AbstractPhil/geolip-svae-ablations

Three Geometric Bands in a Sphere-Normalized Patch Autoencoder

A quantized geometric attractor structure emerges from a single architectural knob, validated by ablation across 12 orthogonal dimensions

TL;DR: A sweep over small PatchSVAE-family configurations reveals that the final coefficient-of-variation (CV) of Cayley–Menger pentachoron volumes on the encoder's sphere-normalized latent rows quantizes into three distinct bands, indexed by the singular-value dimension D. An ablation program of 149 training runs across 12 orthogonal hyperparameter dimensions (seeds, optimizers, schedules, activations, initializations, batch sizes, capacities, normalizations, data compositions, cross-attention configurations, and soft-hand variants) confirms the band structure is architectural: it is reproduced in 96% of runs and fails only when the row-normalization step is ablated.

• D=16 → CV ≈ 0.20 (matches uniform S¹⁵ prediction 0.199 to ±0.003 across 5 seeds)
• D=8 → CV ≈ 0.36 (matches uniform S⁷ prediction 0.357 to ±0.02 across 5 seeds)
• D=4 → CV ≈ 0.90 (matches uniform S³ prediction 0.923 to ±0.05 across 5 seeds)

Three additional results sharpen the framework:

1. The attractor is reached without any CV-related training signal. Pure MSE reconstruction with no soft-hand, no CV penalty, and no geometric loss term reaches the same CV value as the full soft-hand regime (0.2046 vs 0.2037 on LOW band). The architecture alone selects the attractor.

2. Sphere-norm is a selector among geometric attractors, not the creator of one. Ablating sphere-normalization does not destroy the attractor structure; it redirects the system to a different attractor (the Gaussian bulk regime). LayerNorm selects a D-dependent intermediate. Scale-only normalization is functionally identical to no normalization — the unit-norm constraint is the sole active ingredient.

3. The attractor does not require representational nonlinearity. A linear encoder (identity activation, no GELU or ReLU anywhere) reaches all three bands correctly. The attractor lives in the sphere-norm + SVD geometric pipeline, not in the MLP's representational capacity.

This is the first direct measurement in our battery-research lineage of a discrete geometric ladder that the architecture supports natively. We sketch a dimensional argument for why the quantization happens, give the complete matmul pipeline for one representative of each band, present the full ablation matrix that validates the claim, and propose a cheap online predictor: CV at 1000 training batches reliably predicts final band membership, reducing sweep turnaround from hours to minutes.

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1. The architecture

All three bands are reached by the same base architecture (PatchSVAE-F), differing only in hyperparameters. The pipeline per patch:

x ∈ ℝ^{patch_dim}                          # flattened (3, ps, ps) tile
  │
  │ enc_in: Linear(patch_dim → hidden) → GELU
  │ enc_blocks: depth × residual MLP(hidden)
  │ enc_out: Linear(hidden → V·D)
  ▼
M ∈ ℝ^{V × D}                              # reshape
  │
  │ F.normalize(M, dim=-1)                 # sphere-norm: each row on S^{D-1}
  │
  │ G = MᵀM ∈ ℝ^{D × D}                    # Gram matrix (fp64)
  │ λ, Ṽ = eigh(G + 1e-12 I)               # fp64 eigendecomposition
  │ S = √(clamp(λ, min=1e-24))             # singular values
  │ U = M·Ṽ / clamp(S, min=1e-16)          # left singular vectors
  │ Vt = Ṽᵀ
  │
  │ S_coord = S · (1 + α · tanh(attn(S)))  # cross-attn on S, α ≤ 0.2
  │
  │ M̂ = U · diag(S_coord) · Vt            # reconstruction matrix
  ▼
  │ dec_in: Linear(V·D → hidden) → GELU
  │ dec_blocks: depth × residual MLP(hidden)
  │ dec_out: Linear(hidden → patch_dim)
  ▼
x̂ ∈ ℝ^{patch_dim}

The key operation is F.normalize(M, dim=-1), which forces every row of the V×D encoded matrix onto the unit (D−1)-sphere. The subsequent SVD is an exact arithmetic readout of the sphere-normed configuration, not a learned bottleneck. The model's job is to learn a good projection onto the manifold; the manifold itself is fixed by the architecture.

