Kronos: Temporal Embedding as Governed Geometry

1 · Introduction

Every computing system carries timestamps; almost none carry time. A timestamp is a scalar annotation — useful for ordering, useless for geometry. It cannot say that 23:59 and 00:01 are two minutes apart rather than a day apart; it cannot say that this Tuesday morning resembles last Tuesday morning; it cannot make an authorization weaker for having aged. Systems that need those behaviors bolt them on: modular arithmetic for the wraparound, cron for the periodicity, revocation lists for the aging.

The construction described here — Kronos — began from the opposite direction, and the method deserves stating because it shaped the result. The shapes came first: a clock face, seen as a manifold on which times are positions; then a spiral, when the circle's failure to distinguish this midnight from last midnight became visible; then decay, drift, and coherence as further geometric behaviors of the same space; finally a helix, when discrete symbols were placed on the spiral as positions offset from a moving mean. The formalization below was written after the geometry, as a description of what was already visible. We do not present this as anecdote but as method: each section opens with the shape and derives the mathematics from it.

The result differs from prior temporal encodings not in its trigonometry — sine–cosine pairs are eighty years old in signal processing and eight years old in deep learning — but in what the encoding is for. Section 2 reviews the prior approaches. Section 3 constructs the embedding. Section 4 gives it operative semantics: decay, conditioning, and authority. Section 5 generalizes to the helix. Section 6 states the differences precisely; Section 7 describes the reduction to practice; Section 8 records limitations.

One drafting convention governs the whole paper. Where the text fixes a specific choice — the eight-dimensional coordinate, the calendar rhythm set, the exponential discount of the centroid weights (§4.4), a particular arithmetic format or distance measure — it is describing the preferred embodiment: the instance that is implemented, tested, and running. Each such choice is a parameter of the underlying construction, not a boundary of it; §3.4 states the generalization explicitly, and the filed disclosures recite the families.

2 · Prior approaches

Temporal representation in learning systems descends from several lines of work, each of which this construction borrows from and departs from.

Sinusoidal position codes. The transformer's position encoding^[1] maps sequence index p to a vector of sinusoids at geometrically spaced frequencies, giving a model access to relative position through linear combinations. Rotary embeddings^[2] refine this by rotating query–key subspaces, encoding relative displacement as phase difference. Both are index encodings: the coordinate is position-in-sequence, not physical time, and the code exists so attention can consume it.

Learned time representations. Time2Vec^[3] represents scalar time as one linear component plus k learned periodic components — sine activations with learned frequencies — and is explicitly motivated by calendar periodicity. Random Fourier features^[4] and their coordinate-network descendants^[5] establish the general principle that low-dimensional coordinates become learnable through sinusoidal lifting. These works treat the frequencies as either learned or random; periodicity is discovered or imposed statistically.

Feature-engineered cyclical encodings. Practitioners have long encoded hour-of-day and day-of-week as sine–cosine pairs so that boundary-adjacent values are metrically adjacent. This folk practice is the nearest ancestor of the clock face below — with the essential limitation that it produces features for a downstream estimator, with no system semantics of its own.

Continuous-time dynamics and event processes. Neural ordinary differential equations^[6] model hidden-state evolution through continuous time; temporal point processes^[7] model event intensities with self-excitation and decay kernels. These give time a dynamical role, but inside a learned model's state — not as a system coordinate shared by heterogeneous components.

Decay in memory and systems. The exponential forgetting curve dates to Ebbinghaus^[8]; rational analyses of memory^[9] model declarative decay explicitly; distributed systems expire credentials by TTL. The first two are descriptive models of organisms; the third is operative but not geometric — a TTL is a deadline, not a trajectory, and it composes with nothing.

Hysteretic conditioning. The Schmitt trigger^[10] introduced dual-threshold state retention to electronics precisely because single-threshold comparators oscillate at the boundary. Control engineering has used deadbands for a century. Machine-learning temporal systems, by and large, have not.

3 · The construction: time as position

3.1 The clock face

Let a cycle of period P be given — a day, a week, a year. Map instant t to the unit circle:

(1)θ(t) = 2π (t mod P)/P ,    c(t) = ( sin θ, cos θ )

Equation (1) is the folk encoding, and on its own it claims nothing. The construction begins with what is done to it next.

