The Formal Paper — Proofs, Correction Algebra, Postulates
Arc ·
This is the formal telling: the thesis in full academic register, with every equation typeset and every claim cited. It grounds representation theory in three older disciplines — the coordinate reference frames of physical geodesy, the statistical thermodynamics of state erasure, and the electrical grounding of real circuits — and carries their invariance laws into high-dimensional semantic space. At its center sits a two-part proof: that the classification behavior of softmax networks is exactly invariant under translations of representation space, and that cosine similarity — the metric nearly all applied semantics is built on — is not. The gap between those two facts is where representation bias lives, and closing it is a coordinate alignment problem with a known algebra: mean-centering, covariance whitening, metric tensor generalization, per-query standardization.
The paper concludes with three postulates. They stand at the head of this lens because they are the destination the whole series has been walking toward — and because each one, by design, compresses one stretch of the walk.
The semantic coordinates of any representation vector are mathematically undefined in isolation; they exist strictly relative to a localized coordinate origin and basis. Comparing raw cosines across unaligned manifolds is coordinate-equivalent to comparing local zenith angles across a curved manifold without establishing a parallel transport mechanism.
The stability of a high-dimensional virtual coordinate space is bounded by the physical entropy dissipation of the hardware executing the calculations. Logical state centering and whitening carry a physical thermodynamic cost governed by the Landauer limit. Failure to mathematically ground the coordinate space mirrors the physical signal drift of a floating electrical ground in high-frequency circuitry.
A high-dimensional representation space is semantically unbiased and geometrically optimal if and only if its coordinate origin is aligned with the empirical centroid of the conceptual distribution. This alignment neutralizes the translational invariance of the softmax classifier, ensuring that the angular distance metric maps monotonically to operational model probabilities.
Meaning is not a static coordinate on a manifold. It is the property of a semantic space that survives continuous origin alignment — the thing that stays the same when everything arbitrary about the frame is allowed to change.
The founding formulation did not survive
That sentence is the claim this lens was written to defend. It was tested and it broke. The campaign pointed its own instrument at the founding equation and kept the verdict: the frame group is not a group — closure fails at 0.448 — so M = R/G is metaphor rather than algebra. The invariant genuinely exists and converges, and it is poorer: extracted, it loses information against simply choosing a good frame, 0.358 against 0.623. Dividing out the frame costs something.
What the campaign earned instead is narrower and sturdier than what it set out to prove. Neighbourhood structure transfers between independently trained models — nine in ten targets within top-5 against a 0.35% chance floor, surviving a simultaneous change of language and model. Point resolution does not: anchor-density-limited, twice missing a frozen 60% bar at 56.6%. What crosses between two private universes is the neighbourhood. The exact star, about half the time.
A third result belongs here because it corrects a reading rather than a number: coherence is not a truth signal. Measured at the statement level the gap came back at −0.0044 and the win rate at exactly fifty per cent. Coherence measures consensus. Truth needs grounding. The sentence above is retained as written, because a refuted claim kept at full prominence is the evidence that the ones still standing were tested too.