AI answers shouldn’t be fake. It shouldn’t cost the planet. And it shouldn’t take people’s work without paying them.

Embed the question as q and each stored passage as x_i. The demo uses 384-dimensional BGE embeddings, normalized to length one. Its nearest-record distance is:

Smaller d means a closer passage. The recorded gate admits knowledge questions when d < 0.80 and declines at or above 0.80. Holonomy lands at 0.618; the restaurant request lands at 0.883. Identity and conversational turns take a separate persona route before this test.

The engine also exposes a Gaussian kernel support score, written in the technical pitch as:

Each passage contributes a bump centered on its embedding. The bandwidth σ sets how wide that bump is. Nearby passages contribute more; distant ones contribute less. Distance measures the closest support, while density measures support accumulated across records. This unnormalized sum is sensitive to bandwidth and corpus duplication; it is not a calibrated probability of correctness.

Embedding and distance receipts · Technical pitch §2

Retrieval supplies passages; the “weld” combines them. A connection supplies transport maps between their fiber frames. Composing these maps makes the route matter: rotations generally do not commute, so visiting A, B, C can have a different effect from visiting A, C, B.

R_j is a transport rotation, R_w is their product over a window, and I is the identity: no net rotation. The Frobenius norm takes the square root of the sum of squared matrix entries. For a closed loop, the composed map is holonomy; the defect measures how far it is from returning unchanged.

Here n is the fiber-frame dimension. For orthogonal rotations, the defect is at most 2√n, giving this normalized coherence score. The pitch uses A ≥ 0.91 as a composition threshold: a window below it is a signal to cut the weld. This is a geometric diagnostic under the chosen transport rule, not a test of the truth of every sentence.

Capacity, horizon, and the meaning of “curvature”

C = τ / Ks_max = τ / (K ℓ_c)

The pitch defines a capacity dial C and a coherent-horizon dial s_max. τ is the allowed budget, K is the curvature quantity for the region being measured, and ℓ_c is an estimated correlation length. With the other terms fixed and positive, doubling K halves both budgets. The K = 0 limit needs an explicit flat-region policy, rather than division by zero.

The scoped engine dials compute K as the mean of each selected field’s variance divided by its squared range. Their fallback estimate for ℓ_c is the square root of the mean per-coordinate variance. Selecting the embedding fields and retrieval neighborhood therefore matters: including an unrelated timestamp can change the scale dramatically.

GIGI has several operators called curvature. For example, the generic insert-time record score instead averages |value − running mean| divided by the field’s range. These statistical scores are distinct from differential-geometric curvature of a connection. The selected operator, fields, and neighborhood determine what a reported K means.

Coherence implementation · Record geometry · Technical pitch §3

The constellation contains 9,134 passages after merging near-duplicates. Each passage connects to its eight nearest neighbors; an undirected edge is kept if either passage chooses the other. The resulting graph has 55,323 edges and one connected component.

Here A is the graph’s adjacency matrix and D has each node’s degree on its diagonal. L is the normalized graph Laplacian. Its low-frequency eigenvectors vary slowly across strongly connected neighborhoods, which makes them useful coordinates for showing the library’s broad structure.

The layout skips the trivial zero mode and uses D^−½φ₁, D^−½φ₂, D^−½φ₃ as its three axes. It then rescales them, adds a little jitter, and stretches the radial display for legibility. The first nonzero eigenvalue is approximately 0.0040.

Graph, spectrum, and layout receipts

The research compares a closed-form next-token baseline with a learned language model. In the baseline, X holds context features, Y holds target tokens as one-hot rows, and W maps features to token scores. Ridge regression chooses W by balancing fit against coefficient size:

The regularization strength α limits amplification of weakly supported directions. For positive α, the fit can be solved by matrix factorization without iterative gradient training. The reported four-minute A10G run covers the 92.5M-token closed-form fit; it is not the training time for the whole language model.

The spectral solution and the thermal training update

With the thin singular-value decomposition X = USVᵀ, each singular value s_i is filtered separately:

W_α = V diag(s_i / (s_i² + α)) UᵀY

One decomposition supports a sweep over α, selected on held-out accuracy. At larger scale, the experiment accumulates XᵀX and XᵀY in chunks and solves through the resulting Gram matrix.

The separate 126.7M-parameter transformer learns phrasing. During voicing, the reported SGLD experiment adds Gaussian noise to gradient updates:

θ_t+1 = θ_t − η_t∇L(θ_t) + ε_tε_t ∼ N(0, 2η_tT_tI)

θ holds the model weights, η is the learning rate, L is training loss, and T controls the noise temperature. The experiment lowers T on a cosine schedule. The curves below show the measured result: SGLD keeps improving through epoch 16, while AdamW reaches its best held-out perplexity at epoch 6 and then worsens.

Technical pitch §4 · training and experiments