Research log — Hilbert 16(a), degree 8

All sources below were actually opened on 2026-08-21 (by this session or its literature subagents, arXiv HTML/PDF or author PDFs; raw downloads where noted). Forum posts were not used.

The record

• S. Yu. Orevkov, Classification of flexible M-curves of degree 8 up to isotopy, GAFA 12 (2002) 723–755. Author PDF (read, pp. 723–728): https://www.math.univ-toulouse.fr/~orevkov/m8.pdf. Before it: 9 M-schemes open (list (1), p. 723). Thm 1.1/1.2a: two of them, ⟨1⊔1⟨3⟩⊔1⟨16⟩⟩ and ⟨1⊔1⟨6⟩⊔1⟨13⟩⟩, are not realizable even pseudo-holomorphically. Thm 1.2b: the other seven are pseudo-holomorphically realizable. Table 1 (p. 725–726): the 89 pseudo-holomorphically realizable M-schemes; asterisks on the six algebraically open ones. Caveat (Remark 1, p. 724): the seven are not claimed realizable in a single almost complex structure.
• S. Yu. Orevkov, New M-curve of degree 8, Funct. Anal. Appl. 36 (2002) 247–249. Author PDF (read in full): https://www.math.univ-toulouse.fr/~orevkov/m711.pdf. Realizes ⟨7⊔1⟨2⊔1⟨11⟩⟩⟩ algebraically; states "it remains 6 arrangements of 22 ovals whose realizability by M-curves is still unknown."
• The six (also independently listed in Gabard arXiv:1310.1865 Thm 2.2, and still current in arXiv:2602.06888 v3, 2026): ⟨4⊔1⟨2⊔1⟨14⟩⟩⟩, ⟨14⊔1⟨2⊔1⟨4⟩⟩⟩ with (p,n)=(19,3); ⟨1⊔1⟨1⟩⊔1⟨18⟩⟩, ⟨1⊔1⟨4⟩⊔1⟨15⟩⟩, ⟨1⊔1⟨7⟩⊔1⟨12⟩⟩, ⟨1⊔1⟨9⟩⊔1⟨10⟩⟩ with (p,n)=(3,19).
• Geiselmann, Joswig, Kastner, Mundinger, Pokutta, Spiegel, Wack, Zimmer, Limits of Combinatorial Patchworking, arXiv:2602.06888 (v1 6 Feb 2026 as "121 Patchworked Curves of Degree Seven"; v3 27 Jul 2026, major revision). Abs + full HTML v3 (raw download, quotes verbatim): https://arxiv.org/abs/2602.06888, https://arxiv.org/html/2602.06888v3. Current degree-8 record: 2,367 distinct nonempty schemes as T-curves ("as the search is not exhaustive, this is a lower bound"); 38 of the 89 M-schemes T-realized (Table 1 bold; 5/11/14/8 per (p,n) = (19,3)/(15,7)/(11,11)/(7,15)); Theorem 21: maximal degree-8 T-curves have (p,n) ∈ {(19,3),(15,7),(11,11),(7,15)}, so the 12 M-schemes with (p,n)=(3,19) — including four of the six open — are not T-curves; 39 M-schemes undecided as T-curves (Question 39), including the two open (19,3) deep-nests. Degree ≤ 7: every nonempty scheme is a T-curve (Thm 38: four triangulations cover all 121 of degree 7), answering Itenberg–Viro 1996. Obstruction machinery: Haas' zone decompositions (their Section 3), purely combinatorial, no Gudkov–Rokhlin. Corrections they make to Orevkov's Table 1: Wiman scheme is ⟨16⊔3⟨1⟩⟩ (misprinted ⟨17⊔3⟨1⟩⟩), and ⟨17⊔1⟨2⊔1⟨1⟩⟩⟩ is attributed to Hilbert in Viro's 1980 table.
• Same group, Fast Isotopy Computation for T-Curves, arXiv:2604.09221 (10 Apr 2026, ICMS): abs + HTML opened. Reports the earlier sweep state: "exactly 2359 distinct real schemes … of which only 30 are maximum", "53 others have no known T-curve realization". Superseded by 2602.06888 v3 (2,367 / 38). The near-quadratic isotopy algorithm and GPU implementation live here; C++ library https://github.com/polymake/libisotopy (exists, branch master; no scheme data in the repo).