3.2 The spiral

The circle forgets the loop count: this midnight and last midnight are the same point. Add a monotone radial (or axial) term and the circle becomes a spiral — the cycle that never quite returns:

(2)s(t) = ( r(t) sin θ(t),  r(t) cos θ(t) ),    r(t) monotone in t

3.3 Multi-scale composition: the eight-dimensional now

A single cycle is a single rhythm; behavior has several. Compose the phase pairs for the diurnal, weekly, and annual cycles with a linear epoch term and a bias:

(3)T(t) = [ sin θ_d, cos θ_d, sin θ_w, cos θ_w, sin θ_y, cos θ_y, αt, 1 ] ∈ ℝ⁸

Each pair preserves proximity at its own scale — midnight adjacent to 23:59, Sunday to Monday, December to January — while the linear term enforces global monotone order: the embedding never claims that this year is last year, only that they rhyme. T(t) is fixed and parameterized, not learned: two implementations given the same instant produce the same eight numbers, which Section 4.3 will make load-bearing.

3.4 The coordinate is a family: N dimensions

Equation (3) is the preferred embodiment, not the construction. The construction is parameterized by a rhythm set: choose periodicities P₁ … P_K, take the phase pair of each, and append the epoch term and the bias:

(3′)T(t) = [ sin θ₁, cos θ₁, …, sin θ_K, cos θ_K, αt, 1 ] ∈ ℝ^2K+2,    θ_k = 2π (t mod P_k)/P_k

Nothing privileges the calendar. Human behavior turns in days, weeks, and seasons, so the eight-dimensional instance carries those three; machine behavior turns in milliseconds and beneath — scheduler quanta, frame cadences, transport heartbeats, sensor duty cycles — and the construction extends downward as far as the clock resolves: pairs at the millisecond, the microsecond, the nanosecond. The temporal authority of §4.3 already broadcasts the frame stream every component derives its position from, so the fast pairs are computed from the same frames; extending the coordinate costs two dimensions per rhythm and no new machinery. It extends upward the same way — decadal cycles for archival trust, fiscal and settlement cycles for markets, circadian and ultradian cycles for biological signals — each domain printing the coordinate its own world turns by.

The reason to widen it is that every operator of §4 lifts unchanged, per band. Decay already runs per evidence modality; drift conditioning already measures displacement against expected phase — at K rhythms it does so at K scales simultaneously, which yields a discrimination scalar time cannot express even in principle: cross-scale decomposition. A component that keeps phase at the fast bands while wandering at the slow ones is exhibiting clock skew or a schedule change, not a change of identity; the converse — every packet on time, every habit wrong — is a behavioral anomaly under a healthy clock. One is an operations problem and the other is a governance event, and the coordinate renders the difference as geometry: the displacement lives in specific dimensions, and the dimensions say which world moved. The same holds for the helix of §5, whose strand count is likewise open — temporal strands at different resolutions braid into a single object alongside content.

Dimensionality is therefore a governance dial, like the half-life: configured, witnessed, and deterministic at any K — two implementations given the same instant and the same rhythm set produce the same 2K+2 numbers, exactly in the fixed-point core and to documented rounding at the float boundary (§8). The granted claims recite the phase features with open cardinality — “one or more phase features obtained through a time-encoding process” — and the filed family discloses the multi-dimensional coordinate over configured periodicities, with the eight-dimensional instance as the preferred embodiment. Nor is the pair–epoch–bias composition itself a boundary: the pair is the faithful minimal unit for a cycle, and the epoch and bias components are elements of the preferred embodiment rather than requirements of the family. What the composition does not do is discover the rhythms (§8): choosing the set is a design act, made in the open and recorded like everything else.

4 · Operative semantics: what the coordinate does

4.1 Decay: trust with a half-life

Prior decay models describe organisms; this one governs artifacts. Every trust-bearing quantity in the architecture — an evidence weight, an authorization — carries a mass m and a decay constant λ, and its validity is a physical trajectory:

(4)V(t) = m · e^−λ Δt,    λ = ln 2 / t_½

with a governance floor θ: when V(t) crosses below θ, the authorization is dead — everywhere, simultaneously, with no revocation list, no callback, and no network round trip, because every verifier holding the embedded parameters computes the same trajectory. Expiry is not an event that must be distributed; it is a consequence that was always going to happen. The crossing is irreversible in the verifier: nothing re-inflates a decayed mass — renewal is a new issuance, not a resurrection.

The half-life parameterization matters operationally: t_½ is the governance dial — minutes for session-grade trust, days for standing authorization — and the same law applies per evidence modality, so an identity assembled from several signals ages each signal at its own rate.

4.2 Drift, deadband, hysteresis: making comparison governable

A coordinate invites comparison, and naive comparison at a threshold oscillates. The construction conditions temporal comparison — in the preferred embodiment, twice. A deadband subtracts an allowance before drift registers — sub-threshold change is ontological noise, not signal. A hysteresis gate holds state between two thresholds: crossing the strict threshold gains a state; only crossing the relaxed threshold loses it; between them, the system remembers what it last concluded.

