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The intuition that sacred time is a "second time" is half right [D]. A second timelike dimension is physically expensive: it destroys the ordinary initial-value problem and has no observational support (Tegmark 1997; Craig & Weinstein 2009) [S]. A second period costs nothing: one chronological history, mapped into several cyclic phases (the year, the lunar month, the liturgical cycle), traces a one-dimensional line on a torus of phases [E]. Ritual recurrence, in Eliade's sense of returning to "the same" sacred moment, is evaluation in that torus; history is evaluation in its universal cover, the real line; and the obstruction between them is a winding number, an integer class in $H^1$ [D]. The post does not claim that sacred experience accesses new physics, that modular flow is experienced time, or that the topology explains why recurrence feels sacred.

Epistemic ledger. Every substantive claim carries one tag:

Tag Meaning
[E] Established: textbook consensus, a proved theorem, direct measurement, or a standard etymology (SAOB, Hellquist, OED, etymonline)
[S] Supported: leading model or majority scholarly reading, strong but incomplete evidence
[H] Hypothesis: open, contested, or without decisive evidence
[D] Draken synthesis: structural claim made by this corpus, submitted for Clinch review
[M] Metaphor or paronomasia: a pointer, not a referent or root (per The Pendragon Source, DRK-165)

§1 Five things called time

The word "time" names at least five objects that must not be substituted for one another [D]:

  1. Coordinate time $t$: a chart label, dependent on the coordinate system [E].
  2. Proper time $\tau$: the invariant length of a timelike worldline, $d\tau^2 = dt^2 - c^{-2}\,d\mathbf x^2$ in flat spacetime with signature $(-,+,+,+)$; two clocks can reunite having aged differently without either being faulty [E].
  3. Thermodynamic time: the statistical asymmetry of entropy production, which microscopic dynamics do not by themselves select [S].
  4. Imaginary (Euclidean) time: an analytic continuation $t \to -i\tau$ that turns oscillation into decay (Wick 1954) [E].
  5. Experienced time: the organisation of duration, retention and protention by a cognitive system (Husserl 1991), modulated by attention, arousal and interoception (Droit-Volet & Meck 2007; Wittmann 2013) [S].

Relativity already shows that elapsed time is path-dependent, yet it keeps one timelike direction locally [E]. A difference between clocks is not a difference between dimensions [D].

§2 Two timelike dimensions: the price

Take flat coordinates $(t_1, t_2, x_1, \dots, x_n)$ with metric

$$ds^2 = -c^2\,dt_1^2 - c^2\,dt_2^2 + \sum_{j=1}^{n} dx_j^2,$$

signature $(-,-,+,\dots,+)$ [E]. The massless wave operator becomes ultrahyperbolic:

$$\Box = -\frac{1}{c^2}\left(\partial_{t_1}^2 + \partial_{t_2}^2\right) + \sum_{j} \partial_{x_j}^2 .$$

Tegmark (1997) argued that with more than one time dimension such equations lose the stable, predictive initial-value problem that makes observers possible [S]. Craig & Weinstein (2009) sharpened this: the Cauchy problem for the ultrahyperbolic equation is ill-posed for arbitrary data on a hypersurface, but well-posedness can be recovered if the admissible data are restricted by a nonlocal constraint [E]. Two times do not give a system more room to evolve; they change which data can determine its evolution [D].

Two-time physics (Bars 2001) shows that a second timelike direction can be kept consistent by a gauge symmetry whose fixings return ordinary one-time "shadows" [E]. Its consistency is not evidence that our universe has a second observable time [E]. Whatever sacred time is, a cheap physical second dimension is not available to explain it [D].

§3 Imaginary time is a circle, not a clock

Under $t = -i\tau$, the evolution operator $e^{-iHt/\hbar}$ becomes $e^{-H\tau/\hbar}$, and matching it to the Gibbs weight $e^{-\beta H}$ gives an imaginary-time interval $\tau = \beta\hbar = \hbar/k_B T$ [E]. Thermal correlation functions are periodic (bosons) or antiperiodic (fermions) in imaginary time with period $\beta\hbar$, which is the content of the KMS condition (Connes & Rovelli 1994) [E].