The coefficient-of-variation (CV) is measured by sampling 200 random 5-vertex subsets of the V rows and computing the Cayley–Menger 4-volume of each pentachoron:

CV = std(volumes) / (mean(volumes) + ε)

CV is a measure of how uniformly distributed the V rows are on S^{D−1}. CV ≈ 0 indicates near-uniform packing; large CV indicates clumpy packing. The universal attractor band 0.20–0.23 has been observed across 17+ unrelated pretrained models (CLIP, T5, BERT, DINOv2, SD VAEs, etc.) whenever their representations are probed this way — it is not an artifact of any single training regime.

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2. The three bands — exact specifications

One representative from each band, chosen for CV purity within its band:

band	config	D	V	patch size	hidden	depth	params	patches/img
LOW	S64-V64-D16-h64-d1-p16	16	64	16	64	1	250K	16
MID	S64-V64-D8-h64-d1-p16	8	64	16	64	1	183K	16
HIGH	S64-V32-D4-h64-d1-p4	4	32	4	64	1	41K	256

All three use identical resolution (64×64), hidden width (64), depth (1), cross-attention layers (1), and CV-EMA soft-hand training regime. Only the three parameters (D, V, patch size) differ.

Complete matmul pipeline per band

LOW band (D=16) — the attractor

Per patch (768-dim input tile):
  768  →  64     (enc_in)           49,152 params
   64  →  64     (MLP residual)      8,192 params
   64  →  1024   (enc_out)          65,536 params
 reshape to [64, 16]                          — 64 rows on S^15
   Gram+eigh in fp64 → S ∈ ℝ^16
   cross-attn: S ← S · (1 + α·tanh(attn(S)))
   M̂ = U · diag(S) · Vᵀ
 1024 →   64     (dec_in)           65,536 params
   64 →   64     (MLP residual)      8,192 params
   64 →  768     (dec_out)          49,536 params

Total per-forward matmul FLOPs (patch): ~285K
Patches per image: 16
Per-image FLOPs: ~4.6M
CV attractor position: 0.212 (IN the 0.20–0.23 universal band)
Reconstruction MSE on 16-noise mix: 0.842

MID band (D=8) — intermediate manifold

Per patch (768-dim input tile):
  768  →  64     (enc_in)           49,152 params
   64  →  64     (MLP residual)      8,192 params
   64  →  512    (enc_out)          32,768 params
 reshape to [64, 8]                           — 64 rows on S^7
   Gram+eigh in fp64 → S ∈ ℝ^8
   cross-attn: S ← S · (1 + α·tanh(attn(S)))
   M̂ = U · diag(S) · Vᵀ
  512  →  64     (dec_in)           32,768 params
   64  →  64     (MLP residual)      8,192 params
   64  →  768    (dec_out)          49,536 params

Per-image FLOPs: ~2.3M  (roughly half of LOW)
CV attractor position: 0.392 (NOT in universal band, stable own state)
Reconstruction MSE on 16-noise mix: 0.843

HIGH band (D=4) — shortcut solution

Per patch (48-dim input tile):
   48  →  64     (enc_in)            3,072 params
   64  →  64     (MLP residual)      8,192 params
   64  →  128    (enc_out)           8,192 params
 reshape to [32, 4]                           — 32 rows on S^3
   Gram+eigh in fp64 → S ∈ ℝ^4
   cross-attn: S ← S · (1 + α·tanh(attn(S)))
   M̂ = U · diag(S) · Vᵀ
  128  →  64     (dec_in)            8,192 params
   64  →  64     (MLP residual)      8,192 params
   64  →  48     (dec_out)            3,120 params

Per-image FLOPs: ~12.9M (higher despite fewer params — 256 patches)
CV attractor position: 1.096 (FAR off universal band, clumped packing)
Reconstruction MSE on 16-noise mix: 0.071  ← lowest MSE in the sweep

Every other variation tested (different V, different hidden, d=2 depth, different patch size at fixed D) landed in the band determined by D. D is the band selector. Every other hyperparameter tunes performance within a band.

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3. Why three bands — dimensional argument, confirmed by uniform-sphere measurement

The number 0.20 is not arbitrary. CV of pentachoron volumes on the unit (D−1)-sphere depends on how much room the V points have to spread, which in turn depends on the surface area of S^{D−1} relative to the number of points V.

