[#713] Literature retrieval: double-pulse / bilinear kernels vs P398 odd-sector certificate (do not merge)

notes/lit-double-pulse-observability-20260912.md

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+# Literature retrieval: double-pulse / bilinear kernels against the P398 odd-sector certificate
+
+**Date:** 2026-09-12
+**Ticket:** #713 (parent #650). Context: draft #709 on `analysis/p398-double-pulse-20260912`.
+**Type:** retrieval only. No new production, no enumeration, no transfer matrix, no hardware.
+**Does not:** merge, close, or edit `docs/STATUS.md`. No novelty claim is made anywhere below.
+
+Object under discussion, quoting draft #709's own definitions (not re-derived here):
+
+```text
+G = sum_j (J_j + D_j),   R : j -> w-1-j,
+H = J_0 - J_(w-2),       RGR = G,  RHR = -H,
+K(tau) = S H exp(tau G) H F = C_- exp(tau G_-) B_-.
+```
+
+The four questions in #713 are about what earlier literature already contains for this shape.
+Every source is marked **PRIMARY_TEXT_READ**, **ABSTRACT_ONLY**, or **[LIT]** (bibliographic
+pointer; body not retrieved). Quotations below are verbatim from the retrieved text only.
+
+## Tripwire answers (stated first, because they bound the rest)
+
+1. **Bilinear/Hankel prior art claimed as PRIMARY?** No. The only bilinear-Hankel text I read in
+   full is Petreczky, ESAIM: COCV **17** (2011) 446–471, Part II (theorem numbers below). Fliess
+   1974, Isidori 1973 and Ho–Kalman 1966 are **not** upgraded to PRIMARY here: they are
+   ABSTRACT_ONLY or [LIT]. Arbib–Manes 1980 is ABSTRACT_ONLY; it names the Fliess/Isidori Hankel
+   matrices but I did not open those originals.
+2. **Does the P398 double-pulse certificate itself appear in print?** **No.** Nothing retrieved
+   matches the specific scalar observer `Khat(z) = 2(z^2+11z+27)/[(z^2+10z+23)(z^2+11z+26)]`, the
+   4×4 determinant `−16`, or the width-4/5 odd-sector rank attainment on `G = sum_j (J_j+D_j)`
+   with the original `S`, `F`. The expected answer holds. Absence of a retrieved match is **not**
+   an originality certificate — it is the only statement the search supports.
+
+## Evidence legend
+
+| Mark | Meaning in this note |
+|---|---|
+| **PRIMARY_TEXT_READ** | I retrieved and read the full body (arXiv HTML, or publisher/Numdam PDF text extracted with `pdftotext`). Section/equation/theorem numbers are from that body. |
+| **ABSTRACT_ONLY** | Only the abstract and landing-page metadata were retrieved. Equations attributed to the paper are quoted from its own abstract, not reconstructed. |
+| **\[LIT\]** | Bibliographic pointer only (title/venue/year). The body was not retrieved; no statement of its content is made here. |
+
+---
+
+## Q1. Two-pulse / Volterra kernels recovering a hidden sector after a quotient
+
+**What was retrieved and read.**
+
+Lucarini, *Interpretable and Equation-Free Response Theory for Complex Systems*,
+**arXiv:2502.07908v2** HTML, §II.2 (§IV.1.3 for the Volterra remark).
+**PRIMARY_TEXT_READ.** Published version: *Phil. Trans. R. Soc. A* **384** (2026) 20250081,
+DOI 10.1098/rsta.2025.0081 (**ABSTRACT_ONLY**; the DOI landing page was not opened here).
