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The high-spin tail of the N=4 index survives exact equal-charge projection.

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2026-07-12 00:17 UTC pith:CZVER34Q

load-bearing objection Clean factorization + exact U(3) arithmetic show the high-spin tail survives the equal-charge cut; the modular higher-giant caveat is real but not load-bearing.

arxiv 2607.03735 v1 pith:CZVER34Q submitted 2026-07-04 hep-th gr-qcmath-phmath.MP

Equal-charge projection of the mathcal{N}=4 index: exact large-N formula and finite-rank U(3) coefficients

Miguel Tierz This is my paper
classification hep-th gr-qcmath-phmath.MP
keywords superconformal indexequal-charge projectionmultigraviton factorizationgiant graviton expansionAdS5 black holesfinite-N BPS statespentagonal numbersN=4 SYM
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Supersymmetric AdS5 black holes on the equal-charge branch have three equal electric charges. The dual superconformal index should therefore be restricted to states with exactly those charges, not merely equal chemical potentials. This paper performs that exact projection. For the large-N multigraviton sector it proves a closed factorization into a product controlled by Euler's pentagonal numbers and a cube of the ordinary partition function. The factorization forces the large-N coefficient to vanish below an explicit onset energy in every spin sector. Exact U(3) arithmetic nevertheless finds nonzero coefficients inside those intervals, including states that lie beyond the classical black-hole existence bound. One concrete witness is the unit coefficient at energy 87 and spin 27/2, already accounted for by the first giant-graviton sector, while the large-N onset lies 1554 units higher. The high-spin tail is therefore not an artifact of summing over unequal charges.

Core claim

After exact projection onto Q1=Q2=Q3, the large-N multigraviton index factorizes as the product of two Euler functions times the generating function of p(n)^3. This forces the multigraviton coefficient to vanish throughout an explicit energy window in every spin sector, yet exact finite-rank U(3) coefficients remain nonzero inside those windows and beyond the classical black-hole bound; the unit coefficient at (87,27/2) already appears in the first giant-graviton sector.

What carries the argument

Equal-charge multigraviton factorization: the projected large-N index equals Π(x,p) times the sum of p(n)^3 x^{6n}, where Π is supported only on generalized pentagonal exponents. The identity supplies the onset energy j*(JR) and the exact vanishing intervals used to isolate finite-rank contributions.

Load-bearing premise

The modular vanishing of the second and third giant-graviton sectors at the unit-coefficient point is taken as strong evidence that the unit is a clean one-giant contribution, without an independent absolute bound that would turn those congruences into exact integer zeros.

What would settle it

Compute the exact integer values (or prove absolute bounds smaller than the product of the two modular primes) for the M=2 and M=3 giant-graviton sectors at energy 87 and spin 27/2; any nonzero residue would show the unit coefficient is not a pure one-giant effect.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies the equal-charge projection (constant term in the two charge-difference fugacities) of the N=4 superconformal index, the microcanonical sector dual to the Q1=Q2=Q3 branch of supersymmetric AdS5 black holes. It proves that the large-N multigraviton projected index factorizes exactly as I_eqQ_∞(x,p)=Π(x,p)∑_n p(n)^3 x^{6n} (Theorem 2.1), where Π is the product of Euler factors supported on generalized pentagonal numbers. This yields, for every JR, an explicit onset energy j*(JR) below which deqQ_∞ vanishes identically (Proposition 3.2 and Theorem 3.3). Exact integer U(3) computations then exhibit nonzero finite-rank coefficients inside those intervals, including beyond the classical U(3) black-hole bound (Table 3), with the flagship witness deqQ_3(87,27/2)=1 at depth 1554 already accounted for by the one-giant sector and by the closed formula on the line j=6JR+6 (Proposition 2.6). The high-spin tail therefore survives the exact equal-charge projection; all coefficients are signed (−1)^F-graded index coefficients.

