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Overview
An algebraic multiplication rule can remain perfectly well defined while its natural Hilbert-space realization fails. This paper locates that failure in relative multiplication on Connes fusion, proves maximal nonclosability under an explicit orthogonal multiplicity hypothesis, and identifies the dual operator-valued weight through which the operation still survives.
Manuscript status
Research note — not peer reviewed
Reader corrections
Prepared September 8, 2026. These corrections accompany the original PDFs; the PDF bytes have not been changed.
Paper I — appendix result numbers
The body of Infinite-Index Fusion and the Failure of Analytic Descent, version 1.0.0 dated August 8, 2026, has the controlling result numbers. Appendix A and the dependency graph retain two obsolete references:
| Item | Stale reference | Correct body reference |
|---|---|---|
| Orthogonal Jones witnesses | Lemma 4.2 | Lemma 4.4 |
| Maximal fusion-face singularity | Theorem 4.3 | Theorem 4.5 |
Lemma 4.2 in the body is Total graph closure. Definition 4.3 is the expectation-orthonormal unitary family. The website uses the body’s numbering.
Paper II — Remark 6.4
The sentence claiming that a closable operator need not admit witness data satisfying Theorem 6.3 is incorrect when witness data may be chosen freely under the definitions given. For any closable
Then
The graph estimate holds with constant one. Thus the unrestricted existence of such witness data is equivalent to closability. This observation does not give an independent certificate when closability is the property still to be established.
Replace Remark 6.4 with:
Theorem 6.3 supplies a sufficient certificate from specified witness data. With unrestricted witness choice, an already closable operator admits the tautological witness
and decoder . The certificate is informative when its retained data and their closability are established independently, or when a prescribed witness class is part of the reconstruction problem.
Theorem 6.3 remains valid. The nonexistence statement in Corollary 6.8 also remains valid for the nonclosable multiplication operator: any certificate satisfying the hypotheses would imply closability.
Paper II — the additional estimate in Theorem 6.5
The reverse graph estimate in Theorem 6.5 already follows from the bounded decoder in Theorem 6.3:
It may be retained for emphasis, but is redundant in the stated setting. Consequently the graph-domain equality and closure formula of Theorem 6.5 are available under the hypotheses of Theorem 6.3.
The website’s result summaries incorporate these qualifications. These notes concern the identified points, not a new referee report on the full series.
Definitions, results, and open questions in this paper
- Maximal nonclosability — Definition
- Expectation-orthonormal unitary family — Definition
- Descent defect — Definition
- Finite-index bounded regime — Theorem
- Maximal fusion-face singularity — Theorem
- Subgroup index dichotomy — Theorem
- Failure of face-compatible completion — Theorem