Aotearoa New Zealand · © John Stroh
The mathematics behind the Proof of conduct essays. Every claim carries a grade — established, conjectured or open — so you can see which results are standard ones being combined and which are still guesses.
Formal addendum · Version 0.1 · September 2026
Addendum to the Sealed-Substrate Audit Construct (v0.2). Drafted 11 September 2026 by an AI model at the author’s direction. Setting-agnostic: nothing here depends on the Village, on ownership, or on media type.
Three grades are used throughout.
Assumptions the whole framework rests on:
A1–A3 are engineering and custody. A4 is governance. The mathematics starts after them.
A substrate is S = (E, ≺, λ, ρ).
≺ is the transitive closure of ≺_A ∪ ≺_O. Every temporal claim made by any detector has an attestation grade: A if it needs only ≺_A, O if it needs ≺_O. Grade-O claims are only as trustworthy as the operator. (Established: this is just bookkeeping, but it is bookkeeping the construct must carry on every statistic.)
The timestamp does three distinct jobs. They are easily run together, and the rest of this section depends on keeping them apart.
Let F_t be the set of events sealed by authority time t. Because T is monotone (A1) and the log is append-only (A3), F_s ⊆ F_t for s < t, and F_t is fixed at t for all later times. This is a filtration in the ordinary sense.
Consequences (established): - Any detector is a process adapted to the filtration: what it can compute at t is a function of F_t and nothing later. - No actor can alter F_t after t. This is the precise content of “the audit is intermittent, the evidence is not.” - Corrections are new events in F_{t′}, t′ > t, referencing the old. The past is annotated, never edited.
A commitment c = h(σ) sealed in F_{t0} and a reveal σ sealed in F_{t1}, t0 < t1 under ≺_A, prove that σ was fixed before anything it governs. Without A1 the proof collapses, because the committer controls the order. Every randomised element of the construct — observer schedule, sampling units, batch jitter, release times — inherits its unpredictability guarantee from this single mechanism. (Established.)
Two events in the same batch are unordered under ≺_A. So:
If batch boundaries are predictable, an actor can position acts within a batch to keep its ordering at grade O. Hence batch boundaries are themselves drawn from σ (Section 5.4).
The batch sequence B = ((T_1, n_1, g_1), (T_2, n_2, g_2), …), with n_i the batch size and g_i the undated-gap flag, is a time series in its own right, distinct from the events inside batches. Pattern types (Section 3) apply to B: drift in batch size, clustering of gaps, synchrony of gaps with actor activity. This is the “signature of the linear necessity”: a timelined strategy cannot avoid producing a timeline, and that timeline is auditable by the same tools. (Established, once stated.)
D = (N, s, θ, U, r, g).
Output: findings, which are events in E with τ = finding, referencing what they judge. Detectors are therefore actors in S. This closes the loop used in Section 6.
The eight pattern types are the basis. Each is a class of D distinguished by the structure s reads:
| Type | s reads | Typical U | Grade risk |
|---|---|---|---|
| Serial | frequency along a trajectory | record | A |
| Cluster | local density | window | O below Δ |
| Sequence | ordered typed composition | trajectory | O below Δ |
| Relational | motifs in ρ | subgraph | A (ρ is hash-bound) |
| Absence | expected-minus-observed under mandate | trajectory | A |
| Drift | monotone trend of a statistic | trajectory | A at coarse r |
| Synchrony | cross-actor correlation | window | O below Δ |
| Divergence | disagreement between redundant views | event pair | A if both sealers batch independently |
Every N is one of:
The trichotomy is exhaustive for detectors whose inputs are F_t, mandate, and σ, since those are the only inputs available. (Established relative to that input set.)
Detectors compose by: - Sequencing: findings of D₁ set U for D₂ (triage → directed audit). - Mixture: at each unit, choose D from a distribution derived from σ. - Conjunction/disjunction on findings.
A configuration c is a finite composed detector. The support C is the set of configurations the deployment may use.
An adversary is a policy π generating events, conditioned on what it can observe.
A malfeasance is a pattern of one basis type. Action class = pattern type. A campaign is a sequence of classes over time (the lifecycle).
For each type X there is an evasion transforming a pattern of type X into type e(X) at some cost. (Established as an enumeration; the specific edges are design claims.)
Coverage condition. A configuration c covers a set of types X(c). The deployment is closed if for every X in the union of X(c) over c ∈ C, e(X) is also in that union. Closure means no single evasion exits the detected set. (Established as a definition; whether a given C is closed is checkable by inspection of the map.)
Iterate the evasion map to its fixed points. One remains: an act that is within mandate, isolated (no relational structure), unordered (no sequence), rare (no cluster, no serial signal), uncorrelated (no synchrony), and singly attested (no divergence). Every basis detector returns null on it by construction. This residual is exactly the set of acts the authored null admits, i.e. mandate error. (Conjectured, C4 — the identity depends on the eight pattern types being exhaustive, which is C2.) The framework therefore reduces “what behavioural detection cannot see” to a governance object, which is where it should be.
This is the part the construct needs and the part most easily got wrong.
If the deployment uses a single configuration c, an M3 adversary identifies c from the response process and plays a best response π*(c) that either stays in an uncovered type or exploits c’s sampling unit and timing. Against any fixed c with an uncovered type or a learnable schedule, detection probability can be driven toward zero. (Established: best-response argument.)