Data replayed

• https://github.com/dmg-lab/CombinatorialPatchworking — the paper's data repo (Appendix B). deg8.pcoms.txz (125,932 bytes) downloaded from https://raw.githubusercontent.com/dmg-lab/CombinatorialPatchworking/main/deg8.pcoms.txz: exactly 2,367 .pcom certificates deg8/oNN-pNN-nNN/(scheme).pcom, each a polymake JSON with POINTS (the 45 lattice points of T_8), MIN_WEIGHTS (integer lifting), MAXIMAL_CELLS (64 triangles), SIGNS (45 booleans), TYPE (claimed scheme). The 22-oval directories hold exactly 38 files. Replay: compute/replay_census.py (all 2,367 PASS, see ATTACK.md).

Method sources (for the verifier)

• Itenberg–Viro, Patchworking algebraic curves disproves the Ragsdale conjecture, Math. Intelligencer 18:4 (1996). Full HTML: https://www.math.stonybrook.edu/~oleg/math/papers-html/iten-vi/index.html. Construction, sign-replication rule σ_{i,j}σ_{εi,δj}ε^iδ^j = 1, convexity hypothesis, Theorem 1; the 1996 question answered by arXiv:2602.06888.
• O. Viro, Patchworking real algebraic varieties, arXiv:math/0611382 (abs + HTML §1.1 opened): Patchwork Theorem 1.1.D; convexity required, primitivity not (primitivity is what makes the combinatorial T-curve setting).
• I. Itenberg, Construction of real algebraic varieties survey PDF (read): https://webusers.imj-prg.fr/~ilia.itenberg/papers/constr.pdf. Lemma 3.3: Harnack signs on any primitive convex triangulation give the Harnack M-curve. Santos' non-convex piecewise-linear curves are only known pseudo-holomorphic — convexity matters.
• arXiv:2602.06888 v3 §2: modern statement (regular = convex = lifting-induced; unimodular = primitive), diamond A⋄, F₂ sign rule; §4.2 Prop 10: Harnack signs give ⟨18⊔1⟨3⟩⟩ for d=8 on ANY unimodular triangulation (regular or not).
• Joswig–Vater, arXiv:2003.06326 (polymake patchworking, abs).

Negative / corrected lookups

• The task prompt's citation said "2359 realized and 53 non-maximal schemes with no T-curve witness". Wrong on both counts against the papers: the April paper's 53 are maximal (22-oval) schemes with algebraic constructions but no T-curve witness, and the July v3 supersedes 2359→2,367, 30→38, 53→39-undecided+12-impossible.
• Degtyarev–Kharlamov, arXiv:math/0004134 (du côté de chez Rokhlin): full PDF text searched — contains no degree-8 status table. Not a citable source for the six.
• Gabard arXiv:1310.1865 says only three of the six are known pseudo-holomorphic; Orevkov's published Thm 1.2(b) covers all six. Orevkov is authoritative.
• https://www.math.stonybrook.edu/~oleg/math/papers.html — 404.
• De Loera–Wicklin, On the Need of Convexity in Patchworking (Adv. Appl. Math. 20, 1998): ScienceDirect 403, abstract not fetched; known via citations in 2602.06888 (individual degree-7/8 T-curves, non-convex experiments).
• api.github.com unauthenticated: 403 rate-limit (worked around via gh and raw.githubusercontent.com).
• Orevkov's degree-8 papers are not on arXiv; author PDFs used.
• Web searches for any 2003–2026 resolution of one of the six: nothing found.

Haas' theorem (second session, 2026-08-21)

The whole of compute/haas.py and compute/zone_search.py rests on Section 3 of arXiv:2602.06888 v3, which was re-fetched and read in full for this session (raw HTML download, https://arxiv.org/html/2602.06888v3; the first session had only used the abstract-level claims and Theorem 21):

