// =============================================================================
// The Riesz Sector Hierarchy:
// From Newtonian Gravity to Flat Rotation
// =============================================================================
//
// All algebraic identities machine-checked with Z3 via the Kleis language.
// Companion theory file: theories/pot_riesz_sector_signs.kleis
// Complete source: https://kleis.io
//
// =============================================================================

import "stdlib/prelude.kleis"
import "stdlib/templates/arxiv_paper.kleis"

// =============================================================================
// Physical Constants and Frozen Coefficients
// =============================================================================

// All quantities in dimensionless/normalized units for plotting.
// Radii in kpc, velocities in km/s, accelerations in m/s².
// BTFR: v_flat^4 = G*M*a0, so B = v_flat^2 = sqrt(G*M*a0).

define v_flat_kms = 220.0                          // km/s
define B_MW = v_flat_kms * v_flat_kms              // (km/s)^2 = 48400
define D_kms2_per_kpc = 45.1                       // (km/s)^2 / kpc — illustrative value
define r23_kpc = 9.0                               // kpc
define r34_kpc = B_MW / D_kms2_per_kpc             // ~1073 kpc
define A_MW = B_MW * r23_kpc                       // (km/s)^2 * kpc

// B(M) under BTFR: B ∝ √M. Normalize to MW.
define B_from_logM(logM) = B_MW * sqrt(logM / 11.0)
// More precisely: B(M) = B_MW * (M/M_MW)^{1/2}, with log10(M/M_sun)
// B(10^x) / B(10^11) = 10^{(x-11)/2}

// =============================================================================
// Figure Data: Critical Exponent (Fig 1)
// =============================================================================

define crit_x = linspace(0.1, 10.0, 100)
define crit_eps1 = list_map(lambda x . (x - 1.0) / 1.0, crit_x)
define crit_eps05 = list_map(lambda x . (sqrt(x) - 1.0) / 0.5, crit_x)
define crit_eps02 = list_map(lambda x . (exp(0.2 * ln(x)) - 1.0) / 0.2, crit_x)
define crit_eps01 = list_map(lambda x . (exp(0.1 * ln(x)) - 1.0) / 0.1, crit_x)
define crit_eps005 = list_map(lambda x . (exp(0.05 * ln(x)) - 1.0) / 0.05, crit_x)
define crit_ln = list_map(lambda x . ln(x), crit_x)

define fig_critical = ArxivDiagram("fig:critical",
    "Emergence of the logarithm at the critical Riesz exponent. The curves show $(x^epsilon - 1) slash epsilon$ for $x = r slash r_0$ at several values of $epsilon$, converging toward $ln(x)$ (black) as $epsilon arrow 0$. This is the mechanism by which the $R_3^+$ sector produces a logarithmic potential.",
    export_typst_fragment(
        plot(crit_x, crit_eps1, color = "purple", label = "$epsilon = 1.0$"),
        plot(crit_x, crit_eps05, color = "blue", label = "$epsilon = 0.5$"),
        plot(crit_x, crit_eps02, color = "teal", label = "$epsilon = 0.2$"),
        plot(crit_x, crit_eps01, color = "olive", label = "$epsilon = 0.1$"),
        plot(crit_x, crit_eps005, color = "gray", label = "$epsilon = 0.05$"),
        plot(crit_x, crit_ln, color = "black", label = "$ln(x)$ (limit)", stroke = "2.5pt"),
        title = "$(x^epsilon - 1) slash epsilon arrow ln(x)$ at the Critical Exponent",
        xlabel = "$x = r slash r_0$",
        ylabel = "$(x^epsilon - 1) slash epsilon$",
        legend_position = "left + top",
        width = 10
    )
)

// =============================================================================
// Figure Data: Force Hierarchy (Fig 2) — log-log, three sectors
// =============================================================================

define fh_r = [0.1, 0.3, 1.0, 3.0, 10.0, 30.0, 100.0, 300.0, 1000.0, 3000.0, 10000.0, 30000.0, 100000.0]

define fh_a2 = list_map(lambda r . A_MW / (r * r), fh_r)
define fh_a3 = list_map(lambda r . B_MW / r, fh_r)
define fh_a4 = list_map(lambda r . D_kms2_per_kpc, fh_r)

define vline_yrange = [0.001, 100000.0]
define vline_r23 = [r23_kpc, r23_kpc]
define vline_r34 = [r34_kpc, r34_kpc]

define fig_hierarchy = ArxivDiagram("fig:hierarchy",
    "Three-sector force hierarchy for a Milky-Way-scale galaxy ($M = 10^(11) M_sun$, $v_(\"flat\") = 220$ km/s) with illustrative $D approx 45$ (km/s)${}^2$\/kpc. Log-log axes reveal the characteristic slopes: $-2$ (Newtonian), $-1$ (flat rotation), $0$ (constant acceleration). Vertical lines mark transition radii $r_(2 3) approx 9$ kpc, $r_(3 4) = B slash D approx 1$ Mpc (illustrative). The $R_5^+$ sector is not plotted; its force law ($tilde r ln r$) is derived in Section 6.",
    export_typst_fragment(
        plot(fh_r, fh_a2, color = "blue", label = "$|a_2| = A slash r^2$ (Newton)", stroke = "2pt"),
        plot(fh_r, fh_a3, color = "green", label = "$|a_3| = B slash r$ (flat rotation)", stroke = "2pt"),
        plot(fh_r, fh_a4, color = "red", label = "$a_4 = D$ (constant)", stroke = "2pt"),
        plot(vline_r23, vline_yrange, color = "gray", label = "$r_(2 3)$", stroke = "0.7pt"),
        plot(vline_r34, vline_yrange, color = "gray", label = "$r_(3 4)$", stroke = "0.7pt"),
        title = "Three-Sector Force Hierarchy",
        xlabel = "$r$ [kpc]",
        ylabel = "$|a|$ [(km/s)²/kpc]",
        xscale = "log",
        yscale = "log",
        legend_position = "right + top",
        width = 12
    )
)

// =============================================================================
// Figure Data: Rotation Curve in v²_c (Fig 3) — three sectors
// =============================================================================
// v²_c(r) = A/r + B − D·r

define rc_r = linspace(1.0, 2000.0, 100)
define rc_v2_R2 = list_map(lambda r . A_MW / r, rc_r)
define rc_v2_R3 = list_map(lambda r . B_MW, rc_r)
define rc_v2_R4 = list_map(lambda r . 0.0 - D_kms2_per_kpc * r, rc_r)
define rc_v2_full = list_map(lambda r .
    A_MW / r + B_MW - D_kms2_per_kpc * r, rc_r)

define r_cancel_kpc = sqrt(r23_kpc * r34_kpc)
define rc_vline_yrange = [-80000.0, 80000.0]
define rc_vline_cancel = [r_cancel_kpc, r_cancel_kpc]
define rc_vline_r34 = [r34_kpc, r34_kpc]
define rc_zero_line_r = [1.0, 2000.0]
define rc_zero_line_y = [0.0, 0.0]

