The construction proceeds in layers: hardware → causal order → coarse-graining → inference → “now” → arrow → control objective.
0) Hardware layer (the block)
Let the universe be a Lorentzian manifold with fields:
(\mathcal M, g_{\mu\nu}, \{\Phi_a\})
- : 4D manifold of events
- : metric (GR “hardware”)
- : matter and interaction fields
The invariant interval is:
ds^2 = g_{\mu\nu}\,dx^\mu dx^\nu
For a timelike worldline (an embodied agent’s history),
d\tau = \frac{1}{c}\sqrt{g_{\mu\nu}\,dx^\mu dx^\nu}
defines proper time along the organism’s trajectory. This quantity is not yet “experienced time”; it is simply a geometric parameter in the block description.
1) Causality exists without a “Now”
Even in a block universe, a rigid structure exists: the causal partial order.
Define:
p \prec q \quad \Leftrightarrow\quad q \in J^+(p)
meaning event lies in the causal future of , reachable by future-directed causal curves. This relation is invariant and does not require a global present slice.
Thus, “flow” is not a primitive of physics, but causal precedence is.
The critical observation is that the GUI does not create causality; it produces a one-dimensional navigable representation of a partial order.
2) The GUI operator: from 4D block to 1D narrative
An organism cannot operate directly on . Interaction occurs locally through sensory sampling.
Let the sensory interface be represented by a coarse-graining map:
\mathcal C:\ (\mathcal M, g, \Phi)\ \to\ o(\tau)
where is a stream of observations along the worldline. This step compresses the full spacetime description into a thin causal tube surrounding the agent’s trajectory.
Define the narrative or experienced-time construction as an additional mapping:
\mathcal I:\ \{o(\tau)\}_{\tau_0}^{\tau}\ \to\ (\hat t, \hat x, \hat s)
- : experienced time coordinate (the GUI timeline)
- : perceived state
- : story state (memory, identity, goals)
Interpretation: functions as the GUI renderer, producing a stable workspace corresponding to “here,” “now,” and an ordered personal history.
3) “Now” as a moving integration window
In perceptual and decision systems, the present moment behaves as a finite temporal kernel over recent input. This can be modeled as:
\text{Now}(\tau) \equiv W_\Delta(\tau) = \int_{\tau-\Delta}^{\tau} K(\tau-u)\,o(u)\,du
- : integration window (tens to hundreds of milliseconds, context-dependent)
- : weighting kernel encoding recency bias and attention
Thus the present moment is not a cosmic primitive; it is a buffer.
This formulation reflects the fact that the center of attentional focus exists because perceptual input must be bound into control-ready packets.
4) Causal compression: why the GUI becomes one-dimensional
The block universe supplies a partial order of events. Action selection, however, is more tractable on a total order.
Formally:
- The world provides a poset of events.
- The GUI selects a chain (a totally ordered subset) consistent with that poset.
A chain has the form:
e_1 \prec e_2 \prec \cdots \prec e_n
The organism effectively projects the poset onto a chain induced by its embodied trajectory and memory structure.
The experienced timeline is then defined as an ordering index over agent-relevant events:
\hat t:\ \{e_k\}\to \mathbb R,\qquad e_i \prec e_j \Rightarrow \hat t(e_i) < \hat t(e_j)
This defines “GUI time” as a linearization of causal structure for control.
5) Origin of the arrow: entropy production along the agent’s slice
Let the agent maintain an internal probabilistic model of hidden world states. Belief updating requires dissipative computation and is associated with entropy production.
Define the entropy production rate along the agent’s trajectory:
\sigma(\tau) = \frac{dS_{\text{tot}}}{d\tau} \ge 0
Define experienced time density as proportional to update rate or irreversibility budget:
\frac{d\hat t}{d\tau} = \kappa\, \sigma(\tau)
High entropy production corresponds to denser subjective segmentation of time. Low entropy production corresponds to sparser narrative updates.
This formulation does not assert a universal psychophysical law. Instead, it formalizes the claim that the GUI’s directional structure is anchored to irreversible information processing within an entropic gradient.
6) Memory partitions the block into “past” and “future”
The agent’s internal model separates what is treated as fixed from what remains uncertain:
- Past ≈ stored posterior summaries
- Future ≈ predictive distributions
Let denote accumulated data.
Past memory as compressed sufficient statistics:
m(\tau) = \mathcal M(D_{\le \tau})
Future as predictive distribution:
p(x_{\tau+\delta} \mid D_{\le \tau})
Thus:
- “Already happened” corresponds to information committed to memory.
- “Yet to happen” corresponds to remaining probability mass over predictions.
Temporality therefore emerges as a control-friendly data structure consisting of:
- a write-once (or write-hard) memory buffer
- a rolling forecast
7) Decoherence and classical narrative rendering
At the global level, quantum dynamics may be unitary. Locally, agents interact with environments and lose access to phase information, producing effectively classical records.
In GUI terms:
Decoherence functions as the rendering pipeline that converts quantum amplitudes into stable outcome representations suitable for memory and decision-making.
