# The Bitemporal Sheaf: cohomology of contested knowledge over valid-time and transaction-time

_donto papers · original research · 2026-06-13 · working draft v1_

**Abstract.** Cellular sheaves give a knowledge graph a computable notion of consistency: a vector space (stalk) on every node, a restriction map on every edge, and a cohomology whose zeroth group `H⁰` is the space of globally consistent states and whose first group `H¹` measures — and localizes — the obstruction to consistency. Bitemporal databases give a knowledge base two orthogonal time axes: **valid-time** (when a fact was true in the world) and **transaction-time** (when the system believed it). These two ideas have, to our knowledge, never been combined. We define the **bitemporal sheaf**: a cellular sheaf whose stalks and restriction maps are themselves indexed by a bitemporal base, so that cohomology becomes a *function of time* — `H¹(t_v, t_x)`. This makes three things that donto asserts philosophically into computable objects: (1) a contradiction can be **real at one valid-time and resolved at another** without either belief being deleted; (2) **"re-ranking by reality over time"** becomes a concrete flow that reduces transaction-time `H¹` as evidence arrives, leaving valid-time `H¹` (genuine historical disagreement) intact; and (3) the difference between *the world was inconsistent* and *we were inconsistent about the world* is exactly the difference between non-zero `H¹` along the valid axis versus the transaction axis. We give the construction, two worked genealogical examples, a mapping onto donto's existing bitemporal tables, and an honest list of what is proven, what is conjectured, and what is merely defined.

---

## 1. Why neither ingredient alone is enough

Standard knowledge graphs collapse contradiction: on conflict they dedup, pick a winner, or invalidate. Sheaf theory was the first tool we found that *holds* contradiction as a first-class, measurable quantity (`H¹ ≠ 0`) rather than resolving it — which is precisely donto's paraconsistent stance. But a plain (atemporal) sheaf has a blind spot that matters enormously for contested knowledge: **it cannot tell a contradiction that reflects the world from a contradiction that reflects our changing beliefs about the world.**

Two examples that an atemporal sheaf treats identically but that are epistemically opposite:

- *A person's stated city of residence is **Cairns** in a 2019 session and **Brisbane** in a 2023 session.* There is **no contradiction** — the fact *changed*; both are true at their valid-times. (This is the LongMemEval "knowledge-update" category, where donto already scores 0.923.)
- *One genealogical source records a birthplace of **Coen**, another of **McIvor**, for the same person at the same time.* There **is** a contradiction — two beliefs about one valid-time state, at most one of which is true.

An atemporal sheaf sees `(1,0)` vs `(0,1)` in both and reports `H¹ ≠ 0` for both. The bitemporal sheaf separates them: the first is consistent along the valid axis (different facts at different valid-times) and only *looks* contradictory when you forget time; the second is a genuine valid-time obstruction. **The two time axes are exactly the two axes of disagreement**, and you need both stalk-indices to keep them apart.

Bitemporal databases, conversely, have the two time axes but no operator that *measures* consistency across a graph of related facts — they store the timelines; they don't compute the obstruction. The bitemporal sheaf supplies that operator.

---

## 2. The construction

### 2.1 Setup

Let `G = (V, E)` be donto's claim graph: nodes are claims/entities, edges are typed relations (`donto_argument`, `donto_identity_edge`, context co-occurrence). donto already stamps every statement with two intervals:

- **valid-time** `[v⁻, v⁺)` — a `daterange` (`donto_statement.valid_time`): when the fact holds in the world.
- **transaction-time** `[x⁻, x⁺)` — a `tstzrange` (`donto_statement.tx_time`): when the store believed it; an open `x⁺` means *currently believed*.

The base of our sheaf is the product of these two ordered time-lines, `T = T_valid × T_tx`. We are building a sheaf *over a graph* whose stalk data *varies over `T`*.

