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Bott periodicity, Bott Periodicity Chat GPT, Spin representation combinatorics

The Combinatorics of Bott Periodicity

The essence of Bott periodicity is the uniqueness of God, the default, implicit generator (modeled by the real numbers) as opposed to any combination of explicit generators. In particular, we compare the distinguishability of God and "everything" (the pseudoscalar which is the product of all of the generators). Depending on the mental context, God and everything are distinguishable or not. God commutes with all perspectives and their combinations. When n is odd, everything commutes with every perspective because that perspective commutes with itself and it anticommutes with the remaining perspectives, of which there is an even number. Thus the representation for the Clifford algebra splits into two pieces given by God (scalars) and everything (pseudoscalars).

In human's context, where {$n$} is odd, God and everything are equivalent because the pseudoscalar commutes with every product and thus functions like the identity. Thus we have two representations based on {$\frac{1}{2}(1\pm e_k)$}. But in God's context, where {$n$} is even, God and everything are different because the pseudoscalar does not commute with every product. The relationship between God and human is given by twofold complex Bott periodicity.

The Wedderburn-Artin theorem is based on the facts that {$\textrm{End}(R_R)\cong R$} and that for simple {$I_i$} we have that {$\textrm{End}(I_i)$} is a division ring, and that {$\textrm{End}(I_i^{\oplus n_i})=M_{n_i}(\textrm{End}(I_i))$}.

Twofold complex Bott periodicity is based on the fact that two generators (squaring to {$+1$}) yield the two by two matrices.

A perspective is a distinction between what is known and what is not known. A symbolic expression of a perspective (such as {$1_k+0_k$}) is a syntactic distinction regarding the symbol (what is gets expressed whereas what is not does not get expressed) but also a semantic distinction regarding the referent (what is referred to as meaningful). This double distinction grounds the possibility that the opposing syntax (what is not known) is distinguished semantically as relevant.

When a perspective aligns syntax and semantics, then it expresses what is known, and I call it a concept of the answering mind. When a perspective opposes syntax and semantics, then it expresses what is not known, and I call it a context of the questioning mind.

If we don't consider a perspective, then mathematically we indicate that with {$1$}, the generator for the real numbers.

When we multiply {$e_ke_k$} then we are considering {$e_k$} in these two different ways, syntactically and semantically, yielding either {$+1$} (consistency) or {$-1$} (inconsistency).

Algebraically, the (unconscious, direct) concept is given positive sign and the (conscious, indirect) context is given negative sign.

Clifford algebra recursions

The logic seems to be as follows.

A concept expresses consistency, which is the compatiblity (of syntax and semantics). It squares to {$+1$}. A context expresses inconsistency, which is the incompatibility (of syntax and semantics). It squares to {$-1$}. We can say that truth is consistency, falsehood is inconsistency, and truth and falsehood should go together (yielding consistency). Whereas two complementary truths are inconsistent as are two complementary falsehoods.

A concept should be paired with a context. They are compatible. A concept should not be paired with a concept. They are incompatible. Likewise, a context should not be paired with a context.

Two minimal algebras arise.

If we have two contexts, they square to {$-1$}, and then they are incompatible and thus their product squares to {$-1$}, also a context. This gives a three-cycle of contexts, thus the quaternions.

If we have two concepts {$A^2=+1, B^2=+1$}, then they are incompatible and their product {$AB$} yields a context {$(AB)^2-1$}. But this context is compatible with the concepts, and its product with either squares to {$+1$} and is a concept, namely, the other concept. This algebra is the matrices {$M_2(\mathbb{R})$}. The matrices can be written:

{$A= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}, B= \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, AB = \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix} $}

If we have a concept and a context, then together they yield another context, which yields {$M_2(\mathbb{R}$}.

Truths and falsehoods

We can interpret the generators squaring to {$+1$} as knowns, concepts, consistencies, truths and the generators squaring to {$-1$} as unknowns, contexts, inconsistencies, falsehoods.

Recursion relations as tensor products

We can grow our set of truths and falsehoods by

• Adding two truths. Their product is a falsehood. We tensor each existing statement by their product, changing truths to falsehoods and falsehoods to truths. For tensoring creates a context for a statement: "It is false that X" is true if X is false, and false if X is true.
• Adding two falsehoods. Their product is a falsehood. We tensor by the product, switching the truth value.
• Adding a truth and a falsehood. Their product is a truth. We tensor by the product, maintaining the truth value.

Thus mental contexts - divisions of everything - are sets of false statements. The generators all square to {$-1$}. They are excursions into falsehood.

The sets of statements grow by pairs. A pair of statements yields a shift in perspective.

