The factor chart

One convention for placing any adapter or direction in five coordinates - the five oblimin factor directions, Gram-Schmidt orthonormalised in the exact Gram in the fixed order Warmth, Competence, Fearful withdrawal, Arousal, Imagination - which since 2026-09-08 every figure in the project uses.

The factor chart

A chart here is a fixed way of turning an object in weight space - a trait adapter, a steering direction, a hole - into five numbers that different figures can be compared across. Before 2026-09-08 the project's chart was the five leading principal components of the double-centred Gram. Samuel's decision that day made the factor analysis the primary frame, and qwen35/fa_chart.py is the single module every figure is being moved onto, so that the map, the direction cards, the trait pages, the sphere, the hole and the activation-space comparison cannot silently disagree about what "five coordinates" means. As of 2026-09-08 the map, the cards, the trait pages and the activation-space comparison are on it; the sphere sweep and the hole/alien direction are being moved by separate work. See Factor-first migration for the element-by-element state.

What the five directions are

Each factor is a direction in adapter space, not a loading vector: a weighted merge of the 134 stage-one adapters, v_f = sum_i c_fi a_i. The coefficient dictionaries are the ones actually steered in phase 10, qwen35/phase10_runs/steer_spec2_7a.json, under the job names FA_Warmth, FA_Competence, FA_FearfulWithdrawal, FA_Arousal, FA_Imagination. They come from the k=5 centred oblimin solution described on Factor analysis of the adapter Gram.

The order is fixed and it is the solution's own: descending sum of squared oblimin loadings, 10.76210106481124, 8.421873312354228, 7.020338408333558, 6.814511494042316, 5.798806928556024 (qwen35/results/fa_qwen35.json#solutions.centred_k5.ss_loadings.oblimin). fa_chart.FACTOR_ORDER hard-codes that order and ORDER MATTERS is written into the module docstring, because Gram-Schmidt is not order-invariant.

How a coordinate is computed

Inner products come from the exact Gram qwen35/results/gram_sweep.npz, never from coordinates: for two merges with coefficient vectors c and d, <v_c, v_d> = c^T G d.

The five factors are oblique, so they are not an orthonormal frame. The chart is an orthonormal basis of their five-dimensional span, obtained by Gram-Schmidt in the G inner product in the order above and stored as coefficient rows B (5 x 134) satisfying B G B^T = I (asserted in FAChart.__init__ at atol=1e-8).

• an adapter or merge with coefficients c: x = B G c
• an external adapter b - a hole word, an alignment adapter, a second seed - from its cross-Gram column X[:, b] = <a_i, b>: x = B X[:, b]
• the cosine between an object and the chart is |x| / |b|
• chart length is |x|

Because the basis is orthonormal in the Gram, a coordinate is an honest inner product with a unit vector and the five can be compared with each other.

What the chart is oblique about

The basis is orthonormal; the five factor directions it is built from are not. Their pairwise cosines (qwen35/analysis/fa_chart_summary.json#factor_pairwise_cosines, rounded to four places in that file):

Two pairs are strongly oblique: Warmth against Fearful withdrawal at -0.6661 and Competence against Arousal at -0.728. That is a property of the solution, not of the chart; an oblique rotation is allowed to produce correlated factors and this one did. The consequence for reading a figure is that the first basis vector is the Warmth direction itself, but the second is only the part of Competence orthogonal to Warmth, the third only the part of Fearful withdrawal orthogonal to both, and so on. FAChart.factor_chart_coords gives where each oblique direction actually sits in the chart.

How much of an adapter the chart sees

A trait adapter's mean chart length is 0.9369798382181508 against a mean adapter norm of 1.6157416444226869, so the chart captures a mean fraction 0.5784887830866848 of an adapter's weight change - 58 per cent (qwen35/analysis/fa_chart_summary.json#trait_chart_len_mean, #trait_norm_mean, #chart_captures_frac_of_norm_mean). The remaining 42 per cent is real and is not charted. Any statement of the form "this direction is n per cent of a trait adapter" on the blog page is a ratio of chart lengths and inherits that ceiling.

In activation space the same construction, applied to the 134 persona vectors, captures a mean fraction 0.7544359083812503 of a persona vector's norm - more, not less; see Constitution-as-prompt persona vectors vs weight geometry.

The top-3 sphere

fa_chart.SPHERE_FACTORS is the first three of the order - Warmth, Competence, Fearful withdrawal - and the sphere sweep samples the unit sphere of the span of the first three basis vectors. It used to sample the sphere of the top three principal components.

The data file

qwen35/build_viz_data_fa.py writes qwen35/analysis/viz_fa.json, which carries, for all 134 traits in analysis/viz.json's trait order (asserted equal): coords (134 x 5 chart coordinates), loadings (the oblimin pattern loadings, which are a different object - they agree with the coordinates in sign and rank but not in scale), assignment (largest absolute loading), chart_len, adapter_norm, chart_frac_of_norm, communality, uniqueness, the Big Five label and keying carried over from viz.json, factor_coords (the five oblique directions in their own chart) and a map2d block holding the primary two-axis layout, Warmth against Competence. There is no UMAP layout to keep beside it: analysis/umap_test.json stores kNN accuracies only.

Related

• Factor analysis of the adapter Gram - the solution the five directions come from
• PCA, the scree curve and the two nulls - the chart this replaced, now the secondary frame
• Factor-first migration - element-by-element register of what changed
• Constitution-as-prompt persona vectors vs weight geometry - the same chart built in activation space