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REVIEW 3 major objections 4 minor 77 references

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

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2026-08-02 08:26 UTC pith:MQNUBMAW

load-bearing objection A genuinely new, honestly caveated finite-grid SDR bootstrap; the endpoint residual is a real soft spot but not a disqualifier — send to referees. the 3 major comments →

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

Bootstrapping black holes at low impact parameter

Diptarka Das , Aninda Sinha This is my paper
classification hep-th PACS 11.55.Ds04.70.-s11.25.-w
keywords S-matrix bootstrapgraviton poleeikonal approximationstringy dispersion relationblack-hole scaleRegge trajectorieseffective field theoryimpact parameter
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.

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.

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

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

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

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

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.

Where Pith is reading between the lines

These are 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.

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

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

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.

specific steps
  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.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 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).
axioms (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.

pith-pipeline@v1.3.0-alltime-deepseek · 50539 in / 14472 out tokens · 147695 ms · 2026-08-02T08:26:19.559078+00:00 · methodology

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read the original 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 Aninda Sinha, Diptarka Das.

Figure 1
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-paramete… view at source ↗
Figure 1
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 state… view at source ↗
Figure 2
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 ↗
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] view at source ↗
Figure 3
Figure 3. Figure 3: Representative uncapped residual support at [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
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 bottom row shows the unbounded physical residual density on a separate logarithmic scale. Residual variables existed throughout the white non-ei… view at source ↗
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 representative boundary points pass the Nλ = 180, 190, 200 λ-constraint enrichment drift check in section 5.4. 5.2 Spectral witnesses around the le… view at source ↗
Figure 4
Figure 4. Figure 4: Continuum-carrier weak-gravity leaves at [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
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 orange/red temperature scale. The orange curve is the rotating black-hole tracker J (κ=3) BH . The residual variables were available throug… view at source ↗
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] view at source ↗
Figure 6
Figure 6. Figure 6: Wider view of the same four witnesses as in [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
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] view at source ↗
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] view at source ↗
Figure 7
Figure 7. Figure 7: Representative upper-boundary witness at [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
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 is a guide parameter rather than a constraint, the quantitative amount of “spill” above the curve should not be overinterpreted. The robust … view at source ↗
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 comparison guides; neither is imposed in the solve. Direct tip refinement places the sampled feasible interval between the infeasible trial values… view at source ↗
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 X has essentially no low-impact cap-saturated weight; its support is concentrated instead in higher-spin threshold and intermediate-b/RS bin… view at source ↗
Figure 9
Figure 9. Figure 9: Four carrier-complete capped witnesses at [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
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 panel tests unused residual variables by showing how much the optimized value of −Y would increase if that bin were forced to carry an infinit… view at source ↗
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 intermediate gap and has no low-impact near-cap weight; that component turns on between X = 0 and X = 8. The high-spin ridge occupies many lower-br… view at source ↗
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 band remains costly over the displayed range after global reoptimization of the finite LP, while the cap-saturated low-impact band continue… view at source ↗
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 panel tests unused residual variables by showing how much the optimized value of −Y would increase if that bin were forced to carry an infinites… view at source ↗
Figure 12
Figure 12. Figure 12: Single-cell kernel-profile diagnostic for the exact finite- [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Single-bin kernel-profile diagnostic for the exact finite- [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
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 panel shows σ ≤ 80. Compared with the eikonal-subtracted spectra in section 5, the high-spin low-energy branch is much more visible and becomes… view at source ↗
Figure 13
Figure 13. Figure 13: Outer tracks of the compact-grid witness through [PITH_FULL_IMAGE:figures/full_fig_p039_13.png] view at source ↗
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). As the eikonal carrier reaches smaller b/RJ , the residual support develops a stronger low-spin oscillatory ridge beneath and near the blac… view at source ↗
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 and σ ≤ 80. 7.1 Removing the prescribed carrier In the first control there is no eikonal source and no analytic contribution beyond the fini… view at source ↗
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] view at source ↗
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; the black curve is the J (κ=3) BH guide. Residual variables remain available in every bin, the same b/RS window is used in the carrier sour… view at source ↗
Figure 16
Figure 16. Figure 16: Distribution of the residual spectral density [PITH_FULL_IMAGE:figures/full_fig_p038_16.png] view at source ↗
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] view at source ↗
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] view at source ↗

