Turing’s original statement, of theorem, was
Finally, ive understood the last phrase – and how his proof shows its true. Let’s interpret.
The limiting distributions for the quantum operator f are constant – i.e. have uniform probability distribution, which means that all eigenvectors are equally probable, upon entangling the measuring devices with the state of the system, in phase space. There is a degenerate system, in which all vectors correspond to a single eigenvalue.
A certain self-conjugate subgroup of H is H1 – i.e. the world sphere (H1) within the Egyptian pyramid (H). For points on the edge of that H1 planet (i.e. the unit circle/sphere), one can go both forward and back in time, since the self-conjugacy means that the dual space (positivity is looking to the past) is operated on just as in the forespace (positivity is looking towards the future) : in a hermitian manner.
When we model the state of the rotor cage as a sequence of swaps/reflections (U) and settings/rotations (r), we have a unitarity condition, furthermore. In this case, the average of the energy (modeled H1) is conserved (modeled by the zerorule for the rotation sum) – which is a different notion to the energy being conserved.
As susskind says, from the rule conerning the differential of the average value of any operator over time, we can ask what is change in the average of the hamiltonian operator (the energy generator) with respect to time. This is proportioned to the commutator of H with itself – which is zero. In enigma math, we get, equivalently, that the sum of the rotations (applied to each generator term Ur), for each positive and negative direction of rotation is zero.
So, the sum constraint is a type of differential equation, for averages. And this makes it easy to see the connection with limiting distributions which have actually achieved the unitarity condition – once the limitin distribution has reached constant change (in all directions). Obviously, that “limiting” average change in the energy flow is “no-change” –at at which point any futher motion is really about changes of phase, only.
Concerning cosets, now, why is the argument all about “1 coset”?
Well we can see the Turing’s notation does supports this, now.
First he establishes by (4) that the probability is constant through each coset (and can be constant proportion of 1, or 0)
Then, for the particular coset of H1 by some 1 generator term to the mth power (think of M as in Susskinds sense, where L is angular momentum of wheel motion), we know that only in this coset is the generator’s energy conserved – i.e. the operation is closed and the computed point stays within (and indeed on) the normal circle of the geometry.
So just as in expander graphs and zig-sag products, he uses the commutators from the non-abelian components of the geometry to adjust the operator norm, in a monotonic manner, till he reaches the abelian subgroup, at which point the average of further energy change is zero. In Eqpytion thinking, one logically (and only logically, since bricks block light) ) passes from the mortal plane surfaces of visible pyramid onto the surface sphere surrounding the center/tomb, wherein lies the body of the pharoh supporting his dis-embodied spirit (distributed now in non-localized form on the surface of the sphere). Remember, Turing things of heisenburg uncertaintly as being measured by the “amount” of non-commutativity. This is Turing’s version of the Shannon capacity.
It’s a more crypto-centric conception that the American Shannon conception (which hid all the quantum crypto roots of that tradition). Now we see that reducing the gap to capacity is the same as ensuring that the change in all directions is constant – which means that the markov conditions are true within the ciphertext, bit over bit, and are also conditionally true (thinking more in differential cryptanlaysis terms) with respect to the properties of the plaintext and the key.
Ok. Thanks Turing. We see now how differential cryptanalysis, in the era of rotor machines and Friedman-era thinking was thought about – 30/40 years before DES and IBM.
Thanks Egypt too. We can see how the geometry of positioning of the Queen’s tomb works now. Presumably, if we now work in hyperbolic space, we will see how the King’;s tomb is also a center.