Triangulated Best-Judgement Assessment as Constrained Estimation Conservation Structure, Corridor Operators, and the Statistics of Evidentiary Standing
Conservation makes an audited project a control volume; every evidence stream is a boundary measurement that bounds true cost.
The supportable-cost determination is a corridor operator on those bounds — antitone in state evidence, isotone in taxpayer substantiation, idempotent, and 1-Lipschitz: any single input error propagates at most one-for-one. This paper derives the forensic properties Commonwealth evidence law demands.
From a determination to a defensible theorem
This paper supplies the mathematical foundations of the two-pathway indirect audit framework and derives, from them, the forensic properties Commonwealth evidence law demands of a best-judgement assessment.
A best-judgement assessment issued when the taxpayer withholds source documents is only as good as its defensibility. The tribunal will ask whether the estimate is honest, genuine, and rational rather than arbitrary; whether its error rate is known; whether a disputed input can swing the number without limit. Each of those is a mathematical question before it is a legal one — and each has an exact answer here.
An error in any single input propagates to the determination at most one-for-one.
The argument runs in three movements: conservation turns evidence streams into hard bounds; a lattice operator combines them with exact, provable properties; and order statistics convert the framework's stringency settings into a stated, known error rate — the property Daubert-style admissibility analysis requests.
The project as a control volume
Model the project as a control volume V with boundary ∂V. Capital cannot accumulate inside without crossing that boundary, and every crossing is instrumented by an organ of the state. Conservation — not correlation — is what makes each measurement a hard bound.
Imported equipment flux M(t)
The CIF record. Equipment and materials cross the customs boundary, capping the equipment component per category.
Offshore payment flux Fx(t)
Contractor payments clear the banking system; withholding at rate τ meters a sub-flux, an independent measure of the same payments.
Domestic goods flux D(t)
Locally sourced goods observable through the GST system, bounding the in-country supply component.
Labour flux L(t)
Work permits and payroll lodgements meter in-country labour, costed at benchmark rates into a corridor.
Each instrumented flux φi yields an upper bound Ui = φi / σi on true cost X, where σi is the disclosed share the surface can see. Dimensional analysis and engineering scaling supply lower bounds too — and a lower bound exceeding an upper bound is an empty corridor, a contradiction the taxpayer's own claim cannot survive.
The corridor operator and its exact properties
The supportable-cost determination S is a pointwise lattice expression — a composition of min, max, and an affine shift by taxpayer substantiation. Because it is built only from monotone, 1-Lipschitz primitives, its behaviour is provable rather than empirical.
Order structure
Antitone in state evidence (more streams weakly lower S), isotone in taxpayer substantiation — evidence can only tighten, cooperation can only relieve.
1-Lipschitz stability
Any two input tuples differing by δ move S by at most δ. No disputed input swings the determination without limit — a tribunal-fairness property.
Idempotence
Recomputing S on its own output returns S. Path-independent: the order streams arrive in cannot change the answer.
Contradiction test
If a lower bound exceeds an upper bound the corridor is empty — the claim is internally impossible, independent of any parameter choice.
Why this matters in court
A presumption lets an assessor pick a number and shifts the burden to the taxpayer to dislodge it. A corridor bounds the effect of every disputed input a priori — the assessment is the opposite of the arbitrary determination the Trautwein line of authority condemns.
A stated, known error rate
Measurements are noisy, so the corridor's ceilings are noisy. Order statistics turn that noise into an advantage and a guarantee rather than a liability.
Conservative by construction
The minimum of n unbiased ceilings falls below the mean by σ·en (e&sub2;≈0.56) — biased against the taxpayer, so parameter generosity is applied before the min.
The "known error rate"
Setting each of n active ceilings at the (1−α/n) quantile bounds the corridor's exclusion risk — P75 is a screening posture; P90 a litigation one.
Asymmetric-loss choice
Stringency is an explicit quantile decision under asymmetric loss — disclosed, not discretionary, and reproducible on demand.
On the reference case. With three active ceilings per category at P75 the Bonferroni exclusion bound is 75%; the P90 litigation sensitivity discloses a 30% bound and still leaves the corridor USD 2,000m+ below the claim in aggregate. The seeded fabricated ledger (8,400 lines) yields χ² = 493.7 against a 1% critical value of 20.09 — an emphatic Benford failure, reported only in the culpability section, never in the quantum tables.
The escalation protocol is provably near-optimal
Pathway B's ranked "request next" list is not a heuristic ornament. Corridor tightening is a monotone submodular set function of the evidence streams — the marginal gain of each new stream diminishes as evidence accumulates.
Greedy selection under a request budget attains at least (1 − 1/e) ≈ 63% of the maximum attainable tightening.
By the classical result of Nemhauser, Wolsey and Fisher (1978), the console's cost-scaled greedy request-next order is near-optimal with the (1−1/e) guarantee — request the customs extract before the satellite tasking, and interagency requests, with their diplomatic cost, are spent where they tighten most. On the reference case the greedy sequence attains the full available tightening: the guarantee is a floor, not a ceiling.
From mathematics to the courtroom
Each legal standard an indirect assessment must meet maps onto a specific theorem, statistic, or disclosed parameter — closing the distance between the mathematics and the courtroom.
| Legal standard | Formal counterpart | Established in |
|---|---|---|
| Trautwein — honest, genuine, rational (not arbitrary) | Closed-form operator on identified measurements; bounded input effect | §4 · parameter register |
| Civil balance of probabilities | Order-statistic bias + Bonferroni exclusion bound | §5 |
| Makita — expose the reasoning | Conservation structure; disclosed σi and Jacobians | §3–4 |
| Daubert — known error rate, reproducibility | Exact recomputation (Thm 4); propagation Jacobians; P75/P90 | §4–5 |
The standard-to-theorem map. The right-hand column is what the assessment can put before a tribunal — a theorem, a statistic, or a disclosed parameter — in place of an assessor's unexplained judgement.
Limitations
All numerical illustrations use the synthetic reference case; no real taxpayer data is used. The bounds are only as good as the disclosed shares σi and the engineering scalings behind the lower bounds — each a calibration duty the administration must discharge and defend. Digit-distribution evidence is confined by construction to reliability and culpability, and never adjusts quantum.
Selected references: Nemhauser, Wolsey & Fisher (1978) · Benford (1938) · Nigrini (2012) · Trautwein v FCT · Makita (Australia) v Sprowles · Daubert v Merrell Dow · MWP-2026-02.
Reproducibility. Maitras.ai working paper MWP-2026-03 — the mathematical companion to MWP-2026-02. Synthetic reference case only; no live administrative or taxpayer data. Views are the author's and do not represent any revenue administration.
Reference Paper: MCWP-2026-02 · DOI: 10.5281/zenodo.21820511