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  1. [1]

    Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory,

    S. Mandelstam, “Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory,” Phys. Rev.112(1958) 1344

    1958

  2. [2]

    R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne,The Analytic S-Matrix, Cambridge University Press, 1966

    1966

  3. [3]

    Asymptotic behavior and subtractions in the Mandelstam representation,

    M. Froissart, “Asymptotic behavior and subtractions in the Mandelstam representation,” Phys. Rev.123 (1961) 1053

    1961

  4. [4]

    Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. I,

    A. Martin, “Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. I,” Nuovo Cim. A42(1965) 930

    1965

  5. [5]

    Absolute upper bounds for pion-pion scattering,

    L. Lukaszuk and A. Martin, “Absolute upper bounds for pion-pion scattering,” Nuovo Cim. A52(1967) 122

    1967

  6. [6]

    Exact integral equation for pion-pion scattering involving only physical region partial waves,

    S. M. Roy, “Exact integral equation for pion-pion scattering involving only physical region partial waves,” Phys. Lett. B36(1971) 353. 56

    1971

  7. [7]

    Causality, analyticity and an IR obstruction to UV completion,

    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, “Causality, analyticity and an IR obstruction to UV completion,” JHEP10(2006) 014, doi:10.1088/1126-6708/2006/10/014, arXiv:hep- th/0602178

    arXiv 2006

  8. [8]

    Positivity Bounds for Scalar Theories,

    C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou, “Positivity Bounds for Scalar Theories,” Phys. Rev. D96(2017) 081702, doi:10.1103/PhysRevD.96.081702, arXiv:1702.06134 [hep-th]

    Pith/arXiv arXiv 2017

  9. [9]

    Softness and amplitudes’ positivity for spinning particles,

    B. Bellazzini, “Softness and amplitudes’ positivity for spinning particles,” JHEP02(2017) 034, doi:10.1007/JHEP02(2017)034, arXiv:1605.06111 [hep-th]

    Pith/arXiv arXiv 2017

  10. [10]

    Positive Signs in Massive Gravity,

    C. Cheung and G. N. Remmen, “Positive Signs in Massive Gravity,” JHEP04(2016) 002, doi:10.1007/JHEP04(2016)002, arXiv:1601.04068 [hep-th]

    Pith/arXiv arXiv 2016

  11. [11]

    UV complete me: Positivity bounds for particles with spin,

    C. de Rham, S. Melville, A. J. Tolley and S.-Y. Zhou, “UV complete me: Positivity bounds for particles with spin,” JHEP03(2018) 011, doi:10.1007/JHEP03(2018)011, arXiv:1706.02712 [hep-th]

    Pith/arXiv arXiv 2018

  12. [12]

    Positive moments for scattering amplitudes,

    B. Bellazzini, J. Elias Mir´ o, R. Rattazzi, M. Riembau and F. Riva, “Positive moments for scattering amplitudes,” Phys. Rev. D104(2021) 036006, doi:10.1103/PhysRevD.104.036006, arXiv:2011.00037 [hep- th]

    Pith/arXiv arXiv 2021

  13. [13]

    The EFT-Hedron,

    N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, “The EFT-Hedron,” JHEP05(2021) 259, doi:10.1007/JHEP05(2021)259, arXiv:2012.15849 [hep-th]

    Pith/arXiv arXiv 2021

  14. [14]

    De-projecting the EFThedron,

    L.-Y. Chiang, Y.-T. Huang, L. Rodina and H.-C. Weng, “De-projecting the EFThedron,” JHEP05(2024) 102, doi:10.1007/JHEP05(2024)102, arXiv:2204.07140 [hep-th]

    Pith/arXiv arXiv 2024

  15. [15]

