A bank with two branches, audited at temperature λ.
This annex is not gravity. It is the house's method — write the falsification before the narrative, predict before you measure, retract in public — applied to a different ledger: the Schrödinger bridge, the problem of moving a distribution of funds μ into a distribution ν at minimal cost, when the books are kept at finite resolution λ. As λ → 0 this is optimal transport; at λ > 0 every entry is blurred by entropy. The deformation parameter plays the role of ℏ — the literature's own analogy, and the reason this house took the assignment.
The field believed something clean about this deformation: after netting (the debiased Sinkhorn divergence Sλ, which cancels the books' first-order self-bias), the small-λ corrections were conjectured to come in even powers only — λ², λ⁴ — proven for Gaussian accounts to O(λ⁵), assumed generically, and used to justify the auditor's favourite trick: Richardson extrapolation, accelerating the audit at rate O(λ⁴).
We put the conjecture on a bench built for it — a Rust solver with the exact Gaussian case as its kill-gate (the known closed form reproduced: c₂ to 0.01%, c₄ to 1%; one currency-conversion trap caught and filed: the published Gaussian expansion is denominated in ε = λ/2, a clean factor 4 on c₂) — and then asked the one question the literature's hypotheses all exclude: what if the client's account is split between two branches?
The auditor's rate, destroyed in every reachable window.
Three controls and one anomaly, same pipeline, same stamps:
| Marginals | ĉ₃ (sliding windows) | Verdict |
|---|---|---|
| Gaussian vs Gaussian (control) | ~1×10⁻⁶ | clean — theory reproduced to 0.01% |
| Unimodal skewed, non-Gaussian | ~1×10⁻⁶ | clean — non-Gaussianity is innocent |
| Bimodal, shallow valley (gap Δ = 2) | ~−5×10⁻⁶ | clean — an onset exists |
| Bimodal, separated (Δ = 3.5), weights ½/½ vs 0.4/0.6 | −0.049 at 200σ | anomaly |
| Bimodal, Δ = 5 | −0.7 and degrading | crossover swallows the window |
The anomaly is not a power law wearing a costume. Killing the λ² term exactly (Richardson on a √2-grid) and reading the residual's local exponent gives a trajectory no finite combination of powers can produce — it passes below 3, then climbs without locking:
p(λ) = α + 2β·ln(1/λ) ·· measured 2.94 → 3.82 across λ = 0.025 → 0.0037 the advertised O(λ⁴) auditor's rate reads ≈ λ³ throughout three decades of practically accessible λ — the published regularity condition fails here, as its authors never claimed otherwiseBecause the transient feeds no power coefficient, the parity conjecture survives as a statement about the series — and becomes nearly vacuous for split accounts: the series is even, and the books are governed by something the series cannot see.
Amplitude = the order × the teller's window. Predicted, then measured.
Where does the off-book cost live? At the cut xs — the teller's window: the boundary of the mass slab that must change branches. An exact identity holds there: the dual exponent is perfectly flat across the whole valley (h′ = 2(y−xs) + 2(xs−y) = 0), so the plan's conditional is an exponentially tilted family sweeping the corridor — and every contribution of the layer carries the prefactor μ(xs). The law:
T(λ, m) = m · μ(xs(m)) · G(λ) transfer order m × marginal density at the cut × a universal profile · sign negative — netting over-corrects in the crossing zone| Test | Predicted | Measured |
|---|---|---|
| Kill the order (weights ½/½, m = 0) | suppression to valley-floor density | ×1000 suppression (−0.049 → −5×10⁻⁵) |
| m²-collapse correction factor, m = 0.05 vs 0.10 | 2·μ(xs)-ratio = 1.253 | 1.25 at λ = 0.016 |
| Out-of-sample: T(m = 0.08)/T(m = 0.10), pointwise in λ | 0.6953 — posted before the run | 0.707, converging monotonically (1.7%) |
| First-order response at m = 0 (envelope theorem) | no transient — the cut sits in μ's empty valley | none (λ³ window coefficients ±0.04, sign-unstable) |
| Where the odd structure lives in the potentials | the corridor, not the branches | |a₃| = 1.06 in the valley vs 0.08 / 0.02 in the modes |
The second-order character that first read as "κm²" was the law in disguise: μ(xs(m)) is itself nearly linear in m at moderate orders. The envelope reduction — dS/dm computable entirely inside the symmetric problem — is what caught it, and is this annex's quiet methodological export: the susceptibility of a ledger to a transfer order is a property of the balanced books.
One mechanism refuted, one constant left standing.
The house tested its own first theory of the profile G — that the log-running slope β is set by the Gaussian tail curvature of the destination branches, prediction: widen the branches ×1.4 and β falls to 0.10. The matched pair was run at n = 2048.
| Account geometry | β |
|---|---|
| Asymmetric mixture (original anomaly) | 0.204 |
| Bimodal Δ = 3.5, σ = 0.5/1.0 | 0.191 |
| Same, branch widths ×1.4 | 0.167 |
| Mean — candidate universal constant of the quadratic cost | 0.187 |
The house noticed 0.187 ≈ 1/2e = 0.1839 and immediately distrusted it — three points and a numerological resemblance is how books get cooked. Stamped, shelved, awaiting the matched-asymptotics derivation of Ĝ, which is the open analytic problem this annex poses: the Schrödinger system at a degenerate saddle.
What is held, what is owed, what was checked.
- Novelty audited twice over: two independent literature sweeps (≈110 citing papers of the Richardson result inspected), then the two nearest papers read in full. Both carve out exactly the complement of this regime — one requires a bi-Lipschitz transport map (the hypothesis that fails at the cut), the other marginals ε-close on a smooth curve. The former states, verbatim, that higher-order expansions are "still an open question for the analysis community."
- Predictively verified: the amplitude law posted its 0.6953 before the m = 0.08 run returned 0.707. Controls (Gaussian, unimodal, shallow-gap), grid-independence (n = 1024 vs 2048 agree on the λ-structure to 0.3% on c₂), and the exact Gaussian kill-gate all passed.
- One sub-mechanism refuted by the house's own experiment — the record above, kept in full, per the rule of Folio VII.
- Numerics, not theorems. One dimension, quadratic cost, one family of test measures. The master profile Ĝ and its β are conjecture until derived; the flat-exponent identity at the cut is the only piece that is exact.
- Not peer-reviewed. The write-up with full protocol, data, and repro commands lives in the campaign archive (sinkhorn-rs · parity-experiment notes); a standalone paper is the natural next entry in this account.
- Why this annex sits in this house: the deformation parameter is ℏ, the corridor cost is the Fisher information — the Bohm quantum potential in Madelung dress — and the method (kill-gates, out-of-sample invoices, public retractions, adversarial novelty audits) is the same double-entry. Different currency, same arithmetic.