I think I understand pretty well where this is going. I tried to apply that Riemann construction by analogy back in 2015 (from February to April). I remember that in that period I was reading your two Ramanujan-method papers on holomorphic hyperoperations (without fully grasping them). It can be that back than I somehow subconsciously absorbed some concept that was floating in the ether. So in some sense I was not surprised at all when I discovered later that you were working on "composition integral."
Anyways at that time I've momentarily given up that idea because at the time it was exceedingly complex for my mind.
The idea is that there has to be something that fills the following analogies:
Where X/Y are in an analogue relation of Derivation/Integration.
Now, 2021, I feel like after a deep study of your papers I will be in the position to discuss this in a sensible way. But at the moment there is something that feels off with the partition/definition you are giving... I don't grasp what it is yet. My strategy is to understand properly the discrete version first. But my priority is to present you with an extension of the general superfunction trick (SGT) proof to the multivalued case, i.e. the object of your last paper. If I'm successful, I guess we have a solid algebraic foundation where I can understand your results.
Said that, I'm not mad... I'm not going Tate-thesis mode anytime soon... I don't even understand what an Adele is... or an L-function. But with general monoids I merely mean to apply it to (Q,+), (R,+) and (C,+). What I suggest is that the secret key lies, imho, in seeing a monoid like (R,+), that is an abelian group, as a set of points with an arrow \( r\to s \) iff \( r \lt s \) and from that we should rephrase the diagrams of iterated composition In a way that parallels the Riemman construction you are pointing to.
Attached you can see two brief notes I wrote down during an Eureka moment in 2015... I was not even convinced that it made any sense.
![[Image: Delta-Sigma-Calculus-intro.png]](https://i.ibb.co/bXZNtrR/Delta-Sigma-Calculus-intro.png)
![[Image: generalized-Sigma-caculus.png]](https://i.ibb.co/4NP7Y7P/generalized-Sigma-caculus.png)
EDIT: Thank you for these explainations, it will help me alot in moving through your papers.
About your second edit: in the notation \( f\circ g\circ x \) we have that \( x:1\to X \) it's just a function that take as input argument the unique element of the singleton set, thus \( f\circ g\circ x \) is a another function, i.e. x is not a variable/argument but x is A PARTICULAR "element" of X: remember that I want to systematically identify points of X with functions from the singleton to X.
In \( f\bullet g\bullet z \) the z is meant to be an arbitrary \( z\in X \), i.e, an input argument.
For example in your [second iteration, pag 4, first formula] you get the iterated composition evaluated at z=0, implying that \( \Omega_{j=1}^n h_j(s,z)\bullet z \) is actually a bi-indexed family of functions in the z, i.e.
but you just hid the variable in z... or better, you fixed it to be z=0, to make it explicit
About evaluation: evaluation is defined on every function space/set. An evaluation is just a binary operation that evaluates functions.
if we fix the second variable, we get \( {\rm ev}_x:Y^X \to Y \) but if we identify \( X\simeq X^1 \), where the bijection sends a point \( x\in X \) to the function \( \bar{x}:1\to X \), we get that inner composition by elements of \( X^1 \) coincides with evaluation:
Anyways at that time I've momentarily given up that idea because at the time it was exceedingly complex for my mind.
The idea is that there has to be something that fills the following analogies:
Calculus of differences: Differential calculus = Conjugation : X
Discrete Sums: Riemann Integral = Supefunction : Y
Where X/Y are in an analogue relation of Derivation/Integration.
Now, 2021, I feel like after a deep study of your papers I will be in the position to discuss this in a sensible way. But at the moment there is something that feels off with the partition/definition you are giving... I don't grasp what it is yet. My strategy is to understand properly the discrete version first. But my priority is to present you with an extension of the general superfunction trick (SGT) proof to the multivalued case, i.e. the object of your last paper. If I'm successful, I guess we have a solid algebraic foundation where I can understand your results.
Said that, I'm not mad... I'm not going Tate-thesis mode anytime soon... I don't even understand what an Adele is... or an L-function. But with general monoids I merely mean to apply it to (Q,+), (R,+) and (C,+). What I suggest is that the secret key lies, imho, in seeing a monoid like (R,+), that is an abelian group, as a set of points with an arrow \( r\to s \) iff \( r \lt s \) and from that we should rephrase the diagrams of iterated composition In a way that parallels the Riemman construction you are pointing to.
Attached you can see two brief notes I wrote down during an Eureka moment in 2015... I was not even convinced that it made any sense.
EDIT: Thank you for these explainations, it will help me alot in moving through your papers.
About your second edit: in the notation \( f\circ g\circ x \) we have that \( x:1\to X \) it's just a function that take as input argument the unique element of the singleton set, thus \( f\circ g\circ x \) is a another function, i.e. x is not a variable/argument but x is A PARTICULAR "element" of X: remember that I want to systematically identify points of X with functions from the singleton to X.
\( 1 \overset{x}{\rightarrow}X \overset{g}{\rightarrow} Y \overset{f}{\rightarrow} Z \)
In \( f\bullet g\bullet z \) the z is meant to be an arbitrary \( z\in X \), i.e, an input argument.
For example in your [second iteration, pag 4, first formula] you get the iterated composition evaluated at z=0, implying that \( \Omega_{j=1}^n h_j(s,z)\bullet z \) is actually a bi-indexed family of functions in the z, i.e.
\( \Omega_{j=1}^n h_j(s,{-{}})\bullet{-{}}=\phi_n(s,-):X\rightarrow X \)
,\( \phi_n(s,{-{}}):z\mapsto\Omega_{j=1}^nh_j(s,z)\bullet z \)
but you just hid the variable in z... or better, you fixed it to be z=0, to make it explicit
\( \phi_n(s):=\phi_n(s,0) \)
About evaluation: evaluation is defined on every function space/set. An evaluation is just a binary operation that evaluates functions.
\( {\rm ev}:Y^X\times X\to Y \)
if we fix the second variable, we get \( {\rm ev}_x:Y^X \to Y \) but if we identify \( X\simeq X^1 \), where the bijection sends a point \( x\in X \) to the function \( \bar{x}:1\to X \), we get that inner composition by elements of \( X^1 \) coincides with evaluation:
\( \circ :Y^X\times X^1\to Y^1 \)
\( f \circ \bar{x}={\bar{{\rm ev}_x(f)}} \)
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)