The GUI is therefore not incorrect; it is a lossy rendering optimized for robustness and control.
8) Control objective (“user”) without homunculus assumptions
The organism’s objective can be formalized as an optimization problem.
Let denote a policy mapping beliefs to actions. Expected utility is maximized:
\pi^* = \arg\max_\pi\ \mathbb E\Big[\sum_{k=0}^{\infty}\gamma^k\, r(s_{\tau+k}, a_{\tau+k})\Big]
The GUI architecture exists because this optimization is tractable only if:
- causality is linearized into a control loop
- “now” functions as a decision buffer
- uncertainty is projected into probabilistic futures
- memory is stabilized into committed past summaries
This yields the central claim:
Experienced time is the agent’s control interface for acting on a causal partial order under entropic constraints.
Compact “core equation” formulation
A minimal mathematical summary is:
\boxed{
\text{GUI-Time: }\hat t(\tau)
=
\int^{\tau}\! \kappa\,\sigma(u)\,du
\quad\text{and}\quad
\text{Narrative} = \mathcal I\Big(\mathcal C(\mathcal M,g,\Phi)\Big)
}
Physics supplies the block universe and causal structure.
Biology supplies the lossy renderer and predictive machinery.
Temporal “flow” emerges as an internal coordinate indexing irreversible updates, not as a fundamental cosmic parameter.
import numpy as np
# -----------------------------
# Constants
# -----------------------------
c = 1.0 # natural units
kappa = 1.0 # GUI time scaling
dtau = 0.01 # proper time step
WINDOW = 20 # "now" integration width
# -----------------------------
# 0) Hardware Layer (Block World)
# -----------------------------
class Event:
def __init__(self, t, x):
self.t = t
self.x = x
# Minkowski metric signature (-,+)
def interval(p, q):
dt = q.t - p.t
dx = q.x - p.x
return -dt**2 + dx**2
def proper_time_step(velocity):
# dτ = sqrt(dt^2 - dx^2) in c=1 units
return np.sqrt(max(1 - velocity**2, 0)) * dtau
# -----------------------------
# 1) Causal Order
# -----------------------------
def causally_precedes(p, q):
ds2 = interval(p, q)
return (q.t > p.t) and (ds2 <= 0)
# -----------------------------
# 2) Coarse-Graining Operator C
# -----------------------------
def sensory_map(state):
"""
Coarse-grain world state -> observation
"""
noise = np.random.normal(0, 0.05)
return state + noise
# -----------------------------
# 3) Now Window (Integration Kernel)
# -----------------------------
def now_window(observations):
if len(observations) < WINDOW:
return np.mean(observations)
return np.mean(observations[-WINDOW:])
# -----------------------------
# 5) Entropy Production Proxy
# -----------------------------
def entropy_rate(old_belief, new_belief):
"""
Simple KL divergence as irreversibility proxy
"""
eps = 1e-9
p = np.clip(old_belief, eps, 1)
q = np.clip(new_belief, eps, 1)
return np.sum(p * np.log(p / q))
# -----------------------------
# 6) Memory Structure
# -----------------------------
class Memory:
def __init__(self):
self.past = []
self.belief = np.array([0.5, 0.5]) # binary hidden state belief
def update(self, observation):
likelihood = np.array([1 - observation, observation])
new_belief = self.belief * likelihood
new_belief /= np.sum(new_belief)
sigma = entropy_rate(self.belief, new_belief)
self.belief = new_belief
self.past.append(new_belief.copy())
return sigma
# -----------------------------
# 8) GUI-Time Integrator
# -----------------------------
class GUITime:
def __init__(self):
self.t_hat = 0.0
def integrate(self, sigma):
self.t_hat += kappa * sigma
return self.t_hat
# -----------------------------
# Narrative Renderer I
# -----------------------------
class Narrative:
def __init__(self):
self.timeline = []
self.states = []
self.identity = []
def render(self, t_hat, now_state, belief):
self.timeline.append(t_hat)
self.states.append(now_state)
self.identity.append(belief.copy())
# -----------------------------
# Simulation Loop
# -----------------------------
velocity = 0.6
tau = 0.0
memory = Memory()
gui_clock = GUITime()
narrative = Narrative()
observations = []
true_world_state = 0.3
for step in range(500):
# Proper time update
tau += proper_time_step(velocity)
# World evolves
true_world_state += np.random.normal(0, 0.01)
# Sensory coarse-graining
obs = sensory_map(true_world_state)
observations.append(obs)
# "Now" buffer
now_state = now_window(observations)
# Entropy production from belief update
sigma = memory.update(np.clip(now_state, 0, 1))
# GUI time integration
t_hat = gui_clock.integrate(sigma)
# Narrative render
narrative.render(t_hat, now_state, memory.belief)
# -----------------------------
# Output Summary
# -----------------------------
print("Proper time elapsed:", tau)
print("GUI Time elapsed:", gui_clock.t_hat)
print("Memory depth:", len(memory.past))
print("Final belief state:", memory.belief)
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