define fig_rotation = ArxivDiagram("fig:rotation",
    "Rotation curve decomposition in $v_c^2$ for a Milky-Way-scale galaxy with illustrative $D$. Blue: Keplerian contribution $A slash r$. Green: flat $R_3^+$ floor $B = v_(\"flat\")^2$. Red: $R_4^+$ decline $-D r$. Black: full $v_c^2$ (three-sector). Vertical markers: $r_(\"cancel\") approx 100$ kpc ($v_c^2 = B$ exactly), $r_(3 4) = B slash D approx 1$ Mpc (illustrative; $v_c^2 approx 0$ when $A slash r << B$). The exact zero $r_(\"zero\") = (B + sqrt(B^2 + 4 A D)) slash (2 D)$ exceeds $r_(3 4)$ by $approx 0.9%$.",
    export_typst_fragment(
        plot(rc_r, rc_v2_R2, color = "blue", label = "$A slash r$ (Keplerian)"),
        plot(rc_r, rc_v2_R3, color = "green", label = "$B$ (flat, $R_3^+$)"),
        plot(rc_r, rc_v2_R4, color = "red", label = "$-D r$ ($R_4^+$ decline)"),
        plot(rc_r, rc_v2_full, color = "black", label = "Full $v_c^2$", stroke = "2.5pt"),
        plot(rc_zero_line_r, rc_zero_line_y, color = "gray", stroke = "0.5pt"),
        plot(rc_vline_cancel, rc_vline_yrange, color = "purple", label = "$r_(\"cancel\")$", stroke = "0.7pt"),
        plot(rc_vline_r34, rc_vline_yrange, color = "maroon", label = "$r_(3 4)$ ($v_c^2 approx 0$)", stroke = "0.7pt"),
        title = "Three-Sector $v_c^2$ Decomposition",
        xlabel = "$r$ [kpc]",
        ylabel = "$v_c^2$ [(km/s)²]",
        legend_position = "right + top",
        width = 12
    )
)

// =============================================================================
// Figure Data: Mass-Dependent Outer Decline (Fig 4)
// =============================================================================
// v²/B = r₂₃/r + 1 − r/r₃₄ (three-sector approximation)
// Under BTFR: B ∝ √M, so r₃₄ = B/D ∝ √M

define md_r = linspace(10.0, 1000.0, 100)

// For each mass, compute v_c/√B = √(max(v²/B, 0))
// M = 10^8: B = B_MW * 10^{(8-11)/2} = B_MW * 10^{-1.5} ≈ B_MW * 0.0316
// M = 10^9: B = B_MW * 10^{-1.0} = B_MW * 0.1
// M = 10^{10}: B = B_MW * 10^{-0.5} ≈ B_MW * 0.3162
// M = 10^{11}: B = B_MW * 1.0
// M = 10^{12}: B = B_MW * 10^{0.5} ≈ B_MW * 3.162

define md_B8 = B_MW * 0.03162
define md_B9 = B_MW * 0.1
define md_B10 = B_MW * 0.3162
define md_B11 = B_MW
define md_B12 = B_MW * 3.162

define md_A8 = md_B8 * md_B8 / D_kms2_per_kpc / 0.03162
define md_A9 = md_B9 * md_B9 / D_kms2_per_kpc / 0.1
define md_A10 = md_B10 * md_B10 / D_kms2_per_kpc / 0.3162
define md_A11 = A_MW
define md_A12 = md_B12 * md_B12 / D_kms2_per_kpc / 3.162

// Normalized v/v_flat for each mass (simplified: v²/B = 1 − r/r₃₄ in flat regime)
define md_ratio(B, r) = 1.0 - D_kms2_per_kpc * r / B

define md_v11 = list_map(lambda r . sqrt(if md_ratio(md_B11, r) > 0.0 then md_ratio(md_B11, r) else 0.0), md_r)
define md_v12 = list_map(lambda r . sqrt(if md_ratio(md_B12, r) > 0.0 then md_ratio(md_B12, r) else 0.0), md_r)
define md_v10 = list_map(lambda r . sqrt(if md_ratio(md_B10, r) > 0.0 then md_ratio(md_B10, r) else 0.0), md_r)
define md_v9 = list_map(lambda r . sqrt(if md_ratio(md_B9, r) > 0.0 then md_ratio(md_B9, r) else 0.0), md_r)
define md_v8 = list_map(lambda r . sqrt(if md_ratio(md_B8, r) > 0.0 then md_ratio(md_B8, r) else 0.0), md_r)

define fig_mass = ArxivDiagram("fig:mass",
    "Mass-dependent outer decline: $v_c slash sqrt(B)$ versus radius for five galaxy masses under BTFR scaling ($B prop sqrt(M)$) and frozen illustrative $D$. Lower-mass galaxies encounter $R_4^+$ suppression at smaller radii, with $a_4 slash a_3 prop M^(-1\/2)$ [RS-14l2]. The curves reach $v_c^2 approx 0$ near $r_(3 4) = B slash D$; beyond this radius no circular orbit exists in the three-sector approximation.",
    export_typst_fragment(
        plot(md_r, md_v12, color = "blue", label = "$10^(12) M_dot.o$"),
        plot(md_r, md_v11, color = "green", label = "$10^(11) M_dot.o$"),
        plot(md_r, md_v10, color = "orange", label = "$10^(10) M_dot.o$"),
        plot(md_r, md_v9, color = "red", label = "$10^9 M_dot.o$"),
        plot(md_r, md_v8, color = "purple", label = "$10^8 M_dot.o$"),
        title = "Mass-Dependent Outer Decline ($a_4 slash a_3 prop M^(-1 slash 2)$)",
        xlabel = "$r$ [kpc]",
        ylabel = "$v_c slash sqrt(B)$",
        legend_position = "right + top",
        width = 12
    )
)

// =============================================================================
// Figure Data: ESD Correction (Fig 5)
// =============================================================================
// correction = (exact/asymptotic) - 1
// exact = (1 - √(1 - x²)) / (x²/2) where x = R/r_max
// correction_pct = (exact - 1) * 100

define esd_x = linspace(0.01, 0.95, 80)
define esd_corr = list_map(lambda x .
    ((1.0 - sqrt(1.0 - x * x)) / (x * x / 2.0) - 1.0) * 100.0, esd_x)

define fig_esd = ArxivDiagram("fig:esd",
    "Fractional correction of the exact finite-cutoff $R_4^+$ excess surface density relative to the asymptotic limit. At $R slash r_(\"max\") = 0.3$ the correction is $< 3%$ [RS-14h2]. The correction grows rapidly for $R slash r_(\"max\") > 0.5$.",
    export_typst_fragment(
        plot(esd_x, esd_corr, color = "blue", stroke = "2pt"),
        title = "Finite-Cutoff ESD Correction",
        xlabel = "$R slash r_(\"max\")$",
        ylabel = "Correction [%]",
        width = 10
    )
)