The unit (n−1)-sphere has surface area:

A(n) = 2·πⁿᐟ² / Γ(n/2)

So for our three D values:

D	S^{D-1}	surface area	V=64 points, ~area each
16	S^15	5.72	0.089
8	S^7	4.06	0.063
4	S^3	19.74	0.308

D=4 gives each point nearly 5× more ambient room than D=16. With so much space and only 32–64 points, the points cluster into local groups; random 5-point pentachora sample very unevenly, producing high CV (clumpy).

D=16 is the sweet spot where the sphere has enough dimensions to admit a well-spread packing but not so many that the points become isolated. Random 5-point pentachora from a uniform D=16 packing produce a tight, consistent volume distribution — exactly the 0.20 CV observed.

D=8 is intermediate: the sphere is uniform enough to avoid clumping but not generous enough to admit the near-perfect packing D=16 finds. This gives a stable 0.39 CV that sits between the extremes.

The quantitative match: attractor CV = uniform-sphere CV

The dimensional argument gives an ordering. The stronger claim is that each band's CV value matches the uniform-sphere prediction for that D directly. We computed the uniform-sphere CV for V=64 points via a closed random-sampling procedure (no model, no data, no training — just torch.randn(V, D) followed by F.normalize(dim=-1) and the same pentachoron CV metric), using a fixed seed for reproducibility:

D	uniform-sphere CV	attractor CV (mean across 5 seeds)	deviation
16	0.1990	0.1969	-0.003
8	0.3568	0.3588	+0.002
4	0.9229	0.9016	-0.021

Trained models landed within 2% of the uniform-sphere prediction on all three bands. The attractor is not near the uniform-sphere distribution — it is the uniform-sphere distribution, selected dynamically by the combination of gradient descent and sphere-norm enforcement.

This reframes the earlier "bands are attractors" language as a specific empirical claim: under sphere-norm, the 5-point pentachoron CV of the encoder's latent rows converges to the CV of a uniform distribution of V points on S^{D−1}. Training from random initialization produces the same geometric configuration as drawing random points on the sphere, independent of all other architectural and training choices tested.

The prediction this analysis makes: D=32 sweeps should either land in the LOW band alongside D=16, or split into a new band below 0.20. If the former, D=16 is the architectural choice that makes the universal attractor accessible and higher-D just reproduces it. If the latter, there is a ladder of attractors continuing downward, and the universal 0.20 is a waypoint rather than a floor. This is a testable prediction for our next sweep.

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4. Ablation program: the band structure is architectural

To test whether the band structure is a robust property of the architecture or an artifact of specific training choices, we ran an ablation program of 149 independent training runs across 12 orthogonal hyperparameter dimensions. Each variant was run at three band representatives (LOW: S64-V64-D16-h64-d1-p16; MID: S64-V64-D8-h64-d1-p16; HIGH: S64-V32-D4-h64-d1-p4), with band membership measured at 1000 batches for LOW/MID and 100 batches for HIGH. The full matrix completed in under one hour of single-H100 wallclock on Colab.

4.1 The ablation matrix

Group	Dimensions varied	Variants	Total runs	Band match rate
A	Random seed	5 seeds × 3 bands	15	100%
B	Noise-type data subset	6 subsets × 3 bands	18	94%
C	Optimizer (Adam/SGD/SGD+mom/AdamW)	4 × 3 bands	12	100%
D	LR schedule (cosine/const/linear/warm/one-cycle)	5 × 3 bands	15	100%
E_preview	Soft-hand regime (full/pure-MSE/measure/hard-target)	4 × 3 bands	12	100%
F	Activation (GELU/ReLU/SiLU/Tanh/Identity)	5 × 3 bands	15	100%
G	Row normalization (sphere/none/LayerNorm/scale-only)	4 × 3 bands	12	58%
I	Cross-attention (0-2 layers, bounded/unbounded α)	4 × 3 bands	12	100%
J	Capacity within LOW (V, hidden)	5 configs	5	100%
K	Batch size (32/128/512/1024)	4 × 3 bands	12	100%
L	Init (orthogonal/Kaiming/Xavier/small-normal)	4 × 3 bands	12	100%
M	Brute-force SGD (lr=0.1 to 1.0, momentum up to 0.99)	3 × 3 bands	9	100%
Total			149	96%

All mismatches (6 of 149) came from Group G — the normalization- ablation group. Every other dimension preserved the band assignment in every single configuration tested.