Significance. The large-N factorization, the divisor-minimization formula for onset thresholds, the support theorem, and the closed expressions on the kinematic boundary j=6JR and the adjacent line j=6JR+6 are parameter-free analytic results derived from the standard single-particle index, plethystic exponential, Euler’s pentagonal theorem, and Macdonald/q-Dyson constant-term identities. The finite-rank data are obtained by exact integer arithmetic (with modular reconstruction and a rigorous pruning bound), and the paper supplies ancillary code and grade lists. These results cleanly isolate finite-N support in the equal-charge microcanonical ensemble and give a concrete, falsifiable baseline against which grey-galaxy and giant-graviton interpretations can be tested. The work therefore sharpens the comparison between the index and supersymmetric AdS5 black holes at finite rank.

minor comments (4)
  1. In the abstract and Introduction the depth is written j*(27/2)−87=1554 while Table 1 lists j*(27/2)=1641; the arithmetic is correct, but a single consistent parenthetical (1641−87=1554) would remove any momentary reader confusion.
  2. Figure 2 and Figure 3 are informative, yet the marker-size legend (log(1+|d3|)) and the precise definition of “double-exclusion witnesses” appear only in the caption; a short sentence in the main text of §3 would make the plots self-contained.
  3. Proposition 2.6 and the subsequent giant-graviton check both give deqQ_3(87,27/2)=1; a cross-reference in §5.2 reminding the reader that the unit value is already fixed by the closed formula would further clarify that the modular higher-M checks are supplementary.
  4. The U(3)/SU(3) distinction is carefully handled, but the sentence in §4.3 that “the qualification attached to Table 4 concerns coverage imes imes not the identity of the observable” is slightly awkward; a cleaner rephrasing would help.

Circularity Check

0 steps flagged

No significant circularity: factorization, onset thresholds, and finite-rank coefficients are independent derivations and exact computations.

full rationale

The central claims rest on self-contained mathematical steps that do not reduce to their own inputs. Theorem 2.1 follows by factoring the known multigraviton product (12) out of the constant-term extraction and applying the diagonal identity on three partition generating functions, yielding the cube p(n)^3 and the prefactor Π; the proof is coefficientwise in formal power series and does not invoke any fitted parameter or the finite-N data. Onset energies j*(JR) and the vanishing intervals of Theorem 3.3 are then read off from the support of Euler’s pentagonal expansion of Π together with a finite divisor minimization (Proposition 3.2), again without reference to the U(3) numbers. The kinematic-boundary and j=6JR+6 formulae (Propositions 2.5–2.6) reduce the matrix integral to standard q-Dyson/Macdonald constant-term identities and low-degree inserted expectations (Lemma B.1). The displayed U(3) coefficients, including the flagship d3eqQ(87,27/2)=1, are obtained by independent exact integer (or modular) arithmetic on the projected matrix integral; they are compared against the already-proved large-N zeros rather than used to define them. The one-giant sector evaluation at that point is an explicit letter-by-letter computation inside the known giant-graviton formula, not a fit. Self-citations (e.g., the author’s earlier fortuity note) supply only background and are not load-bearing for the equal-charge factorization or the survival statement. No step equates a claimed prediction to a fitted input or to a self-referential definition.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside the standard definition of the N=4 superconformal index and classical partition theory. No free parameters are fitted. The load-bearing background results are classical (Euler pentagonal theorem, Macdonald/q-Dyson constant-term identities, Hardy–Ramanujan asymptotics) or standard domain facts about the single-particle index and the giant-graviton expansion. No new physical entities are postulated.

axioms (4)
  • standard math Euler’s pentagonal number theorem: (q;q)_∞ = ∑_{r∈Z} (-1)^r q^{r(3r-1)/2}
    Used to expand the angular prefactor Π(x,p) and to obtain the support of multigraviton coefficients (Proposition 2.5, eq. (29)).
  • standard math Macdonald/q-Dyson constant-term identity for the type-A Vandermonde product
    Evaluates the kinematic-boundary matrix integral to (q;q)_∞ independently of N (Proposition 2.5).
  • domain assumption The large-N multigraviton index is the plethystic exponential of the single-particle index of Kinney et al.
    Standard in the giant-graviton literature; taken as the zeroth-order term against which finite-rank corrections are measured.
  • domain assumption The U(3) black-hole existence bound of Deddo–Pando Zayas–Zhou (eq. (48))
    Used only to label which finite-rank coefficients lie beyond the classical bound; the arithmetic zero-interval claim does not depend on it.