// =============================================================================
// Figure Data: q₄ Sensitivity (Fig 6)
// =============================================================================
// Left panel: V_c for different q₄ values
// V²_c,lens = B(1 − q₄·r/r₃₄) → V/V_flat = √(1 − q₄·r/r₃₄)

define q4_r = linspace(10.0, 500.0, 80)
define q4_v_00 = list_map(lambda r . v_flat_kms, q4_r)
define q4_v_025 = list_map(lambda r .
    v_flat_kms * sqrt(if 1.0 - 0.25 * r / r34_kpc > 0.0 then 1.0 - 0.25 * r / r34_kpc else 0.0), q4_r)
define q4_v_050 = list_map(lambda r .
    v_flat_kms * sqrt(if 1.0 - 0.50 * r / r34_kpc > 0.0 then 1.0 - 0.50 * r / r34_kpc else 0.0), q4_r)
define q4_v_075 = list_map(lambda r .
    v_flat_kms * sqrt(if 1.0 - 0.75 * r / r34_kpc > 0.0 then 1.0 - 0.75 * r / r34_kpc else 0.0), q4_r)
define q4_v_100 = list_map(lambda r .
    v_flat_kms * sqrt(if 1.0 - 1.00 * r / r34_kpc > 0.0 then 1.0 - 1.00 * r / r34_kpc else 0.0), q4_r)

define fig_q4_lensing = ArxivDiagram("fig:q4-lensing",
    "Lensing $V_c$ profiles for different values of the sector-specific gravitational-slip factor $q_4 = (1 + eta_4) slash 2$, using illustrative $D$. The $q_4 = 0$ curve (sector invisible to lensing) is flat; increasing $q_4$ produces progressively steeper decline. For a Milky-Way-scale galaxy with illustrative $r_(3 4) approx 1$ Mpc, the difference between $q_4 = 0$ and $q_4 = 1$ reaches $approx 30$ km/s at 300 kpc.",
    export_typst_fragment(
        plot(q4_r, q4_v_00, color = "green", label = "$q_4 = 0$ ($eta_4 = -1$)", stroke = "2pt"),
        plot(q4_r, q4_v_025, color = "blue", label = "$q_4 = 0.25$ ($eta_4 = -0.5$)"),
        plot(q4_r, q4_v_050, color = "orange", label = "$q_4 = 0.50$ ($eta_4 = 0$)"),
        plot(q4_r, q4_v_075, color = "red", label = "$q_4 = 0.75$ ($eta_4 = 0.5$)"),
        plot(q4_r, q4_v_100, color = "purple", label = "$q_4 = 1.0$ ($eta_4 = 1$, GR)", stroke = "2pt"),
        title = "Lensing $V_c$ for Different $q_4$",
        xlabel = "$r$ [kpc]",
        ylabel = "$V_(c,\"lens\")$ [km/s]",
        legend_position = "right + top",
        width = 12
    )
)

// Right panel: fractional tightening of exact vs linearized bound
// (q_lin − q_exact) / q_exact × 100% = σ/(2√B − σ) × 100%
// For σ = 15 km/s, √B = 220 km/s: σ/(2·220 − 15) = 15/425 ≈ 3.5%
// This is constant in r! The fractional tightening is r-independent.
// So instead plot % tightening vs σ_V.

define q4t_sig = linspace(1.0, 100.0, 80)
define q4t_pct = list_map(lambda sig .
    sig / (2.0 * v_flat_kms - sig) * 100.0, q4t_sig)

define fig_q4_tightening = ArxivDiagram("fig:q4-tightening",
    "Fractional overestimate of the linearized $q_4$ bound relative to the exact bound: $(q_(4,\"lin\") - q_(4,\"exact\")) slash q_(4,\"exact\") = sigma slash (2 sqrt(B) - sigma)$. This ratio is independent of the observation radius $r_(\"obs\")$ and depends only on $sigma slash sqrt(B)$. For typical weak-lensing uncertainties ($sigma_V approx 15$ km/s with $sqrt(B) = 220$ km/s), the linearized bound overestimates by $approx 3.5%$.",
    export_typst_fragment(
        plot(q4t_sig, q4t_pct, color = "blue", stroke = "2pt"),
        title = "Exact vs Linearized $q_4$ Bound",
        xlabel = "$sigma_V$ [km/s]",
        ylabel = "$(q_(4,\"lin\") - q_(4,\"exact\")) slash q_(4,\"exact\")$ [%]",
        width = 10
    )
)

// =============================================================================
// Paper Metadata
// =============================================================================

define paper_title = "The Riesz Sector Hierarchy: Critical Exponents and Long-Range Gravitational Structure"

define paper_authors = [
    Author("Engin Atik", "1")
]

define paper_affiliations = [
    Affiliation(1, "Kleis Research", "https://kleis.io")
]

define paper_abstract = "The Euclidean Riesz kernel in three dimensions, $I_alpha (r) prop r^(alpha - 3)$, produces a hierarchy of gravitational potentials as $alpha$ ranges over consecutive integers. At $alpha = 2$ the kernel is Newtonian ($1\/r$). At the first critical exponent $alpha = 3$, the normalization has a pole; the finite part is a logarithm, $ln(r\/r_0)$. At $alpha = 4$ the potential is linear ($r$). At the second critical exponent $alpha = 5$, the Riesz recursion produces a static shadow $r^2 ln(r\/r_0) + beta r^2$ rather than a pure $r^2$, with a reference-scale covariance between $r_0$ and $beta$. The force laws from the first three sectors are $r^(-2)$, $r^(-1)$, and $r^0$; the $r^(-1)$ force gives constant circular velocity ($v_c^2 = B$) without inserting a logarithmic potential phenomenologically; the coupling $B$ itself remains empirical.

Assembling the three established sectors into a kernel,
$
Phi(r) = -A\/r + B ln(r\/r_0) - D r,
$
the force hierarchy covers Newtonian gravity ($r^(-2)$), flat galactic rotation ($r^(-1)$), and a constant-acceleration regime ($r^0$). The Newtonian coefficient $A = G M$ is standard. The galactic coefficient $B$ is determined by flat rotation curves and the baryonic Tully-Fisher relation. The constant-acceleration coefficient $D$ is not determined within this paper; numerical illustrations use an illustrative value giving $r_(3 4) = B\/D approx 1$ Mpc. The $R_5^+$ sector's physical coupling, sign, and long-range role are developed in a separate section.

Each regime has a characteristic transition scale: $r_(2 3) = A \/ B$ (Newtonian to galactic), $r_(3 4) = B \/ D$ (galactic to intergalactic). These scales are source-dependent through the coherence functional: compact sources (stars, planets) suppress higher sectors, while extended coherent configurations (galaxies) admit them. Under the assumption of universal $D$ and galaxy-sourced $R_4^+$, the model predicts mass-dependent outer-rotation decline ($a_4 \/ a_3 prop M^(-1\/2)$ at fixed radius). Conditional lensing projections (dependent on the unknown gravitational slip $eta_alpha$) and finite-cutoff corrections are derived as algebraic consequences of the phantom-density structure.

