So let’s assume that symmetrization of a set of linearly dependent vectors is a process in which the mutual information is removed and “instilled” into the similarity transform – now tuned into the statistics of the particular ciphertext.
Lets recall the creative arguments of Susskind and mix in the (technically better) explanations of Bata. lets also reflect back on Turing, and his enigma model of quantum mechanics/computing.
Susskind illuminates some context for lambda in a world of branching networks (rather like those found in a series of enigma rotors…) in which the paths through the space can be componentwise-labeled using p-adic numbers and can be differenced accordingly. Since the differential involving the rate/repeats, gamma, of color changes has a solution in terms of an exponentiating generator series, this translates into the world of operator algebras as a similarity transform … in the “basis” of the solution of that particular series limit, characterized by the unknown lambda.
Now Bata does a really good job of proving identities related to lambda, using the commutator algebras (of Turing’s 7-point geometry, the Fano plane). In particular, she shows how the operator V2 bounds the actions of the raising and lower operators.
minute 36.35, see
We have to remember Turing, next. In his argument concerning a search for enigma alphabets that would produce the 7-point geometry he shows who to identity from suitable alpahabets those generators playing the role of I, j k (Sx, Sy, Sz…) – which are permutation cycles built from several identically-wired rotors, now in sequence. Below we get to see the raising operator, pertaining to the raising of |j, m> to |j, m+1>
Thinking back, we also remember m+ and m-, the upper and lower bounds, which act as annihaltors (value 0) when one raises or lowers too far. Now it makes sense that m+ can be set to be negative m-, which is what we saw in Turing’s picture – as his algebra rotates the labels in what physicists would call phase space:
Now we can see how the upper bound α3 relates to the lower bound α-3.
We see how the cycle does NOT “wrap around”