What messages, gauges, warnings or reports were documented as available to the driver by 9:58?
Companion explainer · Mathematics under review
The idea, the metaphor, and the mathematics.
This note separates the mathematical core of Evidence-Bounded Retrodiction from the wider AI paper so a mathematician can test the definitions, invariants, ordering rules and uncertainty treatment directly.
1 · The idea
Keep three different versions of “what was known” separate.
When an AI reconstructs why something happened, it should distinguish: what a person was actually documented to know at the time; what later evidence shows that person had access to at the time; and everything that becomes known later about what really happened.
The aim is to let an AI learn from history without rewriting history.
2 · MetaphorView 1 — contemporaneous record View 2 — reconstructed historical access View 3 — retrospective evidence The rule
The detective, the driver and the dashcam.
Imagine a car crash at 10:00 a.m. The driver made a decision at 9:58. At noon, a detective investigates.
At noon we may discover that the driver actually saw a warning at 9:57 even though the server archived it only at 11:30.
The detective can use a damaged brake part found at 11:20 even though the driver could not have known about that inspection at 9:58.
The detective may use later evidence to explain the crash. But when judging the 9:58 decision, the system must not secretly give the driver noon's information.
3 · Time variables
Why two timestamps are not enough.
For an evidence item x, EBR distinguishes event time, record time and actor-access evidence.
Iᵥ(x) = event / valid interval
r(x) = system record time
π(x) = provenance · σ(x) = epistemic status
A(a,x,t) = evidence that actor a had access to item x by time t
A past event may be discovered much later. Conversely, a person may see information before the system archives it. So neither event time nor database time alone tells us what an actor could know.
4 · Three core sets
The formal separation.
Let tᵢ be the historical decision time and c a later audit cutoff.
K_att(a,tᵢ) = { x : r(x) ≤ tᵢ and C(a,x,tᵢ)=1 }
K_att is the contemporaneously attested actor state: what the record already supported as accessible by the decision time.
K_rec(a,tᵢ;c) = { x : r(x) ≤ c and H_c(a,x,tᵢ)=1 }
K_rec is a later reconstruction of what the actor had access to by the old decision time. It can legitimately include a late-ingested archive if later provenance demonstrates earlier access.
R(c) = { x : r(x) ≤ c } ∩ S
R(c) is the retrospective evidence available by audit cutoff c within declared retrieval scope S. This is the detective's evidence set; it is not limited to what the historical actor knew.
5 · Invariant
Do not rewrite the past.
Assume append-only transaction history: later records can be added, but they cannot be back-written so that they appear to have existed earlier.
Later insertion after tᵢ ⇒ K_att(a,tᵢ) does not change
Later evidence may expand K_rec or R. It may even show that the actor had seen something earlier. But it may not mutate the contemporaneously attested state.
6 · Candidate histories
Causal links need mechanisms, not just sequence.
For a proposed causal edge u → v, EBR asks what mechanism connects the two events, what evidence supports that mechanism, what intermediate implication should be expected, whether contradictory evidence exists, and whether the bridge is documented, inferred, speculative or unknown.
Similarity can nominate a candidate. Temporal order can make it possible. Neither alone proves causation.
7 · Evidence profilem(h) n_π(h) c(h) u(h)
A profile, not a fake probability.
Φ(h) = ⟨ m(h), n_π(h), c(h), u(h) ⟩
Supported / required mechanism-bridge coverage.
Independent provenance roots after source-lineage deduplication.
Burden of adjudicated contradictions under the domain evidence policy.
Number of unresolved required mechanism bridges.
Φ(h) is deliberately not a probability vector and is not collapsed into one universal weighted score.
8 · Admissibility
Some weaknesses should not be compensated away.
Admissible(h) = 1 only if u(h)=0 and c_hard(h)=0
A missing required bridge should not be “paid for” by accumulating unrelated supporting evidence. Likewise, an adjudicated hard contradiction should not disappear because many correlated reports point the other way.
RankKey(h) = (Admissible(h), −c(h), coverage(h), n_π(h), −u(h))
The prototype uses this lexicographic ordering only as a transparent display convention. A stronger formal treatment may use a partial order or dominance relation so that genuinely incomparable histories remain incomparable.
9 · Open-set uncertainty
What if the true explanation is the one nobody listed?
⊥ = unmodeled-history reserve
If H1, H2 and H3 are the hypotheses currently considered, EBR refuses to assume those three exhaust reality. The explicit reserve ⊥ keeps open the possibility that an unmodeled H4 is the real explanation.
The system can therefore report “H1 is strongest among the admissible histories examined” rather than inventing a closed-set 70% / 20% / 10% distribution. It may also abstain.
10 · Toy result
A timing case designed to break simpler snapshots.
In the synthetic production-cell example, an operator decides at 10:05. A grid warning was actually seen at 09:57 but imported into the archive only at 11:30. A post-event inspection at 11:20 concerns a 09:56 disturbance but obviously was not available to the operator at 10:05.
| Historical view | Leakage | Actor precision | Actor recall |
|---|---|---|---|
| Valid-time only | 1 item | 0.83 | 1.00 |
| Record-time only | 0 | 1.00 | 0.80 |
| Three-view K_rec | 0 | 1.00 | 1.00 |
The result is a deterministic contract check, not evidence of real-world causal accuracy.
11 · Questions for a mathematicianSet semantics and time Non-retroactivity Provenance independence Contradictions Mechanism bridges Candidate ordering Open-set reserve Probabilistic extension
Where the formalization should be attacked.
Should actor access be a relation, event, interval, filtration or another object?
Can invariance of K_att and monotonicity properties of K_rec and R be stated as precise propositions under minimal assumptions?
How should “independent roots” be compared without silently assuming statistical independence?
Would paraconsistent logic, signed evidence, lattices or constraints be cleaner than a contradiction count?
Can admissibility be expressed as graph constraints or proof obligations?
Should the prototype's lexicographic RankKey be replaced by a partial order or Pareto dominance relation?
What is the cleanest formal interpretation of ⊥ when the hypothesis set is intentionally non-exhaustive?
How can calibrated Bayesian or structural-causal probabilities be added in domains that justify them without destroying the three-view separation?
Boundary
What this mathematics is not claiming.
- Chronology alone does not prove causation.
- Φ(h) is not a probability distribution.
- EBR is not presented as a new general causal-identification theorem.
- The unmodeled reserve cannot recover evidence that does not exist.
- Reconstructed actor access is an evidential assertion, not mind-reading.
- The toy prototype does not establish real-world causal accuracy.
The goal is not to make the work look more mathematical. The goal is to discover the cleanest formal structure that actually matches the intended semantics.