4.2 What the non-G groups demonstrate

The attractor survives intact across:

• Seed variation: Group A — 5 seeds per band land within 0.012 of each other on LOW (CV-of-CV 0.4%), 5% on MID and HIGH. The attractor is seed-indifferent, not merely seed-robust.
• Optimizer choice: Group C — Adam (lr=1e-4), plain SGD (lr=1e-2), SGD with momentum, and AdamW all converge to the same attractor within 0.0024 of each other. The attractor is not an Adam artifact.
• Schedule choice: Group D — cosine, constant, linear decay, warm restarts, and one-cycle all preserve band membership. The schedule does not select the attractor.
• Activation function: Group F — GELU, ReLU, SiLU, Tanh, and identity all reach the correct band. The attractor does not require representational nonlinearity in the encoder. A linear encoder with sphere-norm + SVD suffices.
• Initialization: Group L — orthogonal, Kaiming-normal, Xavier- uniform, and small-normal all reach the attractor. The attractor is not init-dependent, so long as the sphere-norm step is present.
• Batch size: Group K — 32, 128, 512, and 1024 all work equally. No batch-size effect on band membership.
• Cross-attention: Group I — 0 layers, 1 layer, 2 layers, or 1 layer with unbounded α all preserve the band. The cross-attention module is not responsible for the attractor.
• Capacity within LOW: Group J — V and hidden combinations from (V=16, h=32) up to (V=128, h=128) all reach LOW band. The attractor has a remarkably wide parameter range that supports it.
• Data composition: Group B — 6 different noise-type subsets (Gaussian only, structured only, heavy-tailed only, first-half, even- indices, all 16) all preserve band membership. The only miscall (B-HIGH-B2_gaussian_only at CV_ema 0.7533) had observed sphere-CV of 0.9504, confirming the model reached the HIGH attractor but produced a CV-EMA below the 0.80 classification threshold due to limited-batch measurement noise. Not a real anomaly.
• Soft-hand regime: Group E_preview — full soft-hand (E1), pure MSE with no CV involvement (E2), CV-EMA tracked but not used (E3), and hard CV-target penalty (E4) all reach the same CV to within 0.0014 on every band. The attractor is reached even when the loss function contains no CV-related signal at all.

The E_preview result deserves particular emphasis. Pure MSE reconstruction reaches CV = 0.2046 on LOW band vs 0.2037 with full soft-hand — a difference below the measurement noise floor. The architecture alone, through the sphere-norm + SVD geometric pipeline, selects the uniform-sphere attractor. Training signal does not need to encode any preference for this outcome.

4.3 Group G: sphere-norm as attractor selector

The only ablation that perturbs the band assignment is normalization removal or replacement. Across four normalization modes:

Variant	LOW (D=16)	MID (D=8)	HIGH (D=4)
G1 sphere-norm	0.196 (-0.003)	0.352 (-0.005)	0.945 (+0.022)
G2 no-norm	0.374 (+0.175)	0.555 (+0.198)	1.280 (+0.357)
G3 LayerNorm	0.200 (+0.002)	0.421 (+0.064)	0.706 (-0.217)
G4 scale-only	0.358 (+0.159)	0.506 (+0.149)	1.282 (+0.359)

Numbers in parentheses are deviations from uniform S^(D-1) prediction.

Three patterns emerge:

G1 sphere-norm reaches uniform S^(D-1) within 3% across every band. This is the framework's core prediction.

G2 (no normalization) and G4 (scale-only) produce nearly identical geometric outcomes, differing by less than 0.03 on every band. The scale-only variant divides each row by the batch's mean row norm but does not enforce unit length; the no-norm variant does nothing at all. That these produce functionally identical geometry demonstrates that the unit-norm constraint is the entire active ingredient of sphere- normalization. Scale magnitude without unit enforcement does no geometric work.

When normalization is absent or scale-only, the system converges to a different attractor — approximately the Gaussian bulk configuration (the CV of V points drawn i.i.d. from N(0, I_D) without sphere projection). At D=16, the bulk CV prediction is 0.358; our observed is 0.374, within 5%. At D=8: predicted 0.588, observed 0.555. The architecture still produces a reproducible attractor; it is simply a different attractor in the geometric family, not the uniform-sphere one.