The algebraic skeleton --- covering transition identities, Poisson consistency, phantom densities, surface-density projections, and observational constraint formulae --- is machine-verified with Z3 via Kleis."

define paper_keywords = "Riesz distributions, gravitational potential hierarchy, flat rotation curves, critical exponent, dark matter, modified gravity, baryonic Tully-Fisher relation, gravitational lensing, Projected Ontology Theory"


// =============================================================================
// Section 1: Introduction (written last, placeholder for now)
// =============================================================================

define sec_intro = ArxivSection("Introduction",
"Two regimes of gravitational phenomenology beyond Newtonian inverse-square gravity are observationally established at galactic and intergalactic scales: approximately flat rotation curves in disk galaxies beyond the visible disk, and a characteristic acceleration scale $a_0 approx 1.2 times 10^(-10)$ m/s${}^2$ separating Newtonian from non-Newtonian dynamics. Large-scale accelerated expansion, described by a cosmological constant $Lambda$, provides a third regime whose theoretical connection to the first two is not established. Each regime is conventionally addressed with independent theoretical machinery: Newtonian gravity, dark matter halos, MOND, and vacuum energy.

Logarithmic gravitational potentials and their connection to flat rotation have been studied in the fractional-Laplacian gravity literature (Giusti 2020, Varieschi 2018, Calcagni and Varieschi 2022). This paper contributes a distinct structural perspective: consecutive members of the causal Riesz operator family, indexed by integer $alpha$, produce a static-sector hierarchy whose critical members yield logarithmic potentials and whose even/odd pattern, transition identities, and projection algebra provide a unified organization of the first two regimes and their mathematical long-range continuation. The Riesz kernel
$
I_alpha (r) prop r^(alpha - 3)
$
evaluated at $alpha = 2, 3, 4$ gives potentials $r^(-1)$, $ln(r\/r_0)$, and $r$, with corresponding forces $r^(-2)$, $r^(-1)$, and $r^0$. The $alpha = 3$ member is exceptional: at this critical exponent, the normalization $C(alpha)$ has a pole and the finite part yields the logarithm rather than a constant potential. The resulting force $prop r^(-1)$ immediately gives constant circular velocity, making flat galactic rotation a structural consequence of the critical Riesz exponent rather than a fitted phenomenological term. At the second critical exponent $alpha = 5$, the same mechanism produces a static shadow $r^2 ln(r\/r_0) + beta r^2$ rather than a pure $r^2$; the mathematical structure and its physical interpretation are developed in Section 6.

The three-sector kernel
$
Phi(r) = -A\/r + B ln(r\/r_0) - D r
$
is governed by three coefficients: $A = G M$ from Newtonian physics, $B = v_(\"flat\")^2$ from flat rotation curves, and $D$ (the constant-acceleration coefficient) which is not determined within the present analysis. The transition scales are algebraically determined: $r_(2 3) = A\/B$, $r_(3 4) = B\/D$. Numerical illustrations throughout use $D approx 45$ (km/s)${}^2$\/kpc, giving $r_(3 4) approx 1$ Mpc for a Milky-Way-scale galaxy; this value is illustrative and its observational determination is an open problem.

A crucial structural feature is that the sector coefficients need not be universal constants. While $D$ may be universal, $B$ is manifestly source-dependent: BTFR scaling gives $B prop sqrt(M)$ for galaxies, but solar-system constraints require extreme suppression of $B$ for compact objects. This is formalized through a coherence functional $cal(C)[rho]$ that determines which sectors are admissible for a given projected source. Sector boundaries belong to projected systems, not to absolute distance.

The paper is organized as follows. Section 2 derives the static Riesz sector hierarchy, with particular attention to the critical logarithmic sector $R_3^+$. Section 3 assembles the gravitational kernel and derives transition scales. Section 4 develops the projection and coherence framework. Section 5 presents galactic phenomenology. Section 6 derives the $R_5^+$ static shadow at the second critical exponent. Section 7 develops weak-lensing phenomenology. Section 8 presents mass-dependent predictions. Section 9 summarizes observational constraints and falsification conditions. Section 10 discusses limitations and open problems.

All algebraic identities are machine-verified with Z3 through the Kleis language. Appendix A provides a mapping between paper equations and verified propositions.")


// =============================================================================
// Section 2: Static Riesz Sector Hierarchy
// =============================================================================

define sec_riesz_hierarchy = ArxivSection("Static Riesz Sector Hierarchy",
"=== The Riesz kernel in three dimensions

The Euclidean Riesz kernel in $n$ dimensions is defined by analytic continuation of the distribution
$
I_alpha (bold(x)) = (Gamma((n - alpha)\/2)) / (2^alpha pi^(n\/2) Gamma(alpha\/2)) dot r^(alpha - n),
$
where $r = |bold(x)|$ and $alpha$ is a complex parameter. Note that $Gamma((n - alpha)\/2)$ appears in the *numerator*: it is this factor that produces poles at $alpha = n, n + 2, dots$ For $n = 3$ and spherical symmetry, the radial dependence is
$
I_alpha (r) = C(alpha) dot r^(alpha - 3), quad C(alpha) = (Gamma((3 - alpha)\/2)) / (2^alpha pi^(3\/2) Gamma(alpha\/2)).
$
Evaluating at consecutive even integers:
$
alpha = 2: quad & I_2(r) prop r^(-1), \
alpha = 4: quad & I_4(r) prop r.
$
The cases $alpha = 3$ and $alpha = 5$ require separate treatment because $C(alpha)$ diverges (the numerator $Gamma((3 - alpha)\/2)$ has poles at $alpha = 3, 5, 7, dots$).

=== The first critical exponent $alpha = 3$

At $alpha = 3$, the numerator $Gamma((3 - alpha)\/2) = Gamma(0)$ has a simple pole, so $C(alpha)$ diverges while $r^(alpha - 3) = r^0 = 1$. To extract the finite radial dependence, set $alpha = 3 + epsilon$ and expand.

*Step 1 (power).* $r^epsilon = e^(epsilon ln(r\/r_0)) = 1 + epsilon ln(r\/r_0) + O(epsilon^2)$, where $r_0$ is an arbitrary reference scale.

*Step 2 (normalization).* Using $Gamma(-epsilon\/2) = -2\/epsilon - gamma_E + O(epsilon)$ and $2^3 pi^(3\/2) Gamma(3\/2) = 4 pi^2$:
$
C(3 + epsilon) = (Gamma(-epsilon\/2)) / (2^(3 + epsilon) pi^(3\/2) Gamma((3 + epsilon)\/2)) = (-2\/epsilon + O(1)) / (4 pi^2 + O(epsilon)) = (-1\/(2 pi^2)) dot 1\/epsilon + O(1).
$
Therefore $c_(-1) = -1\/(2 pi^2)$.

