The claim
For a factorization R ≈ PQᵀ, the change P ↦ PA, Q ↦ QA⁻ᵀ leaves PQᵀ unchanged for every invertible A. A207 §1: “Thus an entire family of distinct latent-coordinate systems predicts exactly the same relationship matrix.” This freedom is the factorization’s gauge. The record already held it in two forms. Park, Choe and Veitch’s Theorem 3.4 (LIT-304) is the same transformation for a softmax model, γ ↦ Aγ, λ ↦ A⁻ᵀλ, and so leaves no inner product identified. A kernel fixes a representation only up to an orthogonal map (THEORY-004).
What the gauge leaves invariant is the predicted matrix and everything computed from it:
- its rank and singular values (A207 §3’s table);
- its column and row spaces, since span(PA) = span(P);
- the item kernel of the predicted matrix, R̂ᵀR̂ = Q(PᵀP)Qᵀ.
The latent Gram matrix QQᵀ is not invariant: it becomes QA⁻ᵀA⁻¹Qᵀ. Nor are latent cosines, or any reading of an axis. So the similarity of two items is gauge-invariant when it is computed from their response profiles, as in A207’s hypothesis Similarity(R_·i, R_·j). It is not invariant when it is computed from their latent vectors. A207’s table does not warn of this, and a relational signature built from latent item vectors would fail it.
Weight decay narrows the gauge without removing it. A207 §1: “orthogonal changes still preserve both predictions and the usual squared-norm penalty.” The minimum of (‖U‖²_F + ‖V‖²_F)/2 over UVᵀ = M is the nuclear norm ‖M‖_* (THEORY-726). Its minimizers are the balanced factorizations U = U_M Σ^½ R and V = V_M Σ^½ R with R orthogonal. The penalty therefore fixes the gauge up to O(k), and it fixes latent inner products to V_M Σ V_Mᵀ. That value is chosen by the penalty, not by the ratings.
R9’s reader checked this numerically on random 6×2 and 5×2 factors. The predicted matrix and Q(PᵀP)Qᵀ were unchanged under a random invertible A, and QQᵀ changed. The Frobenius penalty was unchanged under a random orthogonal A and rose from 17.0 to 73.1 under the general A.
Nonnegative factorization
A210 §2 says NMF has “a simpler form of the gauge freedom”: for any positive diagonal D, WH = (WD)(D⁻¹H), and topics may be permuted. A214 §6 says the same: “Positive diagonal rescalings and topic permutations remain standard ambiguities, with additional nonuniqueness possible.”
Scaled permutations are the freedom every nonnegative factorization always has. The full freedom for a given (W, H) is the set
{A invertible : WA ≥ 0 and A⁻¹H ≥ 0}.
That set depends on the data, is not a group in general, and can be much larger than the scaled permutations, because A itself need not be nonnegative. Nonnegativity reduces the freedom. It does not simplify it.
A worked case. Take W = [[1, .5], [.5, 1], [1, 1]] and H = [[1, .6, .3], [.3, .6, 1]], and A = [[1.2, −.2], [−.2, 1.2]]. Then WA = [[1.1, .4], [.4, 1.1], [1, 1]] and A⁻¹H = [[.9, .6, .4], [.4, .6, .9]]. Both are nonnegative, and they give the same X = WH. A is not a rescaling or a permutation. In this example no word belongs to one topic alone and no document to one topic alone, which is the condition the uniqueness theorems exclude (separability, anchor words). This is THEORY-721’s point in a new place, the core of THEORY-017’s synthesis: the parts are fixed by structure supplied from outside, here nonnegativity in the vocabulary’s own basis plus a condition on the data.
Gauge is not automorphism
A207 §3: “Ordinary matrix factorization has a gauge symmetry, not necessarily a group representation whose character classifies the objects.” A214 §6: “A factorization gauge transformation can leave every predicted observation unchanged without corresponding to any real-world transformation of the communicative situation. A structural automorphism must preserve the specified relational system itself.” A218 Ch5.9 keeps it as a chapter item: “Structural automorphisms versus latent-coordinate gauge.”
A gauge change alters the description and leaves the relata where they are. An automorphism moves relata among structurally equivalent positions. Characters are invariants of the second kind (CLAIM-132), not of the first. The gauge group GL(k), or O(k) under weight decay, leaves every observable fixed, so its characters carry no information about which item is which.
Why the arrows run in this order
A214 §6 boxes a progression, here transcribed from the display: “Observed relationships → Identifiable latent model → Structural automorphisms → Transformation representation → Character”, and says “None of those arrows is automatic.” It says only that gauge and automorphism “can interact”. The interaction is exact.
An automorphism of the data is a pair of permutations (P, Q) of documents and words with PXQ = X. It sends a factorization (W, H) to (PW, HQ), which is another factorization of X. If the factorization is essentially unique, that is unique up to scaled permutations, then (PW, HQ) = (WDΠ, Π⁻¹D⁻¹H) for a unique permutation Π of the topics. So the automorphism group acts on the topics, and that action is a permutation representation, with character χ(g) = the number of topics g fixes. Without identifiability the induced action is defined only up to the gauge, and no representation on the factor space is determined. The order of A214’s arrows is a dependency, not only a sequence. It is the Aut(F) direction of CLAIM-113, which did not reach the manuscript.