• Theorem 13 (their statement of Haas): a T-curve $\mathcal{C}(\mathcal{T},\sigma)$ is maximal if and only if some collection $\mathfrak{S}$ of compatible Harnack splits has a zone decomposition valid for $(\mathcal{T},\sigma)$. No regularity of $\mathcal{T}$ is assumed.
• Section 3.2: a Harnack split is a path of one or two primitive edges joining exactly two boundary points of $d\cdot\Delta_2$, with vertex parities alternating between two distinct classes $\alpha,\beta\in\mathbb{F}_2^2$; simple = one edge, double = two edges through an interior apex. $Z^+$ is the zone containing fewer corners of $d\cdot\Delta_2$; $Z^-$ is the root zone. Compatible = no shared edge and simultaneously realizable in one triangulation.
• Section 3.4, Lemma 14: the surgical twist of a split replaces $\sigma$ by $\epsilon+\sigma\circ s_{ij}$ on $Z^+$, with $\epsilon=\alpha_1\beta_2+\alpha_2\beta_1$, $i=\alpha_2+\beta_2$, $j=\alpha_1+\beta_1$. Since $\sigma(s_{ab}(x,y))=\sigma(x,y)+ax+by$, a twist just adds the affine $\mathbb{F}_2$ function $\epsilon+ix+jy$ on $Z^+$ — so twists commute and a collection acts by a sum of fixed vectors. That is the observation the search is built on.
• Section 3.5: "The notation $\mathcal{C}(\mathfrak{S})$ is independent of $\mathcal{T}$, as every unimodular triangulation that refines $\mathfrak{S}$ will do." So the real scheme of a maximal T-curve is a function of the split collection alone.
• Theorem 17 (Haas): even splits do not change $(p,n)$. Lemma 18 / Remarks 19–20 / Theorem 21: seven odd splits matter in degree 8, each of effect 4 except one of effect 8; nested zones cancel; hence $(p,n)\in\{(19,3),(15,7),(11,11),(7,15)\}$.
• Section 4: families. Prop 25 onion curves (honeycomb + constant signs, maximal nesting depth, maximal only for $d\le3$); Prop 31 nested box curves — the bow tie triangulation $\mathcal{B}_d$ is induced by the nested double $\{(0,0),(1,1)\}$-splits with vertices $\{(i+1,d-i-1),(i,i),(d-i+1,i-1)\}$, and for $d=8$ gives $\langle17\sqcup1\langle2\sqcup1\langle1\rangle\rangle\rangle$, i.e. the $a=17$ member of the deep-nest family; Prop 32 arrowhead curves give $\langle17\sqcup1\langle1\rangle\sqcup1\langle2\rangle\rangle$ and the Wiman scheme $\langle16\sqcup3\langle1\rangle\rangle$.
• Section 5.3 / Question 39: the 39 undecided M-schemes are stated as open; the authors do not run a zone-decomposition enumeration in degree 8 — Theorem 21 is used only for the $(p,n)$ obstruction. That gap is what this session attacks.

Cited but not opened (no copy reachable): B. Haas, Real algebraic curves and combinatorial constructions, 1997 (their [18]) — every use of "Haas' theorem" here is via the statement in arXiv:2602.06888 v3 Theorem 13, which is what we implement and test. Also not opened: B. Bertrand, E. Brugallé, A. Renaudineau, Haas' theorem revisited, Épijournal Géom. Algébrique 1 (2017) Art. 922, DOI 10.46298/epiga.2017.volume1.2030 (their [4], an independent formulation of the same theorem); I. Itenberg, Counter-examples to Ragsdale conjecture and T-curves, Contemp. Math. 182 (1995) 55–72 (their [25], source of the Harnack sign distribution $\eta(x,y)=(x+1)(y+1)$ used here).

2026-08-22 — freshness check

Re-checked the record. Nothing about degree 8 changed. arXiv:2602.06888 is still at v3 (27 Jul 2026) — .../abs/2602.06888v4 returns 404, and the abs page lists exactly v1 (6 Feb 2026), v2 (23 Feb 2026), v3 (27 Jul 2026), still with no journal-ref. arXiv:2604.09221 is still v1 only (10 Apr 2026). The 2,367 degree-8 T-curve count stands: dmg-lab/CombinatorialPatchworking has had no push since 2026-07-21 (HEAD e85b809, merge of PR #1 "data: add degree-eight patchwork certificates"), one closed PR, zero issues, zero releases, two branches (main, feat/degree8-data, same tree), and the README still says 2,367. Orevkov's page (last update 11 Aug 2026) lists nothing on degree 8 since 2002. All six M-schemes are still open: no paper, preprint, or author page found claiming to realize or exclude any of them. Sweeps run: arXiv API on T-curve/patchworking/M-curve/real plane curve, on Hilbert's 16th/octic/22 ovals, and on cat:math.AG AND (T-curves OR patchworking); math.AG 2026-08 listing; Orevkov's page; the dmg-lab org repo list. Three items are new to this log — none of them touches the degree-8 census.