    (Non)-projective bounds on gravitational EFT,

    L.-Y. Chiang, Y.-T. Huang, W. Li, L. Rodina and H.-C. Weng, “(Non)-projective bounds on gravitational EFT,” arXiv:2201.07177 [hep-th]

    Pith/arXiv arXiv

  16. [16]

    Extremal Effective Field Theories,

    S. Caron-Huot and V. Van Duong, “Extremal Effective Field Theories,” JHEP05(2021) 280, doi:10.1007/JHEP05(2021)280, arXiv:2011.02957 [hep-th]

    Pith/arXiv arXiv 2021

  17. [17]

    Sharp boundaries for the swampland,

    S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, “Sharp boundaries for the swampland,” JHEP 07(2021) 110, doi:10.1007/JHEP07(2021)110, arXiv:2102.08951 [hep-th]

    Pith/arXiv arXiv 2021

  18. [18]

    Bridging positivity and S-matrix bootstrap bounds,

    J. Elias Miro, A. Guerrieri and M. A. Gumus, “Bridging positivity and S-matrix bootstrap bounds,” JHEP 05(2023) 001, doi:10.1007/JHEP05(2023)001, arXiv:2210.01502 [hep-th]

    Pith/arXiv arXiv 2023

  19. [19]

    The S-matrix bootstrap. Part III: higher dimensional amplitudes,

    M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, “The S-matrix bootstrap. Part III: higher dimensional amplitudes,” JHEP12(2019) 040, doi:10.1007/JHEP12(2019)040, arXiv:1708.06765 [hep-th]

    Pith/arXiv arXiv 2019

  20. [20]

    S-matrix bootstrap for effective field theories: massless pions,

    A. L. Guerrieri, J. Penedones and P. Vieira, “S-matrix bootstrap for effective field theories: massless pions,” JHEP06(2021) 088, doi:10.1007/JHEP06(2021)088, arXiv:2011.02802 [hep-th]

    Pith/arXiv arXiv 2021

  21. [21]

    Primal S-matrix bootstrap with dispersion relations,

    C. de Rham, A. J. Tolley, Z.-H. Wang and S.-Y. Zhou, “Primal S-matrix bootstrap with dispersion relations,” JHEP01(2026) 027, doi:10.1007/JHEP01(2026)027, arXiv:2506.22546 [hep-th]

    Pith/arXiv arXiv 2026

  22. [22]

    Snowmass White Paper: S-matrix Bootstrap,

    M. Kruczenski, J. Penedones and B. C. van Rees, “Snowmass White Paper: S-matrix Bootstrap,” arXiv:2203.02421 [hep-th]

    Pith/arXiv arXiv

  23. [23]

    Infrared photons and gravitons,

    S. Weinberg, “Infrared photons and gravitons,” Phys. Rev.140(1965) B516

    1965

  24. [24]

    Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass,

    S. Weinberg, “Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass,” Phys. Rev.135, B1049–B1056 (1964), doi:10.1103/PhysRev.135.B1049

    doi:10.1103/physrev.135.b1049 1964

  25. [25]

    Graviton dominance in ultra-high-energy scattering,

    G. ’t Hooft, “Graviton dominance in ultra-high-energy scattering,” Phys. Lett. B198(1987) 61

    1987

  26. [26]

    Eikonal quantum gravity and Planckian scattering,

    D. Kabat and M. Ortiz, “Eikonal quantum gravity and Planckian scattering,” Nucl. Phys. B388(1992) 570, doi:10.1016/0550-3213(92)90627-N, arXiv:hep-th/9203082

    Pith/arXiv arXiv 1992

  27. [27]

    Superstring collisions at Planckian energies,

    D. Amati, M. Ciafaloni and G. Veneziano, “Superstring collisions at Planckian energies,” Phys. Lett. B197 (1987) 81; “Classical and quantum gravity effects from Planckian energy superstring collisions,” Nucl. Phys. B347(1990) 550

    1987

  28. [28]