What it does not say
- It does not say latent axes are meaningless. Constraints that break the gauge, such as nonnegativity under separability, can identify them.
- It does not say the invariant quantities are the communicative kind (TERM-040). They are what the data and the rank-k model determine.
- It does not say observed data have non-trivial automorphisms. Exact permutation symmetries PXQ = X almost never hold in a noisy real-valued matrix. A214 §6 lists “its automorphism group may be trivial” as one possible failure. For measured data it is the generic case. Literal characters then need designed symmetries, stipulated equivalences between realizations, or approximate automorphisms with a tolerance. The second is a new definition, not a theorem of character theory.
- It does not say gauge freedom is unimportant. It is the record’s cleanest case of a representation without identity.
Note of 2026-10-19: the minimizers
THEORY-726 gives the value of the minimum and one factorization that attains it. That every minimizer is balanced, up to an orthogonal R, is this claim’s step, checked for the promotion audit’s batch 015. At a minimizer (U, V), f(UA, VA⁻ᵀ) ≥ f(U, V) for every invertible A, where f(U, V) = (‖U‖²_F + ‖V‖²_F)/2. Taking A = I + tS, the derivative at t = 0 is tr(UᵀU S) − tr(VᵀV S), and it vanishes for every S, so UᵀU = VᵀV. Call it B. Then MMᵀ = UVᵀVUᵀ = UBUᵀ = (UUᵀ)², so UUᵀ = (MMᵀ)^½ = U_M Σ U_Mᵀ, and U = U_M Σ^½ R for some R with orthonormal rows. Likewise VVᵀ = V_M Σ V_Mᵀ, so V = V_M Σ^½ R′. Then UVᵀ = M gives RR′ᵀ = I, which for two matrices with orthonormal rows forces R′ = R. The latent Gram matrix of the items, VVᵀ = V_M Σ V_Mᵀ, is therefore fixed by the penalty.
Note of 2026-10-26: an additive latent-feature model of an organization’s members
T23 of the exchange “Experimental Infrastructure Review” proposed an additive latent-feature model of employees (CASE-165), u = Σ_k B_k a_k W_k o + W^private o, with B a binary availability under a beta–Bernoulli prior and a ∈ [0, 1] an activation. Its own identifiability section names column permutation and the scale trade between W_k and a_k. Both are instances of this claim’s gauge, and the scale trade is exact while the rescaled a stays in [0, 1].
Three more follow from the same algebra, and the exchange missed them.
- B = 1 with a near 0 and B = 0 give nearly the same output, so availability is separated from activation only by the prior and the dynamics, not by the data.
- A feature one employee holds at constant activation c is absorbed into that employee’s private map, W^private + cW_k.
- Features whose B and a columns coincide across employees are identified only through their sum W_k + W_l.
So the distinction T23 calls important, between possessing a capacity and using it, is not identified from the outputs alone. Points 2 and 3 were checked numerically to machine precision; no model was fitted.
Three of the works the exchange cited, read the same day, carry the gauge further than they say.
- The author–topic model (THEORY-972). Its likelihood depends on the topics Φ and the authors’ topic distributions Θ only through ΦΘA, with A fixed by the observed authorship: a three-factor product with one factor known. The paper names the permutations of topics. The likelihood also leaves a continuum of changes of topic basis free whenever every topic gives every word positive probability, which every posterior mean with a positive smoothing parameter does.
- The mixed membership stochastic blockmodel (THEORY-993). Given the memberships, an edge has probability (1 − ρ)π_pᵀBπ_q, so π ↦ Rπ, B ↦ R⁻ᵀBR⁻¹ leaves the likelihood unchanged, a bilinear form of the same gauge, against the paper’s §5, which grants that the two seem “in competition for explaining the data” and says their difference “practically eliminates the issue”. The sparsity ρ enters only through (1 − ρ)B and is not identified apart from B’s scale.
- Correlated latent features (THEORY-1012). A layer of categories over IBP features does not narrow the gauge for features that always occur together: they are separated only where they sometimes occur apart, and in the variant where each observation has a single category the features of one category are never separated.
The claim is unchanged.
Note of 2026-10-29: the neuronal gauge theory’s freedom is the representational kind
Sengupta, Tozzi, Cooray, Douglas and Friston, LIT-1215, read in NOTE-1350, was filed to check whether its gauge freedom is more than the kind this claim holds. It is not. Its gauge freedom is reparameterization of the variational family: the natural gradient is “invariant under re-parameterisation”, and the Riemann gradient “automatically guarantees gauge-invariance by breaking the symmetry within a model” (Fig. 1 caption). That is a change of coordinates on the same distributions, the representational kind this claim separates from a symmetry of what is modelled. The one symmetry that might be more, between models with equal evidence, is proposed in the paper’s conclusions and not constructed. The claim is unchanged.