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REVIEW 3 major objections 4 minor 77 references

Bootstrapping black holes at low impact parameter

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Once the universal eikonal carrier of the graviton pole is supplied, the remaining positive spectrum in six dimensions organizes into a cap-saturated low-impact band near the rotating black-hole scale, a separate high-spin ridge, and an emp

desk verdict A genuinely new, honestly caveated finite-grid SDR bootstrap; the endpoint residual is a real soft spot but not a disqualifier — send to referees. read the letter →

arxiv 2607.05503 v2 pith:MQNUBMAW submitted 2026-07-06 hep-th

Diptarka Das , Aninda Sinha This is my paper · ORCID
classification hep-th PACS 11.55.Ds04.70.-s11.25.-w
keywords S-matrixbootstrapgravitonpoleeikonalapproximationstringydispersionrelationblack-holescaleReggetrajectorieseffectivefieldtheoryimpactparameter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks a precise bootstrap question: in a gravitational effective field theory, if you first supply the universal large-impact-parameter eikonal density that carries the graviton pole, where must the remaining positive spectral weight live? Working in six dimensions, the authors use a crossing-symmetric stringy dispersion relation and solve a finite-dimensional linear program over partial-wave densities. They find that the residual spectrum does not smear across the available non-eikonal region; instead, on their grids, it concentrates into a low-impact, unitarity-cap-saturated band aligned with a rotating black-hole guide, together with a separate high-spin ridge and a mostly empty gap. In the weak-gravity hierarchy the band edge stays near five to six inverse EFT scales and is reproduced by a strict zero-gravity capped problem, making the weak band an intrinsic non-gravitational baseline of the capped extremal problem. A sympathetic reader should care because this is one of the first attempts to say, from dispersion relations alone, where strong-gravity spectral support is forced to sit once the known eikonal piece is accounted for.

What carries the argument

The central object is the stringy dispersion relation (SDR) with auxiliary parameter lambda, a crossing-symmetric representation whose large-energy tail probes fixed-t-like kinematics at momentum transfer roughly lambda, giving a continuous family of sum rules. The paper splits the absorptive partial-wave density into a prescribed eikonal carrier, the elastic density 1-cos(chi) built from the six-dimensional Einstein eikonal phase, and an unknown residual density, then discretizes the residual on a (sigma, J) grid and enforces the sampled lambda constraints as a linear program with the unitarity box 0 <= rho <= 2. The workhorse equation is the carrier-complete sum rule A rho_res - lambda^2 Y

What would settle it

A concrete test is to rerun the carrier-complete problem with steadily finer spectral collocation concentrated near lambda = 0 and near the low-energy onset: if the two-percent dense-residual layer does not shrink and the cap-saturated band, the empty gap, or the wedge changes topology, the central organization is a grid artifact. Alternatively, scan an adjustable string-scale alpha' in the prescribed carrier: if the weak-coupling microscopic edge fails to track sqrt(alpha') while the strict zero-gravity null baseline stays put, the band is set by the threshold and the cap, not by a completion

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that after the complete high-spin continuum tail of the universal eikonal carrier is inserted, the residual positive spectrum required by the SDR is highly organized rather than featureless. In the scale-reversed microscope (Planck mass below the EFT scale) the extremal witnesses develop a cap-saturated low-impact band near an order-one rotating black-hole guide, a distinct high-spin ridge, and eventually parallel far-tail tracks, while the broad available region between the band and the eikonal layer remains almost empty. In the hierarchy-correct weak-gravity regime the low-spin saturated band has an edge that barely moves as Newton's constant shrinks, and

Load-bearing premise

The load-bearing premise is that the finite spectral grids, with up to 1200 sigma-nodes and 120 sampled lambda constraints, together with the prescribed eikonal trust region, faithfully encode the continuum stringy dispersion relation over the scales that decide the structure; since the dense residual reaches about two percent exactly at the lambda-to-0 endpoint layer where the graviton-pole behavior lives, the band, gap, and wedge could in principle be truncation artifacts r

