Pith Number

pith:677P27A3

pith:2026:677P27A3D4LYS5HRX4LD67FH7R

not attested not anchored not stored refs pending

GRALIS: A Unified Canonical Framework for Linear Attribution Methods via Riesz Representation

Raimondo Fanale

Every additive linear continuous attribution method on square-integrable functions has a unique canonical form via the Riesz theorem.

arxiv:2605.05480 v2 · 2026-05-06 · cs.LG · cs.AI · stat.ML

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\pithnumber{677P27A3D4LYS5HRX4LD67FH7R}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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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

every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM.

C2weakest assumption

That the target attribution methods are additive, linear, and continuous functionals on the chosen L^2 space; if any method violates linearity or continuity the canonical representation and all seven theorems cease to apply.

C3one line summary

GRALIS supplies a canonical representation (Q, w, Delta) for every additive linear continuous attribution functional on L^2 via the Riesz Representation Theorem, unifying SHAP, IG, LIME and linearized GradCAM while proving seven simultaneous guarantees including completeness and interaction values.

Receipt and verification

First computed	2026-05-20T01:05:15.376663Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f7fefd7c1b1f178974f1bf163f7ca7fc69ba17d58e7cfbfd027f77ffa6757136

Aliases

arxiv: 2605.05480 · arxiv_version: 2605.05480v2 · doi: 10.48550/arxiv.2605.05480 · pith_short_12: 677P27A3D4LY · pith_short_16: 677P27A3D4LYS5HR · pith_short_8: 677P27A3

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/677P27A3D4LYS5HRX4LD67FH7R \
  | 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: f7fefd7c1b1f178974f1bf163f7ca7fc69ba17d58e7cfbfd027f77ffa6757136

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "45d0a633aa87ce8a5d1eab58b47e41352bfddb12eea786657fb427fd7e1e5cd2",
    "cross_cats_sorted": [
      "cs.AI",
      "stat.ML"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-06T22:01:28Z",
    "title_canon_sha256": "ab3a3022e8854d7fb486c975c0255f7b47ca856254ff6018cabd3accec894c5e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.05480",
    "kind": "arxiv",
    "version": 2
  }
}