Pith Number

pith:C2H6UYBK

pith:2026:C2H6UYBKFJ6AMLI2OVU5D7J32Y

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From Witness-Space Sharpness To Family-Pointwise Exactness For The Solvability Complexity Index

Christopher Sorg

Witness-space sharpness coincides with worst-case exactness but is strictly weaker than family-pointwise exactness for SCI families

arxiv:2604.12750 v2 · 2026-04-14 · math.LO · cs.CC · math.DS · math.FA

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Claims

C1strongest claim

We prove that witness-space sharpness coincides with worst-case exactness but is, in general, strictly weaker than family-pointwise exactness, and give a canonical source-family example witnessing the strictness. We then establish two positive upgrade theorems: an abstract pullback principle and a concrete finite-query criterion guaranteeing that witness-space sharpness upgrades to family-pointwise exactness.

C2weakest assumption

The upgrade theorems and transport results rest on the assumption that the families admit decoder-regular finite-query transports and that the decoder classes satisfy the regularity conditions needed for the pullback principle to apply; this is invoked when moving from the abstract preorder to concrete families such as integration and spectral decision.

C3one line summary

Formalizes a trichotomy of exactness notions for SCI on families, proves witness-space sharpness equals worst-case exactness but is strictly weaker than family-pointwise exactness, supplies upgrade theorems, and illustrates with integration and spectral families.

References

11 extracted · 11 resolved · 2 Pith anchors

[1] Computing spectra--on the solvability complexity index hierarchy and towers of algorithms 2015

[2] Colbrook and Anders C 2023

[3] Davis and Philip Rabinowitz 1984

[4] On the solvability complexity index, the -pseudospectrum and approximations of spectra of operators 2011

[5] Perturbation theory for linear operators 1995

Receipt and verification

First computed	2026-05-20T00:00:37.918960Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

168fea602a2a7c062d1a7569d1fd3bd606c1011f04e19e201f83e2be2cfef77a

Aliases

arxiv: 2604.12750 · arxiv_version: 2604.12750v2 · doi: 10.48550/arxiv.2604.12750 · pith_short_12: C2H6UYBKFJ6A · pith_short_16: C2H6UYBKFJ6AMLI2 · pith_short_8: C2H6UYBK

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/C2H6UYBKFJ6AMLI2OVU5D7J32Y \
  | 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: 168fea602a2a7c062d1a7569d1fd3bd606c1011f04e19e201f83e2be2cfef77a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "474a6d34079f38c7798fa6a6a59b1bdadb24988d63d16b465525f223ded11fce",
    "cross_cats_sorted": [
      "cs.CC",
      "math.DS",
      "math.FA"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2026-04-14T13:59:35Z",
    "title_canon_sha256": "bc6af03c0a69ab356acf64750a863ebcc2762364dc748ae547cf79da63c582e8"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.12750",
    "kind": "arxiv",
    "version": 2
  }
}