OMEGA_BATTERIES_DISCOVERED.md

Omega-class battery potentials — exhaustive catalog

Compiled 2026-04-29 from full scratchpad survey across sessions 000080-000113. Includes every documented config across all sweeps (P/Q/R/S/T plus the A-set verification probes), every published HF model, and the trained substrate prototypes from this week. Both Adam and LBFGS candidates are listed where the sweep ran both.

Definition of omega-class

A trained or candidate battery is omega-class if it satisfies all four:

1. Sphere-solver architecture: PatchSVAE with sphere-norm M tensor, V rows on S^(D-1), output basis on ℝP^(D-1)
2. Projective-clean codebook: |deviation from uniform RP^(D-1) baseline| < 0.05, secondary antipodal pair count ≤ 3, axis utilization > 0.95
3. In its natural CV band for arch class (table below)
4. Codebook-engaged or sphere-engaged: cross-attn coupling moves off floor, recovery curve from random init, geometric stats leave passthrough signature

A "potential" is any trained instance OR sweep candidate that has run through the architecture and produced measurements against the criterion.

Natural CV bands by architecture class

arch class	V	D	natural CV band	source
noise-substrate (Freckles, Johanna)	256	16	0.20-0.23 (sweet spot), 0.13-0.30 (band)	ft1 attractor, ablation Phase 1
h2-class	32	4	0.80-1.05	h2-64 measurement, 000111-112
P-class	32	3	0.029-0.036 (LOW band)	Q-sweep ranks 06/07/09
Phase T D=5 V=16	16	5	~0.04 mean dev	Phase T sweet spot
Phase T D=5 V=32	32	5	varies by noise (partial)	Phase T (qualified ft2 D=5 claim)

Diagnostic signature (for testing any candidate)

dimension	passthrough	engaged (omega-class)
α (cross-attn coupling)	stationary [0.020, 0.030]	rising monotonically off floor
α_std/α_mean	flat ~0.06	climbs (>0.30 in byte-trigram)
row_cv	in arch's natural band	leaves natural band
ratio S0/SD	≈ 1.0 (flat spectrum)	drifts (>1.05)
erank	flat at full rank	dips below
recovery from random init	near-100% from ep 1 (sign-recovery trivial)	curve from ~0% upward
codebook deviation	undefined / not measured	within ±0.05 of uniform RP^(D-1)

---

The load-bearing stack (non-negotiable for any recipe)

Architectural primitives that make the omega regime exist. If any row is absent, the configuration is not an omega recipe — these are not tunable hyperparameters, they are the substrate.

primitive	code	breaks if removed
Sphere-norm M rows	M = F.normalize(M, dim=-1) before SVD/readout	scale-explosion snap (G2/G4 confirm dev +0.16-0.36 across all bands)
fp64 SVD path	_svd_fp64 autocast-disabled, fp64 Gram + sqrt + U-recovery, floors 1e-24/1e-16	discharge spikes, max_grad >1000 (paste 019)
Orthogonal init on encoder readout	nn.init.orthogonal_(enc_out.weight), re-applied after L-group overrides	L2/L3/L4 init variants regress
Bounded-α cross-attn	S_out = S · (1 + α · tanh(...)), α ≤ max_α via sigmoid	unbounded-α (I4) poisons spectrum
BoundarySmooth zero-init conv	identity-at-init smoothing post-stitch	boundary inconsistency under tile stitching
Pure Adam (NEVER AdamW)	torch.optim.Adam	weight decay fights geometric structure
No BatchNorm, no Dropout	—	spectral structure destabilizes
No global average pool	flatten or spatial statistics	accuracy 70% → 29% in geometric encoders
Patch-based forward	all components per-patch except cross-attn	breaks resolution invariance by design

---

The recipes

Each recipe is a named configuration with verified or predicted omega signature. Empirical instances are linked to the tier inventory below.

Recipe A-class — Johanna / Fresnel (workhorse)

matrix_v=256, D=16, patch_size=16, hidden=768, depth=4, n_cross_layers=2
~17M params, full SVD path (svd_mode='svd_fp64')
optimizer=Adam, lr=3e-4

• Omega clean at scale; teachable downstream as a single instance.
• Cost: ~30GB VRAM, ~4hr/epoch. Not stackable.
• Verified: Tier 1 v50_fresnel_64. Pending: Tier 9 Johanna D=16 codebook probe.
• Use when: workhorse encoder; downstream needs single-instance legibility.

Recipe S-class — Freckles (the crown)

matrix_v=48, D=4, patch_size=4, hidden=384, depth=4, n_cross_layers=2
~2.55M params, full SVD path, all defenses stacked
optimizer=Adam, lr=3e-4

• Perfect noise reconstruction (val MSE 5e-6 reporting floor on 16-noise mixture).
• Resolution-invariant by construction — weights at N=256 work at N=4096 (architectural, not empirical).
• Fails to teach as single instance. Requires the full Freckles array for legibility.
• Verified: Tier 1 v40 (64×64), v41 (256×256).
• Use when: noise/texture substrate; downstream consumes the array.

Recipe H2-class — sphere-solver (h2-64 single bank)

matrix_v=32, D=4, patch_size=4, hidden=64, depth=1, n_cross_layers=1
linear_readout=True, svd_mode='none', match_params=True, smooth_mid=16
~57,215 params per bank, optimizer=Adam, lr=3e-3

• The canonical sphere-solver. SVD replaced by learned linear readout; column norms = S, identity = Vt.
• Projective-clean at D=4: 24-27 axes/bank, dev +0.010 ±0.013 across 16 single-noise banks (A2 probe).
• Stackable into 192-bank arrays; teaches via per-bank MSE signature.
• Verified: Tier 1a (full 192-bank h2-64 array). Q-rank08 is the exact same architecture.
• Use when: small-scale, multi-bank, codebook-engaged regime.

Identity invariants for loading h2-64 weights (silent partial-load otherwise):

• smooth_mid=16 (NOT the ps-dependent default 8 at ps<16 — 440 params missing if wrong).
• linear_readout=True, svd_mode='none', match_params=True.
• n_heads divides D — at D=4 with n_heads=4, head_dim=1.

Recipe H2a — minimum H2-class (Q-rank02)

matrix_v=32, D=4, patch_size=4, hidden=64, depth=0, n_cross_layers=0
linear_readout=True, svd_mode='none', match_params=True
~40,227 params, optimizer=Adam, lr=3e-3

• Minimum sphere-solver: depth+n_cross stripped to zero.
• Q-MSE 0.00205 at 1000 batches — best Adam in Q-sweep.
• Use when: floor-finding the sphere-solver capacity envelope at D=4.

Recipe P-class — minimum projective-clean (Q-rank09)

matrix_v=32, D=3, patch_size=4, hidden=64, depth=0, n_cross_layers=0
linear_readout=True, svd_mode='none', match_params=True
~28,899 params, optimizer=Adam, lr=3e-3

• Smallest projective-clean omega-class instance in the catalog.
• LOW-band CV (~0.03), projective-clean on RP², 22 axes (10 pairs + 12 unpaired).
• Use when: minimum parameter budget; D=3 RP² regime is acceptable.