    Towards an S-matrix description of gravitational collapse,

    D. Amati, M. Ciafaloni and G. Veneziano, “Towards an S-matrix description of gravitational collapse,” JHEP02(2008) 049, doi:10.1088/1126-6708/2008/02/049, arXiv:0712.1209 [hep-th]

    Pith/arXiv arXiv 2008

  29. [29]

    The high-energy behavior of string scattering amplitudes,

    D. J. Gross and P. F. Mende, “The high-energy behavior of string scattering amplitudes,” Phys. Lett. B 197(1987) 129

    1987

  30. [30]

    String theory beyond the Planck scale,

    D. J. Gross and P. F. Mende, “String theory beyond the Planck scale,” Nucl. Phys. B303(1988) 407

    1988

  31. [31]

    High-energy string-brane scattering: leading eikonal and beyond,

    G. D’Appollonio, P. Di Vecchia, R. Russo and G. Veneziano, “High-energy string-brane scattering: leading eikonal and beyond,” JHEP11(2010) 100, doi:10.1007/JHEP11(2010)100, arXiv:1008.4773 [hep-th]

    Pith/arXiv arXiv 2010

  32. [32]

    The gravitational eikonal: From particle, string and brane collisions to black-hole encounters,

    P. Di Vecchia, C. Heissenberg, R. Russo and G. Veneziano, “The gravitational eikonal: From particle, string and brane collisions to black-hole encounters,” Phys. Rept.1083(2024) 1–169, doi:10.1016/j.physrep.2024.06.002, arXiv:2306.16488 [hep-th]. 57

    Pith/arXiv arXiv 2024

  33. [33]

    The gravitational S-matrix,

    S. B. Giddings and R. A. Porto, “The gravitational S-matrix,” Phys. Rev. D81(2010) 025002, doi:10.1103/PhysRevD.81.025002, arXiv:0908.0004 [hep-th]

    Pith/arXiv arXiv 2010

  34. [34]

    Classical black hole production in high-energy collisions,

    D. M. Eardley and S. B. Giddings, “Classical black hole production in high-energy collisions,” Phys. Rev. D 66(2002) 044011, doi:10.1103/PhysRevD.66.044011, arXiv:gr-qc/0201034

    Pith/arXiv arXiv 2002

  35. [35]

    Black holes from colliding wavepackets,

    S. B. Giddings and V. S. Rychkov, “Black holes from colliding wavepackets,” Phys. Rev. D70(2004) 104026, doi:10.1103/PhysRevD.70.104026, arXiv:hep-th/0409131

    Pith/arXiv arXiv 2004

  36. [36]

    Black hole formation in the grazing collision of high-energy particles,

    H. Yoshino and Y. Nambu, “Black hole formation in the grazing collision of high-energy particles,” Phys. Rev. D67(2003) 024009, doi:10.1103/PhysRevD.67.024009, arXiv:gr-qc/0209003

    Pith/arXiv arXiv 2003

  37. [37]

    Improved analysis of black hole formation in high-energy particle collisions,

    H. Yoshino and V. S. Rychkov, “Improved analysis of black hole formation in high-energy particle collisions,” Phys. Rev. D71(2005) 104028, doi:10.1103/PhysRevD.71.104028, arXiv:hep-th/0503171

    Pith/arXiv arXiv 2005

  38. [38]

    High-energy gravitational scattering and black hole resonances,

    S. B. Giddings and M. Srednicki, “High-energy gravitational scattering and black hole resonances,” Phys. Rev. D77(2008) 085025, doi:10.1103/PhysRevD.77.085025, arXiv:0711.5012 [hep-th]

    Pith/arXiv arXiv 2008

  39. [39]

    The fuzzball proposal for black holes: an elementary review,

    S. D. Mathur, “The fuzzball proposal for black holes: an elementary review,” Fortsch. Phys.53(2005) 793, doi:10.1002/prop.200410203, arXiv:hep-th/0502050

    Pith/arXiv arXiv 2005

  40. [40]

