TDSL Correction Note

Reconciling the Arity Table and the Δn Convention

This note documents two errors found during an unrelated review of the black hole cases, corrects both, and recomputes every affected result. Both errors were internally consistent within the document that made them — neither is a computational slip — which is exactly why they went unnoticed until the two documents were compared against each other directly.

Error 1: The positive-level arity table in TDSL

TDSL (§4.1) assigns arity values to positive levels using n(k) = 2k+1:

T⁰=1, T¹=3, T²=5, T³=7, T⁴=9

This does not match the arity convention used throughout the rest of the corpus, where positive levels use n(k) = 2k (not 2k+1), and only negative levels use n(k) = 2|k|+1:

T⁰=1, T¹=2, T²=4, T³=6, T⁴=8       (positive: n=2k)
T⁻¹=3, T⁻²=5, T⁻³=7, T⁻⁴=9         (negative: n=2|k|+1)

The corpus-wide convention is the one already load-bearing elsewhere — it is, for instance, the specific basis of the claim that T³ has six phases organized into three closed pairs, which is the entire argument of the article deriving the Dirac equation’s four-component structure. TDSL’s table is corrected to match it. No other part of the corpus is changed; TDSL was the outlier.

Error 2: Two different definitions of Δn, sharing one symbol

TDSL defines Δn as the difference in arity — the number of boundary-condition pairs (independent degrees of freedom) lost between two levels: Δn = |n(k₁) − n(k₂)|.

TDSL’s dimensional-rule addendum, introduced independently to formally verify v3’s level assignments, computed Δn differently: as the raw difference between level indices, Δn = |k₁ − k₂| — never converting through the n(k) formula at all.

These are not the same quantity, and the difference is not small. For the Big Bang case, v3-style accounting (arity difference) and v4-style accounting (index difference) diverge by more than a factor of two. Every one of v4’s 16 verified cases used the index-difference convention; none of them used the corpus’s own arity-difference definition, despite reporting results in the same Δn column as v3 and claiming agreement with it.

Resolution: Δn is defined, corpus-wide, as the difference in arity (boundary-condition pairs lost), matching TDSL’s original definition. This is the version with direct physical motivation — severity should track how many independent degrees of freedom disappear, not how many hierarchy levels are nominally crossed regardless of how much structure each level contains. TDSL’s dimensional rule (3a+2b+c) remains valid and useful for assigning a level to a physical quantity; it is only the subsequent step — computing Δn from two assigned levels — that must now go through the corrected arity table rather than taking the raw index difference.

Recomputation of all 16 TDSL cases

Consequences for the two black hole entries handled separately

The black hole cases treated in the companion article on the information paradox use the same corrected accounting, computed directly rather than read off this table (since both involve T⁴ and T⁰, not covered by the “r→0” row above, which measures a different transition — radius to curvature, not full dimensional collapse):

Central singularity, full evaporation:  T⁴ (n=8) → T⁰ (n=1) → Δn = 7

This also resolves an internal flag in the case catalogue (CASO 29, Hawking radiation): that entry had computed Δn=4 for the same T⁴→T⁰ transition using the index-difference convention, then flagged the result as “partially consistent” because only 2 of a predicted 4 divergent quantities were observed. Under the corrected accounting the predicted Δn is 7, not 4 — and TDSL’s own stated scope (§5.3 of v3) is explicit that Δn was never meant to predict an exact count of divergent quantities, only relative severity. The original flag was checking the theory against a test it never claimed to pass. No further correction to the observational record is needed; the flag is retired as a methodological artifact, not as a resolved discrepancy.

What changes in the severity ordering

TDSL’s central qualitative claim — that larger Δn correlates with more severe structural collapse — is unaffected by this correction; only the ordering of specific cases changes. Most notably, the Big Bang (Δn=5, corrected) is no longer the most severe case in the table. That position is now shared by three cases at Δn=9: the Landau pole, the trans-Planckian problem, and the Unruh effect. The Big Bang’s earlier position as the anchor example of “catastrophic collapse” reflected its familiarity as an example, not a structural maximum — the corrected accounting does not privilege it, and this note treats the reordering as a genuine correction rather than an anomaly to explain away.

What is not affected

The Δn=0 test — TDSL’s strongest and most falsifiable prediction, that no structural loss implies no divergence — is unchanged by this correction in every case where Δn was already 0 (the horizon, the pure mass limit). Nothing about this correction touches the dimensional rule (3a+2b+c) used to assign levels to physical quantities in the first place; only the subsequent step, computing distances between already-assigned levels, is corrected.

For the corrected arity table’s application to the black hole and information paradox cases: “What Comes Out of a Black Hole Is Information Without Space.”
For the corpus-wide arity convention this note brings TDSL into alignment with: “The Arities of Numbers.”

CC BY-SA 4.0 — Diego Luis Tentor, ArXe Research, 2026