Recipe F-class — experimental nursery

matrix_v ∈ {32, 48, 64}, D ∈ {2, 4, 8, 16}, patch_size ∈ {8, 16}
hidden ∈ {32, 64, 128}, depth ∈ {1, 2}, n_cross_layers ∈ {1, 2}
2K – 645K params per config, 1 epoch × 1M samples (triage regime)
optimizer=Adam, lr=3e-4, soft-hand=CV-EMA prox boost (gradient-free)

• Designed to fail often. Collapse is a data point, not a bug.
• Triage signal (1ep × 1M) ≡ 30ep × 200K for keep/kill discrimination at 1/5 wallclock.
• Soft-hand: loss = (1 + boost · prox) · recon_loss; prox is Gaussian on current_cv vs CV-EMA. No penalty term.
• Use when: searching for new viable templates; running large sweeps cheaply.

Recipe T — D=5 sweet spot (Phase T)

matrix_v=16, D=5, hidden=64, depth=1, n_cross_layers=1
linear_readout=True, svd_mode='none', match_params=True
optimizer=Adam, lr=3e-3, 1000 batches

• 62% projective-clean across 16 noise types — only V whose mean dev lands in band at D=5.
• V=32 fails the cross-noise universality test at D=5 (Tier 6 supersedes A3's V=32 reading).
• Use when: extending recipes to D=5 with realistic projective-clean expectation.

Recipe R — polytope-packing (predicted, not yet probed)

(V, D) ∈ {(16, 4) 16-cell, (8, 4) 8-cell, (20, 3) dodecahedron}
hidden=64, depth=0, n_cross_layers=0
linear_readout=True, svd_mode='none', match_params=True
optimizer=Adam, lr=3e-3, 1000 batches

• Hypothesis: V matched to a known regular polytope vertex count for S^(D-1) → static rows, no antipodal pair rotation. Geometric frustration disappears.
• Trained, weights in phaseR_reports/ on HF; codebooks not yet extracted.
• Use when: testing the natural-axis-count framework. Open item from 000101.

Recipe J5 (LOW-band, unverified)

matrix_v=128, D=16, patch_size=16, hidden=128, depth=1, n_cross_layers=1
~1.46M params estimated, optimizer=Adam

• Strongest unverified noise-substrate candidate. LOW-band MSE 0.9595 with dev exactly 0.000 against uniform RP¹⁵ baseline — best of any non-H-group entry at LOW band.
• Worth a U5-style projective probe before committing to long-horizon training.

---

The substrate layer (input encoding rules)

Architecture and encoding are separate hypotheses. Passthrough on one encoding does not invalidate the architecture (lesson from 000113: SP-bit passthrough on h2-class → byte-trigram engaged on the same arch).

Per-patch capacity arithmetic

A (C, ps, ps) patch on the unit sphere supports:

• 4×4 RGB ≈ 1M discriminable positions on S³ at 1° resolution.
• The encoding's information cardinality must approach this for the codebook to do compression work.
• Hard zeros are wasted capacity — every float in every patch should carry signal.

Engagement vs passthrough

The full diagnostic table is the "Diagnostic signature" section above. Shorthand:

• Engaged: α rises monotonically; row_cv leaves natural class band; ratio S0/SD drifts; erank dips below D; recovery curve climbs from low.
• Passthrough: α stationary at init; row_cv in band; ratio ≈ 1.0; erank flat at D; recovery near-100% from ep 1.

Substrate pass/fail history

substrate	architecture probed	result	reason
16-type OmegaNoiseDataset	A, S, F, h2	Engaged	Original substrate; full per-patch density
ImageNet random crops (sublens)	A (Fresnel-64 v50)	Engaged	Natural image structure; full RGB density
Byte-trigram (R,G,B) (000113)	h2	Engaged	256³ cardinality per cell, every float carries signal
Sentencepiece bit content (000112)	h2	Passthrough	1/3-filled patch with 32 hard-zero paddings — model bored
Binary-tree i.i.d. Bernoulli (000111)	h2	Passthrough	Sign-only signal; trivial under linear_readout=True

Substrate design rules

1. Every float in every patch should carry signal. Hard zeros = wasted capacity.
2. Information cardinality per cell should approach 256³ for RGB (or the equivalent for non-RGB channel counts).
3. Spatial coherence within the patch matters — noise/image/byte-trigram all have it; bit-encodings of token IDs do not.
4. Test before training: compute the patch-space cardinality of your encoding. If it's orders of magnitude below per-patch capacity, expect passthrough.

---

The triage protocol (verifying a candidate is potentially an omega)

Run in order. Stop at first failure. Most candidates die at step 3 or 4.

Step 1. Architecture check (free, no training)

• Inspect kwargs against the load-bearing stack. Missing items disqualify before any compute is spent.
• Run the "debug move" from CLAUDE.md: instantiate, print state_dict keys + shapes, compare against a reference checkpoint. Mismatched smooth_mid, n_heads, linear_readout, svd_mode, match_params cause silent partial-loads under strict=False.

Step 2. Substrate check (free, no training)

• Compute per-patch information cardinality of the input encoding.
• If cardinality << per-patch capacity, expect passthrough — fix the encoding before training.

Step 3. 1-epoch triage (~minutes per config on a single GPU)

• 1 epoch × 1M samples, gaussian-only training (noise_types=[0]).
• Score on 16-noise per-noise generalization (256 samples per noise).
• Pass: spectrum bounded (no scale-explosion snap); MSE within 2-3× class baseline.
• Fail: NaN / divergence / snap event / MSE order-of-magnitude worse than class.

Step 4. 1000-batch convergence sweep (~hour per config on a single GPU)

• Full optimizer trajectory at lr=3e-3 (Adam) or lr=1.0 (LBFGS post-fix).
• Pass: MSE leadership in its band; CV stays in natural band; α trajectory shows engagement signature.
• Optimizer regime (000100): Adam dominates ≥500 batches; LBFGS niche is short-budget probing (≤100 batches).

Step 5. Codebook extraction (the omega ratification)

from geolip_svae.inference import (
    InferenceEngine, extract_codebook, make_calibration,
)

calib = make_calibration('sixteen_noise', n=64, size=64)
cb = extract_codebook(
    model, calib,
    model_id='...', calibration_name='sixteen_noise',
)
assert cb.is_projective_clean()         # |dev from uniform RP^(D-1)| < 0.05
assert abs(cb.deviation()) < 0.05

• Pass: projective-clean, axis utilization > 0.95, ≤3 secondary antipodal pairs.
• The candidate is now ratified as omega-class.

Step 6. Vacuum-seal test (cell deployability)

• Freeze candidate parameters (requires_grad=False).
• Train an adapter classifier around it on a downstream task.
• Pass: CV / erank / S0 stay locked under host gradient stress; classifier learns.
• Fail: geometry collapses → not deployable as a cell. (CE on cell internals "ripped the internals to shreds" — paste 023.)

Step 7. Bandwidth probe (downstream legibility)

• Linear head on omega outputs, downstream task.
• Pass: classifier achieves task-meaningful performance.
• Fail: omega outputs lack representational bandwidth at this scale (S-class symptom).

A candidate that passes 1–7 is a deployable omega cell ready for collective integration.

---

Decision tree: which recipe?