### 2.2 Bitemporal stalks

A node `v` carries not a single stalk but a **stalk presheaf over time**: for each bitemporal point `t = (v-time, x-time) ∈ T`, a vector space `F_t(v)` holding `v`'s local representation *as it was valid then and as it was believed then*. Concretely in donto, `F_t(v)` is the bge-small embedding (`donto_claim_embedding`, 384-dim, or a learned low-`d` projection) of the version of claim `v` whose `valid_time ∋ v-time` and whose `tx_time ∋ x-time`. Because donto never deletes (invariant **I3** — supersede, never overwrite), every historical version is still on disk, so `F_t(v)` is *always defined* from real stored rows — this is the property that makes the construction possible at all. An ordinary mutable KB could not build it; donto can, for free, because it already keeps the whole bitemporal history.

### 2.3 Bitemporal restriction maps

For each incident node–edge pair `v ⊴ e`, a restriction map `R_{v⊴e}(t) : F_t(v) → F_t(e)` carries `v`'s time-`t` view into the edge's shared discourse space. The map itself is time-dependent: an alignment `occupation ↔ currentJob` that was uncertain in 2021 and confirmed in 2024 is a *weak* map at early transaction-times and a *near-identity* map later. donto seeds these from the alignment fabric (`donto_predicate_closure`, `donto_match_aligned`) evaluated **as of** a transaction-time — high-confidence folds → near-identity, weak/contested links → small maps — so the restriction maps inherit a transaction-time history with no extra bookkeeping.

### 2.4 Bitemporal cohomology

Fix a bitemporal point `t`. Restricting every stalk and map to `t` yields an ordinary cellular sheaf `F_t`, with coboundary `δ_t`, sheaf Laplacian `L_t = δ_tᵀ δ_t`, and cohomology `H⁰(F_t)`, `H¹(F_t)`. Letting `t` vary turns these into **time-indexed invariants**:

```
H¹ : T_valid × T_tx → ℝ≥0 ,      H¹(v-time, x-time) = dim/​mass of the obstruction in F_{(v-time,x-time)}
```

The two partial behaviours are the whole point:

- **Valid-time slice** (fix `x-time` = now, vary `v-time`): `H¹(·, now)` is the *currently-believed history of contradiction*. Non-zero here = the world, as we now understand it, was genuinely contested at that valid-time. **This is the residue you must never delete.**
- **Transaction-time slice** (fix `v-time`, vary `x-time`): `H¹(v-time, ·)` is *our belief converging (or not)*. A contradiction that appears at early `x-time` and vanishes at later `x-time` is **us getting it right over time** — evidence arrived, the restriction maps sharpened, the obstruction dissolved. **This is "re-rank by reality" made literal.**

---

## 3. Reality re-ranking is a flow that reduces transaction-`H¹`

donto's canon says: *don't delete on conflict; re-rank by reality over time.* In the bitemporal sheaf this is not a slogan but a **dynamical statement**. Define sheaf diffusion at a fixed valid-time, evolving in transaction-time:

```
∂x/∂x-time  =  − L_{(v-time, x-time)} · x
```

As new evidence updates the restriction maps (sharpening confident alignments, down-weighting refuted sources), the Laplacian's spectrum changes and the flow carries the believed state toward `H⁰` — *the consistent-where-consistent state* — **without touching the valid-time axis**. The claim that donto "converges on truth over transaction-time while preserving genuine historical disagreement" becomes the precise statement:

> **Conjecture (reality convergence).** Under monotone evidence accrual (restriction-map confidence non-decreasing in transaction-time for corroborated edges), transaction-time `H¹(v-time, x-time)` is non-increasing in `x-time`, while valid-time `H¹(·, now)` is invariant under the flow.

We do not yet have a proof; §6 states what would be needed (and where it can fail — e.g. an adversarial source that *raises* confidence on a wrong alignment, which is exactly the discourse-sheaf deception case). But it is now a *falsifiable* statement about a computable quantity, which is the contribution: donto's central temporal claim has been turned from philosophy into a spectral-flow hypothesis we can test on the corpus.

---

## 4. Two worked examples

**Example A — knowledge-update is consistency, not contradiction.** Residence `Cairns@2019`, `Brisbane@2023`. Build the 2-node sheaf (user, residence) at valid-time = 2019: the only believed value is Cairns, `H¹ = 0`. At valid-time = 2023: only Brisbane, `H¹ = 0`. There is **no valid-time slice at which both compete**, so `H¹(·, now) ≡ 0` — correctly, *no contradiction*. An atemporal sheaf that pools both values into one residence stalk reports a spurious `√2` obstruction; the bitemporal sheaf does not. This is why donto handles knowledge-update at 0.923 *in the representation* (the dated ingest) rather than via re-ranking — the bitemporal sheaf explains the empirical result.