*Step 3 (product).* Multiplying:
$
C(3 + epsilon) r^epsilon = c_(-1)\/epsilon + c_(-1) ln(r\/r_0) + c_0 + O(epsilon).
$

*Step 4 (finite part).* Subtracting the $r$-independent divergent and constant terms:
$
op(\"FP\")_(alpha = 3) I_alpha (r) = c_(-1) ln(r\/r_0) = -1\/(2 pi^2) ln(r\/r_0).
$
This is the Euclidean finite-part coefficient, consistent with $cal(F)^(-1)(|k|^(-3)) = -1\/(2 pi^2) ln(r\/r_0)$ under the Gel'fand--Shilov convention. The Lorentzian Riesz distribution $R_3^+$ has a distinct normalization: its static shadow is $R_3^+|_(omega = 0) = -1\/(2 pi) ln r$ (BGP convention). The ratio between Euclidean and Lorentzian normalizations is an additional factor that is traced through in the companion paper (Atik 2026a). In the present paper, all normalizations are absorbed into the physical coupling constant $B$:
$
Phi_3(r) = B ln(r\/r_0).
$
The reference scale $r_0$ is not a new physical parameter: changing $r_0 arrow.r r_0'$ adds $ln(r_0\/r_0')$ to the potential, a constant whose gradient vanishes [RS-4d].

=== Lorentzian-to-static bridge

The preceding derivation used the Euclidean Riesz kernel directly. For completeness, we record the Lorentzian route. The causal Riesz distribution $R_alpha^+$ on $(3+1)$-dimensional Minkowski spacetime (BGP convention) has Fourier representation $hat(R)_alpha^+(k) prop (k^2 + i epsilon)^(-alpha\/2)$ with $k^2 = omega^2 - bold(k)^2$. The static shadow is the $omega = 0$ restriction:
$
hat(R)_alpha^+|_(omega = 0) prop |bold(k)|^(-alpha).
$
Inverse spatial Fourier transform then yields the Euclidean Riesz potential. For the two cases of interest:
$
cal(F)^(-1)(|bold(k)|^(-2)) prop 1\/r quad (alpha = 2), quad quad quad cal(F)^(-1)(|bold(k)|^(-3)) = -1\/(2 pi^2) ln(r\/r_0) quad (alpha = 3).
$
The first is the standard Newtonian Green's function. The second is the Gel'fand--Shilov finite part at the critical exponent, with the BGP convention contributing an additional $pi$ to the Lorentzian normalization (see Atik 2026a for the explicit trace). In either case, the Lorentzian route and the direct Euclidean route yield the same static potential up to a normalization constant that is absorbed into $A$ or $B$.

=== Force and circular velocity

Differentiating the four potentials:

#figure(
  table(
    columns: (auto, auto, auto, auto),
    align: (center, center, center, center),
    [$alpha$], [Potential $Phi_alpha (r)$], [Force $a_alpha = -d Phi_alpha \/ d r$], [Structural role],
    [2], [$-A\/r$], [$-A\/r^2$], [Newtonian],
    [3], [$B ln(r\/r_0)$], [$-B\/r$], [First critical static shadow],
    [4], [$-D r$], [$+D$], [Linear potential],
    [5], [$kappa[r^2\/6 thin ln(r\/r_0) - 5r^2\/36]$], [$-kappa r[1\/3 thin ln(r\/r_0) - 1\/9]$], [Second critical static shadow],
  ),
  caption: [Riesz sector hierarchy. The potential at $alpha = 3$ is the finite part at the first critical exponent. At $alpha = 5$, a second pole produces $r^2 ln(r\/r_0)$ rather than pure $r^2$; the entries show the particular solution for fixed $r_0$. The reference-scale covariance $(r_0, beta)$ and its invariant are developed in Section 6. The cosmological interpretation of the $alpha = 5$ sector is not assumed here.]
) <tab:hierarchy>

The critical sector gives flat rotation structurally. For circular motion at radius $r$:
$
v_c^2 / r = |a_3| = B / r quad ==> quad v_c^2 = B = \"const\".
$
This is a consequence of the force law, not a fitted parameter [RS-4d3].

=== Spherical Laplacian and effective density

The spherical Laplacian $nabla^2 f = f'' + (2\/r) f'$ applied to each sector yields the Poisson-equation source that a Newtonian observer would infer [RS-4d4, RS-12a--c]:
$
nabla^2 (B ln r) &= B\/r^2 quad &&==> quad rho_(3,\"eff\") = B\/(4 pi G r^2), \
nabla^2 (-D r) &= -2D\/r quad &&==> quad rho_(4,\"eff\") = -D\/(2 pi G r).
$
The $R_3^+$ effective density is an isothermal sphere --- the standard dark-matter halo profile. The $R_4^+$ phantom density is negative and $prop 1\/r$, producing a net repulsive effect that grows with radius.

=== Growth-rate ordering

The logarithm is the unique monotone function at the boundary between decaying and growing power behaviors. The growth rates satisfy the strict ordering [RS-4d6]
$
d / (d r) (1\/r) < d / (d r) (ln r) < d / (d r) (r) quad \"for\" r > 1,
$
i.e., $-1\/r^2 < 1\/r < 1$. The logarithm grows slower than any positive power of $r$ (hence $R_4^+$ eventually dominates $R_3^+$ [RS-2a]) and faster than any negative power (hence $R_3^+$ eventually dominates $R_2^+$).

")


// =============================================================================
// Section 3: Gravitational Kernel and Transition Scales
// =============================================================================

define sec_kernel = ArxivSection("Three-Sector Gravitational Kernel",
"=== The gravitational kernel

Assembling the first three Riesz sectors with coupling constants $A$, $B$, $D > 0$:
$
Phi(r) = -A\/r + B ln(r\/r_0) - D r,
$
with corresponding total acceleration
$
a(r) = -A\/r^2 - B\/r + D.
$
The Riesz hierarchy determines the *radial dependence* of each sector; the *amplitude and sign* of each physical coupling are not determined by the hierarchy and must be set by phenomenological matching. The sign convention adopted here ($A, B, D > 0$ with the displayed signs) makes $R_2^+$ and $R_3^+$ attractive and $R_4^+$ repulsive. The Euclidean finite-part coefficient $c_(-1) = -1\/(2 pi^2)$ (Section 2) is absorbed into $B$; the physical sign of $B > 0$ comes from phenomenology (attractive $R_3^+$ floor), not from the Riesz mathematics. The $R_5^+$ sector is developed separately in Section 6.

The circular velocity is:
$
v_c^2(r) = A\/r + B - D r.
$
In dimensionless form [RS-14e]:
$
v_c^2(r) = B (r_(2 3)\/r + 1 - r\/r_(3 4)).
$

=== Transition scales

Adjacent sectors have equal force magnitude at the transition radii [RS-9b--c]:
$
r_(2 3) = A\/B, quad r_(3 4) = B\/D.
$
These are exact algebraic identities, not fitted parameters. The ratio $r_(3 4)\/r_(2 3) = B^2\/(A D)$ [RS-14i] is independent of $R_5^+$. A transition between $R_4^+$ and $R_5^+$ exists in principle but its location depends on the physical coupling of the second critical sector (Section 6).