+
+This is the closest retrieved match to Q1's "two-pulse kernel for a Markov chain". For a
+finite-state Markov chain with perturbation `M -> M + eps f(n) m`, the paper's second-order
+response is a genuine two-time kernel. Verbatim from the retrieved HTML:
+
+- second-order measure, Eqs. (13)–(14):
+  `nu^(2)(n) = sum_k sum_p Theta(k)Theta(p) M^k m M^p m nu_inv f(n-k-p-2) f(n-k-1)`,
+  with the text "the Θ's ensure the correct time ordering of the perturbation";
+- second-order Green's function, Eq. (18):
+  `G^(2)_{m,Psi}(k,p) = Theta(k)Theta(p) <m^T (M^T)^p m^T (M^T)^k Psi, nu_inv>`,
+  called a "double convolution sum";
+- **two-mode spectral decomposition**, Eq. (19):
+  `G^(2)_{m,Psi}(k,p) = Theta(k)Theta(p) sum_{i,j=2}^{N} alpha_{ij} lambda_i^k lambda_j^p`,
+  `alpha_{ij} = <m^T Pi_j m^T Pi_i Psi, nu_inv>`;
+- §IV.1.3: "The nonlinear Green functions can be formally seen as **Volterra kernels**".
+
+**What this does and does not supply for P398.**
+
+- It supplies the general, already-published statement that a Markov chain's second-order
+  response is governed by a two-time (Volterra-type) kernel with two-mode spectral weights.
+  P398's `K(tau) = S H exp(tau G) H F` is an instance of this language, **not a new response
+  formalism**. This matches #709's own disclaimer.
+- It does **not** single out a parity/odd sector. Lucarini's decomposition is by Koopman modes
+  `lambda_i`, not by a `C2` (reflection) irrep. The P398 selection rule that an invariant task is
+  blind to `H` at first order is a representation-theoretic statement (trivial irrep in a tensor
+  product, as in #598/#244), and I did not retrieve a primary paper that phrases a two-pulse
+  hidden-sector recovery as a parity quotient of a Markov generator.
+- Mueller, Basu, Sollich, Krueger, *Coarse-grained second-order response theory*, Phys. Rev.
+  Research **2** (2020) 043123, DOI 10.1103/PhysRevResearch.2.043123, remains **ABSTRACT_ONLY**
+  (already so in #709). Its equilibrium hypotheses are not imported here.
+
+**Finding for Q1.** The *machinery* is published (two-time second-order kernels, spectral
+two-mode decomposition, explicit Volterra framing). A primary source that uses a two-pulse kernel
+to recover a *parity-odd hidden sector after an even quotient* in a Markov chain or interacting
+particle system was **not found**. Treat that as a gap in my retrieval, not as a no-go.
+
+---
+
+## Q2. Bilinear realization, Fliess kernels, and Hankel rank of `C A^k B`
+
+### 2a. The verified primary text
+
+Petreczky, *Realization theory for linear and bilinear switched systems: a formal power series
+approach. Part II: Bilinear switched systems*, **ESAIM: COCV 17 (2011) 446–471**, DOI
+10.1051/cocv/2010015. **PRIMARY_TEXT_READ** (Numdam PDF, `pdftotext -layout`).
+
+Word coefficients and the Hankel matrix, §2.1 (verbatim):
+
+```text
+c_f(eps) = C x0,   c_f(j1 j2 ... jk) = C B_jk B_j{k-1} ... B_j1 x0     (2.6)
+```
+
+> "columns and rows of which are indexed by sequences v in Z*_m. The p × 1 block entry of H_f
+> lying on the intersection of the row indexed by v and the column indexed by w equals c(wv). It
+> turns out that the generating series c has a representation of the form (2.6) if and only if the
+> column rank of H_f is finite. That is, f has a realization by a bilinear system if and only if f
+> has a Fliess-series expansion and the column rank of its Hankel-matrix H_f is finite."
+
+Definition 2.8 defines the Hankel matrix `H_Phi` with block entry
+`(H_Phi)_{(v,i),(w,f)} = (T_{f,sigma_{K+1}}(wv))_r`, and rank = dimension of the column span.
+The minimal-realization characterization, **Theorem 2.3(iii)** (verbatim):
+
+> "The dimension of Sigma_min equals the rank of the Hankel-matrix H_Phi of Phi, i.e.
+> dim Sigma_min = rank H_Phi."
+
+**Theorem 2.7** (existence, constrained switching): finite Hankel rank plus generalized
+Fliess-series expansion gives a realization. **Remark 2.5**: the proofs construct the realization
+from the columns of the Hankel matrix.
