ESTABLISHED
Two low-rank findings
UWS and J-Space each exhibit low-dimensional structure in their own domains. [1][2]
Context Jamming Research instrument
Method · Sources · CollaborateCROSS-SPACE TEST: UNRUN
- KNOWN
- Both findings are low-rank. [1][2]
- UNTESTED
- Their bases remain unmeasured.
- REQUIRED
- Transport plus real tensors.
Research instrument · unrun
The Sliver and the Subspace
Two independent results locate intelligence in low-rank structure. This instrument tests the harder claim: whether the static weight sliver and the dynamic broadcast workspace share the same geometry.
Do the dynamic J-Space broadcast directions live inside the static Universal Weight Subspace?
0.28
θ̄ 73°·FVE 9%
Synthetic planted cosine. Changing this control updates the hero and the detailed instrument below.
Lattice = static UWS · copper sliver = dynamic J-Space. α is synthetic planted overlap. Both controls below and above share this value.
01 · Synthetic detector
Dial the planted overlap. Watch the detector.
This browser mock asks whether principal angles and fraction of variance explained can recover structure deliberately planted into a shared basis. It demonstrates the measuring device—not the real UWS/J-Space result.
BROWSER DEMONSTRATION · SHARED BASIS PLANTED BY CONSTRUCTION
Principal angles against a permutation null
SYNTHETIC PLANTED DATA — NO REAL MODEL TENSORS LOADED
z-score vs permutation null4.6σ
black marker = configured 5σ detector threshold
Suggestive planted overlap
The detector sees a synthetic signal, but it remains below the configured strong-evidence threshold.
This validates the detector under planted structure. It does not validate the UWS/J-Space hypothesis.
Define the controls and threshold
α is the cosine of overlap planted into this synthetic shared basis. It is not measured from a model.
Permutation null means rank-matched random orthonormal bases recomputed under the same statistic.
Rank k = 16 is the browser demonstration setting. A real run must sweep k and report the curve.
Synthetic readouts · d=512 · rank k=16
observed mean 72.8°permutation-null mean 82.8°
- Mean principal angle
- 72.8°
- 1.271 rad · lower means closer
- Fraction variance explained
- 8.7%
- FVE · J projected onto UWS
- Null mean angle
- 82.8°
- rank-matched null · ±0.038 rad
- Approximate p-value
- 2.39e-6
- one-sided vs synthetic null
FVE response · expected ≈ α²
Browser status: analytic synthetic demonstration. The separate Python planted-data validation exercises the same detector shape. Neither is the real-model experiment.
02 · The methodological seam
The comparison is undefined until the spaces are connected
Weight-space directions and activation-space directions do not inherit a common coordinate system. Matching rank is not enough. Before a principal angle can mean anything, one representation must be moved into a commensurable space.
SOURCE
Weight tangent space
UWS basis S from model-weight deviations
MISSING OBJECT
Layer Jacobian or local pushforward
The transport choice is part of the hypothesis
TARGET
Activation tangent space
J-Space broadcast directions J
BROWSER MOCK: SHARED BASIS PLANTED BY CONSTRUCTION
The synthetic instrument is allowed to share coordinates because the test designer planted them. A real-model experiment has to earn that coordinate system.
03 · Epistemic audit
What survives a hostile referee
Name the spaces. Separate the evidence. Then test the bridge.
PLAUSIBLE BRIDGE
Output-facing geometry
The workspace construction and the causal concept metric both touch output geometry. That creates a testable bridge—not an identity. [2][3]
INVALID INFERENCE
Isomorphism ≠ identity
k≈16 and approximately 25 active vectors occupy the same rhetorical neighborhood. Similar ranks do not establish shared directions. [1][2]
Universal Weight Subspace
- Domain
- Weight space
- Method
- HOSVD / stacked LoRAs
- Rank signal
- approximately 16 principal directions [1]
- Cross-test
- Unrun
J-Space
- Domain
- Activation / output-effect space
- Method
- Jacobian-lens corpus average
- Rank signal
- approximately 25 active vectors [2]
- Cross-test
- Unrun
04 · Convergence map
Five nodes, one bottleneck
Physics and biology motivate the search. Weight geometry and workspace geometry supply the objects. The unrun cross-space comparison remains the bottleneck.
PRIOR / ANALOGYEMPIRICALLY MEASUREDCROSS-SPACE CLAIM UNTESTED
- PhysicsPRIOR / ANALOGY
Holographic boundary
AdS/CFT motivates the prior that effective degrees of freedom can scale with a boundary rather than a volume.
