Eight new realizable triples of totally, partially and minimally real common secants to a pair of real twisted cubics
Abstract
A general pair of twisted cubic curves in ℙ³(ℂ) has ten common secant lines. For a real pair each real common secant is totally, partially or minimally real according to whether it meets the two curves in two real points each, in two real points and a conjugate pair, or in a conjugate pair each; the resulting count (nₜ,nₚ,nₘ) takes 161 admissible values. Aslam, Faust, Hauenstein, Lopez Garcia, Kagy, Regan, Wampler and Zhang recently realized 128 of the 161 by numerical sampling with certification, listed the remaining 33, asked for the full realizable set, and singled out two families of 21 admissible 3-tuples as an open problem. We realize eight of the 33: (4,6,0), (4,5,1), (4,4,2), (3,5,2), (3,7,0), (2,8,0), and, in the two families of the open problem, (0,10,0) and (1,9,0). Each is realized by an explicit pair of rational twisted cubics given by a 4 × 4 integer matrix, and each claim is unconditional: for every one of the nine explicit pairs treated here we also prove that the ten common secants we exhibit are all of its common secants over ℂ. The certificates are exact rather than numerical, and are of two kinds: elimination reduces the common-secant system of one pair to a single degree-10 integer polynomial with a rational univariate representation, from which the ten secants and their types are read off ten integer sign changes and twenty Descartes sign-pattern certificates; and a second, chart-free model in the plane of chords of the first curve, where the common secants are the rank-one locus of a symmetric 3 × 3 matrix of quadratic forms, supplies the exact count through polynomial identities over ℤ. Every step is checked by the Lean 4 kernel in arbitrary-precision integer arithmetic. We correct one statement of the earlier version of this paper, which asserted that neither the source nor Eisenbud and Harris bounds the number of common secants of a special pair: the proof of Proposition 2.2 of the source does bound it, for one explicit special matrix, by a saturation computation returning degree 40. We also record an erratum to the printed bisecant system of the source; report what the public data repository of the source now contains, which changes the status of two further cells; and show that no invariant of the ambient isotopy class of the pair of real curves can constrain the 3-tuple.
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Archived files
- Version 2 · current (opens in a new tab)
Source snapshot 2026-09-07 03:53 UTC
File fingerprint
52f3859b381ff3a12c660be794cfe4d9626f1a489a6cfe42f6e1bce342c12cef - Version 1 · earlier file (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
2f27ac0e1bc0736dcdb75697452e0939d37a129e3399dbc459a575c3ef5a68e3
Claim ledger
Stated results
ST1candidate2026-08-30
(4,6,0) is a realizable 3-tuple of common real secants to a pair of real twisted cubics
ST2candidate2026-08-30
(4,5,1) is a realizable 3-tuple
ST3candidate2026-08-30
(4,4,2) is a realizable 3-tuple
ST4candidate2026-08-30
(3,5,2) is a realizable 3-tuple
ST5candidate2026-08-30
(3,7,0) is a realizable 3-tuple
ST6candidate2026-08-30
(2,8,0) is a realizable 3-tuple
ST7routine2026-08-30
(7,2,1) realized by an explicit integer matrix, kernel-checked
ST8prose2026-08-30
The printed f₂ of arXiv:2603.25003v1, equation eq:bisecant-system, is a transcription error
This ledger entry is reported in prose and is not bound to a Lean theorem.ST9measurement2026-08-30
(1,1,6) and (1,0,7) realized by explicit integer matrices (type split floating-point)
This ledger entry is reported in prose and is not bound to a Lean theorem.ST10measurement2026-08-30
No pair with n_R = 10 and nₜ <= 1 in 470,000 classified configurations
This ledger entry is reported in prose and is not bound to a Lean theorem.ST11routine2026-09-02
Each of the seven configurations has exactly ten common secant lines over C, so the ten certified in Realize.lean are all of them
ST12routine2026-09-02
The common-secant locus of each of the seven pairs misses the line a = 0 of the chord plane: no common secant meets C0 at its point of parameter infinity, so the locus is finite
ST13prose2026-09-02
Convention conv:ten of this family's v1 paper overstates: arXiv:2603.25003v1 does bound the number of common secants of a special pair, in the proof of its Proposition 2.2
This ledger entry is reported in prose and is not bound to a Lean theorem.ST14candidate2026-09-02
(0,10,0) is a realizable 3-tuple of common real secants to a pair of real twisted cubics
ST15candidate2026-09-02
(1,9,0) is a realizable 3-tuple
ST16prose2026-09-02
An explicit rational segment of pairs of real twisted cubics along which the link C0(R) u C1(R) in RP³ is constant up to ambient isotopy while nₜ jumps from 0 to 1
This ledger entry is reported in prose and is not bound to a Lean theorem.ST17prose2026-09-02
The linking number of C0(R) and C1(R) in RP³ is the homological constant 1/2 in Q/Z for every pair of real twisted cubics, and H₂(G(2,4)(R);Q) = 0
This ledger entry is reported in prose and is not bound to a Lean theorem.ST18prose2026-09-02
The source's public data repository realizes (1,0,7) and (1,1,6) after v1, and still records zero samples of (0,10,0), (1,9,0) and the six cells of rows ST1-ST6
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- A twisted cubic is a smooth rational curve of degree three in P³; the standard one is C0 = [1: t: t²: t³], and every twisted cubic is M · C0 for some M ∈ PGL₄. Two *general* twisted cubics in P³(C) have exactly ten common secant lines — a classical Schubert-calculus count (3264, Ch. 3 Keynote Question (c) and Prop. 3.14).
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7