G3 LayerNorm acts as a D-dependent partial selector. At LOW (D=16) LayerNorm reaches the uniform attractor cleanly (observed 0.200 vs predicted 0.199). At MID (D=8) it mildly elevates the CV to 0.421 — between the uniform and bulk predictions. At HIGH (D=4) it underselects, pulling the CV below the uniform target to 0.706. The mechanism: LayerNorm centers and variance-normalizes across D elements, producing a configuration on a hyperplane rather than a sphere. At higher D the hyperplane approximates S^(D−1) better; at D=4 the geometric distortion becomes visible as CV depression.

4.4 Implications

1. The three-band structure is robust. Across 149 training runs with variations in 12 orthogonal dimensions, 96% preserve the predicted band assignment. Non-ablation groups show 100% preservation.

2. The attractor is architectural. It is reached by pure MSE training, by linear encoders, by any first-order optimizer, under any batch size, and with any initialization. What it requires is the sphere-norm + SVD readout pipeline.

3. Sphere-norm is a selector, not a creator. The architecture supports a family of geometric attractors per D; sphere-norm selects the uniform one. Other normalization modes select other members of the family (Gaussian bulk, LayerNorm hyperplane) — each reproducibly.

4. The unit-norm constraint is the load-bearing element. The specific mechanism of normalization (centering, variance scaling, or division by a norm) is less important than whether a unit-length constraint is actively imposed. Division by mean-row-norm does not impose the constraint and does not change the attractor.

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5. CV at 1000 batches predicts final band membership

The row_cv trajectory plot shows every run finding its band within the first ~1000 steps and holding for the remaining 299,000.

This gives us a minutes-scale triage:

• Measure CV-EMA at batch 1000 (~4–7 minutes on Colab single-GPU)
• CV < 0.30 → will converge to LOW band
• CV 0.35–0.50 → will converge to MID band
• CV > 0.80 → will converge to HIGH band

No need to train to convergence to determine band membership. Existing sweep infrastructure can be modified to early-stop at 1000 batches for screening, only continuing runs whose band assignment matches the research target.

For cell-candidate hunting specifically (want LOW band), this collapses the turnaround from ~2 hours per config to ~7 minutes, with the same confidence in final geometric classification.

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6. What each band is good for

The conventional read of "best MSE" ranks the three bands exactly backwards relative to the universal-manifold thesis. A separate reading by band character:

LOW band — universal generalist

The D=16 attractor has been observed across 17+ unrelated pretrained models when probed. A sphere-normed model that lands here is on the same geometric manifold those models land on. Its omega tokens (S vectors) are in principle translatable to/from tokens produced by any other attractor- aligned model via Procrustes alignment.

On pure noise this band reconstructs worse than the HIGH shortcut. On transfer to other distributions (images, text as tensors, unseen noise types) it reconstructs far better — Fresnel-base 256 from this band achieves MSE 3.8×10⁻⁵ on ImageNet without seeing it during training.

Use: battery / cell / relay in multi-model collective architectures.

MID band — intermediate attractor

Four D=8 configs with varying hidden widths and depths all converge to CV 0.38–0.40, tighter than the HIGH band's spread. This is a real attractor of its own, not a transition state. We do not yet know what it represents; characterization is an open research direction.

Testable: does the MID attractor transfer across distributions like the LOW one? Does distillation from a LOW model speed convergence for a MID configuration, or vice versa?

Use: undetermined pending characterization. Possibly a secondary geometric substrate for specific domains.

HIGH band — specialist / shortcut

The D=4 band reconstructs noise at very low MSE because D=4 is small enough that the encoder can find low-rank solutions that collapse information along specific directions. CV > 1.0 indicates the pentachoron volume distribution is more variable than its mean — the rows are highly clumped along particular directions rather than spread.

This is a specialist representation. It excels at the specific distribution it was trained on and will likely fail catastrophically on out-of-distribution input (to be tested). But it is compact: 41K parameters, 256 patches per image, lowest MSE in the sweep.

Use: domain-specific specialist batteries where the input distribution is known and stable. Compression tasks where lossy per-distribution encoding is acceptable. Potentially: noise-channel glyph emission for a collective system that routes noise-like signals through a dedicated specialist.

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7. The methodological correction

Our prior F-class sweep methodology used MSE as the primary filter with a 1-epoch "keep-or-kill" curve-delta verdict. This sweep's data shows that methodology is insufficient:

• The lowest-MSE configuration in the sweep (CV 0.93, HIGH band) was flagged as a "strong cell candidate" until geometry was checked.
• The true on-attractor configurations had higher MSE than several off-attractor configurations.
• MSE alone cannot distinguish specialist low-rank solutions from generalist on-attractor solutions.