• S. Yu. Orevkov, On arrangements of plane real quartics with respect to three lines, arXiv:2607.19457 (v1 21 Jul 2026, v2 4 Aug 2026), https://arxiv.org/abs/2607.19457; author PDF linked from https://www.math.univ-toulouse.fr/~orevkov/. Degree 4, not 8, so it does not move the census — but it is a live caution for dent (A). Verbatim from the abstract: "There is one arrangement which is realizable pseudoholomorphically but not algebraically. It can be constructed in different ways, in particular, by a combinatorial patchworking on an irregular triangulation. This is the first example of a combinatorial patchworking which produces a PL curve in RP^2 whose arrangement relative to the coordinate axes is algebraically unrealizable." I.e. patchworking on a non-regular triangulation can output a PL curve with no algebraic model. Our verifier already certifies convexity (regularity) in exact rational arithmetic on every witness, which is exactly the hypothesis Viro's theorem needs and exactly what Orevkov's example lacks. Nothing to fix; the requirement is now backed by a published counterexample rather than by Itenberg's survey remark alone.
• K. Ferry, M. Joswig, J. Rambau, Counting symmetric unimodular triangulations, arXiv:2605.14150 (13 May 2026), abs + full HTML opened: https://arxiv.org/abs/2605.14150, https://arxiv.org/html/2605.14150v1. Explicitly "motivated by work on T-curves (Geiselmann et al., 2026)". Counts unimodular triangulations of $d\cdot\Delta_2$ that are invariant under a fixed symmetry axis $\mathfrak{H}$, up to $\mathfrak{H}$-feasible symmetries; their Table 1 gives $\widetilde{F}(8) = 1{,}211{,}875{,}888$ (and $\widetilde{F}(9) = 422{,}664{,}577{,}207$). Sizes the residue in ATTACK.md: we swept 184 census triangulations plus 4,074 more, and the symmetric unimodular triangulations of $8\cdot\Delta_2$ alone already number 1.2 billion — the unrestricted count is not enumerated anywhere. "Exhaustive per triangulation, not exhaustive over triangulations" is not close to exhaustive. No degree-8 scheme counts in the paper; it does not mention 2,367.
• G. Maletto, Hilbert's 16th problem for arrangements of curves on a surface, arXiv:2606.21449 (19 Jun 2026), https://arxiv.org/abs/2606.21449. The preprint Orevkov's quartic paper completes. Arrangements of three lines with a cubic (complete) and with a quartic (partial) on compact real surfaces; Bézout obstructions, Viro patchworking, a Julia library NWT. Not degree 8.

2026-08-27 — freshness check (q1)

Re-fetched the record before any new search. Nothing about degree 8 moved. python3 scripts/arxiv_fetch.py 2602.06888v3 returns v3, 27 Jul 2026, title Limits of combinatorial patchworking, 43 pages, "major revision". Abs https://arxiv.org/abs/2602.06888 and full HTML https://arxiv.org/html/2602.06888v3 re-read: §5.3 still says 2,367 nonempty degree-8 T-curve schemes, "as the search is not exhaustive, this is a lower bound"; 38 of 89 M-schemes T-realised; Theorem 21 still restricts maximal T-curves to $(p,n)\in\{(19,3),(15,7),(11,11),(7,15)\}$; Question 39 still leaves 39 M-schemes undecided. A request for https://arxiv.org/abs/2602.06888v4 does not produce a v4 paper (the fetcher landed on the old v1 title). arXiv:2604.09221 is still v1 (10 Apr 2026), https://arxiv.org/abs/2604.09221. dmg-lab/CombinatorialPatchworking HEAD is still e85b809 (2026-07-21, merge of PR #1); no later commit. Parent sh run_all.sh this session: 2,367/2,367 certificates replay, 17/17 outside the census, Haas 38/38, C/Python agreement on ranks 6, 10, 12, 13, 16.

A DOML 2026 slide deck by Spiegel, https://christophspiegel.berlin/assets/slides/DOML_2026.pdf, prints "at least 2,368 isotopy types of degree 8" including 38 M-curves. That is a talk slide, not a paper. The v3 text says 2,367 nonempty schemes; counting the empty scheme would make 2,368. Not treated as a census update. No new paper found that realises or excludes any of the six algebraically open M-schemes.

