Research-Stack/4-Infrastructure/NoDupeLabs/research-stack/models/abstract_cot_integration_v0.md

Abstract-CoT Paper Integration v0

Status

status: HOLD
source_type: paper_equation_integration
paper: Thinking Without Words: Efficient Latent Reasoning with Abstract Chain-of-Thought
route_signature: models/abstract-cot/integration/v0
authority_scope: external_literature_context_and_candidate_equation_pack
proof_status: sketch_until_local_files_verified

Integration summary

The Abstract-CoT paper contributes six equation families to the Research Stack equation map:

ID	Equation	Role
Abstract_CoT_Marginal_Likelihood	`p(y	x) = Σ_z p(y
Abstract_CoT_Constrained_Decoding	`πθ^abs(a	h) = πθ(a
Abstract_CoT_Information_Bottleneck	I(Y;C) ≤ I(Y;Z_abs) ≤ I(Z_abs;C)	latent bottleneck / data-processing guard
Abstract_CoT_Power_Law_Distribution	p(token) ∝ token^{-α}	symbolic-token heavy-tail prior
Abstract_CoT_GRPO_Advantage	A_k = (R_k - μ) / σ	normalized group-relative reward signal
Abstract_CoT_Compression_Ratio	`compression_ratio = E[	c_verbal

Forest interpretation

verbal chain-of-thought
→ compressed latent reasoning state
→ constrained abstract action space
→ bottlenecked information channel
→ decoded answer / policy output

The strongest bridge into the existing Research Stack is the information bottleneck road:

C = verbal/context carrier
Z_abs = compressed abstract latent chain
Y = answer/task target

C → Z_abs → Y

If this Markov structure holds, the data-processing inequality gives the intended guard:

I(Y;C) ≤ I(Y;Z_abs) ≤ I(Z_abs;C)

This should remain a HOLD until the Lean formalization verifies the assumptions actually encoded in EntropyMeasures.lean.

Lean placement

Expected Lean file integration:

EntropyMeasures.lean

Expected additions:

mutualInformation
informationBottleneck

Expected dependency relation:

KL divergence / entropy / JSD
→ mutual information
→ data processing inequality
→ information bottleneck guard

Research Stack routes

Route 1 — Latent marginalization

latent variable z
→ marginalize paths
→ answer likelihood p(y|x)

Outcome: HOLD.

Route 2 — Constrained abstract decoding

full policy πθ
→ abstract allowed action set A
→ renormalized constrained policy πθ_abs

Outcome: HOLD.

Route 3 — Bottleneck / compression

context C
→ abstract latent Z_abs
→ target Y
→ mutual-information inequalities

Outcome: high-priority HOLD because it directly connects compression, latent reasoning, and entropy measures.

Route 4 — Power-law token prior

token rank / token symbol
→ heavy-tailed probability mass
→ compression asymmetry

Outcome: HOLD; requires baseline checks to avoid false numeric pattern attraction.

Route 5 — GRPO advantage

reward samples R_k
→ group mean μ
→ group standard deviation σ
→ normalized advantage A_k

Outcome: HOLD; useful bridge into route weighting and policy update logic.

Route 6 — Compression ratio

verbal CoT length
→ latent abstract token count
→ compression_ratio

Outcome: HOLD; useful measurement candidate for Hutter/compression route.

Authority boundary

paper equation → external light source
Lean formalization → candidate proof object only if assumptions are explicit
numeric similarity → no basin
compression gain → no truth claim

The paper can illuminate the map. It cannot by itself promote a basin.

Immediate validation gates

1. Verify MATH_MODEL_MAP.tsv contains the six equation IDs.
2. Verify EntropyMeasures.lean compiles after adding mutualInformation and informationBottleneck.
3. Check whether the Lean bottleneck theorem assumes an explicit Markov chain C → Z_abs → Y.
4. Route the six equations through Semantic Number Pattern Search.
5. Add the bottleneck road to the forest as HOLD, not basin.
6. If compilation fails, classify as SCAR: abstract_cot_entropy_formalization_compile_failure.

Notes

This integration is important because it bridges:

compression
latent reasoning
policy restriction
entropy / information measures
route weighting

That makes it relevant to the Hutter/compression road, but only after the assumptions are made explicit.