=== Coefficient provenance

#figure(
  table(
    columns: (auto, auto, auto),
    align: (center, left, left),
    [Coefficient], [Physical meaning], [Determination],
    [$A = G M$], [Newtonian coupling], [Laboratory/solar-system $G$],
    [$B = v_(\"flat\")^2$], [Galactic rotation coupling], [Flat rotation curves + BTFR],
    [$D$], [Constant-acceleration coupling], [Not determined here; illustrative value used (Section 10)],
  ),
  caption: [Coefficient provenance. Two of three coefficients ($A$, $B$) are independently anchored. $D$ is not determined within this paper; all numerical illustrations use $D approx 45$ (km/s)${}^2$\/kpc.]
) <tab:coefficients>

=== The value-cancellation radius

The Keplerian ($R_2^+$) and constant-acceleration ($R_4^+$) contributions to $v_c^2$ have opposite signs. Their *values* cancel exactly at [RS-14g]
$
r_(\"cancel\") = sqrt(A\/D) = sqrt(r_(2 3) r_(3 4)),
$
the geometric mean of the first two transition scales, giving $v_c^2(r_(\"cancel\")) = B$ exactly [RS-14g3]. Note that this is a value cancellation, not a slope cancellation: $d v_c^2 \/ d r = -A\/r^2 - D < 0$ for all $r > 0$ (both corrections steepen the decline), so the curve is monotonically decreasing and has no flatness extremum at $r_(\"cancel\")$.")


// =============================================================================
// Section 4: Projection, Coherence, and Sector Truncation
// =============================================================================

define sec_coherence = ArxivSection("Projection, Coherence, and Sector Truncation",
"=== The coherence problem

The Newtonian coefficient $A = G M$ is universal: the same $G$ applies to all gravitating systems. The question is whether $B$ and $D$ are similarly universal.

A naive application of BTFR scaling ($B prop sqrt(M)$) to the Sun gives $B_dot.o approx 1.5 times 10^5$ m${}^2$ s${}^{-2}$. At 1 AU, the resulting anomalous acceleration would be
$
a_3\/a_N = B_dot.o r\/(G M_dot.o) approx 2 times 10^(-4),
$
a 0.02% correction to solar gravity. This is ruled out by planetary ephemerides, which constrain deviations from inverse-square gravity at the $10^(-6)$ level.

Therefore [RS-10]:
$
B[rho] != b_0 sqrt(M[rho]) quad \"universally\".
$
BTFR is an emergent relation within the galactic class, not a universal source-charge law.

=== The coherence functional

The proper formulation replaces universal $B$ with a source-dependent coupling:
$
B[rho] = B_(\"gal\")(M) dot cal(C)[rho],
$
where $cal(C)[rho]$ is a coherence functional satisfying
$
cal(C)[rho_(\"gal\")] approx 1, quad cal(C)[rho_(\"compact\")] << 1.
$
This makes the transition scale source-dependent [RS-10c]:
$
r_(2 3)[rho] = (G M[rho]) / (B_(\"gal\")(M) dot cal(C)[rho]).
$
As $cal(C)[rho] arrow 0$, the transition radius $r_(2 3) arrow infinity$: the $R_3^+$ boundary recedes continuously. Sector truncation is parameterized by coherence suppression; deriving $cal(C)[rho]$ from the underlying projected structure is an open problem (Section 10). A further open question is compositionality: if the Riesz operator is linear while $B[rho]$ is a nonlinear configuration functional, the theory must establish whether $Phi[rho_1 + rho_2] = Phi[rho_1] + Phi[rho_2]$ and how $cal(C)$ behaves under source superposition.

=== Solar-system constraint

For the compact-source limit, the ephemeris bound requires [RS-10b]
$
B_dot.o < epsilon_(\"SS\") dot (G M_dot.o) / r,
$
where $epsilon_(\"SS\")$ is the observational upper limit on anomalous $1\/r$ acceleration. The naive BTFR extrapolation exceeds this bound, requiring
$
cal(C)_dot.o lt.tilde 6 times 10^(-3)
$
--- at least two orders of magnitude suppression relative to the galactic branch.

=== Galactic consistency window

Analogously, the $R_4^+$ sector must not intrude too strongly within the observed flat-rotation regime. The contamination ratio [RS-11a] is
$
a_4\/a_3 = r\/r_(3 4).
$
For a Milky-Way-scale galaxy at 50 kpc, $a_4\/a_3 approx 5%$. The three-sector galactic analysis assumes that the physical $R_5^+$ coupling $lambda_5$ is sufficiently small that $|a_5| = |lambda_5| r |1\/3 thin ln(r\/r_0) - 1\/9| << B\/r$ over the radii considered. This assumption must be checked once the $R_5^+$ coupling is derived. The domain of validity for the flat-rotation approximation is [RS-11c]
$
r_(2 3) lt.tilde r << r_(3 4),
$
numerically $approx 9$ kpc $lt.tilde r << r_(3 4)$ (illustratively $approx 1$ Mpc).

=== Source-dependent versus universal sectors

The hierarchy naturally separates into source-dependent sectors ($R_2^+$, $R_3^+$: coefficients $A$, $B$ depend on $M$ and $cal(C)[rho]$) and a potentially universal sector ($R_4^+$: coefficient $D$ may be independent of the source). The transition scales $r_(2 3)$ and $r_(3 4)$ are source-dependent. The $R_5^+$ sector's coupling and its transition to $R_4^+$ depend on the physical interpretation of the second critical static shadow (Section 6).")


// =============================================================================
// Section 5: Galactic Dynamics
// =============================================================================

define sec_galactic = ArxivSection("Galactic Dynamics",
"=== Rotation curves with full corrections

The three-sector circular velocity is
$
v_c^2(r) = A\/r + B - D r.
$
Under the assumption that the $R_5^+$ physical coupling is sufficiently suppressed at galactic radii (Section 4), the three-sector circular velocity is an adequate approximation.
In the flat-rotation regime ($r_(2 3) << r << r_(3 4)$), the Keplerian term is a perturbation:
$
v_c^2(r) approx B (1 + r_(2 3)\/r - r\/r_(3 4)) quad \"[RS-14f]\".
$
The leading correction to flat rotation is the $R_4^+$ decline $-D r$, giving
$
v_c^2 approx B(1 - r\/r_(3 4)).
$
This predicts a gradual velocity decline in the outer regions. The exact zero of $v_c^2$ occurs at $r_(\"zero\") = (B + sqrt(B^2 + 4 A D))\/(2 D) approx r_(3 4)$ when $A\/r_(3 4) << B$ [RS-11f]. For the illustrative value $r_(3 4) approx 1$ Mpc, the slope is
$
d v_c \/ d r approx -v_(\"flat\") \/ (2 r_(3 4)) approx -0.1 \"km/s/kpc\",
$
likely undetectable in most current surveys but potentially accessible through satellite kinematics and weak lensing at $100$--$500$ kpc.

=== Baryonic Tully-Fisher relation

Since $v_c^2 = B$ in the flat regime and $B = v_(\"flat\")^2$, the BTFR $M_(\"bar\") prop v_(\"flat\")^4$ constrains the relationship between $B$ and source mass:
$
B prop sqrt(M_(\"bar\")),
$
within the galactic class. This is not assumed as a universal law but derived as a property of the galactic coherence branch.")