+
+Petreczky, Wisniewski, Leth, *Moment matching for bilinear systems with nice selections*,
+**arXiv:1605.04414v1**, IFAC-PapersOnLine 49(18):838–843 (2016), DOI
+10.1016/j.ifacol.2016.10.270. **PRIMARY_TEXT_READ** (arXiv HTML). This is the paper #709 already
+cites. It gives the bilinear system (1a)–(1b), the matrix words `A_w = A_{q_k}...A_{q_1}` with
+`A_eps = I`, the coefficient identity `c_f(w) = C A_w x0`, the Fliess operator/series (Eq. (4)),
+and the `gamma`-partial-realization construction (Def. 2, Theorems 3–4).
+
+Important negative reading, verified in the same body: **this 2016 paper does not state a
+Hankel-rank criterion.** Its minimality test is algebraic — "Sigma is a minimal realization of f
+if and only if Sigma is a realization of f, and it is span-reachable and observable", with
+observability `intersection_w ker C A_w = {0}` and span-reachability `Span{A_w x0 | w in Q*} = R^n`.
+So it is not the place to attribute the rank theorem to; the 2011 Part II is.
+
+### 2b. The classical linear case (`K(tau) = C exp(tau A) B`)
+
+The shape `H_i^j = C A^{i+j} B` and the equality "minimal order = Hankel rank" are the classical
+linear realization theorem.
+
+- Arbib, Manes, *Generalized Hankel Matrices and System Realization*, **SIAM J. Math. Anal. 11(3)
+  (1980) 405–424**, DOI 10.1137/0511038. **ABSTRACT_ONLY.** The abstract states verbatim: "Our
+  definition of the Hankel matrix unifies the familiar `H_i^j = CA^{i+j}B` of linear system theory
+  ... with the bilinear Hankel matrix of A. Isidori ... and the Hankel matrix of M. Fliers
+  (Matrices de Hankel, J. Math. Pure Appl., 53 (1974), pp. 197–224)." It also states a
+  realisability theorem, a partial-realization theorem, and a canonical-realization theorem for
+  finite Hankel blocks.
+- Fliess, *Matrices de Hankel*, J. Math. Pures Appl. (9) **53** (1974) 197–222 (the Arbib–Manes
+  abstract prints the range as 197–224). **[LIT]** — body not retrieved.
+- Isidori, *Direct construction of minimal bilinear realizations from nonlinear input-output maps*,
+  IEEE Trans. Autom. Control **AC-18** (1973) 626–631. **ABSTRACT_ONLY** (IEEE/scilit landing
+  pages; the bilinear Hankel matrix is attributed to it by Arbib–Manes, not read here).
+- Ho, Kalman, *Effective construction of linear state-variable models from input/output functions*,
+  Regelungstechnik **14** (1966) 545–548. **[LIT]** — not retrieved; named for attribution only.
+
+### 2c. What is *not* claimed as new if we only certify a finite instance
+
+A finite 4×4 or 16×16 nonzero Hankel minor is an **instance** of Theorem 2.3(iii)/Theorem 2.7 of
+Petreczky Part II, specialised to `C A^k B` with `A = G_-`, `B = B_-`, `C = C_-`. If #709's
+deliverable is read as "certified a finite instance", then all of the following are prior art and
+must not be presented as new:
+
+- Fliess-series / word-coefficient representation `c(w) = C A_w x0`;
+- the Hankel-matrix construction and the finite-rank ⇔ realizable equivalence;
+- the identity "minimum realization order = rank of the Hankel matrix";
+- partial-realization-from-Hankel-blocks.
+
+The only candidate for a new statement is the **model-specific** one: that P398's *particular*
+`S`, `F`, `G`, `H` at `w = 4, 5` attain the parity upper bounds with a nonzero integer minor, and
+that the **equal-time** mixed derivative is zero while the delay kernel has full odd rank. Per the
+tripwire, that specific certificate is not in print (Q4/tripwire).
+
+---
+
+## Q3. Noncrossing partitions / Temperley–Lieb / join-split generators as CTMCs
+
+Two named families were retrieved; neither is P398's generator.
+
+### 3a. Temperley–Lieb stochastic processes (closest named relative)
+
+Pearce, Rittenberg, de Gier, Nienhuis, *Temperley-Lieb Stochastic Processes*,
+**arXiv:math-ph/0209017v2**, J. Phys. A **35** (2002) L661–L668, DOI
+10.1088/0305-4470/35/45/105. **PRIMARY_TEXT_READ** (arXiv HTML).