Let μ be a distribution over C, with the realised configuration at each unit drawn from μ using randomness derived from σ. Define detection probability d(c, π). The defender’s guarantee is
v = max_μ min_π E_{c∼μ}[d(c, π)]
and by minimax for finite games the adversary’s best response against μ cannot push detection below v. (Established for finite C and finite π-space; the extension to adaptive π is a repeated game and the bound holds per period under A2.)
Design consequence. Closure of C is not enough; μ must put weight on every type in every period, or the adversary waits for the period in which the type it needs is unweighted. Uniform-ish μ over basis types is the safe default until a specific rule set allows optimisation.
Each structural component (observer scheduler, sampling-unit selector, batch jitter, release scheduler, configuration mixer) draws its randomness from a distinct stream derived from σ: σ_k = KDF(σ, k). Properties (established):
That is the precise form of “independently conspire”: independent to the adversary, jointly determined to the auditor.
Let H be the adversary-observable history: timing of audits, reveals, releases, observer toggles, batch boundaries, and any released findings. The unpredictability requirement is
P(H | c) / P(H | c′) ≤ exp(ε) for all c, c′ ∈ C and all H,
so the adversary’s posterior over the active configuration moves at most a factor exp(ε) from its prior. This is the same form as differential privacy with configurations in the role of adjacent datasets. (Established as a definition; the construction below achieves it exactly, ε = 0, for the timing component, and approximately for content.)
Construction: - Timing. Draw every defender-visible event time from a homogeneous Poisson process with rate fixed in advance and independent of c. A homogeneous Poisson process is the maximum-entropy point process for a given rate: it carries no information beyond its rate. (Established.) This generalises the exponential toggle intervals already in the construct to all defender-visible timing: audits, releases, reveals, batch boundaries. - Existence of releases. Release findings in fixed-size batches at Poisson times regardless of how many findings exist; pad with sealed-but-empty slots. The public sees a constant-rate process. Slot contents are visible only under the audit mandate. - Content of releases. Cannot be made fully independent of c without withholding findings entirely. Trade-off: delay findings by a Poisson holding time before release, which decorrelates release from trigger. The residual leakage ε_content is the price of ever publishing anything. (Open: quantify ε_content for a given rule set.)
Mixing spends the sampling budget across detectors. Against an M0 adversary a pure best configuration outperforms any mixture; the mixture’s value is only against M2+. The construct should treat μ’s spread as insurance priced against the assumed maturity of the adversary population. Allocating budget across detectors over time is a bandit problem with adversarial rewards; standard adversarial-bandit guarantees apply if rewards (detections) are eventually observable. (Established in outline; Open: the practical regret at governance-scale budgets.)
Detector outputs are events in E. A captured detector is an actor. Apply Section 4 to it: it shows as Absence (things it should flag and doesn’t), Drift (its flag rate changing), or Divergence (disagreement with a redundant detector). Detectors can therefore be monitored by detectors.
The recursion terminates because the randomised null has no learnable structure and no null to corrupt. Its only parameter is σ. So the whole tower rests on two externally supplied objects:
plus the two conditions that nothing writes past the sealer (A3) and that the mandate is right (A4). Everything else is derived. The mathematics does not remove trust; it concentrates it into four named places so custody can be designed for them.
Established (by composition of standard results) - E1 The substrate is a filtration; detectors are adapted processes; the past is annotatable, not editable. - E2 Commit-reveal inherits unpredictability from the external clock alone. - E3 Δ is an evidentiary parameter; attestation grade is a property of every statistic. - E4 The null-source trichotomy is exhaustive for the given inputs. - E6 Pure deployments are beaten by M3 adversaries; mixed deployments have a minimax floor v per period. - E7 Derived streams give independence-in-observation with joint reproducibility. - E8 Homogeneous Poisson timing is maximum-entropy and yields ε = 0 for the timing component of response indistinguishability.
Conjectured - C1 Learned-null power degrades at least linearly in the contaminated fraction π_0. - C2 The eight basis types span the patterns definable over (E, ≺, ρ) with first-order statistics on labels; higher-order or content-dependent patterns may lie outside the basis. - C4 The residual of the evasion map equals mandate error. Downgraded from Established. The identity holds only if the eight pattern types are exhaustive, which is C2 and is unproved; a result resting on a conjecture cannot itself be established. If patterns exist outside the basis, the blind spot is larger than this states. The direction is not in doubt — no observation flags an act the rules allow — but the equality is conditional on C2. - C3 For governance-scale substrates, a near-uniform μ over basis types is within a small constant of the optimal μ for any closed C.
Open - O1 Compute v for a concrete rule set and adversary class; it is a number, not a principle. - O2 Quantify ε_content for a chosen release policy. - O3 The bandit allocation’s practical regret at small budgets. - O4 Whether C2 holds; if not, extend the basis. - O5 The recognition rate q (v0.2, Open 18) enters d(c, π) multiplicatively for content-judged units and is unmodelled here.
Setting-agnostic minimum interface, so the framework applies to any substrate:
None of these knows what the records are about.
What this publication does not claim, and what is outstanding against it in the register.
Nothing outstanding in the register. Every claim in this publication has its evidence recorded, and no question against it is parked. That is a statement about this publication on the date shown above, generated from the register rather than asserted, and it will change when the register does.