// =============================================================================
// Section 6: Second Critical Sector
// =============================================================================

define sec_cosmological = ArxivSection("The Second Critical Sector: $R_5^+$ Static Shadow",
"=== The $alpha = 5$ critical exponent

The Euclidean Riesz normalization $C(alpha) = Gamma((3-alpha)\/2) \/ (2^alpha pi^(3\/2) Gamma(alpha\/2))$ has poles at all odd $alpha >= 3$ in three spatial dimensions: the numerator $Gamma((3-alpha)\/2)$ diverges at $alpha = 3, 5, 7, dots$ The $alpha = 3$ pole produces the logarithmic potential $ln(r\/r_0)$ (Section 2). The $alpha = 5$ pole requires the same finite-part treatment.

The Riesz recursion $square.stroked R_(alpha+2)^+ = R_alpha^+$ gives $square.stroked R_5^+ = R_3^+$. In the static limit ($omega = 0$), this becomes the Poisson equation
$
nabla^2 phi_5 = K ln(r\/r_0), quad K = 1\/(2 pi^2),
$
where the right-hand side is the Euclidean static shadow of $R_3^+$ with its Gel'fand--Shilov normalization (the physical coupling is absorbed later).

=== Particular solution

With spherical symmetry, $(r^2 phi_5')' = K r^2 ln(r\/r_0)$. Integrating twice:
$
phi_5^((\"part\"))(r) = K [r^2\/6 thin ln(r\/r_0) - 5r^2\/36].
$
*Verification:* $nabla^2(r^2 ln r) = 6 ln r + 5$ and $nabla^2(r^2) = 6$, so $nabla^2[r^2\/6 thin ln r - 5r^2\/36] = ln r$. $checkmark$

Note that the $-5r^2\/36$ term is *not* a freely adjustable piece: it is required by the particular solution to cancel the $+5$ generated by $nabla^2(r^2 ln r)$. Nor is $r^2$ a homogeneous solution of the spherical Laplacian ($nabla^2 r^2 = 6 eq.not 0$).

=== Fixed-source solution and boundary conditions

For fixed $r_0$, the PDE $nabla^2 phi_5 = K ln(r\/r_0)$ has a unique particular solution (up to the Poisson homogeneous freedom $C_1\/r + C_2$):
$
phi_5(r) = K [r^2\/6 thin ln(r\/r_0) - 5r^2\/36] + C_1\/r + C_2.
$
The coefficient $beta = -5\/36$ is *forced* by the PDE: substituting $K[r^2\/6 thin ln(r\/r_0) + beta r^2]$ into the Laplacian gives $K[ln(r\/r_0) + 5\/6 + 6 beta]$, which equals $K ln(r\/r_0)$ only when $5\/6 + 6 beta = 0$. The $C_1\/r$ term is a Newtonian potential (absorbed into $A$); the constant $C_2$ has no dynamical effect. These are the standard boundary-condition freedoms.

=== Reference-scale covariance

Two distinct operations involve the reference scale $r_0$:

*Change of finite-part scale.* Replacing $r_0 arrow.r r_0'$ in the PDE changes the source:
$
nabla^2 phi_5' = K ln(r\/r_0').
$
This is a *different* PDE with a *different* canonical particular solution: $beta' = -5\/36$ again (the canonical value is always $-5\/36$). The physical potential $phi_5'$ differs from $phi_5$ by the Poisson solution of the constant source $K ln(r_0\/r_0')$, which is $prop r^2$.

*Same-potential reparameterization.* One may instead represent the *same* physical potential using the scale $r_0'$ by writing
$
phi_5(r) = K [r^2\/6 thin ln(r\/r_0') + beta' r^2] + C_1\/r + C_2,
$
where
$
beta' = beta + 1\/6 thin ln(r_0'\/r_0).
$
This compensating shift of $beta$ absorbs the $r^2$ term generated by rewriting $ln(r\/r_0) = ln(r\/r_0') + ln(r_0'\/r_0)$. Crucially, the Laplacian of the reparameterized form still equals $K ln(r\/r_0)$ (the original source), not $K ln(r\/r_0')$: the extra $6 beta' - 6 beta = ln(r_0'\/r_0)$ term under the Laplacian exactly compensates the rewriting of the logarithm.

*Invariant.* Under the same-potential reparameterization, neither $r_0$ nor $beta$ is individually invariant; the invariant combination is
$
1\/6 thin ln(r\/r_0) + beta.
$
This is *not* a homogeneous-solution ambiguity (the Poisson homogeneous solutions are $1\/r$ and constants, not $r^2$). It is a covariance of the finite-part representation.

*Sanity check:* In the canonical representation, $beta = -5\/36$. The same-potential reparameterization to $r_0'$ gives $beta' = -5\/36 + 1\/6 thin ln(r_0'\/r_0)$. The invariant $1\/6 thin ln(r\/r_0) + beta = 1\/6 thin ln(r\/r_0') + beta'$ is unchanged.

=== Force law

The force from the fixed-source solution (dropping the Newtonian piece $C_1\/r$) is
$
a_5(r) = -phi_5'(r) = -K r [1\/3 thin ln(r\/r_0) + 2 beta + 1\/6].
$
With $beta = -5\/36$: $2 beta + 1\/6 = -5\/18 + 3\/18 = -1\/9$, recovering $a_5 = -K r[1\/3 thin ln(r\/r_0) - 1\/9]$. The force grows as $|a_5| tilde r ln r$ asymptotically, not as a pure linear force $prop r$. The $(r_0, beta)$ reparameterization changes the sub-leading coefficient but not the asymptotic $r ln r$ behavior.

=== Comparison with earlier pure-$r^2$ approximation

An earlier version of this analysis treated the $R_5^+$ static shadow as a pure $r^2$, leading to the identification $F_5 = Lambda c^2\/6$ and $H_Lambda^2 = Lambda c^2\/3$ (de Sitter). The exact recursion calculation shows this is not the $R_5^+$ static shadow: the $r^2 ln(r\/r_0)$ term is structurally forced by the pole at $alpha = 5$. The effective density from $r^2 ln r$ is not uniform, and the cosmological ODE does not reduce to exponential growth. The physical coupling sign and the resulting dynamics are the subject of separate analysis.

=== Even/odd pattern

The calculation reveals a structural pattern in the static shadows:
$
&R_2^+ arrow.r r^(-1), quad R_4^+ arrow.r r, quad R_6^+ arrow.r r^3, quad dots quad \"(even: clean powers)\"\
&R_3^+ arrow.r ln r, quad R_5^+ arrow.r r^2\/6 thin ln(r\/r_0) - 5r^2\/36, quad dots quad \"(odd: logarithmic)\".
$
Odd-order sectors, which sit at poles of $Gamma((3-alpha)\/2)$, acquire logarithmic factors through the finite-part mechanism. Even-order sectors produce ordinary powers. This alternation is a consequence of the Gamma function's pole structure.

