Very boring essay. Such a filling of the maximum allowed length for so little... maybe I'm just already too familiar with the fact that mathematics is the study of structures to have any interest reading it again, but...
Despite your try to develop a visual metaphor to illustrate the question of discovered vs. invented, you did not even come up with any decent answer to this question. Your illustration by points of a line is more confusing than explaining, since if you take a physical line made of aligned atoms, a physical point in this line as defined by a particular atom can only encode a few words, not even an ordinary sentence. To encode whole texts you need a mathematical line, far from anything concrete or visual (would you qualify this as visual ? I don't). There are not even enough atoms in the visible Universe to be in bijection with all possible meaningful texts of 1 page length. But a decent answer is possible as I explained in my essay : by making the difference between mathematical existence (where all possibilities exist but can be lost in a huge pool of alternatives as soon as they are a bit complex) and the conscious act (or contingent event) of pointing out a particular possibility here and now (which is what the notion of "brilliant text" is actually about).
"Some may hope that there are things in the universe which can't be described by mathematics. But can you name those things? To name them, you would have to provide a list of their properties, of propositions which hold for them. If the universe is describable by a list of propositions, then there is a mathematical structure describable by the same propositions. But then, couldn't we find something to say which is true about our universe, but not about the mathematical structure? The answer is no."
Example : the sensation of the red color. I can name it (as I just did). I can list its properties : this is the empty list, since it does not have any mathematical structure. Or maybe I can say that I do not like this color : would you classify it a property of this sensation ? We can also say that it is the color of blood. However, if the sensation of the red color is a property of the sight of blood, I doubt the relevance of this link as a description of the sensation of red itself. So, some things can be said about the sensation of the red color, which does not mean that it admits any mathematical description as a mathematical structure.
"If we can't [describe it completely by a list of true propositions], it's only because of practical limitations."
Disagree: if I can't describe the sensation of the red color completely by a list of true propositions, it is not because of any practical limitations, but on the contrary because the list of mathematical structures it is made of is trivial: an empty list.
"We know what a feeling is: some chemistry of the brain"
Disagree. The sensation of the red color surely has neural correlates which can be described mathematically, however these mathematical structures will never account for what this sensation actually is. And it is found in NDEs that many feelings occur far away from any chemistry of the brain. As for the idea that a feeling is some chemistry of the brain, it is still pure speculation from a scientific viewpoint. Feelings must have correlates in the brain of course, but these concepts of chemical correlates in the brain did not achieve any actual scientific understanding of what feelings are and how they work, and I think they never will (if psychiatrists think they do, they are just hallucinating).
"any kind of world, as long as it is free of contradictions, is isomorphic to a mathematical structure"
Unfortunately, this claim looks much less like an expression of amazement at how deep mathematical concepts are involved in physics, than like an expression of lack of imagination to consider any other possibility. In my essay I explained how I consider the world as not a mere mathematical structure since it includes the non-mathematical component of consciousness, even if mathematics takes a large part in it.
"...maybe the universe obeys two or even more sets of laws. This doesn't make much sense, since if the universe obeys two or even more independent sets of laws, there must be two or more disconnected mathematical structures modeling them. But we can't live simultaneously in two disconnected worlds."
Looks like you never heard of any intermediate possibility for 2 sets between being equal or disjoint. I see no contradiction in having a world made of a combination of the fundamentally different ingredients of maths and consciousness, as I described in my essay.
You don't even seem to understand what is Godel's incompleteness theorem actually saying. You wrote: "To obtain an inconsistency, we should make the physical laws assert their own undecidability". What the incompleteness theorem says, is that "To obtain an inconsistency, we should make a mathematical theory able to express arithmetic, stating (among its theorems) its own consistency". I admit that your claim is rigorously correct, since, actually, and still according to this incompleteness theorem, the claims of consistency and incompleteness (in a theory able to express arithmetic) are logically equivalent. But... the reason for the correctness of your formulation is so indirect that it makes things even more twisted to figure out than they basically are. Especially given your previous sentence: "If a man states the undecidability of some problem in physics, would this introduce an inconsistency in the universe? No, since the statement can simply be wrong". If consistent theories able to express arithmetic are unable to prove the claim of their own undecidability, it is not because this claim can be wrong, but on the contrary because it is right: these theories are undecidable, which is why they are unable to prove some true facts, such as the fact of their own undecidability.
And I see no justification for your implicit assumption that the laws of physics should be able to express arithmetic, as I disagree with this claim (see my essay).
"the only prediction made by the mathematical universe hypothesis is that our universe has to be Turing complete"
I answered about the issue of predictions of the MUH in reply to Marc Séguin's essay.
In your previous essay "Flowing with a Frozen River" you described a possibility for free-will in a way that seems to be the same as I support, except that you seem to attribute free will to a retroactive effect on the initial conditions. Maybe because you use your own variant of quantum mechanics, "Smooth quantum mechanics", while I accept quantum mechanics in its standard formulation, and I attribute randomness to the event of wavefunction collapse by conscious observation, which I qualify not as a physical event but a metaphysical one (so that the discontinuity is not something physical, it is not located in the physical space-time). It looks like your interpretation is just a twisted rewording of the "discontinuous collapse of the wave function" into a "retroactive discontinuous collapse of the initial conditions" from which the present state would be a posteriori re-determined by the continuous quantum evolution, and which is just mathematically equivalent to the former, but only coming as an illusory way to deny that anything here is discontinuous.
So it looks like, your special way of wording your interpretation through this mathematical reformulation is just hiding the fact that we have essentially the same interpretation, which I invite you to read in my essay.
But... no, there is still a difference, by which I would qualify your version as incoherent: the problem I see with your interpretation is that the new observations do not just complete the previous ones, but can also be incompatible with them (in the sense of non-commutation). Thus, the new initial conditions are turning past states into thermodynamically incoherent states (such as with the story of "liar states" which I commented there), retrospectively changing past clear observations into indeterminate ones, and finally, changing the big bang into a combination of initial conditions mixing the smooth big bang that normally explains things (the thermodynamic time arrow) with highly chaotic initial states with multiple singularities and so on, where the thermodynamic time arrow cannot be found anymore.
In your text "Modern physics, determinism and free will", you make a distinction between "branching time" and "choice time". But what is the role of a "branching time", that would not be the same as "choice time" ? I cannot see a role for it. Nothing in the formalism of quantum physics, speaks about "branching" as a fundamental event. Proponents of the Many-worlds interpretation saw it well, as they just dismissed the existence of any branching, to conclude that different possible measurement results keep coexisting, not really as branches from any specific branching event (as might be intuitively said for approximative descriptions), but as emergently separable components of the unitarily evolving state, which remains a unique physical state that only happens to be equal to a combination of these practical measurement results (without being directly affected by this fact). As David Wallace wrote : there is no such a thing as a well-defined "number of branches". Instead, in my view, all what plays the role of a branching time, is what you call the choice time; it needs not be retroactive, but only non-local (see details). So, since other interpretations (Bohm and Many-worlds) just deny the existence of any special measurement times at all, I think that introducing 2 different special times, one for branching, the other for choice, is a bit too much.