Growing well founded logic

A truth and a falsehood together yield a truth. This system grounds {$M_2(\mathbb{R})$}.

Then we add an additional truth and a falsehood by tensoring {$M_2(\mathbb{R})$}.

The new collection is: the new truth, the new falsehood, and their product (which is true) applied to the previous statements: "It is true (as the new truth complements the new falsehood) that X (from before)".

Growing the divisions of everything

We add two new contexts (inconsistencies, falsehoods) and their product is also a context (inconsistency, falsehood). Then that product is tensored with a set of truths, transforming them into falsehoods: It is false (as by the inconsistency of the two falsehoods) that X".

X may be an isolated truth as with {$\mathbb{R}\oplus\mathbb{R}$}. This yields {$\mathbb{H}\oplus\mathbb{H}$} and the threesome, whereby the first {$\mathbb{H}$} expresses the pair of falsehoods and the second {$\mathbb{H}$} expresses the falsehood (by the product) of the isolated truth. The latter remakes the product as an independent generator by applying it to an independently grounded truth.

Or there may be a pair of truths as with {$M_2(\mathbb{R})$}. This is a statement of their tension, which means that as complements they generate a falsehood. Then we can add a pair of falsehoods (contexts) and their product (context) tensors with the pair of truths. The latter become two contexts. So this describes the nature of the foursome, how two contexts open the way, through their product (their tension) to remake clashing truths as clashing contexts.

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Creating the linear complex structures {$J_1, J_2, J_3, \cdots$}

The recursion relations give rise to the linear complex structures {$J_1, J_2, J_3, \cdots$}.

{$J_1=i$} in the quaternions

{$J_2=j$} in the quaternions

{$J_3=k\otimes \begin{pmatrix} 1 & \\ & -1 \\ \end{pmatrix}$}

{$J_4=k\otimes \begin{pmatrix} & 1 \\ 1 & \\ \end{pmatrix}$}

{$J_5=k\otimes \begin{pmatrix} & -1 \\ 1 & \\ \end{pmatrix} \otimes i_2$} where {$i_2 = \begin{pmatrix} & 1 & & \\ -1 & & & \\ & & & -1 \\ & & 1 & \\ \end{pmatrix} $}

{$J_6=k\otimes \begin{pmatrix} & -1 \\ 1 & \\ \end{pmatrix} \otimes j_2$} where {$j_2 = \begin{pmatrix} & & -1 & \\ & & & -1 \\ 1 & & & \\ & 1 & & \\ \end{pmatrix} $}

{$J_7=k\otimes \begin{pmatrix} & -1 \\ 1 & \\ \end{pmatrix} \otimes k_2 \otimes \begin{pmatrix} 1 & \\ & -1 \\ \end{pmatrix} $} where {$k_2 = \begin{pmatrix} & & & -1 \\ & & 1 & \\ & 1 & & \\ -1 & & & \\ \end{pmatrix} $}

{$J_8=k\otimes \begin{pmatrix} & -1 \\ 1 & \\ \end{pmatrix} \otimes k_2 \otimes \begin{pmatrix} & 1 \\ 1 & \\ \end{pmatrix} $}

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John Harland's recursion

Define {$A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}, Z=\begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}, X=\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} $}

Let {$K_0= \begin{pmatrix} 1 \\ \end{pmatrix}$}. Define {$K_k = \begin{pmatrix} K_{k-1} & 0 \\ 0 & -K_{k-1} \\ \end{pmatrix}$} for {$k\in\mathbb{N}, k>0$}.

Define {$M_k= \begin{pmatrix} 0 & -K_k^T \\ K_k & 0 \\ \end{pmatrix}$}.

Define {$J_k=\textrm{diag}[M_k]$} as the infinite block diagonal matrix with block {$M_k$}.

We can write {$J_k=A\otimes Z \otimesZ \otimes \cdots \otimes Z \otimes Z = A \otimes Z^{\otimes k}$} and we can embed this as {$J_j=I^{\otimes (k-j)}\otimes A \otimes Z^{\otimes j}$}.

If we multiply {$J_k$} and {$J_j$} then at each place the factors commute except at one place where the factor {$A$} from {$J_j$} is multiplied by a factor {$Z$} from {$J_k$}. At that position, {$AZ=-ZA$}.

Show that {$J_k, k=1,...,n$}, are a set of generators for the Clifford algebra {$Cl_{0,n}$}.

The positive-sign generators of {$Cl_{n,0}$} are given by {$P_k=I^{\otimes (n-k)}\otimes X \otimes Z^{\otimes k}$} and the negative-sign generators of {$Cl_{0,n}$} are given by {$J_k=I^{\otimes (n-k)}\otimes A \otimes Z^{\otimes k}$}.