Checked and not relevant to degree 8, recorded so the next check can skip them: E. Pasquereau, On the topology of T-manifolds of higher codimension, arXiv:2602.14988 (16 Feb 2026), https://arxiv.org/abs/2602.14988 — new component bounds, but for high codimension and $\mathbb{RP}^3$, nothing on ovals in $\mathbb{RP}^2$. Also scanned and off-topic: Akyar–Shkolnikov arXiv:2603.08094 (non-Abelian patchworking), Demory arXiv:2601.07751 (T-polynomials, dilated triangulations), Ichikawa arXiv:2601.18093 (dimers, M-curves of a Riemann surface, not plane schemes), Janasz–Pokora arXiv:2512.24707 (M-arrangements of conics and lines), Frühbis-Krüger–Joswig–Kastner arXiv:2603.12985 (drawing real plane curves in OSCAR). No forum sources were used.

2026-08-27 — freshness check (q2)

Re-fetched before the leftover high-rank thicken. Nothing about degree 8 moved. python3 scripts/arxiv_fetch.py 2602.06888v3 returns v3, 27 Jul 2026. Abs https://arxiv.org/abs/2602.06888 and full HTML https://arxiv.org/html/2602.06888v3 re-read: §5.3 still 2,367 nonempty degree-8 T-schemes, "as the search is not exhaustive, this is a lower bound". https://arxiv.org/abs/2602.06888v4 is 404. Parent 17/17 and q1 replay still green. Bound still ≥ 2,384.

2026-08-27 — freshness check (q3)

Re-fetched before the leftover ranks 22–26 thicken. Nothing about degree 8 moved. python3 scripts/arxiv_fetch.py 2602.06888v3 returns v3, 27 Jul 2026. Abs https://arxiv.org/abs/2602.06888 and full HTML https://arxiv.org/html/2602.06888v3 re-read: §5.3 still 2,367 nonempty degree-8 T-schemes, "as the search is not exhaustive, this is a lower bound". https://arxiv.org/abs/2602.06888v4 is 404. Parent 17/17 still the folder bound. Rank 21 leftover finished on main. Bound still ≥ 2,384.

2026-08-29 — freshness check (q4)

Re-fetched after the leftover ranks 22–26 thicken merged. Nothing about degree 8 moved. python3 scripts/arxiv_fetch.py 2602.06888v3 returns v3, 27 Jul 2026. Abs https://arxiv.org/abs/2602.06888 and full HTML https://arxiv.org/html/2602.06888v3 re-read: §5.3 still 2,367 nonempty degree-8 T-schemes, "as the search is not exhaustive, this is a lower bound". https://arxiv.org/abs/2602.06888v4 is 404. Parent 17/17 still the folder bound. Leftover ranks 22–26 finished on main. Bound still ≥ 2,384.

2026-08-29 — freshness check (q5)

Re-fetched after the leftover (19,3) neighbourhood wrap merged as e6fe948 (PR #151). Nothing about degree 8 moved. python3 scripts/arxiv_fetch.py 2602.06888v3 returns v3, 27 Jul 2026. Abs https://arxiv.org/abs/2602.06888 and full HTML https://arxiv.org/html/2602.06888v3 re-read: §5.3 still 2,367 nonempty degree-8 T-schemes, "as the search is not exhaustive, this is a lower bound". https://arxiv.org/abs/2602.06888v4 is 404. Parent 17/17 still the folder bound. q4 finished the depth-3 three-split 12/12 and left the even-split BFS queue at 1,167,098. Bound still ≥ 2,384.

2026-08-29 — freshness check (q5 resume)

Re-fetched after the leftover walk was interrupted. Nothing about degree 8 moved. python3 scripts/arxiv_fetch.py 2602.06888v3 returns v3, 27 Jul 2026. Abs https://arxiv.org/abs/2602.06888 and full HTML https://arxiv.org/html/2602.06888v3 re-read: §5.3 still 2,367 nonempty degree-8 T-schemes. https://arxiv.org/abs/2602.06888v4 is 404. Parent 17/17 still the folder bound. Bound still ≥ 2,384.

