Fudan University  ·  Shanghai Innovation Institute  ·  arXiv:2605.14619

Does the same answer mean the same reasoning?

SliceGraph — an activation-key atlas of many CoT runs, where same-answer correct trajectories split into distinct process routes we call isomers

SliceGraph·same answer, different routes

One answer. Many minds.

Kang Chen∗  ·  Junjie Nian∗  ·  Yixin Cao†  ·  Yugang Jiang

Watch the film The idea Read the paper

The one idea

A vote keeps one answer. It throws away how the runs got there.

Test-time sampling now draws many chains of thought per problem, and self-consistency keeps only the majority string. But the N runs of one problem are not a bag of independent samples: they share intermediate computation, fork at decision points, and rejoin through common subroutines. Once internal states are linked by similarity, consistency can be lifted from final answers to processes — and the question becomes whether correct solutions converge on one route or split into several.

Two chains of thought reach the same final answer, but their intermediate computation traces follow visibly different routes through the graph — illustrating that same-answer runs can belong to distinct process families.
Fig. 1Same-answer runs can follow distinct process families. The final string is identical; the route through intermediate computation is not.
01

One scalar hides the process

A majority vote ranks answers but says nothing about whether two correct runs reached it through the same computation or through separate routes that merely agree at the end.

02

Slices, not whole runs

Whole-run similarity is too coarse: two runs can share a long trunk and fork once, or reach the same answer by different routes. SliceGraph compares 256-token slices instead.

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Process isomers

Two correct runs are process isomers when they reach the same normalized answer but belong to different route families — same destination, different scaffold.

How SliceGraph works

Slice the runs. Link by activation overlap. Read the atlas.

SliceGraph is a measurement object, not a decoding program. It is built after sampling, directly from sparse neuron-activation keys — no projection, no text embedding — and every structural unit is validated by blinded human annotation before any geometric claim is made.

The SliceGraph pipeline: for one problem-model cell, many sampled CoTs are cut into 256-token slices summarized by top-k MLP activation keys; slices are linked by mutual-kNN Jaccard similarity into a per-problem graph; the graph is decomposed into biconnected components with block roles and clustered into process families; a label-seeded reward field and family-specific typed-state transition kernels are overlaid on the fixed scaffold.
Fig. 2The SliceGraph pipeline. Many sampled CoTs for one problem-model cell become an activation-key atlas; families are read from the fixed graph, and label-seeded value cores and navigation kernels are layered on top.
  1. slice

    Slice & link by activation keys

    Each 256-token slice is a binary set of the most-active MLP neuron keys — no magnitudes, no projection. Two slices connect by Jaccard overlap, kept only when the neighbourhood is mutual (mutual-kNN). The result is one per-problem graph of cross-run local similarity.

  2. atlas

    Decompose into a reasoning atlas

    Biconnected components are shared reasoning-state units; articulation points are the forks where the run population branches. Each non-trivial block gets one of five roles — trunk, answer basin, weak basin, decision point, intermediate.

  3. family

    Cluster runs into process families

    Reduce each correct run to the blocks it visits, weight rare blocks up, and partition by Louvain. Same-answer runs in different families are process isomers. A label-seeded reward field and typed-state kernels are separate overlays — never used to define families.

    J(r,r′) = ∑b∈Sr∩Sr′ ωb∑b∈Sr∪Sr′ ωb

The graph, its blocks, and the family partition never see answer labels or the reward field. The route scaffold is answer-free; correctness and value enter only as interpretive overlays on a structure validated in its own right.

Is the atlas real?

Validate the units before reading geometry off them.

Before any isomer claim, two questions: do the graph's units match human notions of shared thinking, and does the activation-key scaffold preserve process geometry rather than inventing it? Two blinded annotation studies and a representation ablation answer both.

Three readouts of a single SliceGraph for one problem (BRUMO25-10, 64 runs, 60 correct). Left: biconnected-component block roles — trunk, basin, decision, weak, other. Middle: correct-only process families in three colours, with incorrect runs marked by crosses. Right: the propagated reward field with stars marking the centroids of eleven high-value-core components. Family colours and core locations split differently.
Fig. 3Three readouts of one atlas (BRUMO25-10, 64 runs). Left: BCC block roles. Middle: correct-only process families (× = incorrect runs). Right: reward field with ★ high-value-core centroids — route families and value cores are related but non-identical layers.
①

Units match human judgement

On blinded pairs, annotators recover the same-BCC / across-BCC split at 88.9% and 90.3% (κ = 0.97); within-family pairs are judged same-strategy in 94.4% of decisive cases — the units are semantic, not artefacts.