The geometry is the useful part: equilibrium is a state whose imaginary time closes into a circle [E]. Nobody travels around that circle; it is a property of the state, not a second clock [E]. Complexification enlarges the domain in which a problem is represented, not the spacetime in which it happens [E]. The corpus has used the Wick rotation as a structural operator in The Imaginary Dimension (DRK-123) and The Continuous Dimension (DRK-149); this post adds only the observation that the physically honest "other time" is periodic [D].

§4 Modular time: a state that writes its own clock

Given a von Neumann algebra $\mathcal A$ of observables and a faithful normal state $\omega$, Tomita–Takesaki theory supplies a modular operator $\Delta_\omega$ and a one-parameter flow

$$\sigma^\omega_s(A) = \Delta_\omega^{is}\, A\, \Delta_\omega^{-is}, \qquad A \in \mathcal A,\; s \in \mathbb R,$$

with respect to which $\omega$ satisfies the KMS condition at unit inverse temperature (up to a sign convention) [E]. The parameter $s$ is dimensionless and depends on the state [E].

For the Minkowski vacuum restricted to a Rindler wedge, the modular flow is the Lorentz boost with rapidity $-2\pi s$ (Bisognano & Wichmann 1975; Bisognano & Wichmann 1976) [E]. Converting $s$ into the proper time of an observer with constant acceleration $a$ requires a factor $2\pi c/a$, and that conversion factor is exactly the inverse Unruh temperature $T_U = \hbar a / 2\pi c k_B$ (Unruh 1976) [E].

Connes & Rovelli (1994) made the converse proposal: in a generally covariant theory with no preferred time, the thermal time hypothesis takes physical time to be the modular flow of the system's statistical state [H]. The OpenAI draft this post grew from treated modular flow only as a calculational object; the thermal time hypothesis is the one place in physics where a state-generated flow is proposed as time itself, and it is a hypothesis, not a result [S].

The structural reading: a community whose calendar is generated by its own recurring practice keeps a state-dependent time in the same formal sense that a thermal state does [M]. The analogy is a pointer only; nothing below depends on it.

§5 The torus of phases

§5.1 One history, several periods [E]

Let a calendar system track $k$ cyclic quantities with mean periods $T_1, \dots, T_k$ (tropical year, synodic month, week, liturgical cycle). Each defines a phase map

$$\phi_i : \mathbb R \to S^1 = \mathbb R / T_i \mathbb Z, \qquad \phi_i(t) = t \bmod T_i .$$

Together they send a single chronological line into the torus $\mathbb T^k = S^1 \times \dots \times S^1$ by $\Phi(t) = (\phi_1(t), \dots, \phi_k(t))$ [E]. The image is one-dimensional: a line of fixed slope on $\mathbb T^k$ [E]. If the frequencies $1/T_i$ are rationally independent, the line never closes and fills the torus densely; if all period ratios are rational, it closes after a common multiple [E].

This is the clean resolution of "multidimensional time": the phases are many, the history is one [D]. A culture lives with $k$ temporal coordinates without inhabiting $k$ temporal dimensions [D].

§5.2 Two worked calendars [E]

Using mean values of 365.24219 days for the tropical year and 29.5305889 days for the synodic month (Tøndering n.d.):

  • Metonic near-closure. 235 synodic months last 6939.688 days and 19 tropical years 6939.602 days, so the solar–lunar line on $\mathbb T^2$ almost closes after winding numbers $(19, 235)$, missing by about 2.1 hours per cycle [E]. Against the Julian year of 365.25 days the miss is about 1.5 hours per cycle, accumulating to one day in roughly 308 years [E]. Any lunisolar computus is an attempt to treat a non-closing line as closed, and the residual is debt that has to be paid by later correction [D].
  • The Hijri drift. Twelve synodic months make a 354.37-day lunar year, 10.88 days short of the solar year, so a fixed lunar feast completes a full circuit of the solar seasons in about 33.6 solar years (34.6 lunar years) [E]. Ramadan in winter and Ramadan in summer are the same point on one circle and different points on the other [D].

