CAL-L3 — Tensor Volume Layer

Part of: CAL — Cognitive Abstraction Layers — the research starts at CAL; L3 is its Tensor Volume Layer.
Author: Juan Pablo Chancay · Aural Syncro
Status: Characterization closed (~95%, 2026-06) — operator C characterized; causal conservation = structural sparsity preservation. See L3_CLOSURE.md.
Target venue: NeurIPS / ICML
License: CC BY-NC 4.0 (docs) · AGPL-3.0 (src)

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What is L3?

L3 is the Tensor Volume Layer of the CAL architecture. It defines the operator that compresses a sequence of cognitive tensor snapshots (L2 sessions) into a single unified volume, preserving the causal relationships that make downstream meta-inference tractable.

L3:  V = C({T⁽ˢ⁾}_{s=1}^{n})

Where:

• T⁽ˢ⁾ ∈ ℝⁿˣˢˣᵃˣᵗ — cognitive tensor from L2 session s (dimension × stage × agent × time)
• C — composition operator (open problem, this repo)
• V — tensor volume; input to L4 meta-inference

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The Central Problem: Operator C (characterized)

C must satisfy four properties (§5.2 CAL pre-paper): causal preservation, temporal coherence, dimensional stability, tractability. L3 characterized what causal preservation actually requires — the headline result:

Reconstruction fidelity ≠ causal conservation. Low-rank Tucker preserves 98% of variance yet destroys causal structure (S3-bis: reconstruction causal F1 = 0.135 against a perfect raw control). Variance-optimal compression and causality-preserving compression are different objectives.

This reordered the framework to the validation order Causality ≻ Topology ≻ Reconstruction, and built C as:

C = C_causal ∘ C_compress
    C_compress = Tucker   (validated tractable: κ(V) sub-linear in n, 195.6× compression)
    C_causal   = structural prune to the causal support

Operative property (the definition L3 closed on): an operator is causally conservative iff it preserves the observational invariants Ω₀ = (R, C, S) — reachability, coverage, consistency — under compression, not reconstruction error.

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What L3 found (the characterization, closed)

Result	Finding
S2	Tucker is tractable: κ(V)=1296, 195.6× compression, sub-linear in n (Property 4 holds).
S4	Governance manifold confirmed: dim(M_gov)≈2–3, trustworthiness ≥0.96. Static (time-averaged).
S3-bis	Property 1 refuted for plain Tucker: reconstruction ≠ causality (F1=0.135 vs raw 1.000).
TCI	A ground-truth-free causal metric U (PCMCI val-matrix flow) validated as both instrument and objective.
Proxy audit	Causality is structural, not magnitude: no differentiable magnitude proxy reproduces U's ordering.
Q_L3.2A	Tucker's failure is spurious-edge fabrication, not loss: coverage + sign held, reachability/|E| exploded.
Form 1	A structural prune to the raw causal support recovers 75% of the causal-conservation headroom with no ground truth (U 0.441→0.862, raw↔GT gap = 0.000).

Full narrative + formal definition + provenance: L3_CLOSURE.md.

Downstream — the residual frontier, now characterized (L4-B0, 2026-06). Form 1's prune leaves a residual ΔU≈0.138 (the gap from U=0.862 to 1.0), declared by L3 as a frontier (flow magnitude / higher-order structure). L4 characterized it: only ~19% is linear-edge-representable (magnitude + sign); ~81% is non-linearity a single linear volume cannot carry (lag>1 ≈0%). The frontier is therefore non-linear, not a tuning gap — which is why L4 keeps the dual (V_Tucker, G_pruned) as terminal and does not pursue a single linear V′ (L4 §5.5). L3's "structural sparsity preservation" thesis is unaffected; the residual it declared is now explained.

Governance Manifold Hypothesis (§5.6) — confirmed, but demoted to descriptor

dim(M_gov)≈2–3 is confirmed. However L3 found this manifold is static (reconstructible with the time axis averaged out) while causality is temporal — so M_gov is retained as a descriptor, not as a projection driver (Π_gov suspended). Whether a low-dimensional causal manifold exists is an open L4 question (Q_L3.2).

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Experimental record (all pre-registered, all run)

Every step committed its pre-registration before code; negatives reported honestly.

Step	Task	Result
S4	Manifold test: dim(M_gov) << ambient?	✅ dim≈2–3, tw≥0.96 (confirmed on S1 and S1-bis)
S1 / S1-bis	Synthetic corpus with known causal graph	✅ 3 graphs × 30 sessions, t=48
S2	C = Tucker over {T⁽ˢ⁾}; κ(V)	✅ κ=1296, 195.6×, tractable
S3 / S3-bis	Does the reconstruction recover the causal graph?	❌ Property 1 refuted (clean) → reframing
TCI · proxy audit · operator search · Ω · Q_L3.2A · Form 1	Characterize causal conservation	✅ Closed — sparsity-preservation thesis confirmed
S5	O(n²) flat vs O(κ) cost contrast on MI300X	✅ Run & frozen on MI300X (D(n)→52.8× @4k, mechanism confirmed) — via L4-A / AMD

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Repository Structure

L3/
├── README.md
├── paper/                  ← L3 paper (in development)
├── src/
│   └── composition/        ← operator C implementations (Tucker, sparse, manifold)
├── benchmarks/
│   └── synthetic/          ← corpus with known causal ground truth
└── experiments/
    ├── manifold_test/      ← S4: intrinsic dimension of M_gov
    └── causal_conservation/ ← S3: does V preserve causal structure?

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Dependencies

• Consumes: L2 validated corpus (T⁽ˢ⁾ from scenarios S1–S5 in TCO-L2)
• Unblocks: L4 — with C characterized and κ(V) concrete, the L4-A operator delivers the dual volume (V_Tucker, G_pruned) that L4 / AMD's κ vs n² contrast consumes (gate-C closed)
• AMD-Instinct collaboration: the O(n²) vs O(κ) contrast runs on MI300X; L3 supplies the O(κ) side (κ=1296)

Full collaboration context: CAL collaboration doc

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Related Repos

Repo	Role
CAL	Framework root — pre-paper, architecture
L2 — TCO	Produces T⁽ˢ⁾ corpus that L3 consumes
L4 — Meta-Inference	Consumes V produced by C