Möbius Bonding I: The Ubiquity of Bo
Year: 2026
Type: article
Abstract: In A. L. Baker's algebra the bo, bo = (1 + 𝐤)/√2, is a quarter turn, and two bos make the half turn bo · bo = 𝐤. A bo is equivalent to a (-Bo). A Bo is equivalent to an electron. Acting on any vector v perpendicular to 𝐤, the half turn gives 𝐤 v 𝐤⁻¹ = −v: a vector carried once through a half twist returns reversed. This is the Möbius property, and it is the bonding operation called here Möbius bonding. The same operator recurs at every scale the author's work has reached. Physics. The bo is the half-charge pole of the strong-force series. Two bos turned against each other, bo · bo = 𝐤, form the flipped complex that carries the proton–neutron bond. The bo's own powers, bo⁴ = −1 and bo⁸ = +1, are the rule that a frame must turn twice to return. Chemistry. The same half turn is the closure operator of a ring of orbitals. It decides whether a ring is stable at 4n + 2 or at 4n electrons. It also decides which pericyclic reactions run thermally and which only under light. Genetics. A nucleotide is the bo of its base, (1 + b)/√2 = b¹ᐟ², just as bo = 𝐤¹ᐟ². Base pairing is the half turn 𝐤 b 𝐤⁻¹, so every Watson–Crick rung is a local Möbius bond. Baker's mean reversion 𝐢, half of the reversion that the bo · bo half turn produces (𝐢² = −1), is the half step that sets DNA's 10.5 base pairs per turn. One operator, a turn by two bos, thus bonds nucleons, closes aromatic rings, governs reaction pathways and pairs the bases of DNA. Five results are reported. (1) The closure operator decides aromaticity. A ring of N coupled orbitals is closed by an operator c: the amplitude after one circuit is multiplied by c. Computed for every even ring from 4 to 30 sites, closure c = +1 gives closed bonding shells at 4n + 2 electrons, Hückel's rule. Closure c = −1, the bo · bo half turn, gives closed shells at 4n, the Möbius rule. For a ring with L half twists the closure is (𝐤 · 𝐤⁻¹)ᴸ, i.e., (−1)ᴸ, so the electron rule is set by the parity of L alone. (2) The rule matches demonstrated chemistry. Planar annulenes follow 4n + 2; the Möbius hydrocarbon of Herges (2003) has 16 π electrons (4n); a Hückel–Möbius switching porphyrinoid adopts its single-half-twist conformer at 28 π electrons (4n); and the double-twist hexaphyrins (L = 2) are aromatic at 26 π electrons (4n + 2) and antiaromatic at 28 (4n), exactly as (−1)ᴸ requires. (3) Möbius closure and base pairing are one operation. The half turn that closes a Möbius ring, 𝐤 v 𝐤⁻¹ = −v, is the operation that takes every DNA base to its Watson–Crick partner in the author's genome theory (A = +𝐢 ↦ −𝐢 = T, G = +𝐣 ↦ −𝐣 = C). The rung of the double helix is a local Möbius bond. (4) The pericyclic selection rules follow from the same closure. A pericyclic transition state is a ring whose antarafacial components each insert one half twist. Registered as an either/or test, the rule "thermally allowed iff (−1)ᴸ matches the electron count; photochemically, excitation adds one reversion" agrees with all 21 electrocyclic, cycloaddition, sigmatropic and cheletropic reactions scored. Photo-excitation is one Baker reversion. (5) The mean-reversion closure is a third class. Closure by Baker's mean reversion, c = 𝐢, removes every degeneracy: a ring closed this way has a closed shell at every even electron count and no aromatic alternation. In DNA the same mean reversion is the half step that turns 10 into 10.5 base pairs per turn. All quantities are pure numbers; every relation is dimensionally homogeneous.
Author: Willie Johnson Jr.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225792352
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