Editorial extensions

If this is right

  • If the central claim is right, the weak-coupling low-impact band is a property of the capped dispersion-relation problem itself, not evidence for a string-to-black-hole crossover; gravity's visible role is to reorganize the support toward the rotating black-hole scale as the coupling grows.
  • The high-energy cap-saturated support forms a wedge with an outer envelope J_out + 3/2 ~ C GN^{1/3} sigma^{2/3}, a clean six-dimensional rotating black-hole homogeneity with an order-one coefficient, plus a nearly linear lower envelope whose slope is compatible with GN^{-1/3}, suggesting a Regge-like organization.
  • In the strong-gravity microscope, a cap-saturated low-impact band near the rotating black-hole guide coexists with a separate high-spin ridge, while most available residual bins in between stay unoccupied; reduced costs show that forcing weight into that gap is expensive and that relaxing the cap on the selected band improves the objective.
  • The low-impact band is branch-dependent: it is prominent on the upper boundary and the positive-X lower boundary but essentially absent on the negative-X lower boundary, showing the organization is a genuine selection by the extremal problem rather than a generic artifact of cap saturation.
  • At the far-energy tail the extremal spectra contain series of nearly parallel, equally spaced tracks at threshold-defined edges, reminiscent of leading and daughter Regge trajectories in dual amplitudes, although they are edges of a continuous density rather than pole locations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the strict zero-gravity null already reproduces the weak-coupling band, the decisive next test is whether that band tracks an independent completion scale such as sqrt(alpha'); if it does not, the band is a threshold artifact of the capped SDR and any string/black-hole reading should be dropped.
  • The residual dense error peaks around two percent precisely at the lambda-to-0 endpoint layer where the graviton-pole behavior lives; a targeted refinement of spectral collocation near that layer could settle whether the band, gap, and wedge survive in the continuum, a question the paper explicitly leaves open.
  • The single-bin kernel-profile sign diagnostic could be exported to other crossing-symmetric bootstrap setups as a cheap pre-solver predictor of where extremal support will land; the paper only gestures at this comparison.
  • If the proposed Ericson-type fluctuation width were measured at amplitude level, it would furnish a dynamical, density-independent test of whether the low-impact band behaves like an absorbing black-hole region, which the positive density alone cannot establish.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the D=6 stringy dispersion relation (SDR) with a prescribed high-energy, large-impact-parameter eikonal carrier and asks where the remaining positive spectral density is placed. The authors formulate a finite-dimensional linear program over residual partial-wave densities and report several organized structures: a weak-coupling band whose physical impact-parameter edge is nearly fixed as G_N varies and is reproduced by a strict G_N=0 null; a high-energy wedge whose outer envelope scales as G_N^{1/3}\sigma^{2/3} and whose lower envelope is approximately linear with slope compatible with G_N^{-1/3}; and, in a deliberately scale-reversed 'spectral microscope', a cap-saturated low-impact band near a rotating Giddings--Porto black-hole guide, a mostly empty gap, a Regge-like ridge, and far-tail parallel tracks. The paper repeatedly and honestly labels the outputs as finite-grid primal witnesses rather than continuum theorems.

Significance. The question addressed is novel and important: after supplying the universal eikonal carrier, where does the remaining positive spectral weight actually sit? The paper makes several methodical contributions: a carrier-complete source with an analytic high-spin continuum tail, multiple independent source checks (Mathematica and Python), exact agreement between HiGHS dual-simplex and interior-point solves, and ancillary code for reproducibility. The strict G_N=0 null is a clean control that correctly separates non-gravitational baseline effects from genuine strong-gravity organization. If the reported band/gap/wedge structures survive a properly endpoint-resolved calculation, this would be a striking spectral-organization result for gravitational EFT completions. The main caveat is that the load-bearing structures are established only on finite collocation grids, with a percent-level residual concentrated at the small-\lambda endpoint where the carrier cancellation is most singular.