Goal	Recipe	Why
Stable workhorse encoder, single-instance teachable	A-class (Johanna/Fresnel)	Omega clean at scale; teaches downstream
Best noise reconstruction, multi-bank deployment	S-class (Freckles)	Perfect recon; needs full array for legibility
Small-scale, multi-bank ensemble, text/vision substrate	H2-class (h2-64)	57K params/bank, projective-clean, stackable
Minimum sphere-solver footprint at D=4	H2a (Q-rank02)	40K params, depth=0, n_cross=0, MSE 0.00205
Minimum projective-clean footprint at D=3	P-class (Q-rank09)	28.9K params, RP², MSE 0.028
Search for new templates cheaply	F-class triage	1ep × 1M, designed to fail often
D=5 representative	T (V=16)	Only V whose mean dev lands in band at D=5
Test natural-axis-count hypothesis	R polytope-packing	Predicts static-row H2-LIKE; codebooks unprobed
LOW-band scale-up (unverified)	J5	dev exactly 0.000 vs uniform RP¹⁵ baseline

---

Empirical catalog (trained and in-progress instances)

The sections below are the empirical record backing the recipes above. Each tier represents a class of trained or in-progress instances with measured signatures.

---

TIER 1 — Trained, verified omega-class on HuggingFace

HF path	arch class	params	D	V	optimizer	natural CV	n_axes	dev	training content	verified by
AbstractPhil/geolip-SVAE/v40 (Freckles 64×64)	Freckles	2,557,539	4	48	Adam	~0.20-0.23	—	<0.05	16-noise mixture	ft1, U5 cross-band
AbstractPhil/geolip-SVAE/v41 (Freckles 256×256)	Freckles	2,557,539	4	48	Adam	same	—	—	resolution-scaled v40	continuation of v40
AbstractPhil/geolip-SVAE/v50_fresnel_64	Fresnel-base	16,942,419	4	—	Adam	—	—	—	140M+ ImageNet random crops, sublens	streaming run, "phenomenal MSE recon"
AbstractPhil/geolip-svae-h2-64 (192-bank array)	H2_linear_matched	57,215 × 64 batteries × 3 phases	4	32	Adam	0.80-1.05	24-27/bank	+0.010 mean, ±0.013 std	per-bank, see Tier 1a	A2 probe (ft2), 192-bank cosine sweep (000109)
AbstractPhil/geolip-svae-implicit-solver-experiments/G-Cand	H2-class	28,899-45,852	3	32	Adam	0.03 (LOW)	22 (10 pairs + 12 unpaired)	−0.004	gaussian-only	A0 probe (ft2)
AbstractPhil/geolip-svae-implicit-solver-experiments/H2a	H2_linear_matched	40,227-57,215	4	32	Adam	0.80-1.10	26 (6 pairs + 20 unpaired)	+0.002	gaussian-only	A1 probe (ft2)
AbstractPhil/geolip-svae-implicit-solver-experiments/A3 (3 runs, qualified)	H2 variant	varies	5	16/32/64	Adam	varies	16/29/51	−0.015 / +0.016 / +0.019	gaussian-only single-arch	A3 probe (ft2). QUALIFIED by 000106 — single-arch single-noise. Phase T showed D=5 partial.
AbstractPhil/geolip-SVAE/byte_trigram_proto_v1	h2-class	57,215	4	32	Adam	left band ep 6 → 1.31	TBD	TBD	wikitext-103 byte trigrams	000113 — engaged signature confirmed; codebook investigation pending

TIER 1a — h2-64 array decomposition (192 banks = 64 batteries × 3 phases)

All 16 single-noise experts (Group 1) verified PROJECTIVE-CLEAN in A2. Remaining 48 batteries share architecture and were measured in 192-bank cosine sweep but not individually probed against projective threshold.

Group 1 — 16 single-noise experts (banks 0-15)

bank	noise type	special	training distribution
0	gaussian	universal-pull center	standard normal noise
1	uniform	—	uniform[-1,1] noise
2	uniform_scaled	—	scaled uniform
3	poisson	—	poisson-distributed
4	pink	clone-pair with bank 5	1/f spectrum
5	brown	clone-pair with bank 4	1/f² spectrum
6	salt_pepper	hardest reconstruction (S-sweep), cleanest projective at D=5 (Phase T)	impulse-style
7	sparse_impulses	cosine outlier — heavy-tailed	sparse extreme-value
8	block_upsampled	—	block-correlated
9	gradient_gaussian	most isolated battery on S³ — only non-stationary noise	spatial gradient
10	checker	structured noise	checkerboard
11	gauss_uniform_mix	—	mixture
12	four_quadrant	0% projective-clean across all Phase T archs	structured
13	cauchy	heavy-tailed	cauchy-distributed
14	exponential	—	exponential
15	laplace	heavy-tailed	laplace-distributed

Group 2 — gaussian+one pairs + generalist (banks 16-31)

15 pair-trained banks (gaussian + each of noises 1-15), plus 1 generalist trained on all 16. Notable: pairs (19, 20) = (gaussian+pink, gaussian+brown) are clone-pair on S³ for the same noise-family adjacency reason as (4, 5). Pair (16, 26) = (gaussian+uniform, gaussian+gauss_uniform_mix) is a clone-pair (gaussian dominates the residual; both bounded distributions).

Group 3 — gaussian-balanced quads (banks 32-47)

16 (gaussian, easy, medium, hard) covers via stride-7 deterministic enumeration over the EASY (uniform/uniform_scaled/cauchy/exponential/laplace) × MEDIUM (poisson/salt_pepper/sparse_impulses/gauss_uniform_mix) × HARD (pink/brown/block_upsampled/gradient_gaussian/checker/four_quadrant) product. All contain gaussian.

Group 4 — no-gaussian quads (banks 48-63)

16 (easy, medium, hard, hard) covers via stride-19 deterministic enumeration. No gaussian seen during training. Banks 48 and 51 are kNN top-5 outliers because they solve the sphere problem from a fundamentally different starting distribution than Groups 1-3 (which all see gaussian as universal pull-toward-interior).

---

TIER 2 — Q-sweep candidates (10 top-P configs × 1000 batches)

After the LBFGS Hessian-corruption fix (000099), the Q-sweep ran clean on 10/10 configs. All 10 are listed; both Adam and LBFGS variants where present.

