The Memory Clock: A Counting Law for Subjective Time, Three Routes to "Fast and Sparse", and the Shared Clock One counting law, two propositions, one group-level quantity and a set of registered predictions

Abstract Why are some years long in retrospect while others pass in a blur? This paper writes the retrospective clock as a count: the length of a period in memory is set by the number of distinguishable "firsts" in it. The unit is the switch of process context — what one is doing, where, with whom and in what role — not items of content and not depth of attention. Within a stable regime this count follows the Ewens formula E[Kn] = α[ψ(α+n) − ψ(α)] ≈ α·ln(1 + n/α). Doubling the intake adds only about α·ln 2 firsts; a regime switch resets the count. A regime switch is a memoryless reset, one of the three anchors that pin a multiplicative process. At illustrative parameters, doubling the intake raises the annual number of firsts by 9.7 percent, while doubling the diversity of contexts or the switching rate raises it by about 80 percent each; doubling both raises it by 219.8 percent, more than the 159.8 percent sum of the separate doublings. Two propositions follow. Proposition 1: the price of an hour in retrospect equals the new tables it opens directly minus the switches it crowds out. The first term falls to zero in a mature regime as α/(α+n), so any positive crowding makes the price negative; the lever that changes the sign is the regime, not the intensity. Proposition 2: "fast and sparse" has three routes — input poverty, encoding collapse and the decay of switching. The three read the same on temporal fidelity Fτ = I(τ; m)/H(τ) and are separated only by two perturbations: adding a new context, or withdrawing the feed. At the group level the paper defines the shared clock: the number of landmarks remembered in common by a substantial share of a cohort. It is not an average of individual readings; personalization can drive it close to zero while individual counts stay unchanged. From this follow a ratio prediction for the form of religion during involution, and a set of hypotheses for the period after AGI substitutes for work: price, meaning and time converge on synergistic occasions that require presence. Eight registered predictions close the paper. One point must be stated at the outset: the model's core empirical assumption comes from laboratory studies measured in minutes, and its extrapolation to autobiographical memory measured in years is untested; the registered predictions exist to test that step.