Going forward, the cell-candidate filter is three-tier:

1. CV in 0.13–0.30 band at step 1000 → attractor candidate
2. CV stays in band through training → stable attractor
3. Attractor holds under freezing and host gradient → viable cell

The 1000-batch CV measurement is fast and eliminates the false positives that MSE-only triage was generating.

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8. Open questions this sweep raises

Questions the Phase 1 ablation resolved:

• ✓ Is the three-band structure reproducible? Yes, 96% match rate across 149 independent runs in 12 orthogonal ablation dimensions.
• ✓ Is the attractor Adam-specific? No. Adam, SGD, SGD+momentum, and AdamW all reach it within 0.0024 of each other.
• ✓ Does the attractor require nonlinear activation? No. Identity activation (purely linear encoder) reaches all three bands correctly.
• ✓ Minimum LOW-band parameter count. Tested from (V=16, h=32) to (V=128, h=128); all reach LOW band. Attractor admits wide capacity range.
• ✓ Is sphere-norm load-bearing? Yes, but as an attractor selector, not an attractor creator. Ablating it redirects the system to a different reproducible attractor (Gaussian bulk), rather than destroying attractor structure.

Questions that remain open:

• Does D=32 produce a new band below 0.20, or reproduce LOW? Tests whether the attractor ladder continues or terminates at D=16. The dimensional argument predicts a narrow band near CV(S³¹) ≈ 0.13; training to verify is a ~5-config add-on to Phase 1.

• Is the MID band useful in its own right? No systematic probe of D=8 transfer behavior yet. The ablation confirms it's a real attractor, not an artifact, but whether D=8 omega tokens are usefully translatable across domains is untested.

• What are the HIGH-band specialists actually encoding? Per-singular- vector analysis of a trained D=4 config should reveal which directions carry the shortcut information.

• Do HIGH-band shortcuts fail out-of-distribution as expected? Run the universal diagnostic (16 noise types, text, images) on the HIGH-band champion. Confirms or falsifies the specialist-vs-generalist reading.

• What is the LayerNorm-at-D=4 undershoot measuring? The G3 variant at HIGH band reached CV 0.706, significantly below uniform S³ prediction 0.923. This is a distortion specific to the centering+variance-norm combination at low D. Characterization would clarify exactly which geometric property of sphere-norm is irreplaceable by the standard normalization layers.

• Within-attractor reconstruction MSE: Phase 1 was a band-classification sweep, not a reconstruction-quality sweep. The within-attractor MSE data needed to characterize reconstruction floors at each band requires full 30-epoch runs. Phase 2 was planned for this; initial results suggest LBFGS reaches meaningfully lower MSE than Adam within the HIGH attractor (0.0644 vs 0.072 at 100 batches vs 30 epochs), pointing at optimizer-dependent within-attractor structure that a Phase 2 program should map.

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9. Artifacts

All 149 Phase 1 ablation runs are preserved on HuggingFace under AbstractPhil/geolip-svae-ablations with per-run final_report.json files and TensorBoard event files. The aggregated analysis (band_matrix.csv, anomalies.csv, group_summaries.csv, uniformity_diagnostic.csv, snapshot_meta.json) is under _analysis/{timestamp}/ within the same repo.

The 13 configurations from the original three-band sweep are preserved under AbstractPhil/geolip-svae-batteries with full TensorBoard logs, checkpoints every 5 epochs, and final reports.

Training code: johanna_F_trainer.py (base trainer), ablation_trainer.py (ablation adapter with PatchSVAE_F_Ablation subclass), ablation_configs.py (explicit matrix of 149 Phase 1 and 174 Phase 2 variants), ablation_orchestrator.py (Colab cell for sequential execution with HF resume logic), aggregate_results.py (snapshot aggregator writing timestamped analyses).

Formula catalogue (every load-bearing equation in the architecture): johanna_F_formula_catalogue.md.

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This finding emerged from two complementary sweeps: an original F-class (miniature battery) exploration over approximately 40 hours of A100 time that produced the three-band hypothesis, followed by a 149-run ablation program completed in under one hour on a single H100 that validated the hypothesis across 12 orthogonal dimensions of variation. The sphere-norm- as-selector finding and the linear-encoder result were emergent findings from the ablation, not anticipated by the original sweep.