Q-rank	variant	params	optim	G-MSE	CV	D	V	depth	n_cross	class	notes
01	Q_rank01_h64_V32_D4_dp1_nx0_lbfgs	57,123	LBFGS	0.00421	0.954	4	32	1	0	H2a	LBFGS Q-sweep best
02	Q_rank02_h64_V32_D4_dp0_nx0_adam	40,227	Adam	0.00205	0.862	4	32	0	0	H2a	Smallest H2a, canonical sphere-solver. ≈ 1 h2-64 bank's capacity at 70% the params.
03	Q_rank03_h64_V32_D4_dp0_nx1_adam	40,319	Adam	0.00250	0.890	4	32	0	1	H2a	Adam-vs-LBFGS twin of rank 04
04	Q_rank04_h64_V32_D4_dp0_nx1_lbfgs	40,319	LBFGS	0.00391	0.893	4	32	0	1	H2a	LBFGS twin of rank 03; Adam wins 36% lower MSE
05	Q_rank05_h64_V16_D4_dp1_nx1_lbfgs	36,607	LBFGS	0.03117	1.069	4	16	1	1	H2b	Only V=16 candidate; CV slightly above HIGH ceiling. Underexplored size class.
06	Q_rank06_h64_V32_D3_dp1_nx1_adam	45,852	Adam	0.02497	0.029	3	32	1	1	P-class (D=3)	LOW-band; originally framed "polynomial," now confirmed projective-clean on RP²
07	Q_rank07_h64_V32_D3_dp0_nx1_adam	28,956	Adam	0.03151	0.036	3	32	0	1	P-class (D=3)	Smaller P-class variant
08	Q_rank08_h64_V32_D4_dp1_nx1_adam	57,215	Adam	0.00231	0.960	4	32	1	1	H2a	Exact h2-64 single-bank arch (depth=1+n_cross=1)
09	Q_rank09_h64_V32_D3_dp0_nx0_adam	28,899	Adam	0.02782	0.035	3	32	0	0	P-class (D=3)	Smallest projective-clean omega-class candidate. Under 30K params. 30% smaller than H2a at ~14× MSE cost.
10	Q_rank10_h64_V32_D2_dp0_nx1_adam	19,649	Adam	0.16139	0.000	2	32	0	1	EXCLUDED	D=2 cannot form pentachoron (needs ≥5 points), CV undefined

Per-architecture optimizer comparison (where direct twin exists):

arch	Adam Q-rank	Adam MSE	LBFGS Q-rank	LBFGS MSE	winner
h64_V32_D4_dp0_nx0	02	0.00205	(no twin)	—	Adam
h64_V32_D4_dp0_nx1	03	0.00250	04	0.00391	Adam by 36%
h64_V32_D4_dp1_nx0	(no twin)	—	01	0.00421	LBFGS only
h64_V32_D4_dp1_nx1	08	0.00231	(no twin)	—	Adam

Optimizer regime guidance (000100, post-LBFGS-fix): Adam @ lr=3e-3 dominates at 1000-batch budgets. LBFGS retains niche for short-budget probing (≤100 batches) and floor-finding sweeps. For sphere-solver canonical training: Adam is the recommended default since 2026-04-24.

---

TIER 3 — Phase 1 ablation grid (233 configs across 15 groups × 3 bands × seeds)

The Phase 1 ablation predates the P-sweep. A through M groups, each varying one architectural axis at a time, all three CV bands (LOW D=16, MID D=8, HIGH D=4), seed-replicated where statistically meaningful. 233 total entries; 227 band-match (97%); 229 params-finite (98%); 223 valid (band-match + finite). Data from omega_inventory.csv. Columns: D / V / hidden / depth / patch_size = dp/ps / n_seeds / mean test MSE / mean CV / band deviation (observed_sphere_cv − uniform_RP^(D-1)_baseline) / n_finite seeds.

HIGH band (D=4, V=32, hidden=64, ps=4) — sphere-solver candidates

group	variant	n	mse	cv	dev	fin	notes
A	baseline	5	1.4591	0.902	+0.018	5	seed-replicated reference
B	B1_all16	1	1.4709	0.858	+0.021	1	16-noise mixture
B	B2_gaussian_only	1	0.9590	0.753	+0.027	1	gaussian-only train
B	B3_structured	1	1.7076	0.968	−0.036	1	structured-noise train
B	B4_heavy_tailed	1	1.7982	0.820	+0.064	1	heavy-tailed train
B	B5_first_half	1	1.4395	0.944	+0.025	1	noises 0-7
B	B6_even_indices	1	1.9918	0.880	+0.023	1	noises 0,2,4,...,14
C	C1_adam	1	1.4667	0.860	+0.022	1	Adam @ default lr
C	C2_sgd	1	1.5670	0.858	+0.019	1	plain SGD
C	C3_sgd_momentum	1	1.2006	0.860	+0.013	1	SGD with momentum
C	C4_adamw	1	1.4706	0.859	+0.023	1	AdamW
C	C5_lbfgs	1	nan	0.943	−0.923	1	LBFGS Hessian-corruption — pre-fix bug
D	D1_cosine	1	1.4712	0.858	+0.020	1	cosine LR
D	D2_constant	1	1.3028	0.858	+0.028	1	constant LR
D	D3_linear_decay	1	1.4652	0.858	+0.021	1	linear decay
D	D4_warm_restart	1	1.3216	0.858	+0.016	1	warm restart
D	D5_one_cycle	1	1.4656	0.858	+0.019	1	one-cycle LR
E	E1_full_softhand	3	0.4420	0.907	+0.026	3	soft-hand CV regularizer
E	E2_pure_mse	3	0.4692	0.904	+0.037	3	pure MSE, no regularizer
E	E3_measure_only	3	0.4517	0.908	−0.011	3	measure CV but don't regularize
E	E4_hard_cv_penalty	3	0.4504	0.902	+0.002	3	hard CV penalty
F	F1_gelu	1	1.4669	0.858	+0.022	1	GELU activation
F	F2_relu	1	1.4744	0.855	+0.025	1	ReLU
F	F3_silu	1	1.4702	0.901	+0.022	1	SiLU
F	F4_tanh	1	1.4550	0.806	+0.027	1	tanh
F	F5_identity	1	1.4522	0.866	+0.031	1	identity (no activation)
G	G1_sphere_norm	1	1.4668	0.858	+0.022	1	reference (sphere-norm M)
G	G2_no_norm	1	1.4680	1.112	+0.357	1	REMOVES sphere-norm — geometry breaks
G	G3_layer_norm	1	1.4482	0.701	−0.217	1	layer-norm instead — geometry breaks
G	G4_scale_only	1	1.4588	1.112	+0.359	1	scale-only — geometry breaks
H	H1_svd_fp64	3	0.4198	0.906	+0.006	3	full-fp64 SVD reference
H	H2_linear_matched	3	0.0456	0.908	+0.005	3	CANONICAL — H2-class baseline. 9× lower MSE than H1 fp64.
H	H3_linear_unmatched	3	0.0948	0.911	+0.001	3	linear readout, mismatched dims
H	H5_batch_shared_svd	2	0.2799	0.918	+0.056	2	shared-SVD across batch
H	H6_no_svd_direct	1	0.4606	0.902	+0.085	1	direct readout, no SVD
I	I1_1layer	1	1.4733	0.859	+0.023	1	1 cross-attn layer
I	I2_0layers	1	1.8687	0.910	+0.087	1	0 cross-attn layers
I	I3_2layers	1	1.6295	0.924	−0.002	1	2 cross-attn layers
I	I4_unbounded_alpha	1	1.4803	0.859	+0.022	1	no clip on α
K	K1_bs128	1	1.4653	0.858	+0.021	1	batch_size=128
K	K2_bs32	1	1.4276	0.858	+0.024	1	batch_size=32
K	K3_bs512	1	1.5732	0.859	+0.024	1	batch_size=512
K	K4_bs1024	1	1.5428	0.859	+0.017	1	batch_size=1024
L	L1_orthogonal	1	1.4802	0.858	+0.018	1	orthogonal init
L	L2_kaiming	1	2.4802	0.912	−0.009	1	Kaiming init
L	L3_xavier	1	1.6138	0.952	+0.025	1	Xavier init
L	L4_normal_small	1	1.5360	0.942	−0.019	1	normal init small std
L2	L2_lbfgs_pure_mse	3	0.0058	0.878	−0.605	1	LBFGS+pure MSE: BEST MSE ON HIGH but only 1/3 finite, geometry broken
M	M1_sgd_aggressive	1	1.0770	0.858	+0.009	1	SGD aggressive
M	M2_sgd_huge_lr	1	0.6798	0.858	+0.052	1	SGD huge lr
M	M3_sgd_high_momentum	1	1.2939	0.859	+0.013	1	SGD high momentum
E_preview	E1-E4 (4 variants × 1 seed each)	4	~1.47	~0.858	+0.020 to +0.028	4	1-seed preview of Group E

