Dear Philip,
Thank you for reading and commenting my essay.
Your question "Symmetry of space and time means that the laws of physics are unchanging over space and time. If that were not the case it would be hard to do science. Does this mean that symmetry must be fundamental?聽" is linked to the approach you have in your essay.
"symmetry is agebraic" you say, so it is part of a "language" that is an intermediate between thinking and reality (both emerging phenomena). In my approach fine-tuning is an essential result of the Reality Loop the agent is part of. (if its was not fine-tuned the agent would be a different agent in a different reality loop. One of the languages agents are using to explain this fine-tuned reality is emerging algebra (symmetry).
You argue "I expect to find this symmetry in a pre-geometric meta-law that transcends spacetime,taking a purely algebraic form, only beyond that point will it be emergent, rising from immutable relationships between systems of information." Indeed in this approach symmetry transcends space-time because space and time are (dimensional) restrictions (emerging from total simultaneity), and algebra/symmetry/thinking are not limited by these restrictions because they are the "cause" through consciousness of reality. The what you are calling "immutable relationships between systems of information" is maybe too strictly bound to our emerging reality. My approach places the "rising" outside our reality, so even more foundational.
"If those leaders say that symmetry is unimportant because it is emergent or that geometry is more fundamental than algebra, other possibilities may be neglected."
Fully agreed, every emerging phenomenon is essential in a specific reality. Geometry is a description methodology, to be compared to filling in data in a computer, it is the software (thinking) that is concluding.
Best regards Philip nd good luck in the contest.
Wilhelmus