+
+The abstract generator is Eq. (2.1): `H = sum_a c_a (1 - w_a)`, `c_a >= 0`, an intensity matrix
+satisfying the master equation (2.2) `dP_a/dt = -sum_b H_ab P_b`. For the Temperley–Lieb algebra
+the generator is Eq. (2.6) `H = sum_{j=1}^{L-1} (1 - e_j)`; the **cylindrical** version (Eq.
+(2.15)) adds the closed bond, `H = sum_{i=1}^{L} (1 - e_i)`. The state space is the set of
+**link patterns / connectivities** (words in an ideal of the TL algebra), dimension given by (2.9)
+with the Catalan count (2.13).
+
+Crucially, verbatim: "**the terms in the Hamiltonian may connect disconnected lines but it is not
+possible to have the reverse process**". So this generator is a **join-only** (irreversible)
+dynamics on planar connectivities.
+
+**Resemblance and difference for P398.** The *cylindrical* TL generator `sum_{i=1}^{L}(1-e_i)` is
+structurally the nearest published object to a sum of local generators on *cyclic* planar
+connectivities, which is why it is worth naming. But it is not P398's `G`:
+
+- TL acts on link patterns (noncrossing **matchings**), P398 on noncrossing **partitions** of `w`
+  cyclic points;
+- TL's `sum (1-e_j)` is **join-only**, while P398's `G = sum_j (J_j + D_j)` contains the detach
+  half `D_j` as well. The two are therefore different generators, and the TL result does not
+  transfer as a statement about `G`.
+
+### 3b. Exchangeable fragmentation–coagulation (both directions, but exchangeable/nonlocal)
+
+Bertoin, *Two-parameter Poisson-Dirichlet measures and reversible exchangeable
+fragmentation-coalescence processes*, **arXiv:0704.3122v1** (2007). **PRIMARY_TEXT_READ**
+(arXiv HTML, §2.2).
+
+An EFC process is a `P_N`-valued exchangeable Markov process whose restriction to `P_[n]` "only
+evolves by fragmentation of one block or by coagulation"; the allowed first jumps are "obtained
+from `pi^[n]` by **splitting exactly one of its blocks** ... or by **merging at least two of its
+non-empty blocks**". This *does* have both join and detach, and can be reversible. But it is
+**exchangeable** (non-local, all partitions, invariant under relabelling), so it is not the
+planar/local object P398 uses.
+
+### 3c. Answer: is `G = sum_j (J_j + D_j)` a named process?
+
+**No named process matching P398's generator was found.** The retrieved named relatives are:
+TL stochastic processes (join-only, matchings, elliptic/local but irreversible) and EFC /
+coalescent–fragmentation (join+detach and reversible, but exchangeable and nonlocal). P398's
+combination — **adjacent join plus point-detach, on cyclic noncrossing partitions, both
+directions, local** — was not identified with a name in the retrieved literature. Pitman and
+Diaconis appear in the surrounding noncrossing-partition and coalescent literature, but no primary
+noncrossing-partition CTMC with this join+detach generator was retrieved; those remain **[LIT]**
+pointers only. **Do not call `G` "the Temperley–Lieb process" or "an EFC process."**
+
+Consistent with repo policy: an unretrieved name is not evidence that the process is unnamed, and
+it is not an originality claim.
+
+---
+
+## Q4. A published "zero delay misses a delay-mode" example of the same shape
+
+**Linear-systems side (verified shape, not a special example).** For `K(tau) = C e^{tau A} B`, the
+first Hankel block is `C B` and later blocks are `C A^{i+j} B` (Arbib–Manes **ABSTRACT_ONLY**,
+quoted above). `K(0) = 0` with nonzero Hankel rank is the ordinary **strictly proper / relative
+degree >= 1** case: minimal order equals the full Hankel rank regardless of the vanishing of the
+first Markov parameter (Ho–Kalman **[LIT]**; Petreczky Part II **PRIMARY_TEXT_READ** for the
+bilinear version). In other words, "zero at zero delay, nonzero rank afterwards" is **generic**, not
+a named counterexample, and certifying one such instance carries no novelty by itself.