HIGH-band omega-class candidates (band-match + finite + |dev|<0.05): 49 of 51 HIGH entries. The two failures: G2_no_norm (+0.357 — confirms sphere-norm is non-negotiable for omega-class) and G4_scale_only (+0.359 — same finding). G3_layer_norm at −0.217 is the third sphere-norm-disruption case. C5_lbfgs is the pre-fix Hessian-corruption casualty.

HIGH-band MSE leader: H2_linear_matched at MSE 0.0456 with dev +0.005. This is the canonical sphere-solver template that h2-64 was built from — 9× lower MSE than the fp64-SVD baseline (H1) at the same dev tolerance. The L2_lbfgs_pure_mse achieves MSE 0.0058 (8× better still) but only 1 of 3 seeds converged finite and geometry deviation hits −0.605, so it's not omega-class despite the MSE win.

MID band (D=8, V=64, hidden=64, ps=16) — bulk-Gaussian attractor

group	variant	n	mse	cv	dev	fin
A	baseline	5	1.0115	0.359	+0.005	5
B	B1_all16	1	1.0088	0.380	−0.004	1
B	B2_gaussian_only	1	0.9530	0.369	−0.017	1
B	B3_structured	1	0.7076	0.347	−0.007	1
B	B4_heavy_tailed	1	1.6415	0.352	+0.007	1
B	B5_first_half	1	0.9441	0.380	−0.004	1
B	B6_even_indices	1	1.1708	0.352	−0.010	1
C	C1_adam	1	1.0093	0.380	−0.004	1
C	C2_sgd	1	1.7354	0.381	−0.018	1
C	C3_sgd_momentum	1	1.0390	0.380	+0.001	1
C	C4_adamw	1	1.0095	0.379	−0.005	1
C	C5_lbfgs	1	nan	0.337	−0.357	1
D	D1-D5 (5 variants)	5	~1.00	~0.380	−0.002 to −0.009	5
E	E1_full_softhand	3	0.9441	0.361	+0.008	3
E	E2_pure_mse	3	0.9416	0.362	+0.010	3
E	E3_measure_only	3	0.9430	0.363	+0.009	3
E	E4_hard_cv_penalty	3	0.9447	0.362	+0.011	3
F	F1-F5 (5 variants)	5	~1.01	0.375-0.388	−0.002 to −0.009	5
G	G1_sphere_norm	1	1.0105	0.381	−0.005	1
G	G2_no_norm	1	0.9935	0.560	+0.198	1
G	G3_layer_norm	1	1.0093	0.428	+0.064	1
G	G4_scale_only	1	1.0103	0.556	+0.149	1
H	H1_svd_fp64	3	0.9429	0.362	+0.009	3
H	H2_linear_matched	3	0.9195	0.365	+0.015	3
H	H3_linear_unmatched	3	0.9343	0.359	+0.018	3
H	H4_svd_fp32	2	0.9434	0.368	+0.015	2
H	H5_batch_shared_svd	2	0.9439	0.367	+0.016	2
H	H6_no_svd_direct	1	0.9453	0.368	−0.002	1
I	I1-I4 (4 variants)	4	1.007-1.011	0.355-0.380	−0.005 to +0.004	4
K	K1-K4 (4 variants)	4	0.999-1.038	~0.380	−0.002 to −0.012	4
L	L1-L4 (4 variants)	4	1.011-1.340	0.336-0.383	+0.006 to +0.011	4
L2	L2_lbfgs_pure_mse	3	0.8924	0.359	−0.233	1
M	M1-M3 (3 variants)	3	0.976-1.024	~0.380	+0.002 to +0.006	3
E_preview	E1-E4	4	~1.01	~0.380	−0.004 to −0.006	4

MID-band omega-class candidates: 73 of 78 MID entries. Same sphere-norm-disruption failures (G2, G4). MSE leader at H2_linear_matched 0.9195 with dev +0.015.

LOW band (D=16, V=64, hidden=64, ps=16) — noise-substrate attractor

group	variant	n	mse	cv	dev	fin
A	baseline	5	1.0080	0.197	−0.000	5
B	B1_all16	1	1.0104	0.203	−0.003	1
B	B2_gaussian_only	1	0.9389	0.210	+0.001	1
B	B3_structured	1	0.7051	0.207	+0.006	1
B	B4_heavy_tailed	1	1.6091	0.191	+0.009	1
B	B5_first_half	1	0.9401	0.201	−0.005	1
B	B6_even_indices	1	1.1678	0.211	−0.001	1
C	C1-C4 (4 finite)	4	1.008-1.626	~0.204	−0.004 to +0.008	4
C	C5_lbfgs	1	nan (excluded)	—	—	0
D	D1-D5	5	0.974-1.010	0.203-0.205	−0.000 to −0.003	5
E	E1_full_softhand	3	0.9338	0.199	+0.002	3
E	E2_pure_mse	3	0.9340	0.200	+0.001	3
E	E3_measure_only	3	0.9342	0.200	+0.002	3
E	E4_hard_cv_penalty	3	0.9336	0.199	+0.001	3
F	F1-F5	5	1.003-1.011	0.199-0.204	−0.001 to −0.004	5
G	G1_sphere_norm	1	1.0085	0.204	−0.003	1
G	G2_no_norm	1	0.9914	0.353	+0.174	1
G	G3_layer_norm	1	1.0048	0.211	+0.001	1
G	G4_scale_only	1	1.0092	0.347	+0.159	1
H	H1_svd_fp64	3	0.9345	0.200	+0.002	3
H	H2_linear_matched	3	0.9128	0.204	+0.005	3
H	H3_linear_unmatched	3	0.9195	0.202	+0.010	3
H	H4_svd_fp32	2	0.9327	0.202	+0.002	2
H	H5_batch_shared_svd	2	0.9331	0.202	+0.003	2
H	H6_no_svd_direct	1	0.9359	0.202	+0.003	1
I	I1-I4	4	1.004-1.014	0.201-0.210	−0.001 to +0.012	4
J	J1_V64_h64	1	1.0094	0.204	−0.002	1
J	J2_V32_h32	1	1.1314	0.205	−0.028	1
J	J3_V16_h32	1	1.1862	0.234	−0.002	1
J	J4_V64_h32	1	1.0957	0.208	−0.008	1
J	J5_V128_h128	1	0.9595	0.199	−0.000	1
K	K1-K4	4	0.997-1.047	~0.204	−0.003 to −0.006	4
L	L1-L4	4	1.009-1.242	0.197-0.204	+0.002 to +0.012	4
M	M1-M3	3	0.977-1.011	0.204	+0.002 to +0.006	3
E_preview	E1-E4	4	~1.01	~0.204	−0.001 to −0.003	4

LOW-band omega-class candidates: 76 of 79 LOW entries. Same sphere-norm-disruption failures (G2, G4). MSE leader: H2_linear_matched at 0.9128 with dev +0.005 — the same H2 template that wins on every band.

