From 308f66c7c6e34af7feb4fb689f27dfccea3eefea Mon Sep 17 00:00:00 2001 From: workbuddy Date: Sat, 12 Sep 2026 21:13:34 +0800 Subject: [PATCH] Add #713 literature retrieval: double-pulse / bilinear kernels vs P398 odd-sector certificate --- ...lit-double-pulse-observability-20260912.md | 275 ++++++++++++++++++ 1 file changed, 275 insertions(+) create mode 100644 notes/lit-double-pulse-observability-20260912.md diff --git a/notes/lit-double-pulse-observability-20260912.md b/notes/lit-double-pulse-observability-20260912.md new file mode 100644 index 00000000..b1b0b609 --- /dev/null +++ b/notes/lit-double-pulse-observability-20260912.md @@ -0,0 +1,275 @@ +# 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) `, + 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} = `; +- §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.