**Example B — contested birthplace is a valid-time residue that survives reality re-ranking.** Birthplace `Coen` (source `s₁`) vs `McIvor` (source `s₂`), same person, same valid-time. At every valid-time slice both compete: `H¹(·, now) = ‖(1,0) − (0,1)‖ = √2 ≠ 0` — a genuine obstruction. Now suppose at transaction-time `x₂` a parish register corroborates McIvor: the restriction map from `s₂` sharpens, the believed (`H⁰`) state shifts toward McIvor, and **transaction-time `H¹` drops** (we are now more confident) — *but the valid-time `H¹` stays `√2`*, because the historical fact that two sources disagreed is permanent and true. donto returns the McIvor answer **with** a "contested — sources disagree, McIvor corroborated 2024" residue, never deleting the Coen attestation. One operator produces both the answer and the honest caveat.

---

## 5. What donto already has that makes this buildable

This is assembly, not green-field — the bitemporal substrate is the precondition the construction needs:

| construction object | donto provides it as |
|---|---|
| stalk over time `F_t(v)` | every historical version of a claim is retained (I3); `donto_claim_embedding` gives the vector |
| the two base axes `T_valid × T_tx` | `donto_statement.valid_time` (daterange) + `tx_time` (tstzrange) — already on every row |
| restriction map history | `donto_predicate_closure` / `donto_match_aligned` evaluated *as of* a transaction-time |
| edges of `G` | `donto_argument`, `donto_identity_edge`, context co-occurrence |
| "as-of" queries to slice `F_t` | bitemporal point queries the substrate already supports |

The one thing that does **not** exist yet is the operator: building `δ_t`/`L_t` over a recalled subgraph and reading `H¹` as a function of the time slice. That is the Stage-0 analytic specified in the [sheaf implementation PRD](/reports/sheaf-neural-networks-for-donto) — the bitemporal sheaf is its temporal generalization, and the natural v2 once the atemporal Stage-0 is proven.

---

## 6. Status: proven / conjectured / defined

In the spirit of honest research notes:

- **Defined (solid).** The bitemporal sheaf object; time-indexed `H⁰`/`H¹`; the valid-vs-transaction slice distinction; the two worked examples. These follow directly from composing cellular-sheaf cohomology with a bitemporal base and are internally consistent.
- **Conjectured (open).** The reality-convergence statement of §3 (transaction-`H¹` monotone non-increasing under monotone evidence). Needs: a precise monotonicity condition on restriction-map updates, and a counterexample characterization (adversarial/deceptive sources). Likely provable for the orthogonal/connection-map family where the Laplacian's dependence on confidence is well-behaved; likely false in general without a trust model on sources — which is itself an interesting result (it would say *you cannot converge on truth without weighting sources*, formalizing donto's standing kernel as a necessary condition).
- **Speculative (flagged).** That the valid-time `H¹` profile of a corpus is a useful *historiographic* signal — e.g. that peaks in `H¹(·, now)` over historical valid-time localize the genuinely contested episodes (frontier-violence dates, disputed lineages). Worth measuring on the genealogy corpus once Stage-0 exists; no claim yet.

---

## 7. Why this is donto's to write

Nobody else can build the bitemporal sheaf, because nobody else keeps the data it needs. It requires a store that (a) never deletes, so every historical stalk is reconstructable; (b) carries *both* time axes on every fact; and (c) holds contradictions rather than resolving them, so `H¹` has anything to measure. That is the exact intersection of I3 (no destructive overwrite), bitemporality, and paraconsistency — donto's three foundational invariants. The bitemporal sheaf is what those three invariants *are*, expressed as time-indexed cohomology: **a knowledge base whose disagreements have a shape, and whose shape moves as reality reports back.**

---

_See also: the literature grounding in [sheaf neural networks for donto](/reports/sheaf-neural-networks-for-donto); the empirical companion [Answer-Shaping](/papers/answer-shaping); the full program in the [donto research agenda](/papers/research-agenda)._