1. Centre before fitting Procrustes

Status: Accepted · Date: 2026-08 · Evidence: docs/m0-findings.md, section 4

Decision

auto fits centred Procrustes by default: subtract each space's mean before solving for the rotation, restore it after. Plain Procrustes stays in the candidate list.

Context

The preprocessing step was originally ℓ2 normalisation and nothing else. That is the conventional choice and it has a conventional justification: cosine similarity is scale-free, so normalise and rotate.

The justification is incomplete. A rotation is a linear map that fixes the origin, so it cannot express a translation — and two embedding spaces do not share an origin. Whatever offset separates them, the rotation has to absorb, and absorbing it costs alignment everywhere else.

Evidence

Across 4 corpora and 3 model pairs (M0, 24 measurements):

	Mean ARR gain from centering
T0 (document proxies)	+0.166
T1 (real queries)	+0.260
Best case	+0.75

It hurt in 3 of 24 measurements, always by ≤0.018 — inside the ±0.024 confidence interval measured for ARR at these sample sizes.

Centred Procrustes then matched the residual MLP's quality at half the memory and a third of the latency, which changed the default candidate ordering too.

Consequences

• The largest single quality gain in the project came from one subtraction.
• Plain Procrustes is kept so the report can show what centering bought. Removing it would save one fit and lose the ability to answer "is this worth it?"
• The adapter stores both means, so apply is a translate–rotate–translate. That is two extra vector adds on the query path, measured at well under the 15 µs budget.

Centring alone is not a principle — a footnote

The reasoning above is "a rotation cannot express a translation", and it is sound as far as it goes. What it does not license is the stronger claim that centring brings two spaces closer together on its own.

Maystre et al., When Embedding Models Meet, give a two-dimensional counter-example (their Appendix B.3): two spaces where centring makes the alignment worse — X̃ᵀỸ = 0 after subtracting the means, while a suitable orthogonal matrix aligns the originals perfectly. Their measured conclusion is that alignment with centring is best and centring on its own is not principled (their Figure 7).

That is exactly the shape of the decision recorded here: centring is a step inside a fit for a rotation, never a substitute for one. It also gives the three regressions above something to be. All three are e5→bge, and a systematic loss confined to one model pair is what a counter-example looks like when it shows up in practice rather than in two dimensions.

Nothing changes. The default was set on 24 measurements and holds; what changes is the claim made for it, which is now the measured one rather than a general argument the literature does not support.

Alternatives

Normalisation only, as specified. Rejected on the measurement.

Centre as a separate adapter type rather than a default. Considered; it is what the candidate list does in effect. Making it the default is the part that matters — a user who never reads the candidate table still gets it.