- BiologyPRIOR / ANALOGY
The watt and the manifold
Sparse coding and low-dimensional cortical manifolds motivate the compression analogy under an energy constraint.
- AI · WeightEMPIRICALLY MEASUREDOBJECT UNDER TEST
Universal Weight Subspace
Kaushik et al. report a shared low-rank structure across weight deviations. This is one of the two objects under comparison. [1]
- AI · ActivationEMPIRICALLY MEASURED
Linear representations and manifolds
Park and Gurnee supply measured activation-space geometry and a candidate causal metric. [3] [4]
- AI · WorkspaceEMPIRICALLY MEASUREDOBJECT UNDER TEST
J-Space (Global Workspace)
The Jacobian lens isolates capacity-limited broadcast directions. This is the second object under comparison. [2]
CROSS-SPACE CLAIM UNTESTED
Five nodes establish a recurring low-rank motif. Only nodes 3 and 5 supply the proposed comparison—and no result yet shows that they share directions.
05 · Falsifiable protocol
From two bases to one verdict
- 01
Extract UWS basis S
Build the top-k orthonormal basis from centered, stacked weight deviations.
- 02
Extract J-Space directions J
Use the Jacobian lens to recover output-facing broadcast directions.
- 03
Transport into one space
Apply an explicit layer Jacobian, local pushforward, or other defensible transport.
- 04
Apply optional causal whitening
Test whether output anisotropy explains the apparent overlap.
- 05
Compute overlap
Measure FVE and the complete principal-angle spectrum.
- 06
Build permutation nulls
Compare against thousands of rank-matched random orthonormal bases.
- 07
Sweep rank k
Report the full curve rather than selecting a favorable integer.
BROWSER
Demonstration
Analytic response under a planted shared basis.
PYTHON
Detector validation
Planted-data permutation test outside the browser.
REAL MODEL
Not run
Transported UWS and J-Space tensors from compatible checkpoints.
06 · Why it matters
Either result improves the science
If the two AI results share transported directions, Architectural Determinism gains a measurable joint—not another analogy. Static adaptation and dynamic broadcast would become two views of one constrained geometry.
If they remain orthogonal, the result is equally sharp. Low rank survives. The proposed identity does not. The unification story breaks at a named seam and can be rewritten around the failure.
The point of a falsifiable instrument is that you do not get to choose which result you like.
07 · Sources
Primary literature and claim map
- [1]WEIGHT GEOMETRYRANK SIGNALThe Universal Weight Subspace Hypothesis
Kaushik, Chaudhari, Vaidya, Chellappa, Yuille · arXiv 2512.05117 · Dec 2025
HOSVD across 1,100+ models; approximately 16 shared directions; PEFT compression.
- [2]GLOBAL WORKSPACEACTIVE-VECTOR ESTIMATEVerbalizable Representations Form a Global Workspace in Language Models
Gurnee, Lindsey, et al. · Anthropic · Transformer Circuits · July 2026
Jacobian lens; J-Space as a capacity-limited broadcast workspace; approximately 25 active vectors.
- [3]CAUSAL METRICOUTPUT GEOMETRYThe Linear Representation Hypothesis and the Geometry of LLMs
Park, Choe & Veitch · ICML 2024 · arXiv 2311.03658
Concepts as directions; causal inner product; embedding and unembedding unification.
- [4]ACTIVATION MANIFOLDSWhen Models Manipulate Manifolds: The Geometry of a Counting Task
Gurnee, Ameisen, Kauvar, Tarng, Pearce, Olah, Batson · Transformer Circuits · 2025
Dynamic manifold geometry of computation in activation space.
- [5]HISTORICAL PRIORGenerative Models for Discovering Sparse Distributed Representations
Hinton & Ghahramani · 1997 · RGBN
Sparse distributed representations; explaining-away; generative mixture genealogy.
Epistemic note: quantitative details of the July 2026 workspace paper and related synthesis notes should be checked against the primary source before formal citation. This page does not claim that cross-space overlap has been measured.
08 · Next experiment
Run the cross-space test on an open model
The useful inputs are concrete: LoRA collections or multi-seed finetunes, compatible checkpoints, J-lens or Jacobian extraction infrastructure, and researchers able to evaluate transport choices.
- LoRA collections or multi-seed finetunes
- Compatible open-model checkpoints
- J-lens or Jacobian extraction infrastructure
- Researchers evaluating transport choices