non-monotone prefix" pathology.

#600 Phase C (sibling compute ticket)

Expected result — nonzero task Hankel data identical between the Catalan space and the R-orbit quotient; R-odd sector exactly uncontrollable/unobservable — is the Kalman decomposition of a C2-equivariant realization with even B and even C (Wonham). Not a new theorem. The odd-width halves_linked trap is the teeth. Pointer posted on #600.

X keyword search this round: no hits. Watanabe–Wolfer acceptance tweet (G. Wolfer, 14 March 2026) found via the web, not via X search.

Owner may close this ticket against #602 once both notes are accepted. I am not auto-closing.

[LightChainr]

#599 probe pack on #604 occupies the Q3 language slot with citation discipline already applied in its C1 snapshot: Schur / Wigner–Eckart for the vanishing; Kalman / Wonham for C2 even-task factorization; claimed remainder = Markov-generator / Duhamel-pointwise form + order-ℓ tensor criterion + C2-accident (V_triv = Fun(X/K) is the orbit quotient only for C2).

Does not fill Q2 (Gate 1 stays a five-width instance). Does not close this ticket. Q1 remains Callan–Smiley.

Note: notes/probe-markov-selection-rule-and-l2pi-fibre-20260906.md.

[LightChainr]

Queued for Go-quota refill. Deep retrieval, notes only. arXiv may be blocked in the CLI session — NOT FOUND is valid. Do not duplicate #592. Draft PR against main when assigned.

[LightChainr]

ASSIGNED_MACHINE: local

Deep retrieval, notes only. arXiv may be blocked — NOT FOUND is valid. Do not duplicate #592. Draft PR against main. No STATUS. Do not close #598/#601.

[LightChainr]

PR: #689

Round-3 retrieval. Q1 published source is CJS 2004. Ticket stays open. Not merging. No new dispatch until the remaining cluster exits.