2026-08-29 — leftover wrap (q5)

Stopped the even-split BFS remainder and the three odd size-5 shards before either neighbourhood finished. Nothing about degree 8 moved. No q5/certs/new_schemes.json. Combined odd progress 18,340,000 of 37,632,123 evaluations; even-split queue still open at 4,619,029 after 3,554,000 remainder evaluations. Same twelve published M-schemes as the size-4 sweep. Incomplete search, not a lower bound. Bound still ≥ 2,384. ATTACK.md and the README Problems row were left unchanged.

2026-09-04 — freshness check (q7)

The paper moved from v3 to v4 on 31 Aug 2026, but its degree-8 record did not move. Fetched this session:

• abs https://arxiv.org/abs/2602.06888 — v4, 31 Aug 2026, 37 pages, “major revision”
• PDF https://arxiv.org/pdf/2602.06888v4 (saved locally and pdftotext’d)
• HTML https://arxiv.org/html/2602.06888v4 did not expose a usable text extract in this session; the PDF is the source for the numbers below
• data repo https://github.com/dmg-lab/CombinatorialPatchworking and https://github.com/dmg-lab/CombinatorialPatchworking/commits/main

Section 4.3 still gives 2,367 distinct nonempty degree-8 T-schemes and still says the search is not exhaustive, so that figure is a lower bound. 38 of the 89 M-schemes are T-curves; Theorem 21 is unchanged (maximal degree-8 T-curves have $(p,n)\in\{(19,3),(15,7),(11,11),(7,15)\}$); the concluding question is now Question 31 and still leaves 39 of the 89 maximal schemes undecided. The certificate repository main branch still ends at e85b809 (21 Jul 2026, merge of PR #1), so the archived 2,367 certificates are unchanged. Parent verify_new.py still has 17/17 outside that archive. Working lower bound still ≥ 2,384.

2026-09-04 — leftover wrap (q7)

Finished the q5 odd size-5 leftover in C (37,632,123/37,632,123, twelve published M-schemes, neither open nest) and the two tractable (19,3) even components. Further enumerable even components of published M-collections, including the (7,15) neighbour of the hole-map target ⟨5⊔1⟨6⟩⊔1⟨9⟩⟩, stayed on known schemes. No q7/certs/new_schemes.json. The Harnack / Wiman / ⟨17⊔1⟨1⟩⊔1⟨2⟩⟩ even components remain unenumerated (counts $8.8\times 10^{10}$, $2.5\times 10^{11}$, $2.0\times 10^{11}$). Compatible odd 6-tuples number 174,457,166 (count only). Incomplete search, not a lower bound. Bound still ≥ 2,384.

2026-09-08 — q8 record replay

• Opened https://arxiv.org/abs/2602.06888: still v4, 31 Aug 2026.
• Fetched https://arxiv.org/pdf/2602.06888v4 with scripts/arxiv_fetch.py, saved the PDF and read its extracted text. Section 4.3 still reports 2,367 nonempty schemes as a lower bound; Table 1 marks the same six algebraically undecided schemes, and Question 31 leaves 39 maximal T-schemes undecided. Theorem 21 still permits only the four listed parity pairs for maximal T-curves.
• https://arxiv.org/html/2602.06888v4 returned 404; the PDF supplied the full text.
• Opened https://github.com/dmg-lab/CombinatorialPatchworking and downloaded https://raw.githubusercontent.com/dmg-lab/CombinatorialPatchworking/main/deg8.pcoms.txz. The archive matches the local copy byte for byte, SHA-256 1c48052be79722b775f533df27f8604afee03e31f78dc3a5765af7ebd7b94afd. This checks the cited baseline; it is not a claim to have surveyed every later paper on real octics.

The q8 parent replay completed successfully: 2,367/2,367 published certificates and 17/17 prior additions. The new Python/Rust pair also accepted every member of that 2,384 baseline. The new scheme ⟨3 ⊔ 1⟨3⟩ ⊔ 1⟨12⟩⟩ passes both verifiers and is absent from the archive's TYPE fields and filenames and from the seventeen additions. Thus the verified notebook count is now at least 2,385, improving the paper's 2,367 by eighteen in total and the prior notebook count by one. Neither open maximal deep nest is decided.