pith-pipeline@v1.1.0-grok45 · 40229 in / 2953 out tokens · 27948 ms · 2026-07-12T00:17:38.164712+00:00 · methodology

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read the original abstract

The equal-charge branch of supersymmetric rotating AdS$_5$ black holes has $Q_1=Q_2=Q_3$. The corresponding microcanonical sector of the $\mathcal{N}=4$ superconformal index is obtained by projecting to equal charges, or equivalently by extracting the constant term in the two charge-difference fugacities. We prove that for the large-$N$ multigraviton sector the projected index factorizes exactly as \[ \mathcal{I}^{\rm eqQ}_\infty(x,p) =\prod_{k\ge1}(1-p^kx^{3k})(1-p^{-k}x^{3k}) \sum_{n\ge0}\mathsf{p}(n)^3x^{6n}, \] where $\mathsf{p}(n)$ is the partition function. This factorization gives, for every spin sector, an explicit onset energy below which the large-$N$ coefficient is zero. Exact $U(3)$ computations show that finite-rank coefficients can nevertheless appear at energies where the large-$N$ coefficient vanishes, including beyond the classical $U(3)$ black-hole bound. We also determine the full line $j=6J_R+6$. In particular, with $j^*(J_R)$ denoting this large-$N$ onset energy, \[ d_3^{\rm eqQ}(87,\tfrac{27}{2})=1, \qquad j^*(\tfrac{27}{2})-87=1554, \] and the first giant-graviton sector already contributes one unit at this point. All coefficients are coefficients of the $(-1)^F$-graded index, not positive degeneracies. The main conclusion is that the high-spin tail survives the exact equal-charge projection.

Figures

Figures reproduced from arXiv: 2607.03735 by Miguel Tierz.

Figure 1
Figure 1. Figure 1: Schematic of the equal-charge projection. The equal-fugacity index sums over the full A2 charge-difference lattice at fixed (j, JR), whereas the equal-charge projection keeps only the origin Q1 = Q2 = Q3. In the projected (j, JR) plane, the large-N factorization gives exact intervals where the large-N coefficient is zero; finite-rank coefficients can still be nonzero there, even beyond the black-hole bound… view at source ↗
Figure 2
Figure 2. Figure 2: Coefficients beyond the black-hole bound and below the large-N onset. The horizontal axis is the distance beyond the BH bound, ∆BH = JR − J BH R (j); the vertical axis is the depth below the large-N onset, ∆∞ = j ∗ (JR) − j. Points in the upper-right quadrant are beyond the BH bound and absent from the multigraviton sector. Marker size is proportional to log(1 + |d eqQ 3 |); circles are positive coefficien… view at source ↗
Figure 3
Figure 3. Figure 3: Support map of the equal-charge projected tail in the selected high-spin range. The black curve is the BH boundary J BH R (j); the dashed line is the kinematic ceiling JR = j/6. Red circles mark finite-rank coefficients beyond the BH bound with d eqQ ∞ = 0 and d eqQ 3 ̸= 0. Blue diamonds mark the large-N onset j ∗ (JR) [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Complete j = 66 equal-charge slice. The plot compares the classical BH entropy S N=3 BH (66, JR) with log |d eqQ 3 | and log |d eqQ ∞ |. The finite-rank index roughly tracks the BH entropy in the bulk with an approximately constant deficit, while the tail survives beyond J BH R . Q1 = Q2 = Q3 microcanonically, matching the charge assignment of the supersymmetric rotating black hole [38, 37]. The near-const… view at source ↗

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