②

The scaffold preserves geometry

Under a fixed mutual-kNN framework, unprojected activation-key Jaccard keeps the richest, most stable family multiplicity. PCA-50 compresses ~1 family per cell; slice-text MiniLM collapses the atlas (21 / 954 valid cells).

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Then, and only then, geometry

With validated units and a preserved scaffold, the multi-route claim rests on structure, not on a projection artefact. The isomer statistics that follow are measured on this vetted atlas.

What the atlas shows

The same answer is usually not one route — it is a set of them.

On 954 problem-model cells — 318 problems, three primary 4B/8B models, 64 runs each, on AIME24/25, BRUMO25, HMMT25, and GPQA — same-answer correct CoTs routinely split into multiple process families, and the value landscape splits with them.

85.5%of cells have >1 correct process family
76.6%of same-answer run pairs are process isomers
60,622analysed trajectories · 954 cells · 6 models

In 85.5% of 954 cells, correct CoTs sharing the same normalized answer split into multiple process families; the average pairwise isomer rate is 76.6% (problem-clustered bootstrap 95% CI 75.6–77.5). Split-half resampling keeps both halves multi-family in 97.2% of cells, and a held-out atlas retains 91% — SliceGraph recovers multi-route structure when present rather than forcing it.

Three population-geometry panels. Left: ECDF of pairwise family-isomer rates over 834 cells with bootstrap 95% confidence interval lines. Middle: scatter of correct-family count versus reward-core component count over 479 cells, Spearman rho equals 0.08, showing the two layers are non-redundant in count. Right: family-core specialization over 324 multi-core cells, pooled mean 0.59, showing families spatially specialize over distinct core footprints.
Fig. 4Population geometry. (a) ECDF of pairwise isomer rates (CI marked). (b) Families vs. value cores count weakly (ρ = 0.08) — non-redundant layers. (c) Yet families spatially specialize over distinct core footprints (mean 0.59).

A multi-basin value landscape

A label-seeded reward field, diffused over the fixed graph, splits into disjoint high-value cores in 67.6% of reward-evaluable cells. Route families specialize over those cores (specialization 0.59) and are individually coverage-critical — value refines the route map without redefining it.

Families navigate differently

Lifting each run to a typed-state transition model, between-family total variation exceeds a size-preserving label-shuffle null in 80.9% of cells (median z = 3.14). Restricting to shared states keeps 77.4% — families differ not just in where they concentrate but in how they move.

An atlas, not a decoder

SliceGraph is a post-hoc measurement object: the reward field is descriptive (label permutations stress-test that layer, not one-sided nulls), and claims are scoped to non-code math and science. It audits route diversity, collapse, and over-concentration — using it inside decoding or early-stopping is left for future work.

Built from 954 problem-model cells · 318 problems · 6 models R1-8B · Qwen3-4B Think / Inst (primary) · Llama-8B · Qwen3-32B · Qwen2.5-Math-72B (replications) · AIME24 · AIME25 · BRUMO25 · HMMT25 · GPQA · κ=0.97 & 94.4% blinded validation

Watch

One answer, many minds — a visual essay

A silent vector essay, drawn live in your browser — no video file. Sixty-four runs, one answer; link their slices into an atlas; the correct runs split into families — same answer, different route — with disjoint value cores and distinct navigation.

Cite

Reference

Kang Chen1,∗, Junjie Nian1,∗, Yixin Cao1,2,†, Yugang Jiang1

1 Fudan University    2 Shanghai Innovation Institute    ∗ Equal contribution    † Corresponding author

Correspondence: yxcao@fudan.edu.cn

Links: Paper (PDF)  ·  arXiv  ·  Code

@misc{chen2026slicegraph,
  title   = {SliceGraph: Mapping Process Isomers in
             Multi-Run Chain-of-Thought Reasoning},
  author  = {Chen, Kang and Nian, Junjie and Cao, Yixin and Jiang, Yugang},
  year    = {2026},
  eprint  = {2605.14619},
  archivePrefix = {arXiv},
  primaryClass = {cs.AI},
  note    = {SliceGraph — a process atlas for multi-run CoT},
  url     = {https://arxiv.org/abs/2605.14619},
}

Same answer. Different minds — a process atlas, not a decoder.