Group J — V × hidden capacity sweep at D=16: only 5 configs (one of the smallest groups), but the only group that varies V away from the standard {32, 64} options at the LOW-band. Confirms V=128 (J5) drives dev to exactly 0.000 and produces the lowest MSE in the LOW group at 0.9595. V=16 (J3) at h=32 still maintains dev −0.002 but loses 23% on MSE relative to V=128. J5_V128_h128 is the strongest unverified noise-substrate candidate in the catalog — params 1.46M (estimated from V=128, h=128), dev exactly zero against the uniform-RP^15 baseline, lowest MSE in the LOW band among non-H-group entries. Worth a U5-style projective probe.

Phase 1 ablation — engineering invariants surfaced

• Sphere-norm (Group G) is non-negotiable: removing it (G2, G4) blows up dev across all three bands by 0.15-0.36. Confirmed independently in HIGH/MID/LOW.
• H2_linear_matched is the canonical template: best MSE in 2 of 3 bands (LOW + HIGH), within 0.003 of best in MID. Forms the architectural template behind h2-64, all H2a/H2b Q-sweep candidates, and the substrate prototypes (bintree/SP-bit/byte-trigram).
• L2_lbfgs_pure_mse achieves the lowest MSE on HIGH (0.0058) at the cost of 67% non-convergence and broken geometry — the pre-fix Hessian-corruption pattern that 000099 diagnosed.
• CV bands quantize cleanly by D: HIGH ≈ 0.86 (D=4), MID ≈ 0.36 (D=8), LOW ≈ 0.20 (D=16). Bulk of variants land within ±0.05 of band center; the failures are sphere-norm disruption + LBFGS Hessian corruption.
• Group A 5-seed reference establishes within-config variance: HIGH dev variance ±0.018, MID ±0.005, LOW ±0.000. Within-config noise is comparable to or smaller than the |dev|<0.05 projective-clean threshold, so single-seed entries are still informative for the omega-class criterion.

Phase 1 ablation — direct architecture comparisons

comparison	HIGH	MID	LOW
H2_linear_matched (canonical) MSE	0.0456 ± 0.0076	0.9195 ± 0.0026	0.9128 ± 0.0043
H1_svd_fp64 (full SVD) MSE	0.4198 ± 0.0154	0.9429 ± 0.0041	0.9345 ± 0.0023
H2 advantage over H1	9.2× lower MSE	2.5% lower	2.3% lower
H3_linear_unmatched MSE	0.0948 ± 0.0123	0.9343 ± 0.0025	0.9195 ± 0.0011
H6_no_svd_direct MSE	0.4606	0.9453	0.9359

The HIGH-band H2 advantage (9.2× lower MSE than full-SVD H1) is what made H2_linear_matched the canonical template. At MID and LOW the H-group converges (linear-matched, full-SVD, and direct-readout all within 3% of each other), so the H2 win is specifically a HIGH-band phenomenon. This matches the architectural reading: at D=4 the SVD dimension is small enough that linear readout matches the spectral bandwidth without information loss; at D=8 and D=16 the spectral residual matters.

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TIER 4 — P-sweep small-battery floor grid (600 configs at 20 batches)

group_P_small_battery_floor from ablation_configs.py line 615. Full product: 5 × 5 × 3 × 2 × 2 × 2 = 600 configs. This is the parent grid the Q-sweep top-10 came out of. It runs after Phase 1 ablation establishes H2_linear_matched as the canonical template, and varies the architecture axes around that template at a budget-minimal 20 batches per config.

P-sweep grid axes

axis	values	count
hidden	{4, 8, 16, 32, 64}	5
V	{2, 4, 8, 16, 32}	5
D	{2, 3, 4}	3
depth	{0, 1}	2
n_cross	{0, 1}	2
optimizer	{Adam, LBFGS}	2

Pins (H2_linear_matched baseline)

• svd='none', linear_readout=True, match_params=True (the H2 ablation winner)
• HIGH band: patch_size=4, img_size=64
• batch_size=256, batch_limit=20 (~5,120 samples seen per config)
• n_heads=1 (since D varies down to 2, default n_heads=4 would fail)
• grad_clip=1.0 (defensive — see TIER 4a)
• soft_hand=False, cv_measure_every=2
• Adam: lr=3e-3 (Phase-2 default scaled to 20-batch budget)
• LBFGS: lr=1.0 (default unit-Wolfe-step, lib's own line search)
• Training: gaussian-only (noise_types=[0])
• Testing: 16-noise per-noise generalization (test_noise_types=list(range(16)), 256 samples per noise)

P-sweep outcomes by optimizer split

optimizer	total configs	finite	NaN/divergent	Q-sweep top-10 representation
Adam	300	300 (100%)	0	6 of top-10 (Q-ranks 02, 03, 06, 07, 08, 09, 10)
LBFGS	300	291 (97%)	9	3 of top-10 (Q-ranks 01, 04, 05)
Total	600	591	9	10 advanced to Q-sweep at 1000 batches

The 9 NaN/divergent LBFGS configs all matched the 000099 Hessian-corruption profile: depth=1 + n_cross=1 architectures where gradient clipping inside the LBFGS closure caused the (s_k, y_k) Hessian approximation to underestimate, generating runaway H⁻¹ steps over enough iterations. 20 batches is the threshold below which divergence is incipient but not yet catastrophic; the 1000-batch Q-sweep would have surfaced 30 LBFGS divergences had it not been pre-fixed.

P-sweep geometric attractor split (observed across all 600 configs, confirmed in Q)

D class	typical CV (post-20-batch)	attractor	members
D=4	0.86 - 1.07 (HIGH band)	sphere-solver (H2 family)	200 configs (5 hidden × 5 V × 2 depth × 2 n_cross × 2 opt)
D=3	~0.03 (LOW band)	projective-clean on RP² (P-class, originally framed "polynomial")	200 configs
D=2	undefined (V<5 cannot form pentachoron)	failed geometric validity	200 configs

200 D=4 configs are theoretically all H2 candidates. Survival of the 6 Adam + 3 LBFGS into Q-sweep top-10 depends on continued_training_potential, which combines convergence-rate-at-20-batches with extrapolated-MSE-at-1000-batches. Q-sweep's full table (Tier 2) gives the actual 1000-batch outcomes for those 10.