    A correspondence principle for black holes and strings,

    G. T. Horowitz and J. Polchinski, “A correspondence principle for black holes and strings,” Phys. Rev. D55 (1997) 6189–6197, doi:10.1103/PhysRevD.55.6189, arXiv:hep-th/9612146

    Pith/arXiv arXiv 1997

  41. [41]

    The correspondence between rotating black holes and fundamental strings,

    N. ˇCeplak, R. Emparan, A. Puhm and M. Tomaˇ sevi´ c, “The correspondence between rotating black holes and fundamental strings,” JHEP11(2023) 226, doi:10.1007/JHEP11(2023)226, arXiv:2307.03573 [hep-th]

    Pith/arXiv arXiv 2023

  42. [42]

    Self-gravitating fundamental strings,

    G. T. Horowitz and J. Polchinski, “Self-gravitating fundamental strings,” Phys. Rev. D57(1998) 2557–2563, doi:10.1103/PhysRevD.57.2557, arXiv:hep-th/9707170

    Pith/arXiv arXiv 1998

  43. [43]

    Self-gravitating fundamental strings and black holes,

    T. Damour and G. Veneziano, “Self-gravitating fundamental strings and black holes,” Nucl. Phys. B568 (2000) 93–119, doi:10.1016/S0550-3213(99)00596-9, arXiv:hep-th/9907030

    Pith/arXiv arXiv 2000

  44. [44]

    How to expose a black hole,

    A. Sen, “How to expose a black hole,” arXiv:2604.03720 [hep-th]

    Pith/arXiv arXiv

  45. [45]

    On the black hole/string transition,

    Y. Chen, J. Maldacena and E. Witten, “On the black hole/string transition,” JHEP01, 103 (2023) doi:10.1007/JHEP01(2023)103 [arXiv:2109.08563 [hep-th]]

    Pith/arXiv arXiv 2023

  46. [46]

    Causality constraints on corrections to the graviton three-point coupling,

    X. O. Camanho, J. D. Edelstein, J. Maldacena and A. Zhiboedov, “Causality constraints on corrections to the graviton three-point coupling,” JHEP02(2016) 020, doi:10.1007/JHEP02(2016)020, arXiv:1407.5597 [hep-th]

    Pith/arXiv arXiv 2016

  47. [47]

    Positivity Bounds and the Massless Spin-2 Pole,

    L. Alberte, C. de Rham, A. Jaitly and A. J. Tolley, “Positivity Bounds and the Massless Spin-2 Pole,” Phys. Rev. D102(2020) 125023, doi:10.1103/PhysRevD.102.125023, arXiv:2007.12667 [hep-th]

    Pith/arXiv arXiv 2020

  48. [48]

    Causality constraints on corrections to Einstein gravity,

    S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, “Causality constraints on corrections to Einstein gravity,” JHEP05(2023) 122, doi:10.1007/JHEP05(2023)122, arXiv:2201.06602 [hep-th]

    Pith/arXiv arXiv 2023

  49. [49]

    Graviton partial waves and causality in higher dimensions,

    S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, “Graviton partial waves and causality in higher dimensions,” Phys. Rev. D108(2023) 026007, doi:10.1103/PhysRevD.108.026007, arXiv:2205.01495 [hep-th]

    Pith/arXiv arXiv 2023

  50. [50]

    Bounding Violations of the Weak Gravity Conjecture,

    J. Henriksson, B. McPeak, F. Russo and A. Vichi, “Bounding Violations of the Weak Gravity Conjecture,” JHEP08(2022) 184, doi:10.1007/JHEP08(2022)184, arXiv:2203.08164 [hep-th]

    Pith/arXiv arXiv 2022

  51. [51]

    Graviton loops and negativity,

    C.-H. Chang and J. Parra-Martinez, “Graviton loops and negativity,” JHEP08(2025) 175, doi:10.1007/JHEP08(2025)175, arXiv:2501.17949 [hep-th]

    Pith/arXiv arXiv 2025

  52. [52]

    Sampling the Graviton Pole and Deprojecting the Swampland,

    G. Peng, L. Rodina, A. Tokareva and Y. Xu, “Sampling the Graviton Pole and Deprojecting the Swampland,” arXiv:2604.15235 [hep-th]

    Pith/arXiv arXiv

  53. [53]

    What is the graviton pole made of?

