Pith Number

pith:WDJKIJBY

pith:2026:WDJKIJBYP3JRFVFMTM7OODH5WO

not attested not anchored not stored refs resolved

TIER: Trajectory-Invariant Execution Rewards for Multi-Step Tool Composition

Anay Kulkarni, ChiaEn Lu, Dheeraj Mekala, Gaowen Liu, Jayanth Srinivasa, Jingbo Shang

Rewards derived from tool execution and schemas let models maintain high accuracy on tasks requiring up to six sequential tool calls.

arxiv:2605.16790 v1 · 2026-05-16 · cs.LG · cs.AI · cs.CL

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\pithnumber{WDJKIJBYP3JRFVFMTM7OODH5WO}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

On DepthBench, a compositional benchmark stratified by depth (1 to 6 steps), TIER achieves >90% accuracy across steps, where trajectory-supervised rewards collapse beyond step-4.

C2weakest assumption

The paper assumes that runtime execution feedback and schema verification can be obtained reliably and at low cost for every candidate step without introducing new errors or requiring additional human annotation, and that this feedback is sufficient to guide learning across all valid alternative paths.

C3one line summary

TIER creates trajectory-invariant rewards from tool schemas and execution results for multi-step LLM tool use, reaching over 90 percent accuracy on DepthBench up to six steps where reference-trajectory methods fail after four.

References

29 extracted · 29 resolved · 0 Pith anchors

[1] (one output)

[2] (all outputs) 18

[3] (one output)

[4] You respond with ONE API call at a time and wait for the environment to execute it and provide you with the tool response

[5] Each turn may require multi-step API calls (as described in scenario 4)

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:03:22.180674Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

b0d2a424387ed312d4ac9b3ee70cfdb3abc00e6a9dca29e30b79700db47885c0

Aliases

arxiv: 2605.16790 · arxiv_version: 2605.16790v1 · doi: 10.48550/arxiv.2605.16790 · pith_short_12: WDJKIJBYP3JR · pith_short_16: WDJKIJBYP3JRFVFM · pith_short_8: WDJKIJBY

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/WDJKIJBYP3JRFVFMTM7OODH5WO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: b0d2a424387ed312d4ac9b3ee70cfdb3abc00e6a9dca29e30b79700db47885c0

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9d807c834812c3f883301705a8601c6529e184687a16cdd975d545bdec2e9f58",
    "cross_cats_sorted": [
      "cs.AI",
      "cs.CL"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-16T03:47:26Z",
    "title_canon_sha256": "37384d239678b03f3437dc69881db8e2462b06b1f7655e907515ab99fd0089fe"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16790",
    "kind": "arxiv",
    "version": 1
  }
}