TIER 4a — P-sweep extrapolation rankings (the top-10 source data)

Each Q-sweep entry started as a P-sweep entry. The "P-MSE" column from group_Q_h2_candidates shows the MSE that ranked it at 20 batches:

Q-rank	source P config	P-MSE (20 batch)	Q-MSE (1000 batch)	improvement	optimizer regime change
01	h64_V32_D4_dp1_nx0_lbfgs	0.053	0.00421	13×	LBFGS clean at 1000 (post-fix)
02	h64_V32_D4_dp0_nx0_adam	0.572	0.00205	279×	strongest extrapolation
03	h64_V32_D4_dp0_nx1_adam	0.584	0.00250	234×
04	h64_V32_D4_dp0_nx1_lbfgs	0.041	0.00391	10×	LBFGS started low, gained little
05	h64_V16_D4_dp1_nx1_lbfgs	0.115	0.03117	4×	V=16 hits H2b ceiling
06	h64_V32_D3_dp1_nx1_adam	0.656	0.02497	26×	D=3 P-class
07	h64_V32_D3_dp0_nx1_adam	0.641	0.03151	20×	D=3 P-class
08	h64_V32_D4_dp1_nx1_adam	0.620	0.00231	268×	h2-64 single-bank arch
09	h64_V32_D3_dp0_nx0_adam	0.638	0.02782	23×	D=3 P-class smallest
10	h64_V32_D2_dp0_nx1_adam	0.736	0.16139	5×	D=2 — failed geometric validity

Reading: Adam configs started P-sweep with high MSE (0.57-0.74) and gained 20-280× by 1000 batches. LBFGS configs started P-sweep with low MSE (0.04-0.12) and gained only 4-13×. This confirms the optimizer regime shift that 000100 logged: LBFGS dominates at short budgets (≤20 batches), Adam dominates at long budgets (≥500 batches). The architectural template (h64_V32_D4 with depth/n_cross combinations) is unchanged across optimizers; only the optimization trajectory differs.

P-sweep coverage gaps

• 9 NaN configs never re-run with the post-fix trainer (parked open item from 000100). Could surface 9 additional Tier 2 candidates at the LBFGS+depth=1+n_cross=1 architecture frontier.
• V<32 LBFGS coverage is thin — only Q-rank05 represented from 300 LBFGS configs across V ∈ {2, 4, 8, 16, 32}. The full P-sweep contains 60 LBFGS configs at each V; their ranking among each other is not surfaced into the catalog.
• hidden < 64 LBFGS coverage is sparse — Q-sweep's top-10 are all hidden=64. A dedicated lower-hidden LBFGS sub-sweep was never run.

Test of the natural-axis-count hypothesis: V matched to known polytope vertex counts on S^(D-1) should produce static sphere-solver rows (no rotating antipodal frame).

variant	V	D	polytope	predicted	params
R_h64_V16_D4_16cell_orthoplex_adam	16	4	16-cell (4-orthoplex)	H2-LIKE static	—
R_h64_V8_D4_8cell_or_16cell_subset_adam	8	4	8-cell (tesseract)	H2-LIKE static	—
R_h64_V20_D3_dodecahedron_adam	20	3	dodecahedron	H2-LIKE static	—

Pins: same H2_linear_matched baseline as Q, Adam @ lr=3e-3, depth=0, n_cross=0, 1000 batches, gaussian-only training, 16-noise per-noise test.

Status: trained (in phaseR_reports/ on HF), results not surfaced into the projective-clean catalog yet. Worth probing against the omega-class criterion since natural-axis-count framework predicts they should land cleanly.

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TIER 4b — R-sweep polytope packing test (3 configs at 1000 batches)

group_R_packed_polytope_test from ablation_configs.py. Predicted-H2-LIKE configs where V is matched to known polytope vertex counts on S^(D-1). Same H2_linear_matched baseline as P/Q, Adam @ lr=3e-3, 1000 batches, gaussian-only training. Ran 2026-04-24 alongside Q-sweep.

variant	V	D	polytope	predicted	architecturally implies
R_h64_V16_D4_16cell_orthoplex_adam	16	4	16-cell (4-orthoplex)	H2-LIKE static rows	natural axis count for D=4 = 16
R_h64_V8_D4_8cell_or_16cell_subset_adam	8	4	8-cell (tesseract)	H2-LIKE static rows	sub-polytope vertex count
R_h64_V20_D3_dodecahedron_adam	20	3	dodecahedron	H2-LIKE static rows	natural axis count for D=3 = 20

Hypothesis tested: when V matches a known regular polytope vertex count for S^(D-1), training should produce static sphere-solver rows (no antipodal pair rotation). Phil's framing (000100): the 32-row × D=3 G-Class behavior emerged because 32 points cannot be uniformly arranged on S² — geometric frustration. Match V to polytope, frustration disappears.

Status: trained, weights in phaseR_reports/ on HF, but codebooks were never extracted and probed against the projective-clean threshold. Worth running through extract_codebook to surface them into the verified-omega-class tier; the natural-axis-count framework (ft2 §9.1) predicts they should land cleanly. Open item from 000101.

TIER 5 — Phase S D=5 architecture floor map (1600 configs at 20 batches)

axis	values	count
hidden	{4, 8, 16, 32, 64}	5
V	{2, 4, 8, 16, 32}	5
D	{5}	1
depth	{0, 1}	2
n_cross	{0, 1}	2
noise_type	{0..15}	16
optimizer	{Adam}	1 (LBFGS too slow for sweep)

Total: 5 × 5 × 1 × 2 × 2 × 16 × 1 = 1600 runs.

Headline finding (000105): cross-noise rank correlation +0.954. Architectures rank near-identically across all 16 noise types — what changes per noise is achievable floor MSE, not which model achieves it. Top-4 universal architectures all hidden=64, V=32.

Top-4 architectures from S analysis (mean rank across 16 noise types):

rank	architecture	mean rank
1	h64_V32_dp0_nx1_D5	1.1
2	h64_V32_dp1_nx0_D5	(close to 1.1)
3	h64_V32_dp1_nx1_D5	1.9
5	h64_V16_dp1_nx1_D5	(the V<32 entry)

Note: the 1391 individual config directories were lost to HF rate-limiting (87% of submitted commits failed). The 1600-config aggregate JSON survived; per-config artifacts mostly did not. Engineering invariant logged (000108): batch-sync uploads from this point forward.

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TIER 6 — Phase T D=5 convergence sweep (64 configs at 1000 batches)

Top-4 S architectures × 16 noise types, run at A3-reference budget. The D=5 walk-back (000106) lives here.

arch	hidden	V	depth	n_cross	optimizer	% projective-clean across 16 noises
h64_V16_dp1_nx1	64	16	1	1	Adam	62% (10/16) — D=5 sweet spot
h64_V32_dp0_nx1	64	32	0	1	Adam	50% (8/16)
h64_V8_dp1_nx0	64	8	1	0	Adam	25% (4/16)
h64_V32_dp1_nx1	64	32	1	1	Adam	19% (3/16)

Headline: 23/64 (~36%) configs converged within ±0.05 of uniform RP⁴ baseline. V=16 was the geometric sweet spot at D=5, not V=32 — overturning the V=32 universality reading from A3's three runs.

Per-V deviation summary:

V	mean dev	p25-p75	in band?
8	+0.115	[0.07, 0.18]	No
16	+0.040	[0.01, 0.07]	Yes (only V whose mean lands inside)
32	+0.057	[0.04, 0.07]	Just outside

Salt_pepper anomaly (000106): 100% projective-clean across all 4 archs in Phase T despite being the worst noise to reconstruct in S-sweep (best MSE 2.51, ~34× worse than pink). Geometry decouples from MSE — a bank that fails to reconstruct can still produce a clean projective codebook. This matters for downstream cross-bank analysis: "the worst-fitting bank" might still produce the most useful projective representation.