2026-09-12 — q9 record replay

• Opened https://arxiv.org/abs/2602.06888: still v4, 31 Aug 2026. Comments still "major revision, 37 pages". The abs page does not restate the 2,367 figure; that number lives in the PDF body.
• https://arxiv.org/html/2602.06888v4 returned "No HTML for 2602.06888v4". The v3 HTML remains the last HTML conversion and is not this version.
• Downloaded https://raw.githubusercontent.com/dmg-lab/CombinatorialPatchworking/main/deg8.pcoms.txz (125,932 bytes). SHA-256 1c48052be79722b775f533df27f8604afee03e31f78dc3a5765af7ebd7b94afd, matching the hash recorded in q8 coverage provenance. This checks the cited baseline; it is not a claim to have surveyed every later paper on real octics.

The q8 certificate still verifies, so the notebook count starts at 2,385. Nine further schemes in compute/q9/certs/new_schemes.json are absent from the archive TYPE fields and filenames and from the seventeen plus Q, which is why the verified count is now at least 2,394. Neither open maximal deep nest is decided.

2026-09-20 — q10 record replay

• Opened https://arxiv.org/abs/2602.06888: still v4, 31 Aug 2026. Comments still "major revision, 37 pages". The abs page does not restate the 2,367 figure; that number lives in the PDF body.
• Opened https://arxiv.org/pdf/2602.06888v4. Section 4.3 still reports "2,367 distinct nonempty real schemes as T-curves" as a lower bound. HTML for v4 is not offered.
• Downloaded https://raw.githubusercontent.com/dmg-lab/CombinatorialPatchworking/main/deg8.pcoms.txz (125,932 bytes). SHA-256 1c48052be79722b775f533df27f8604afee03e31f78dc3a5765af7ebd7b94afd, matching the hash recorded in q8/q9 coverage provenance. The archive contains 2,367 .pcom files, 38 of them 22-oval. This checks the cited baseline; it is not a claim to have surveyed every later paper on real octics.

The notebook count starts at 2,394. The selected leaf is the unsearched one-flip neighbourhood of the four q9 followup certificates.

2026-09-22 — q11 record replay

• Opened https://arxiv.org/abs/2602.06888: still v4, 31 Aug 2026. Comments still "major revision, 37 pages". The abs page does not restate the 2,367 figure; that number lives in the PDF body.
• Opened https://arxiv.org/pdf/2602.06888v4. Section 4.3 still reports "2,367 distinct nonempty real schemes as T-curves" as a lower bound. HTML for v4 is not offered; the v3 HTML still has the same 2,367 figure in §5.3.
• Copied deg8.pcoms.txz (125,932 bytes) from the q10 worktree and checked SHA-256 1c48052be79722b775f533df27f8604afee03e31f78dc3a5765af7ebd7b94afd. The archive contains 2,367 .pcom files, 38 of them 22-oval. This checks the cited baseline; it is not a claim to have surveyed every later paper on real octics.

The notebook count starts at 2,407. The selected leaf is the unsearched one-flip neighbourhood of the seven q10 followup certificates.

Eight schemes in compute/q11/certs/new_schemes.json are absent from the archive TYPE fields and filenames and from the seventeen plus Q, N, and the thirteen q10 schemes, which is why the verified count is now at least 2,415. Neither open maximal deep nest is decided.

2026-09-26 — q12 record replay

• Opened https://arxiv.org/abs/2602.06888: still v4, 31 Aug 2026. Comments still "major revision, 37 pages". The abs page does not restate the 2,367 figure; that number lives in the PDF body.
• Opened https://arxiv.org/html/2602.06888v4: "No HTML for 2602.06888v4".
• Opened https://arxiv.org/pdf/2602.06888v4. Section 4.3 still reports "2,367 distinct nonempty real schemes as T-curves" as a lower bound ("as the search is not exhaustive, this is a lower bound"). Theorem 21 is unchanged.
• Copied deg8.pcoms.txz (125,932 bytes) from the q11 worktree and checked SHA-256 1c48052be79722b775f533df27f8604afee03e31f78dc3a5765af7ebd7b94afd. The archive contains 2,367 .pcom files, 38 of them 22-oval. This checks the cited baseline; it is not a claim to have surveyed every later paper on real octics.

The notebook count starts at 2,415. The selected leaf is the unsearched one-flip neighbourhood of the three q11 followup certificates.

All 93 leftover balls finished (1,416,018 evaluations). No scheme outside the 2,415 appeared. Neither open maximal deep nest is decided. The bound is unchanged.