+
+**Two-pulse / Markov-chain side.** I found **no published example** of exactly the P398 shape — a
+two-pulse Markov-chain delay kernel with `K(0) = 0` and Hankel rank 4 on an odd sector — in the
+retrieved literature. The nearest physical analogue I noticed in search (a two-photon/dark-state
+transition with zero one-photon matrix element, i.e. a selection-rule analogue) was **not retrieved
+as a primary text** and is listed here only as a **[LIT]** pointer, not a claim.
+
+**Finding for Q4.** The *linear* shape is standard and prior; the *P398-specific* two-pulse
+certificate is not in print.
+
+---
+
+## Sources, with marks
+
+| # | Source | Mark | Retrieved |
+|---|---|---|---|
+| 1 | Lucarini, *Interpretable and Equation-Free Response Theory for Complex Systems*, arXiv:2502.07908v2; Phil. Trans. R. Soc. A 384 (2026) 20250081 | **PRIMARY_TEXT_READ** (arXiv HTML; §II.2 Eqs. 10–19, §IV.1.3) | https://arxiv.org/html/2502.07908v2 |
+| 2 | Lucarini, journal version, DOI 10.1098/rsta.2025.0081 | **ABSTRACT_ONLY** | (via search result; not opened) |
+| 3 | Petreczky, *… Part II: Bilinear switched systems*, ESAIM: COCV 17 (2011) 446–471 | **PRIMARY_TEXT_READ** (§2.1 Eqs. 2.6; Def. 2.8; Thm 2.3(iii); Thm 2.7; Rem. 2.5) | https://www.numdam.org/item/COCV_2011__17_2_446_0.pdf |
+| 4 | Petreczky, Wisniewski, Leth, *Moment matching for bilinear systems with nice selections*, arXiv:1605.04414v1 / IFAC-POL 49(18):838–843 | **PRIMARY_TEXT_READ** (§2.1–2.2, §3 Def. 2, §4 Thms 3–4; no Hankel criterion) | https://arxiv.org/html/1605.04414v1 |
+| 5 | Pearce, Rittenberg, de Gier, Nienhuis, *Temperley-Lieb Stochastic Processes*, arXiv:math-ph/0209017v2; J. Phys. A 35 (2002) L661 | **PRIMARY_TEXT_READ** (Eqs. 2.1–2.3, 2.6, 2.9, 2.13, 2.15) | https://arxiv.org/html/math-ph/0209017v2 |
+| 6 | Bertoin, *Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes*, arXiv:0704.3122v1 (2007) | **PRIMARY_TEXT_READ** (§2.2) | https://arxiv.org/html/0704.3122v1 |
+| 7 | Arbib, Manes, *Generalized Hankel Matrices and System Realization*, SIAM J. Math. Anal. 11(3) (1980) 405–424 | **ABSTRACT_ONLY** | https://epubs.siam.org/doi/10.1137/0511038 |
+| 8 | Isidori, *Direct construction of minimal bilinear realizations…*, IEEE TAC AC-18 (1973) 626–631 | **ABSTRACT_ONLY** | https://ieeexplore.ieee.org/document/1100424 |
+| 9 | Fliess, *Matrices de Hankel*, J. Math. Pures Appl. (9) 53 (1974) 197–222 | **[LIT]** | (cited via #7's abstract) |
+| 10 | Ho, Kalman, *Effective construction of linear state-variable models from input/output functions*, Regelungstechnik 14 (1966) 545–548 | **[LIT]** | https://ntrs.nasa.gov/citations/19670049337 (pointer) |
+| 11 | Mueller, Basu, Sollich, Krueger, *Coarse-grained second-order response theory*, PRR 2 (2020) 043123 | **ABSTRACT_ONLY** (as in #709) | https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.043123 |
+
+## What this note does not do
+
+- No novelty claim, in either direction. A gap in my retrieval is not a no-go theorem, and an
+  unretrieved name is not an originality certificate.
+- No new computation, no width campaign, no transfer matrix, no hardware.
+- No `docs/STATUS.md` edit; no merge; no close of #713 or #709.
+- The unique P398 science (the certificate itself) stays in the #709 GitHub PR, not here.