major comments (3)
  1. [Section 6.4, Table 5, Eq. (60)] The dense off-grid residual \Delta_{\rm dense} saturates at ~0.020 for every refined grid, and the text locates the maximum below the first collocation point \lambda_{\min}=1.07\times10^{-5} (Eq. (32)). This is exactly the layer where the carrier cancellation in Eq. (33) is most singular: F_{\rm eik} carries 1/\lambda and 1/\lambda^2 terms that must cancel against the residual density to reproduce the low-energy EFT target. A 2% mismatch there is not a harmless tail effect. The band/gap/wedge diagnostics (Table 4, Figs. 9-13) are computed from the same finite spectral grid, so they could shift if this endpoint were properly resolved. I would need an endpoint-refined collocation run—e.g., adding \lambda constraints below \lambda_{\min} together with matched spectral endpoint enrichment—before accepting the empty gap and the wedge shape as properties of the full SDR rather than artifacts o
  2. [Section 6.8] The far-tail 'series of Regge trajectories' is defined by threshold-defined edges at J_{\rm edge}+2k, k=0,...,98. The reported R^2=0.99386 and the exact 2-unit spin separation are therefore built into the construction, not emergent properties of the spectrum. The abstract's claim that 'series of Regge trajectories emerge' overstates what is demonstrated. Please either extract tracks without imposing the 2-unit spacing and show that the spacing emerges independently, or explicitly state that the parallel curves are threshold-defined edges at fixed spacing by definition. As written, this part of the central picture is circular in construction.
  3. [Section 5.4, Eq. (55)] The lower-envelope exponent is quoted as a(G_N)\propto G_N^{-0.35\pm0.02}, with the systematic range spanning window-dependent fits from -0.360 to -0.348. This is compatible with G_N^{-1/3} but does not establish it. Since the wedge structure is one of the paper's headline results, the abstract and conclusions should state the exponent as 'compatible with -1/3 within fit uncertainty' rather than implying a measured G_N^{-1/3} law. Additional data or a more controlled fitting procedure would be needed to sharpen this point.
minor comments (4)
  1. [Section 6.8, first paragraph] The sentence 'Here we solve the G_N=4\pi^2, X=20, Y_{\max} point was on 1\le\sigma<\infty' is ungrammatical; it should read 'was solved on the interval 1\le\sigma<\infty' or similar.
  2. [Eq. (55)] The notation 'G^{-0.35\pm0.02}' is ambiguous about the base of the exponent. Please write the exponent explicitly as -0.35\pm0.02 with base G_N, and state which quantity (e.g., the slope a(G_N)M_{\rm EFT}^2) is being fit.
  3. [References] Reference [58] contains the DOI '10.1103/ljzx-q254', which appears to be a placeholder or invalid DOI. Please verify and correct.
  4. [Section 6.4] The phrase 'phase I requires a nonzero equality slack' likely refers to the Phase I of the simplex method; please spell this out to avoid confusion with the eikonal phase variables used elsewhere.

Circularity Check

2 steps flagged · score 6.0 of 10

Wedge outer envelope and far-tail 'Regge tracks' are partly fixed by the paper's own definitions; the G_N=0 null and external tracker provide independent content, so the circularity is partial.

  1. self definitional [Section 2.3, Eq. (28), with Section 5.4, Eq. (54)]
    "The active set used in the scans quoted below is E(χ max) ={(σ, J) : √σ≥4, J≥20, b≥2, b/R_S(σ)≥R_min, 0≤χ(σ, J)<χ max}, R min = 3, χ max = 30. ... On E we prescribe the eikonal density and switch off the residual variable. Outside E we set ρphys_eik,i = 0 and allow a residual variable."

    Residual variables exist only outside the eikonal trust region E, so every residual bin must satisfy b/R_S < 3. Using the paper's own conventions b = 2(J+3/2)/√σ (Eq. (25)) and R_S = (3G_N/(2π))^{1/3}σ^{1/6} (Eq. (29)), the mask boundary b/R_S = 3 is exactly J+3/2 = (3/2)(3/(2π))^{1/3} G_N^{1/3}σ^{2/3} ≈ 1.17 G_N^{1/3}σ^{2/3}. Section 5.4 then 'finds' J_out+3/2 = C_out G_N^{1/3}σ^{2/3} with C_out = 1.15–1.17 and presents this as 'a clean rotating black-hole homogeneity.' The scaling and even the order-one coefficient are inherited from the trust-region cut, which is an input, not an LP output; only the support filling up to that mask edge is dynamical.

  2. self definitional [Section 6.8, paragraph after Fig. 13]
    "Here, however, the curves are threshold-defined edges of a continuous positive spectral density rather than pole locations with factorizing residues, and their separation by two units of spin is built into the construction Jedge + 2k. They therefore provide evidence for Regge-like organization of the extremal support, but do not yet identify its microscopic constituents."

    The 'series of Regge trajectories' is an enumeration J_edge, J_edge+2, J_edge+4, ... of the even-spin lattice inside a contiguous cap-saturated block. Parallel straight tracks separated by two units of spin are automatic from the even-spin grid and the labeling, so the far-tail 'series of Regge trajectories emerge' is a renaming of contiguous support rather than a new spectral prediction. The paper itself concedes the spacing is built into the construction.

