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Abstract. It summarizes a minimal mathematical model that is meant to unpack the consistency among three often conflicting ideas: (1) a block universe view of spacetime, (2) the lived experience of temporal flow, and (3) a form of free will conceived of as genuine openness of future alternatives at the level of experience. The point is not to provide a physical theory, but to demonstrate that such concepts are not structurally inconsistent. 1. The Model. Let x_n be a state variable defined as the interval [0, 1], which gives an abstract measure of “progress” of a universe along a path. Define the initial condition:
x_0 = 0. It evolves according to a fixed rule in the system: x_{n+1} = x_n + (1 - x_n) R_n where each R_n is a value in the interval (0, 1). There is no change of the rule with time. 2. Structural Properties This rule has many important formal characteristics: First of all, x_n is monotonically increasing. Since (1 - x_n) is a non-negative value and R_n is positive, each step moves the state forward without overshooting 1. Second, x_n remains bounded above by 1. The state, under continued iteration, approaches 1 asymptotically. Third, the rule is invariant in itself. The entire evolution is governed by the same law, with no time-related alteration. 3. Block Universe Interpretation, a complete history of the system is defined by the complete sequence:
(x_0, R_0, x_1, R_1, x_2, R_2, ...). If all R-values are fixed, the sequence of x-values has been decided. The history may therefore be represented as one mathematical object: one element of the set of all sequences consistent with the evolution rule. In this sense the model allows for a block-universe reading, where the entire temporal structure is fixed “at once” as an answer to a global constraint. 4. Temporal Experience. Its evolution over time is sequential, from inside a history. The state x_{n+1} is simply a function of the previous state x_n and a new input R_n. An agent with access only to past and current x-values may not infer future x-values uniquely. This creates a structural analogue of temporal flow: a succession of states, memory of earlier states, and uncertainty about future states. 5. Free Will Analogue. The openness of the future in this model comes from the fact that the evolution rule does not specify x_{n+1} from x_n alone. We require additional inputs R_n at every step. These inputs can be interpreted in many different ways: as stochastic variables and hidden parameters, or as agent-level decisions, or as deeper laws not captured in the coarse-grained state description. The model is neutral about their ontology. Formally, they are no more than degrees of freedom independent of the previous macroscopic state. 6. Conclusion. This minimal construction indicates that a single, time-invariant rule can produce a space of complete histories (which supports a block universe view), and each history shows sequential dependence and local underdetermination of the future (supporting temporal experience, as well as a compatibilist analogue of free will). The model does not provide answers for empirical or metaphysical questions concerning the nature of R_n terms, but shows that the three conceptual frameworks are not logically incompatible.
Abstract. It summarizes a minimal mathematical model that is meant to unpack the consistency among three often conflicting ideas: (1) a block universe view of spacetime, (2) the lived experience of temporal flow, and (3) a form of free will conceived of as genuine openness of future alternatives at the level of experience. The point is not to provide a physical theory, but to demonstrate that such concepts are not structurally inconsistent.
1. The Model. Let x_n be a state variable defined as the interval [0, 1], which gives an abstract measure of “progress” of a universe along a path. Define the initial condition:
x_0 = 0. It evolves according to a fixed rule in the system:
x_{n+1} = x_n + (1 - x_n) R_n where each R_n is a value in the interval (0, 1). There is no change of the rule with time.
2. Structural Properties
This rule has many important formal characteristics:
First of all, x_n is monotonically increasing. Since (1 - x_n) is a non-negative value and R_n is positive, each step moves the state forward without overshooting 1. Second, x_n remains bounded above by 1. The state, under continued iteration, approaches 1 asymptotically. Third, the rule is invariant in itself. The entire evolution is governed by the same law, with no time-related alteration.
3. Block Universe Interpretation,
a complete history of the system is defined by the complete sequence:
(x_0, R_0, x_1, R_1, x_2, R_2, ...). If all R-values are fixed, the sequence of x-values has been decided. The history may therefore be represented as one mathematical object: one element of the set of all sequences consistent with the evolution rule. In this sense the model allows for a block-universe reading, where the entire temporal structure is fixed “at once” as an answer to a global constraint.
4. Temporal Experience. Its evolution over time is sequential, from inside a history. The state x_{n+1} is simply a function of the previous state x_n and a new input R_n. An agent with access only to past and current x-values may not infer future x-values uniquely. This creates a structural analogue of temporal flow: a succession of states, memory of earlier states, and uncertainty about future states.
5. Free Will Analogue. The openness of the future in this model comes from the fact that the evolution rule does not specify x_{n+1} from x_n alone. We require additional inputs R_n at every step. These inputs can be interpreted in many different ways: as stochastic variables and hidden parameters, or as agent-level decisions, or as deeper laws not captured in the coarse-grained state description. The model is neutral about their ontology. Formally, they are no more than degrees of freedom independent of the previous macroscopic state.
6. Conclusion. This minimal construction indicates that a single, time-invariant rule can produce a space of complete histories (which supports a block universe view), and each history shows sequential dependence and local underdetermination of the future (supporting temporal experience, as well as a compatibilist analogue of free will). The model does not provide answers for empirical or metaphysical questions concerning the nature of R_n terms, but shows that the three conceptual frameworks are not logically incompatible.