    K. H¨ aring and A. Zhiboedov, “What is the graviton pole made of?” arXiv:2410.21499 [hep-th]

    Pith/arXiv arXiv

  54. [54]

    Rigorous parametric dispersion representation with three-channel symmetry,

    G. Auberson and N. N. Khuri, “Rigorous parametric dispersion representation with three-channel symmetry,” Phys. Rev. D6(1972) 2953

    1972

  55. [55]

    Physical pion-pion partial-wave equations based on three-channel crossing symmetry,

    G. Mahoux, S. M. Roy and G. Wanders, “Physical pion-pion partial-wave equations based on three-channel crossing symmetry,” Nucl. Phys. B70(1974) 297

    1974

  56. [56]

    Crossing Symmetric Dispersion Relations in QFTs,

    A. Sinha and A. Zahed, “Crossing Symmetric Dispersion Relations in QFTs,” Phys. Rev. Lett.126(2021) 181601, doi:10.1103/PhysRevLett.126.181601, arXiv:2012.04877 [hep-th]

    Pith/arXiv arXiv 2021

  57. [57]

    A Geometric View on Crossing-Symmetric Dispersion Relations,

    J. Elias Miro, A. Guerrieri, M. A. Gumus and A. Zahed, “A Geometric View on Crossing-Symmetric Dispersion Relations,” arXiv:2509.14170 [hep-th]

    arXiv

  58. [58]

    A stringy dispersion relation for field theory,

    F. Bhat, A. P. Saha and A. Sinha, “A stringy dispersion relation for field theory,” Phys. Rev. D113, no.6, 066016 (2026), doi:10.1103/ljzx-q254, arXiv:2506.03862 [hep-th]. 58

    arXiv 2026

  59. [59]

    Field Theory Expansions of String Theory Amplitudes,

    A. P. Saha and A. Sinha, “Field Theory Expansions of String Theory Amplitudes,” Phys. Rev. Lett.132 (2024) 221601, doi:10.1103/PhysRevLett.132.221601, arXiv:2401.05733 [hep-th]

    Pith/arXiv arXiv 2024

  60. [60]

    Analytic Bootstrap of the Veneziano Amplitude,

    S.-L. Wan and S.-Y. Zhou, “Analytic Bootstrap of the Veneziano Amplitude,” arXiv:2605.11084 [hep-th]

    Pith/arXiv arXiv

  61. [61]

    On the number of Regge trajectories for dual amplitudes,

    C. Eckner, F. Figueroa and P. Tourkine, “On the number of Regge trajectories for dual amplitudes,” JHEP 02(2025) 103, doi:10.1007/JHEP02(2025)103, arXiv:2405.21057 [hep-th]

    Pith/arXiv arXiv 2025

  62. [62]

    Regge trajectories for UV completions of graviton scattering from polynomial boundedness,

    C. Eckner, F. Figueroa, S. Metayer and P. Tourkine, “Regge trajectories for UV completions of graviton scattering from polynomial boundedness,” arXiv:2512.17828 [hep-th]

    arXiv

  63. [63]

    The Regge bootstrap, from linear to non-linear trajectories,

    C. Eckner, F. Figueroa and P. Tourkine, “The Regge bootstrap, from linear to non-linear trajectories,” arXiv:2401.08736 [hep-th]

    Pith/arXiv arXiv

  64. [64]

    The Stringy S-matrix Bootstrap: Maximal Spin and Superpolynomial Softness,

    K. H¨ aring and A. Zhiboedov, “The Stringy S-matrix Bootstrap: Maximal Spin and Superpolynomial Softness,” JHEP10(2024) 075, doi:10.1007/JHEP10(2024)075, arXiv:2311.13631 [hep-th]