full rationale

The paper contains genuine independent checks: the strict G_N=0 capped-SDR null is a clean control that reproduces the weak-coupling microscopic edge and occupied support; the rotating Giddings–Porto tracker is an externally defined, unfitted geometric guide; and the branch asymmetry, gap emptiness, and reduced-cost structure are LP outputs rather than inputs. It also honestly labels its results as finite-grid witnesses and repeatedly disclaims continuum certification. The main circularity concerns are the two construction-level reductions above. The high-energy wedge's outer envelope has the scaling and normalization of the b/R_S=3 boundary of the active eikonal mask, outside which residual variables are switched off by definition, so presenting it as a rotating black-hole homogeneity attributes to the spectrum a curve that was put in as the residual-region boundary. Similarly, the far-tail parallel 'Regge tracks' are the even-spin ladder J_edge+2k of a contiguous block, so their existence and spacing are definitional. These are not mere numerical caveats; they affect two of the paper's advertised structural findings. The independent null and external tracker prevent the circularity from being total, but the wedge outer envelope and the Regge-track series reduce by construction, warranting a score of 6 rather than a lower score.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper postulates no new particles or forces. The load-bearing inputs are the SDR of Ref. [58], the prescribed eikonal carrier with hand-chosen trust cuts, and the finite-grid collocation trust. The fitted quantities are the wedge envelope constants and the lower-edge exponent, plus the deliberately chosen microscope normalization.

free parameters (5)
  • eikonal trust-region cuts = χmax=30, Rmin=3, J≥20, b≥2, √σ≥4, b/RS≥3
    Hand-chosen cuts (Eq. 28) define the active eikonal set; the residual spectrum and the observed band/gap depend on them. Rmin=3 is explicitly a reference-coupling trust cut, not a black-hole threshold.
  • rotating-guide tracker constant κ = κ=3
    Order-one parameter in the Giddings–Porto tracker J_BH^(κ) (Eq. 59); used as a visual comparison, not fitted to the LP data. Changing κ changes where the band appears to 'match' the guide.
  • lower-envelope exponent = -0.35 ± 0.02
    Fit to four LP solutions at a fixed g2 slice (Eq. 55); the paper concludes it is 'compatible with' G^{-1/3}, but the exponent is measured, not derived.
  • outer-envelope prefactor C_out = 1.15–1.17
    Fitted constant in J_out + 3/2 ≃ C_out G^{1/3} σ^{2/3} (Eq. 54); about 2.1 times larger than the κ=3 tracker coefficient 0.544.
  • scale-reversed normalization = G_N = 4π², M_Pl/M_EFT ≃ 0.18
    Deliberate toy normalization for the strong-coupling microscope, bringing the trans-Planckian window into a tractable σ range; not a weak-gravity EFT construction (Section 6).
assumptions (5)
  • domain assumption The SDR of Ref. [58] with Regge growth M(s,t)=O(s^{2−ε}) is a valid dispersion relation; λ in (0,1/3) is an admissible sampled interval.
    Invoked in §2.1; the exact SDR holds only for −λ in the analyticity/growth domain D. The subinterval is a numerical conditioning choice, not certified beyond.
  • domain assumption The large-impact-parameter absorptive density is exactly 1−cosχ with χ = G_N σ/(π b^2) on the active eikonal set.
    Equations (26)–(28); the eikonal phase is a semiclassical model, not derived from the SDR. The answer to 'where the remaining spectrum goes' depends on this prescription.
  • standard math Physical partial-wave unitarity gives 0≤ρ≤2 for the full density, and the same cap applies to residual density bins.
    Standard S-matrix unitarity; the ρ=2 endpoint is reflective, ρ≈1 absorptive (Section 2.6).
  • domain assumption The finite collocation with the stated grids and sampled λ constraints faithfully represents the continuum SDR over the resolved scales.
    Convergence ladder (Table 5) shows about 2% dense residual at the endpoint; the paper explicitly avoids claiming a continuum-certified bound. This is a key load-bearing assumption.
  • domain assumption The large-spin continuum tail of the carrier uses the Bessel/impact-parameter asymptotics with J* = 320 handover, and the H(x) integral representation is exact.
    Equations (35)–(39) and Appendix C. The source is checked by quadrature and by varying J*, but the continuum form is an approximation to the exact Gegenbauer kernel at large J.