(5)state(x) = high if x ≥ τ_strict;  low if x ≤ τ_relaxed;  unchanged otherwise

The consequence for governance is audit legibility: a decision boundary that flaps converts the audit chain into a record of noise. The dead zone is the price of a meaningful history, paid deliberately.

Conditioning has a second, geometric dimension that scalar approaches cannot express: phase. Because every event carries its position on the nested cycles, drift is measured not merely as elapsed time but as displacement from the expected position on the cycle — showing up is not the same as showing up at your usual point on the dial. Phase-coherent recurrence registers as proximity even across long gaps; phase-inconsistent activity registers as displacement even across short ones. This inverts an assumption every TTL and every decay-only trust model shares — that silence can only cost you. Under phase-conditioned drift, absence at the expected phase is itself consistent behavior: the Tuesday-morning user who was silent all weekend has not drifted at all, while a perfectly timely credential presented at three a.m. from the wrong rhythm has. Elapsed time penalizes everyone equally; phase geometry knows the difference between a weekend and an anomaly.

4.3 The temporal authority: time as a broadcast frame

Every construction above assumes agreement about t — and uncoordinated per-component clock reads forfeit it. In the implemented system a single temporal authority broadcasts atomic frames at fixed frequency; every component derives its temporal position exclusively from received frames. Time becomes a substrate service with three consequences: heterogeneous components share one coordinate frame; temporal computation becomes deterministic — replaying the frames replays the decisions bit-for-bit; and the clock itself falls inside the audit perimeter, because the frame stream is recorded evidence like everything else. This is the step that separates a temporal feature from a temporal citizen: the coordinate is issued, witnessed, and replayable.

4.4 The per-entity origin: the temporal centroid

Section 4.2 measured drift against “the expected position on the cycle” and deferred a question: expected by whom? A configured expectation would be one more global constant — every entity judged against the same nine-to-five, which is the scalar timeline’s arbitrariness returned at a higher altitude. The construction derives the expectation instead. Let T(t₁), …, T(t_n) be the temporal coordinates of an entity’s previously admitted observations — a record the architecture already keeps, since every score of §4.2 is computed by comparison against prior state. Their decay-weighted mean

(6)c = Σ_i w_i T(t_i) / Σ_i w_i,    w_i = e^{−λ_c(t_n−t_i)},    t_n the most recent admission

is the entity’s temporal centroid: a single point of the coordinate space that corresponds to no hour on any wall clock, was chosen by no one, and is particular to the entity that generated it. The decay constant plays a different role here than in §4.1. Because the weights are normalized, a shared factor of elapsed time cancels — the origin does not fade between observations; it moves on admission. Each new observation re-centers the frame toward the recent past, with λ_c setting how steeply older evidence is discounted against newer, and between admissions the origin is a constant of the record: the frame of judgment changes only when the record does. The exponential form is the preferred embodiment — §4.1’s own law applied to evidence age; any monotone discount of evidence age serves, and the properties used below survive the substitution.

Because the coordinate is built of phase pairs, the centroid’s geometry carries two readings at once. Restricted to rhythm k, the observations are points on a unit circle and their mean c_k falls on or inside it. Its direction — defined whenever r_k > 0 — is the entity’s preferred phase at that rhythm: the usual hour, the usual weekday. Its length r_k = ‖c_k‖ ∈ [0, 1] is the decay-weighted mean resultant length of directional statistics^[11], and it measures how concentrated the entity’s phase is at that scale: r_k near 1, tightly periodic; r_k near 0, no single preferred phase — arrhythmic, or split between opposed phases. The split case is real: a rhythm kept at nine and twenty-one o’clock cancels itself on the daily pair and hides from r_k; it is caught by configuring its harmonic — a twelve-hour pair — which is one more use for the open dimension count of §3.4. The complement

(7)V_k = 1 − r_k

is the weighted circular variance — the entity’s own dispersion at rhythm k, computed from its own history. Both quantities are deterministic functions of the admitted set: every verifier holding the same record derives the same origin, the same preferred phases, the same variances. The frame of judgment is replayable, which places it inside the audit perimeter exactly as §4.3 placed the clock there.