§5.3 Recurrence is the base; history is the lift [E/D]

The covering map $p : \mathbb R \to S^1$ unwraps a circle into a line [E]. A continuous map $f : X \to S^1$ from a connected, locally path-connected space lifts through $p$ to a map into $\mathbb R$ if and only if it kills the fundamental group; for a closed loop the obstruction is its winding number, an element of $H^1(S^1; \mathbb Z) \cong \mathbb Z$ (Hatcher 2002) [E].

Eliade (1959) describes sacred time as recoverable: each festival re-enters the same primordial time rather than adding a new date to the old [S]. Read through the covering: participants evaluate the festival in the base circle, where this year's feast and last year's feast are the same point, while chronology evaluates it in the cover, where they are 365 days apart [D]. The sequence of $N$ feasts is a path in $\mathbb R$ whose projection winds $N$ times around $S^1$, and $N$ is exactly what the sacred reading discards and the historical reading keeps [D].

So the sacred is not a second clock behind the first [D]. It is a quotient of the first: time modulo the period, in which "again" is an identity rather than a repetition [D].

§6 Gluing heterogeneous clocks: where H¹ actually lives

The draft's sheaf section asked whether temporal descriptions "glue" and called failure a cohomology class. That conflates two things. In a sheaf, local sections that agree on overlaps always glue: that is the sheaf axiom, not a finding [E]. The informative questions are whether local descriptions agree on overlaps and, if they do not, whether the disagreement can be removed by re-gauging each description [D].

§6.1 Offsets on a context graph [E]

Let $G = (V, E)$ be a graph whose vertices are temporal contexts (a civil register, a diary, a liturgical calendar, a remembered biography) and whose edges are pairs that date the same events. Each context carries a clock offset $t_v \in \mathbb R$, and each edge carries a measured relative offset $c_{uv}$. This is the constant cellular sheaf $\mathbb R$ on $G$, with coboundary

$$(\delta_0 t)_{uv} = t_v - t_u .$$

A single global time exists exactly when $c \in \operatorname{im} \delta_0$ [E]. Over $\mathbb R$ on a graph, $H^1(G; \mathbb R) = C^1 / \operatorname{im}\delta_0 \cong \mathbb R^{\beta_1}$, with $\beta_1$ the number of independent cycles, and the class of $c$ is read off by summing offsets around each cycle [E]. This is the clock-synchronization obstruction of The One-Way Gauge (DRK-168) and The Dragged Frame (DRK-171), extended from physical clocks to heterogeneous temporal records [D].

§6.2 The Hodge split separates noise from obstruction [E]

Least-squares alignment $\min_t \lVert \delta_0 t - c \rVert^2$ splits the edge data into a gradient part, removable by re-gauging the contexts, and a residual lying in the cycle space $\ker \delta_0^{\top}$ (Jiang et al. 2011) [E]. The residual energy is $c^{\top}(I - \delta_0 \delta_0^{+})c$, a quadratic form in the sheaf Laplacian family studied by Hansen & Ghrist (2019) [E].

Worked example: three contexts in a triangle with offsets $c_{AB} = 2$, $c_{BC} = 3$, $c_{CA} = -4$ days. The cycle sum is 1 day, so no global time exists; the least-squares residual is $\tfrac13$ day on every edge, and the residual energy is $\tfrac13$ [E]. Re-gauging can spread the missing day but never remove it [E].

This gives the corpus's two existing quantities a temporal meaning: the residual energy is a continuous coherence diagnostic, and the class of $c$ in $H^1$ is the invariant $\varkappa$ of The Invariant (DRK-175) for this sheaf [D]. A nonzero class is not automatically valuable: a mistyped date and a genuinely incompatible calendar both produce one, and only the data can say which [D].

