The corpus's nano-scale row leans on string field theory and on a self-dual point at R = 1. Four passages in Kaku's book and three papers from a workshop in his honour are what can be held about the man behind both — one credit corroborated, one still resting on a collaborator's word.
Two layers, kept apart. The first is Michio Kaku's account in Parallel Worlds, which is a collaborator's. The book's own index lists Keiji Kikkawa on four stretches of pages, and the script confirms the sentences below are on them. Page numbers are the book's; the PDF page is the book page plus 19.
| Book page | What Kaku says he did |
|---|---|
| 190 | At the University of Wisconsin, with Bunji Sakita and Miguel Virasoro, realized that the S-matrix could be seen as an infinite series of terms, with the Veneziano model only its first term. |
| 191 | In 1974, with Kaku, extracted the field theory of strings; the book gives him as “of Osaka University”. |
| 209–210 | With Kaku, using string field theory, catalogued the interactions of type I strings — five — and showed closed strings need only one; and showed that theories of closed strings alone are self-consistent, which the book says are today called type II. |
| 237 | In 1984, with his student Masami Yamasaki, first found the duality between a circle of radius R and one of radius 1/R. |
The second layer is three papers the author supplied, all presented at a workshop in Osaka, 14–16 December 1995, in honour of Kikkawa's 60th birthday. They are independent of Kaku and of one another in what they cite:
| Paper | What it holds about him |
|---|---|
| J. H. Schwarz, M Theory Extensions of T Duality, hep-th/9601077 | Kikkawa and Yamasaki discovered T duality, for closed strings compactified on a circle (Phys. Lett. 149B (1984) 357). The 1969 Kikkawa–Sakita–Virasoro paper (Phys. Rev. 184, 1701) introduced regarding dual-resonance n-point functions as the tree approximation of a unitary quantum theory. Schwarz and Kikkawa studied the Regge behaviour of the loop amplitudes of that model (Phys. Rev. D1 (1970) 724). Schwarz: on several occasions Kikkawa pioneered concepts that led to advances in string theory. |
| H. Ooguri, String Solitons and Singularities of K3, LBL-38591 | Cites the same 1984 paper (with the surname misspelt in its list). |
| T. Kubota, The Rubakov–Callan Effect and Black Holes, hep-th/9603079 | Writes from the Department of Physics, Osaka University; was Kikkawa's research associate there from spring 1983; cites Kikkawa, Kubota and Song, Prog. Theor. Phys. 71 (1984) 1346, on the finite-temperature monopole condensate. |
A namesake, so nobody merges the two: book6/ch07-microtubule-fibonacci cites an M. Kikkawa, a J. Cell Biol. paper on
the microtubule seam. Same surname, different field; the script checks the citation is there and this page does not connect them.
The F row of the nano-scale column in chGravity-scales.html says: string interaction vertex: A₁ =
emission, A₂ = splitting, A₃ = four-string junction; ADE singularity hierarchy in scattering amplitudes.
That names three vertices. The book above says type I strings have five interactions and closed strings one. Three, five and
one are three different lists.
They do not have to conflict. The corpus's A₁–A₃ are singularity types in a fold hierarchy; the book's five are terms in a field-theory action, and one is not obliged to be the other. But nothing on the corpus page says which it means, and the mapping is exactly the kind a reader will assume. Under R9 a count that collides is recorded and left to the author, so this chapter does not choose:
The same row's column header says the theory has a self-dual point at R = 1. Kaku's description is one line: wrap a dimension into a circle of radius R, or one of radius 1/R, and the two theories are the same. Schwarz is more careful about what it relates: type IIB on a circle with type IIA on a circle of reciprocal radius, and the two heterotic strings likewise. For those pairs the dual of a theory is a different theory, so the nano row's “self-dual point” needs to say which string it means; the row does not. The script confirms, for the standard closed-string zero-mode formula (CITED; Schwarz states only the reciprocal-radius statement), that the spectrum at R equals the spectrum at 1/R with the two integers swapped, for 49 ratios, and that R = 1/R has one positive solution.
On the log axis that is a reflection: $\log R \mapsto -\log R$, an involution with exactly one fixed point. A scale ladder such as Kardashev's is the opposite kind of map: one rung is $+10$ in $\log_{10}$ of power (the paper's own are 14 and 11), it has no fixed point and it only ever moves one way. The scales column carries a ladder and a self-dual point on the same nano column; whether they are one structure is a model question and is not claimed here. The unit is also open: the script's R = 1 is the string length, and Parallel Worlds states the circle sizes against the Planck length (OPEN).
Kaku is the chapter that found the gap and holds the level-N counts the same row needs; Kardashev is the ladder side of Part III; the row itself is chGravity-scales. The three chapters share one measurement — before them, no page of the corpus named Keiji Kikkawa or the Kardashev scale — and one open question about what the nano-scale column's rungs are.
| Held | M. Kaku, Parallel Worlds — PDF, 447 pages, md5 55ac8f9f…; the four passages above. |
| Held | J. H. Schwarz, hep-th/9601077 (md5 faf6ab6a…); H. Ooguri, LBL-38591 (e06e75e4…); T. Kubota, hep-th/9603079 (10f05247…). |
| Cited by the held papers, not held | Kikkawa, Sakita, Virasoro, Phys. Rev. 184 (1969) 1701 · Kikkawa, Schwarz, Phys. Rev. D1 (1970) 724 · Kikkawa, Yamasaki, Phys. Lett. 149B (1984) 357 · Kikkawa, Kubota, Song, Prog. Theor. Phys. 71 (1984) 1346. |
| Wanted | The 1974 string-field-theory papers Kaku describes — no held source cites them. |
| Wanted | A textbook for the compactified closed-string mass formula. |
| Corpus | chGravity-scales.html — the F row and the R = 1 header. ch-kaku, ch-kardashev. |
| Script | ch-kikkawa-verify.py — every sentence and page above; SKIP, not PASS, without a source. |