The operative payoff is that the entity’s own regularity prices its own anomalies. Define the coherent temporal envelope as the region within per-band tolerance of the preferred phase — the centroid’s direction carried back out to the rim, p_k = c_k/r_k — with the tolerance scaled by the band’s circular variance:

(8)admissible(T)  ⇔  ‖T_k − p_k‖ ≤ ρ(V_k)  for every configured band k with r_k > 0,   ρ increasing

A rhythm the entity keeps tightly (V_k small) is strict — small displacement there is large evidence. A rhythm the entity never kept (V_k near 1) is cheap — displacement there says almost nothing, and the envelope knows not to charge for it; a band with r_k = 0 has no preferred phase and imposes no constraint at all. The chord metric is likewise a choice of convenience — angular distance to the preferred phase serves identically, the two being monotonically related on the circle — and ρ is itself a governance parameter, configured rather than prescribed. Membership is a bounded predicate like every other witness in the architecture, evaluated when an authorization is issued and re-evaluated when it is exercised; in this architecture a coordinate that leaves the envelope does not degrade the decision — it refuses, fail-closed. A new entity has no centroid, so the envelope activates only once the admitted record reaches a configured minimum; until then authorization rests on the architecture’s other witnesses — the frame must be earned before it can judge. Drift, in this frame, is not distance from midnight. It is distance from yourself.

None of this is a new statistic. The mean resultant length is classical — chronobiology and signal detection have used it descriptively for decades^[11]. What is new is, once more, the operative role, and the architecture of the comparison is already granted: U.S. Patent 12,688,267 B1 scores coherence, trust, and drift “by comparison to one or more previously authenticated Wisps” (claim 1) and detects drift by “comparing at least a temporal vector or a location vector of the Wisp to corresponding vectors of previously authenticated Wisps” (claim 6). In this architecture that reference set is the entity’s own record, never a global profile, and the centroid is the comparison’s geometric limit: n prior observations collapsed into one governed origin, with the dispersion around it retained as the price sheet. The narrative companion arrives at the same point on foot and calls it the hour nobody chose.

5 · The helix: when time meets content

The spiral of §3.2 is a curve through an empty space. Give the space content: wrap the temporal spiral on a cylinder and embed discrete symbols — characters, tokens, events — as positions offset from the moving mean of the spiral at their moment of occurrence:

(9)h(s, t) = s(t) + δ(s),    δ(s) the symbol’s displacement from the temporal mean

Two strands with a fixed angular offset give the dual-helix form used for identity: one strand carries syntactic structure, the other a cryptographic rolling commitment, and their inter-strand geometry localizes tampering to the position where coherence breaks. The N-strand generalization treats every temporal-geometric encoding above as a special case of one family — the registry's unified form — which is why the same drift, decay, and conditioning laws apply at every level from a character stream to an identity trajectory.

6 · What is different, precisely

The compact statement: prior work makes time legible; Kronos makes time operative. The encoding is upstream of authorization, not upstream of a loss function.

7 · Reduction to practice

The construction is not proposed; it is running. The temporal subvector of §3 — elapsed time, the decay coefficient λ = ln 2/t_½, and Fourier/cosine phase features — is recited in each of the three independent claims (1, 16, and 17) of granted U.S. Patent 12,688,267 B1 — in which, per the claims as granted, it is computed during identity fusion — and in the implemented system it is carried into every downstream governance decision. The reference implementation's temporal engine implements the fixed-point arithmetic (Q16.16 with widened intermediates), the hysteresis gate, and the deadband as tested modules; the decay law runs in the warrant path with the dual kill conditions (mass exhaustion, temporal decay) tested at genesis, mid-decay, threshold-crossing, and zero-friction; and the twenty-test claim suite replays the granted pipeline against captured observations, one test per claim. The centroid path of §4.4 is likewise running: the temporal engine computes the eight-dimensional centroid of a coordinate set and the centered encoding that re-expresses a coordinate in the entity’s own frame, both as tested pure functions, and the governance functions maintain the streaming mean and variance the baseline constructions rely on. The multi-scale embedding is broadcast in every frame of the system's temporal authority at fixed frequency. The mathematics is cataloged in the signature registry accompanying the sovOS omnibus filing, where the temporal family and its N-strand generalization are formally rendered.

8 · Limitations and open work

Stated per house practice — the limitations are part of the record. The rhythm set is configured, not discovered — which is the point — but it means non-calendar periodicities (shift schedules, market sessions) require an explicit design act; the current composition does not discover them. Cross-implementation bit-identity of the full temporal path is an obligation the fixed-point core meets but the float ingress/egress boundary can forfeit under careless callers — a known delta, disclosed. Two decay implementations exist with different precisions and time bases; consolidation is scheduled. The helix's tamper-localization property is proven at the construction level and exercised in tests; adversarial characterization at scale is future work. The per-band resultant weighting of §4.4 likewise runs ahead of its instrumentation: the centroid and the centered coordinate are computed and tested, and streaming mean and variance are maintained, but envelope tolerances are not yet priced by circular variance per band — the formalization names the target before the code reaches it. And the deadband widths are per-chip-tier calibrated on some substrates — an honest measurement problem, documented where it occurs, not hidden in a slack constant.

References

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