The reflection {$Z$} expresses consistency and is tensored {$k$} times for the {$k$}-the perspective.

The factor for swapping {$X$} or for rotating {$A$} ensure that we have anticommutativity. Also, {$A$} ensures that we have negative sign upon squaring.

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There are two copies of {$\mathbb{H}$}, the left action and the right action, that commute to yield {$\mathbb{H}\otimes\mathbb{H}\cong M_4(\mathbb{R})$}. Note also that {$M_2(\mathbb{R})\otimes M_2(\mathbb{R}\cong M_4(\mathbb{R})$}.

We have commutation {$i_2i_1=i_1i_2$} and {$i_2j_1=j_1i_2$} as below:

{$\begin{pmatrix} & 1 & & \\ -1 & & & \\ & & & -1 \\ & & 1 & \\ \end{pmatrix} \begin{pmatrix} & & 1 & \\ & & & -1 \\ -1 & & & \\ & 1 & & \\ \end{pmatrix}= \begin{pmatrix} & & & -1 \\ & & -1 & \\ & -1 & & \\ -1 & & & \\ \end{pmatrix} = \begin{pmatrix} & & 1 & \\ & & & -1 \\ -1 & & & \\ & 1 & & \\ \end{pmatrix} \begin{pmatrix} & 1 & & \\ -1 & & & \\ & & & -1 \\ & & 1 & \\ \end{pmatrix} $}

And {$k_1i_2$} gives

{$\begin{pmatrix} & & & 1 \\ & & 1 & \\ & -1 & & \\ -1 & & & \\ \end{pmatrix} \begin{pmatrix} & 1 & & \\ -1 & & & \\ & & & -1 \\ & & 1 & \\ \end{pmatrix} = \begin{pmatrix} & & 1 & \\ & & & -1 \\ 1 & & & \\ & -1 & & \\ \end{pmatrix} $}

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The number of basis elements that square to {$+1$} minus the number that square to {$-1$}

Consider the Clifford algebras {$Cl_{0,n}$} with {$n$} generators {$e_k$} which all square to {$-1$}. The sequence {$D_n$} gives the number of basis elements that square to {$+1$} minus the number that square to {$-1$}.

Consider the Clifford algebras {$Cl_{n,0}$} with {$n$} generators {$e_k$} which all square to {$+1$}. The sequence {$E_n$} gives the number of basis elements that square to {$+1$} minus the number that square to {$-1$}.

{$$D_n=\sqrt{2}^{n+1} \cos ((n+1)\frac{\pi}{4})$$}

{$$D_n= \sqrt{2}^n( \cos \frac{\pi}{4}n - \sin \frac{\pi}{4}n)$$}

{$$D_n=\textrm{Tr}(\sigma)=\sum_{k=0}^n \binom{n}{k}(-1)^{k(k+1)/2}$$} where {$\sigma$} is conjugation

{$D_n$} is the {$+1$}-eigenspace dimension minus the {$-1$}-eigenspace dimension.

{$$E_n=\sqrt{2}^{n+1} \sin ((n+1)\frac{\pi}{4})$$}

{$$E_n= \sqrt{2}^n( \cos \frac{\pi}{4}n + \sin \frac{\pi}{4}n)$$}

{$$E_n=\textrm{Tr}(\tau)=\sum_{k=0}^n \binom{n}{k}(-1)^{k(k-1)/2}$$} where {$\tau$} is reversion

{$\begin{pmatrix} D_{n+1} \\ E_{n+1} \\ \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ 1 & 1 \\ \end{pmatrix}\begin{pmatrix} D_{n} \\ E_{n} \\ \end{pmatrix}$}

This is the matrix form of multiplication by {$1+i$}. It rotates the vector by {$45^\circ$} and multiplies the length by {$\sqrt{2}$}.

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Pairing a concept with a context

Real Bott periodicity comes from the constraint of pairing each concept (expressed as {$1_{k_1}$}) with a context (expressed as {$0_{k_0}$}). This is to say that a concept is not paired with a concept, and a context is not paired with a context. Combinatorially, this means that 00 and 11 have opposite signs and are paired and cancelled away. What survives has the form 10 or 01.

This means that subsets of {$1_1,\dots,1_n$}, each {$1_k$} contributes multiplicative weight {$i$}. Thus there are two tracks, off by one, odd numbers of {$1$} and even numbers of {$1$}, which alternate in sign. This means that the signs are {$+,-,-,+,+,-,-,+,\dots$} or {$+,+,-,-,+,+,-,-,\dots$} or their negatives.