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Cite this review

Pith. "Pith review of Bootstrapping black holes at low impact parameter." pith.science (2026). https://pith.science/paper/MQNUBMAW

@misc{pith2026260705503,
  author       = {Pith},
  title        = {Pith review of: Bootstrapping black holes at low impact parameter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQNUBMAW}},
  note         = {Machine review of arXiv:2607.05503}
}
abstract

We use the stringy dispersion relation (SDR) to ask the following question about gravitational effective field theories: once the universal large-impact-parameter eikonal carrier is supplied, where does the remaining positive spectrum go? Working in six dimensions for concreteness, we include the complete high-spin continuum tail of the carrier and first work at weak coupling, where $M_{\rm Pl}>M_{\rm EFT}$. The extremal spectra contain a saturated low-impact band whose outer edge stays at roughly five to six times the inverse EFT scale even as the gravitational radius shrinks. The same edge is reproduced by a strict $G_N=0$ capped-SDR problem on the present grids. Thus the weak-coupling band approaches an intrinsic non-gravitational baseline of the capped extremal problem. On a common high-energy grid, a coupling ladder crossing $M_{\rm Pl}=M_{\rm EFT}$ resolves the cap-saturated support as a wedge: its outer envelope has the rotating black-hole homogeneity $G_N^{1/3}E^ {4/3}$, while its approximately linear lower envelope flattens as $G_N$ grows. We then use the deliberately reversed hierarchy $M_{\rm Pl}<M_ {\rm EFT}$ as a microscope for strong-gravity structure. In this regime a cap-saturated low-impact band follows an order-one Giddings-Porto rotating black-hole scale, a separate Regge-like ridge appears at high spin and low energies, the broad available region between these structures and the eikonal layer remains mostly empty, while at the far-tail of energy, series of Regge trajectories emerge.

Figures

Figures reproduced from arXiv: 2607.05503 by the authors.

