This is closely related to this post. Conundrum of the self?
And this one. Can a persons continuity of consciousness be broken and a new person arise?
This is the philosophic question of personal identity.
As far as objections to the argument (substituting your definitions):
P1. [The continuous feeling of subjective identity] is static in [the continuous unfolding of experience in time].
P2. [The continuous unfolding of experience in time] cannot be [the continuous feeling of subjective identity] otherwise the [the continuous feeling of subjective identity] wouldn't be static.
C. Therefore, [the continuous unfolding of experience in time] is [is a cause of] a [continuous feeling of subjective identity] is identifiable.
The best objection:
- The continuous feeling of subjective identity actually can't be static. Most people who accept who we are and how we identify changes over time. (See existentialists like Sartre on Being and Nothingnessor Maslow on self-actualization) Do you really identify the same as your 8-year old self? Doubtful.
That derails the entire chain of reasoning, since if P1 is false, and P2 purports to be true conditionally on P1 (and not independently), then one doesn't have a sound or valid argument to reach C.
Also, C can be objected to on the [premise that experience over time] is neither necessary nor sufficient for subjective identity. Perhaps more is required than mere experience over time, for instance if experience which implies consciousness is not structured in a certain way, for instance conceptually, then consciousness will fail to achieve a subjective identity. You invoke 'feeling', but perhaps 'thought' would be a broader term as we are more than our feelings. Concepts and inferences saturate our experience, and the argument fails to list them as a necessary condition. The very notion of "subjective identity is identifiable" cries out for suitable definition. Here too might another necessary condition be found. What does (presumably) subjective identity identifying self mean? Is that metacognition, and not mere cognition? Is therefore besides conceptualization self-awareness required? It would go a long way to explaining how there is a gap between us and our fellow hominids intellectually. But then again, perhaps not.
What about the possibility that subjective identity actually is responsible for experiencing time? That causality also extends in the other direction, perhaps, each process reinforcing the other? Phenomenologists tend to see subjective experience as the foundation for all else, unlike those of us who tend towards analytical philosophy and tend to reduce the mental to the physical as far as it is appropriate? The infamous Horseman of New Athiesm Daniel Dennett famously reduces all mental to the physical in a form of eliminative materialism. Remember, your argument presupposes metaphysical assertions.