Definition 1.1 (chrono-debt spectrum)

The survival functions of Lemma 1 induce a defined object that travels alongside the lemma as the framework's signature instrument. Fix a baseline policy ${\pi }_{\circ }$ inducing survival function $S(\tau ;{\theta }_{\circ })$. The chrono-debt at scale $\tau$ at time $t$ is the difference between the two survival functions; the spectrum $D(t,\cdot )$ is the function of $\tau$.

The aggregate chrono-debt is the spectrum integrated against a weighting function $w$ — by default $w(\tau )\propto \tau$, giving longer foreclosed windows proportionally more weight.

What the spectrum reveals depends on which scales the weighting prioritizes. At short scales ($\tau$ comparable to ${\delta }_{\min }$), both survival functions are near unity and the debt is small. At intermediate scales (${\delta }_{\min }\ll \tau \lesssim {\theta }_{\mathrm{ref}}$), the closed loop has driven $S(\tau ;{\theta }_t)$ down sharply while the baseline retains substantial mass — the debt is maximal here. At long scales ($\tau \gg {\theta }_{\mathrm{ref}}$), both survival functions are small and the debt asymptotes.

The reader can watch the spectrum form across scales at the chrono-debt-spectrum plate. The plate exposes the parameter ${\theta }_{\mathrm{ref}}/{\delta }_{\min }$ and the contraction modulus ${\kappa }_{\Phi }=(\beta /\alpha )L_{\mathcal{D}}$ through its three constituents; the reader can see how the debt distribution shifts across scales as those parameters change.

The instrument's substantive content: what the closure removes is a spectrum of foreclosed intervals, foreclosed time resolved scale by scale. A foreclosed second is recoverable. A foreclosed adolescence is lost for good. The weighting function encodes which scales of foreclosed survival the analysis cares about; the framework's default linear weighting encodes the claim that a foreclosed adolescence weighs more than a foreclosed second by the relevant timescale ratio.

What gets foreclosed at which scale matters. At seconds, the foreclosure removes the interval at which a complete thought lands — the small pause between perception and response in which system-2 deliberation operates. At minutes, the timescale of sustained attention to a problem, the inner duration in which a worry resolves into something workable or names itself a worry. At hours, the structural condition for sustained engagement with a project — what writers and craftspeople call flow, what therapists call the holding interval. At days, the timescale of mood and weather, the diurnal rhythm against which a self registers its own variations. At years, the timescale of an extended biographical project — the developmental window analyzed in Proposition 8 as the cohort-gradient claim. The spectrum's defining work is to differentiate across timescales: what is foreclosed at a second carries different political weight from what is foreclosed at a decade.

Stiegler's analysis of tertiary retention — the technically-recorded traces that exteriorize memory — provides one of the older critical traditions' most-developed treatments of chrono-politics. Stiegler argued that techniques of recording (writing, photography, phonography, video) progressively proletarianize the inner duration through which a self maintains continuity, with the technically-mediated traces displacing what they were supposed to extend. Lemma 1 and Definition 1.1 give the structural form of what Stiegler diagnosed phenomenologically. The chrono-debt spectrum names which scales are now colonized, and at what rate.

The baseline policy ${\pi }_{\circ }$ is a modeling choice. The framework specifies what chrono-debt is given ${\pi }_{\circ }$; the choice of ${\pi }_{\circ }$ depends on what the analysis reads the debt against — the pre-platform regime, a regulated regime under Mode A, or another reference policy.