Figure 1
Figure 1. Cartoon of the regimes expected in 2 ↔ 2 gravitational scattering for D > 4. The brown regions indicate positive spectral density selected by the bootstrap, while the intervening white regions indicate available bins that remain largely unoccupied in the finite-grid witnesses. There is a substantial body of work on gravitational EFT bounds using fixed momentum￾transfer dispersion relations, impact-parameter causalit… view at source ↗
Figure 1
Figure 1. Cartoon of the regimes expected in 2 → 2 gravitational scattering for D > 4. The brown regions indicate positive spectral density selected by the bootstrap, while the intervening white regions indicate available bins that remain largely unoccupied in the finite-grid witnesses. representations of string amplitudes, not an assumption that the unknown completion is pertur￾bative string theory. Under the stated analytic… view at source ↗
Figure 2
Figure 2. Bounds obtained from the uncapped residual cone after supplying the eikonal input. [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figures from the paper (30 more)
Figure 2
Figure 2. Figure 2: Bounds obtained from the carrier-complete uncapped residual cone. The calculation [PITH_FULL_IMAGE:figures/full_fig_p019_2.png]
Figure 3
Figure 3. Figure 3: Representative uncapped residual support at [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]
Figure 3
Figure 3. Figure 3: Representative uncapped support at X = 8 on the (900, 256, 120) grid. The columns are the lower and upper sampled boundaries. The top row locates finite samples of the known carrier; the complete high-spin continuation is included analytically in the sum rule. The bott…
Figure 4
Figure 4. Figure 4: Capped GN = 4π 2 fixed-grid leaf at (Nσ, Jmax, Nλ, Ndense) = (2160, 560, 200, 600). The orange and blue curves are the upper and lower capped SDR boundaries. The dashed black lines are fixed-t guides, and the gray band is the fixed-a/CSDR comparison region. Six represe…
Figure 4
Figure 4. Figure 4: Continuum-carrier weak-gravity leaves at [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Low-σ view of four capped GN = 4π 2 spectral witnesses from the leaf in figure 4. The top panels show upper-boundary witnesses at X = −8 and X = 20; the bottom panels show lower-boundary witnesses at X = −8 and X = 24. Eikonal-active bins and residual bins use the same…
Figure 5
Figure 5. Figure 5: The contiguous 90%-cap edge at two matched upper-boundary positions. The colored [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Wider view of the same four witnesses as in [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 6
Figure 6. Figure 6: The high-energy wedge across the coupling ladder. Top: residual support in the ( [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Upper-boundary evolution of the black-hole-guide band in the low- [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 7
Figure 7. Figure 7: Representative upper-boundary witness at [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Wider version of figure 7, with σ ≤ 80 and a local compressed 0 ≤ J ≤ 55 axis in each slice. The positive-X witnesses show a visible broadening of the cap-saturated band above the κ = 3 tracker, while the negative-X witnesses sit more tightly below or near it. Since κ …
Figure 8
Figure 8. Figure 8: Carrier-complete capped leaf at GN = 4π 2 on the (Nσ, Jmax, Nλ, Ndense) = (300, 160, 80, 801) grid. The orange and blue curves are the upper and lower sampled SDR bound￾aries. The dashed black lines and gray region are, respectively, the fixed-t and fixed-a/CSDR compar…
Figure 9
Figure 9. Figure 9: Boundary morphology of the fixed-grid GN = 4π 2 leaf. Solid curves denote the upper boundary and dashed curves denote the lower boundary. The upper boundary is dominated by low-b/RS, near-cap support over a broad range of X. By contrast, the lower branch near negative …
Figure 9
Figure 9. Figure 9: Four carrier-complete capped witnesses at [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: Reduced costs and cap marginals for the GN = 4π 2 , X = 20, Ymax boundary point on the (Nσ, Jmax, Nλ) = (2160, 560, 200) grid, shown for σ ≤ 20. The left panel is the primal witness: it shows where the optimizer actually places residual spectral density. The middle pa…
Figure 10
Figure 10. Figure 10: Carrier-complete boundary morphology at GN = 4π 2 . Solid curves denote the upper boundary and dashed curves denote the lower boundary. On the upper branch, nearly all residual weight is in low-impact near-cap bins. The negative-X lower branch instead uses the interme…
Figure 11
Figure 11. Figure 11: Wider version of figure 10, with σ ≤ 80. It shows the same three diagnostics: primal support, lower-bound reduced costs for empty residual spectral bins, and upper-bound marginals for cap-saturated bins. The wider panel shows that the empty region above the low-impact…
Figure 11
Figure 11. Figure 11: Reduced costs and cap marginals for the GN = 4π 2 , X = 20, Ymax boundary point on the (Nσ, Jmax, Nλ) = (600, 240, 80) grid, shown for σ ≤ 20. The left panel is the primal witness: it shows where the optimizer actually places residual spectral density. The middle pane…
Figure 12
Figure 12. Figure 12: Single-cell kernel-profile diagnostic for the exact finite- [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]
Figure 12
Figure 12. Figure 12: Single-bin kernel-profile diagnostic for the exact finite- [PITH_FULL_IMAGE:figures/full_fig_p037_12.png]
Figure 13
Figure 13. Figure 13: No-eikonal finite-cutoff control at GN = 4π 2 , X = 6.5, (Nσ, Jmax, Nλ) = (1200, 320, 120). All finite-grid bins with σ ≤ 1200 are available, ρ phys eik = 0 everywhere, and the only box constraint is 0 ≤ ρ phys res ≤ 2. The left panel shows σ < 20, while the right pan…
Figure 13
Figure 13. Figure 13: Outer tracks of the compact-grid witness through [PITH_FULL_IMAGE:figures/full_fig_p039_13.png]
Figure 14
Figure 14. Figure 14: Residual spectral density ρ phys res for the extended eikonal-carrier scan. The panels show the nonzero residual support in the (σ, J) grid for b/RJ > 0.9, 0.5, 0.25, 0.1, with a common color scale for ρ phys res . The orange curve is the black-hole spin curve JBH(σ).…
Figure 14
Figure 14. Figure 14: No-eikonal control at (Nσ, Jmax, Nλ) = (1200, 320, 120). The low-energy high-spin branch becomes the dominant finite-grid carrier of the pole sum rule. A low-impact component remains visible, but it carries little of the leading small-λ budget. The panels show σ < 20 …
Figure 15
Figure 15. Figure 15: Residual-density oscillations as the eikonal carrier is extended to smaller impact [PITH_FULL_IMAGE:figures/full_fig_p037_15.png]
Figure 15
Figure 15. Figure 15: Carrier-complete b/RS inward-continuation control at X = 6.5 on the refined (Nσ, Jmax, Nλ) = (1200, 220, 100) residual grid. The panels show the optimized residual density after moving the prescribed carrier from b/RS ≥ 0.9 to b/RS ≥ 0.75, 0.5, 0.25, with χmax = 1000;…
Figure 16
Figure 16. Figure 16: Distribution of the residual spectral density [PITH_FULL_IMAGE:figures/full_fig_p038_16.png]
Figure 16
Figure 16. Figure 16: Fixed-spin residual-density profiles for the four carrier-complete deformations in [PITH_FULL_IMAGE:figures/full_fig_p043_16.png]
Figure 17
Figure 17. Figure 17: Toy string-spreading broadening of the allowed leaf in the [PITH_FULL_IMAGE:figures/full_fig_p049_17.png]

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