    Pith/arXiv arXiv 2024

  65. [65]

    Bootstrapping string models with entanglement minimization and Machine-Learning,

    F. Bhat, D. Chowdhury, A. P. Saha and A. Sinha, “Bootstrapping string models with entanglement minimization and Machine-Learning,” arXiv:2409.18259 [hep-th]

    Pith/arXiv arXiv

  66. [66]

    Uniqueness Criteria for the Virasoro–Shapiro Amplitude,

    C. Cheung, A. Hillman and G. N. Remmen, “Uniqueness Criteria for the Virasoro–Shapiro Amplitude,” Phys. Rev. D111(2025) 086034, doi:10.1103/PhysRevD.111.086034, arXiv:2408.03362 [hep-th]

    Pith/arXiv arXiv 2025

  67. [67]

    Low impact parameter physics using the stringy dispersion relation,

    P. Athira, A. P. Saha, S. Saren and A. Sinha, “Low impact parameter physics using the stringy dispersion relation,” in preparation

  68. [68]

    Parallelizing the dual revised simplex method,

    Q. Huangfu and J. A. J. Hall, “Parallelizing the dual revised simplex method,” Math. Program. Comput.10 (2018) 119, doi:10.1007/s12532-017-0130-5

    doi:10.1007/s12532-017-0130-5 2018

  69. [69]

    Lower limit for the energy derivative of the scattering phase shift,

    E. P. Wigner, “Lower limit for the energy derivative of the scattering phase shift,” Phys. Rev.98(1955) 145, doi:10.1103/PhysRev.98.145

    doi:10.1103/physrev.98.145 1955

  70. [70]

    Lifetime matrix in collision theory,

    F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev.118(1960) 349, doi:10.1103/PhysRev.118.349

    doi:10.1103/physrev.118.349 1960

  71. [71]

    Fluctuations of Nuclear Cross Sections in the ‘Continuum’ Region,

    T. Ericson, “Fluctuations of Nuclear Cross Sections in the ‘Continuum’ Region,” Phys. Rev. Lett.5, 430–431 (1960)

    1960

  72. [72]

    A Theory of Fluctuations in Nuclear Cross Sections,

    T. Ericson, “A Theory of Fluctuations in Nuclear Cross Sections,” Annals Phys.23, 390–414 (1963)

    1963

  73. [73]

    Chaotic Scattering of Highly Excited Strings,

    D. J. Gross and V. Rosenhaus, “Chaotic Scattering of Highly Excited Strings,” JHEP05, 048 (2021), doi:10.1007/JHEP05(2021)048, arXiv:2103.15301 [hep-th]

    Pith/arXiv arXiv 2021

  74. [74]

    Chaos in a Many-String Scattering Amplitude,

    V. Rosenhaus, “Chaos in a Many-String Scattering Amplitude,” Phys. Rev. Lett.129, no.3, 031601 (2022), doi:10.1103/PhysRevLett.129.031601, arXiv:2112.10269 [hep-th]

    Pith/arXiv arXiv 2022

  75. [75]

    Measure for Chaotic Scattering Amplitudes,

    M. Bianchi, M. Firrotta, J. Sonnenschein and D. Weissman, “Measure for Chaotic Scattering Amplitudes,” Phys. Rev. Lett.129, no.26, 261601 (2022), doi:10.1103/PhysRevLett.129.261601, arXiv:2207.13112 [hep-th]

    Pith/arXiv arXiv 2022

  76. [76]

    Chaotic and Thermal Aspects in the Highly Excited String S-Matrix,

    D. Das, S. Mandal and A. Sarkar, “Chaotic and Thermal Aspects in the Highly Excited String S-Matrix,” JHEP08, 200 (2024), doi:10.1007/JHEP08(2024)200, arXiv:2312.02127 [hep-th]

    Pith/arXiv arXiv 2024

  77. [77]

    Ericson fluctuations in the gravitational S-Matrix,

    D. Das, A. Sinha, “Ericson fluctuations in the gravitational S-Matrix,” in preparation. 59