Four_quadrant anomaly: 0% projective-clean across all 4 archs. Spatially structured noise where no architecture in T converged. Open in ft3 §10 as a deeper-probe candidate.

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TIER 7 — A-set verification probes (the 19-model count from ft2)

These are the projective-codebook verification runs that produced the n=19 count cited in ft2's Section 5 table. All entries are individual probes with explicit deviation measurements.

probe	model	D	V	n_axes	pairs	mean projective angle	uniform baseline	dev	result
A0	G-Cand	3	32	22	10	1.011	1.015	−0.004	PROJECTIVE-CLEAN
A1	H2a	4	32	26	6	1.116	1.114	+0.002	PROJECTIVE-CLEAN
A2 (×16 banks)	h2-64 single-noise (banks 0-15)	4	32	24-27	5-8	mean 1.115	1.105	+0.010 mean, ±0.013	ALL 16 PROJECTIVE-CLEAN
A3 (×3 runs)	A3 D=5 (single-arch single-noise)	5	16/32/64	16/29/51	0/3/13	varies	varies	-0.015 / +0.016 / +0.019	PROJECTIVE-CLEAN at A3, QUALIFIED by Phase T — generalization fails

Probe count: 1 (A0) + 1 (A1) + 16 (A2) + 3 (A3) = 21 individual probe runs. ft2 cited "19 models" because the A3 three runs were treated as one architectural data point. Phase T (000106) re-classified A3 as a single-arch test, leaving 17 projective-clean instances at D=3/4 robustness level.

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TIER 8 — Substrate prototype trained models (this week's runs)

HF path	params	D	V	content	regime	result
AbstractPhil/geolip-SVAE/bintree_proto_v1	57,215	4	32	depth-4 binary tree, i.i.d. Bernoulli ±1, BFS-encoded	PASSTHROUGH	best test_mse 3.5e-5 ep 20, 100% bits/trees from ep 1, CV 0.80-1.00, erank 4.00, ratio 1.00
AbstractPhil/geolip-SVAE/sentencepiece_proto_v1	57,215	4	32	t5-base SP token IDs as 16-bit ±1 floats	PASSTHROUGH	best test_mse 5.78e-6 ep 18, 100% bits/tokens from ep 1, α=0.023 throughout, CV 0.85-1.11, erank 4.00, ratio 0.99
AbstractPhil/geolip-SVAE/byte_trigram_proto_v1	57,215	4	32	UTF-8 byte trigrams as RGB pixels at 256×256	ENGAGED	best test_mse 1.7e-5 ep 19, 83.9% byte / 61.3% trigram from 0% floor, α 0.024→0.043, CV left band ep 6, ratio 1.07, erank 3.9955 dip

The bintree + SP-bit pair establishes the passthrough control. Byte-trigram is the first text-engaged omega-class candidate but its codebook hasn't been formally probed against the projective threshold yet. Pending follow-up: byte_trigram_proto_128 (img_size decision pending, 100M sample-view target).

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TIER 9 — Architectural templates with no measured instance

Documented architectures the catalog does not yet cover. Each is omega-class eligible under the architecture criterion but lacks a verification probe.

template	arch	params	D	V	status	what's needed
Johanna D=16	PatchSVAE-F	8.7M (estimate)	16	256	not yet U5-tested	run extract_codebook on Johanna checkpoint, compute deviation from uniform RP^15
Grandmaster omega tokens	concept	—	—	—	paper-level reference	needs trained instance + verification
geolip-svae-nosvd-ablation repo	svd_mode='none' variants	varies	varies	varies	independent repo, omega verification not surfaced into main catalog	inventory the trained checkpoints, run U5 across them
D=6, D=7, D=8 with V matched to natural axis count	predicted by Phase T framework	—	6/7/8	~22/28/34 (predicted)	not run	sweep with natural-axis-count V matching

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TIER 10 — Explicitly excluded from omega-class

exclusion	reason	source
D=2 configs	Cannot form pentachoron (needs ≥5 points), CV undefined	Q-rank 10
Q-rank 10 (h64_V32_D2_dp0_nx1_adam)	D=2, MSE 0.16 = essentially failed reconstruction	Q-sweep
9 P-sweep NaN configs	LBFGS Hessian-corruption casualties (000099 bug profile)	P-sweep, never re-run
bintree_proto_v1	Passthrough regime, codebook not engaged	000111
sentencepiece_proto_v1	Passthrough regime, cross-attn idle	000112
Phase T D=5 V=32 cells (most)	V over-counted vs natural axis count ~16, fails projective-clean	000106
Phase T D=5 four_quadrant (all 4 archs)	0% projective-clean, spatial-structured noise	000106
ablation Group H, M, L SVD-removal variants	Spectrum-degenerate; not sphere-solvers in proper sense	Phase 1 ablation

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Smallest-instance benchmarks across the catalog

For when minimal-parameter operation matters:

tier	smallest config	params	D	V	MSE	regime	use case
absolute smallest projective-clean	Q-rank09 (P-class)	28,899	3	32	0.02782	LOW-band projective-clean on RP²	minimum-parameter omega
smallest H2a (canonical sphere-solver)	Q-rank02	40,227	4	32	0.00205	HIGH-band sphere-solver on RP³	canonical sphere-solver baseline
Phase T D=5 V=16 sweet spot	h64_V16_dp1_nx1_D5	~36,607	5	16	varies	D=5 partial projective-clean	D=5 representative
h2-64 single bank (production)	bank_idx 0..63	57,215	4	32	varies	per-noise sphere-solver	bank-level training composition

The 28,899-param P-class candidate (Q-rank09) is the absolute floor for projective-clean. The 40,227-param H2a (Q-rank02) is the floor for canonical sphere-solver behavior at D=4. Under 30K and under 41K respectively.

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What's missing from this catalog

1. byte_trigram_proto_v1 codebook investigation — engagement signature confirmed in trajectory, but extract_codebook against the trained checkpoint hasn't been run for the formal projective-clean verification. Tier 1 entry should move to "verified" once this is done.
2. byte_trigram_proto_128 — pending img_size=64-vs-128 decision + 100M-sample-view run completion.
3. 9 P-sweep NaN re-runs — never executed with the LBFGS-fixed trainer (parked open item from 000100). Could surface 9 additional Tier 2 candidates.
4. R-sweep results probed against projective-clean criterion — the polytope-packing predictions (16-cell, 8-cell, dodecahedron) trained but their codebooks weren't surfaced into the catalog.
5. Johanna D=16 verification — the only large-D representative of the noise-substrate line; not yet probed.
6. Cross-substrate kNN graph — bintree, SP-bit, byte-trigram, h2-64-noise codebook-similarity matrix. The "what survives the universal-substrate-hope death" finding (000111) requires this measurement.
7. Disproof candidates for Omega — the methodological pivot from 000108 demands negative-result candidates the catalog doesn't yet contain (non-spherical bottleneck variants, no-spatial-coherence content, byte-misaligned content).