Furthermore, when there is an extra perspective, adding either {$0_k$} or {$1_k$}, then the number of perspectives is odd. Since we can also choose no perspectives, the number of choices {$n+1$} is even, thus divisible by two. The row in Pascal's triangle is symmetric and the number of choices for {$\frac{n+1}{2}$} and {$\frac{n+1}{2}-1=\frac{n-1}{2}$} have the same value. They have either the same sign in the sequence (doubling their individual contributions with constructive interference) or opposite sign (cancelling out their individual contributions, yielding zero through destructive interference).

Thus suppose we have the sign sequence {$+,-,-,+,+,-,-,+,\dots$} where we start with zero.

• The zeroth perspective is the empty string and has positive sign by the sign sequence.
• The first perspective is either {$1$} or {$0$}. The options {$\binom{1}{0}$} and {$\binom{1}{1}$} have opposite sign because {$0$} has positive sign and {$1$} has negative sign. So they cancel away and the total is zero.
• The second perspective has a negative sign. We have {$01$} or {$10$} and they have the same negative sign. Whereas {$00$} and {$11$} cancel away.
• The third perspective yields the options {$\binom{3}{1}$} and {$\binom{3}{2}$}, both with positive sign given by {$1$} and {$2$} in the sequence.
• The fourth perspective has a positive sign.
• The fifth perspective yields {$\binom{5}{2}$} and {$\binom{5}{3}$}, with positive sign given by {$2$} and negative sign by {$3$}, thus cancelling away.
• The sixth perspective has a negative sign.
• The seventh perspective yields {$\binom{7}{3}$} and {$\binom{7}{4}$}, both with negative sign given by {$3$} and {$4$}.
• The eighth perspective has a positive sign.

Thus eightfold Bott periodicity actually arises from the fourfold periodicity of the odd perspectives. The fourfold periodicity comes from considering the two approximate halves of {$k$}, namely {$\frac{k-1}{2}$} and {$\frac{k+1}{2}$}, which are adjacent integers, and considering whether they have the same sign in the sequence (interfering constructively, yielding positive or negative) or opposite signs (cancelling destructively). Which is to say, the fourfold periodicity comes from the ambiguity of whether the extra perspective is attributed to concepts or to contexts, and whether that changes the sign or not, and whether the sign is positive or negative. So it is a fourfold periodicity in the analysis of that ambiguity.

Minimal pattern

The basic pattern is given by {$\binom{3}{0},\binom{2}{1},\binom{1}{2},\binom{0}{3}$} which is {$1,3,3,1$} as {$+,-,-,+$}. So it is rooted in the three minds but also the foursome. But this pattern regards the axes, which is to say, pairs of perspectives.

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Compare and interpret {$(1\pm 1)^n$} and {$(1\pm i)^n$}.

{$1\pm i$} expresses the twosome, the division of everything into two perspectives: opposites coexist {$\pm i$} and all is the same {$1$}.

Bott periodicity expresses the combinatorics of the eighth root of unity {$e^{\pm\frac{\pi}{4}i}=\frac{\sqrt{2}}{2}(1\pm i)$}.

{$e^{\pm n\frac{\pi}{4}i}=\frac{\sqrt{2}}{2}(1 \pm i)^n$}

{$$D_n=\underset{k\equiv 0,3(mod 4)}{\sum_{k}}\binom{n}{k}-\underset{k\equiv 1,2(mod 4)}{\sum_{k}}\binom{n}{k} = \sum_{k=0}^n\binom{n}{k}(-1)^{\frac{k(k+1)}{2}}=\sqrt{2}^{n+1} \cos ((n+1)\frac{\pi}{4})$$}

{$$D_n= \sqrt{2}^n( \cos \frac{\pi}{4}n - \sin \frac{\pi}{4}n)$$}

Explain the cos formula.

{$D_n$} sums the {$n$}-th row of Pascal's triangle weighting the entries with signs {$+,-,-,+,+,-,-,+\cdots$}.

The first values are

{$n$}	{$D_n$}
0	1
1	0
2	-2
3	-4
4	-4
5	0
6	8
7	16
8	16
9	0

We have {$D_{n+4}=-4D_n$} and {$D_{n+8}=16D_n$}.

We also have that {$|D_{2n}|=\sqrt {2^n}$}. The combinatorics swaps sequences with pairs {$00$} and {$11$} leaving {$01$} or {$10$}. This suggests that a concept must come with a context and vice versa. Two concepts would get swapped with two contexts.

I need to explain the sign of {$D_{2n}$}.

Additionally, I need to show that {$D_{4n+1}=0$} and that {$D_{4n+3}=D_{4n+4}$}.