# Actual-model parameter clouds: frozen grouping and null protocol

Assets: retained Qwen3.5-2B original15852e8c16360a2fea060d615a32b45270f8a8fc,
and pure GSM8K GRPO LoRA seed0 at HF038d8be9bd0edd3883fa019651b02bdf65609468.
Use the established read-only FP32 W+2BA wrapper. No training, GPU, new models,
weight mutation, inference requirement or deletion. Every retained asset stays.

## What constitutes a distribution

A model has no unique canonical number of parameter distributions. Enumerate a
fixed family, not "all possible distributions":
1. Each ndim>=2 tensor flattened as [axis0, product(other axes)]: row vectors
   and column vectors, when point count>=16 and feature dimension>=32.
2. Decoder language/MTP full-attention Q and interleaved gate rows per query
   head; K/V rows per KV head. GDN concatenated Q/K/V rows per corresponding head.
   No claim that these groups are disjoint or iid; whole-matrix views overlap.
3. Primary sample-size stratum has >=64 available points;16..63 is small-sample
   descriptive only. One-dimensional norm/bias vectors have no cloud of vectors;
   GDN kernel row dimension4 is low dimensional and excluded with explicit reason.
   Vision parameters are covered by matrix/flattened views, not vision-head splits.

Include all checkpoint components (language, vision and MTP). Per view choose
up to512 point indices without replacement, split deterministically into calibration
and held-out halves. Use full vector coordinates up to dimension8192; only larger
vectors use a fixed random coordinate subset, labelled projected rather than full.
Use identical indices, coordinates and null random draws for original/FT.
No overlap-group parameter savings or total distribution counts are additive.

## Centers and radius/scaling

Analyze origin and calibration-mean centers separately. Fit R_max using calibration
norms only; do not silently clip held-out points outside R_max. Also fit calibration
RMS radius. Store held-out maximum, outside-radius mass and radial spread.

Null A: uniform d-dimensional ball. Simulate the SAME center-fitting and calibration
maximum/RMS fitting before held-out evaluation. For origin, also retain exact
expected max-fitted null CDF a^d*n_cal/(n_cal+1), a<=1; log probabilities remain
available and zero Monte Carlo hits are not exact zeros.
Null B: independent coordinate-uniform box bounded by calibration min/max,
including asymmetric coordinates. Fit its center/radii using its own calibration
sample by exactly the same procedure. Record energy/range mismatch and held-out
coordinate violations; this null is conditional on fitted bounds, not a statement
that all real vectors lie inside that box.
Null C: independent coordinate-uniform box with calibration mean mu and standard
deviation sigma, bounds mu +/- sqrt(3)*sigma (sample variance ddof1). This matches
first and diagonal second moments in its generating law. Use its own calibration
and held-out center/max/RMS fits too; RMS-normalized comparisons target shape
rather than a change of global scale. Matching diagonal moments is not covariance
matching and does not preserve correlations.

Use512 validation null draws (not iid model observations), same calibration count
as each data cloud. Reuse isotropic-ball nulls only for identical dimension/count
and center-fitting rules. Monte Carlo resolution/error must be stated. Fit both
centers from the same raw draws where applicable; paired FT uses common randomness.

## Metrics and interpretation

Retain measured norm arrays, empirical CDF grids, max-normalized interior fractions
at0.5/0.75/0.9/0.95/0.99 and outside1, RMS-normalized CDF from0.5 to1.5,
median radial ratios and radial spread. Prefer probability-point differences over
astronomical ratios against tiny a^d. Inspect positive AND negative CDF gaps:
more inner tail plus more outer tail is not global inward stochastic dominance.

Record exact zeros, coordinate energy participation (sum squares)^2/sum fourths,
coordinate-variance participation, empirical covariance/second-moment participation
via calibration Gram matrices. Label sample rank ceiling n_cal-1 (mean centered)
or n_cal (origin); compare to equally sampled nulls, do not interpret the raw
finite-sample participation as true intrinsic dimension.

Report both full-radius and moment-controlled results for each center, whole-view
and head subgroups, primary versus small sample/projection strata, original and
changed72-LoRA-target groups. Unchanged parameter views can reuse identical results
with explicit references rather than re-read/recompute. Results describe parameters,
not activations, model quality, functional duplicates, or pruning gains.

Execution budget: one CPU, 512 MiB RSS ceiling, 7200 seconds per invocation;
checkpoint every group, resume without replacing completed groups. Any incomplete
coverage is reported before interpretation. No numerical analysis runs locally.

Illustrative radial CDFs use these fixed original/origin views, selected by module
role before the census completes, not by the resulting similarity/interior score:
language layer0 linear-attention Q head0, layer3 full-attention Q head0, and layer0
MLP up-projection rows. Their plots are illustrations, not independent replications.