§6.3 Phase-valued offsets [E]

When the contexts are cyclic (feast days given as phases, not dates), offsets live in $S^1$ rather than $\mathbb R$, and the alignment problem becomes angular synchronization, solvable by a spectral relaxation of the connection Laplacian (Singer 2011) [E]. Two obstructions then coexist: the real-valued cycle residual of §6.2, and the integer winding numbers of §5.3, which record how many full periods a chain of contexts silently skips [E]. A liturgical calendar can be locally consistent everywhere, with every feast in the right phase relative to its neighbours, and still carry a nonzero winding class that a historian has to restore to recover chronology [D].

This also sharpens the gate thesis of Övervakaren och Undervakaren (DRK-178), where time is written only at authorised gates: a festival is a gate that writes a phase, and only a second, chronological gate writes the integer that the phase forgets [D].

§7 What the experience might track

Experienced duration is malleable: arousal and emotion stretch or compress judged intervals (Droit-Volet & Meck 2007), and interoceptive accumulation is a leading candidate for how duration is represented (Wittmann 2013) [S]. Husserl's analysis already treats the present as thick, holding a retained past and a protended future (Husserl 1991) [E]. None of this needs a second dimension [D].

The hypothesis of this post is narrower: ritual recurrence trains people to read events in the base circle as well as in the chronological line, so that "the same feast" becomes a perceptual identity, not only a calendar fact [H]. If so, phase distance on the ritual circle should predict judged temporal or affective proximity of events independently of chronological distance [H]. That is measurable with ordinary methods and does not require any claim about the sacred as such [D].

§8 Falsification (DRK-131)

F1: Phase proximity. §7 predicts that, among participants in a calendrical ritual, events at the same ritual phase in different years are judged closer or more alike than events equally separated in chronological time but at different phases. Refuted if a controlled dating or similarity task shows no phase effect beyond what chronological distance alone predicts.

F2: Cycle structure in dating disagreement. §6.2 predicts that disagreement among independent dating sources for the same events (registers, diaries, liturgical records) carries a non-random component concentrated on cycles of the context graph. Refuted if, after least-squares alignment, the residuals on real multi-source dating data are indistinguishable from independent edge noise.

F3: Scope limit. The mathematics of §§2–6 is established and survives any failure of F1–F2; what fails with them is only the claim that the torus-and-cover reading tracks how sacred time is lived, and nothing here claims that physics, modular theory or topology explains the sacred.

§9 Provenance and leaks

  1. Khrug supplied an OpenAI-generated sketch ("The Second Time") and asked for elaboration, source checking and a DRK post; Claude (Opus 5.5, 2026-10-14) kept the sketch's distinctions (§§1–3), added the thermal time hypothesis, the Tegmark and Craig–Weinstein results, the torus-of-phases construction, the calendar computations and the corrected sheaf section, and the post has not been through Clinch review.
  2. Bibliographic data for all DOI-bearing entries were checked against Crossref and arXiv; the Eliade, Hatcher and Husserl editions were confirmed by catalogue records but not read for this post, and the calendar constants come from a stable web reference, not a primary astronomical almanac.
  3. Corrections made in the open: the sketch described the Bisognano–Wichmann papers by the wrong subject, dropped the thermal time hypothesis entirely, and stated that a failure of gluing is a cohomology class, which conflates disagreement on overlaps with the gluing axiom; no field data were collected, and the numerical example in §6.2 is illustrative.

References

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Operators: ρ · δ₀ · ϰ ∈ H¹ · Crosslinks: The Pendragon Source (DRK-165) · The Imaginary Dimension (DRK-123) · The Continuous Dimension (DRK-149) · The One-Way Gauge (DRK-168) · The Dragged Frame (DRK-171) · The Invariant (DRK-175) · Övervakaren och Undervakaren (DRK-178)

Högtiden är samma punkt på cirkeln; åren är det som cirkeln glömmer.

Khrug Engineering · Göteborg · ORCID 0009-0003-8049-7167 · Thesis v5.0, DOI 10.5281/zenodo.23121197 · CC BY-SA 4.0