")


// =============================================================================
// Section 7: Effective Densities and Weak Lensing
// =============================================================================

define sec_lensing = ArxivSection("Effective Densities and Weak-Lensing Projections",
"=== Phantom densities and enclosed masses

A Newtonian observer interpreting each sector through the Poisson equation $nabla^2 Phi = 4 pi G rho$ infers phantom densities (Section 2). The enclosed phantom masses are [RS-12d--g]:
$
M_(3,\"ph\")(r) = B r \/ G, quad M_(4,\"ph\")(r) = -D r^2 \/ G.
$
These are consistent with the orbital accelerations: $G M_(3,\"ph\")\/r^2 = B\/r$ (flat rotation), $G M_(4,\"ph\")\/r^2 = -D$ (constant repulsive) [RS-12h].

=== Surface density and excess surface density

For gravitational lensing, the relevant quantity is the excess surface density (ESD):
$
Delta Sigma(R) = overline(Sigma)(< R) - Sigma(R),
$
where $R$ is the projected radius.

*$R_3^+$ lensing.* The isothermal phantom density $rho_3 prop 1\/r^2$ projects to $Sigma_3 prop 1\/R$, yielding [RS-13a, RS-13b]
$
Delta Sigma_3(R) = B \/ (4 G R).
$
The resulting lensing circular velocity is $V_(c,\"lens\")^2 = B$ --- flat and $R$-independent [RS-13c], conditional on the gravitational slip $eta_3 = 1$.

*$R_4^+$ lensing.* The phantom density $rho_4 prop -1\/r$ projects through Abel integration to give, in the limit $r_(\"max\") >> R$ [RS-13d]:
$
Delta Sigma_(4,infinity) = -D \/ (2 pi G),
$
which is both $R$-independent and cutoff-independent [RS-13e, RS-13f]. This is remarkable: the $R_4^+$ sector contributes a constant ESD in the asymptotic limit, unlike the $1\/R$ profile of $R_3^+$.

At finite cutoff, the exact ESD is [RS-13d2]:
$
Delta Sigma_(4,\"exact\")(R, r_(\"max\")) = alpha (r_(\"max\")^2 - r_(\"max\") sqrt(r_(\"max\")^2 - R^2)) / R^2,
$
where $alpha = -D\/(pi G)$. This exceeds the asymptotic limit in magnitude, with a correction that is always positive and scales as $O(R^2\/r_(\"max\")^2)$ [RS-14h]. At $R\/r_(\"max\") = 0.3$ the correction is $< 3%$ [RS-14h2].

=== Two-layer lensing structure

The lensing predictions have a two-layer structure:

*Layer 1* (Z3-verified): mathematical projection of phantom densities through the Abel transform, yielding $Sigma(R)$ and $Delta Sigma(R)$ as algebraic identities.

*Layer 2* (open): mapping from POT potentials to photon deflection requires derivation of the gravitational slip $eta_alpha = Psi_alpha \/ Phi_alpha$ for each sector. In GR, $eta = 1$ and lensing mass equals dynamical mass. POT must independently derive its $eta$; until then, all lensing comparisons are conditional on $eta = 1$.")


// =============================================================================
// Section 8: Gravitational Slip and Mass-Dependent Predictions
// =============================================================================

define sec_predictions = ArxivSection("Sector-Specific Gravitational Slip and Mass-Dependent Predictions",
"=== The $q_alpha$ parameterization

In a metric theory, the lensing potential is $Phi_(\"lens\") = (Phi + Psi)\/2$, where $Psi$ is the spatial-curvature potential. The lensing efficiency of each sector is parameterized by [RS-14a--d]
$
q_alpha = (1 + eta_alpha) / 2,
$
where $eta_alpha = Psi_alpha \/ Phi_alpha$. The normalization is:
- $q_alpha = 1$: GR-like lensing ($eta = 1$, no gravitational slip),
- $q_alpha = 0$: sector invisible to lensing ($eta = -1$),
- $q_alpha = 1\/2$: no spatial curvature from this sector ($eta = 0$).

The lensing circular velocity with sector-specific factors is
$
V_(c,\"lens\")^2(r) = q_2 A\/r + q_3 B - q_4 D r.
$
The $R_5^+$ contribution to lensing is omitted under the same suppression assumption (Section 4). For the Newtonian sector, $q_2 = 1$ is required by the weak-field limit. For $R_3^+$, the companion lensing analysis (Atik 2026d) derives $gamma_(\"PPN\") = 1$ from the scalar structure of the POT coherence function (equal time and space metric perturbations). In the present notation this corresponds to $Psi_3 = Phi_3$, i.e. $eta_3 = 1$ and $q_3 = 1$: the $R_3^+$ sector has GR-like lensing as a structural consequence of scalar coherence, not as an assumption. The scalar-coherence argument has been verified only for $R_3^+$; for $R_4^+$, $q_4$ is genuinely open.

In the galactic flat-rotation regime with $q_2 = q_3 = 1$ [RS-14d]:
$
V_(c,\"lens\")^2 approx B(1 - q_4 r\/r_(3 4)).
$

=== Exact observational bound on $q_4$

Given a lensing measurement $V_(\"obs\")$ at radius $r_(\"obs\")$ with uncertainty $sigma$, the exact constraint on $q_4$ is [RS-14k4a]:
$
q_4 < r_(3 4) / r_(\"obs\") (2 sigma / sqrt(B) - sigma^2 / B).
$
The linearized approximation $q_4 < 2 sigma r_(3 4) \/ (sqrt(B) r_(\"obs\"))$ [RS-14k] overestimates the allowed region; the exact bound is strictly tighter [RS-14k4a] by a correction $sigma^2 r_(3 4) \/ (B r_(\"obs\"))$ [RS-14k4b].

When the exact bound is saturated, $V_(c,\"lens\") = sqrt(B) - sigma$, i.e., the velocity deficit equals $sigma$ exactly [RS-14k4d].

=== Mass-dependent contamination

Under BTFR scaling $B prop sqrt(M)$ and universal $D$, the $R_4^+$ contamination ratio at fixed radius is [RS-14l2]
$
a_4\/a_3 = D r \/ B(M) prop M^(-1\/2).
$
A galaxy 100 times less massive experiences 10 times stronger contamination at the same physical radius. This is probably the sharpest differential prediction of the frozen-$D$ model.

The transition scale also inherits mass dependence [RS-14l3]:
$
r_(3 4)(M) = B(M)\/D prop sqrt(M),
$
and the cancellation radius scales identically [RS-14l5]: $r_(\"cancel\") prop sqrt(M)$.

This gives an immediate observational test: mass-binned weak-lensing stacks should show mass-dependent decline in outer circular velocity, with lower-mass galaxies declining more rapidly [RS-14l4].")


// =============================================================================
// Section 9: Observational Constraints and Falsification
// =============================================================================

define sec_observations = ArxivSection("Observational Constraints and Falsification Conditions",
"#figure(
  table(
    columns: (auto, auto, auto, auto, auto),
    align: (left, left, left, left, left),
    [Scale], [Observable], [POT prediction], [Status], [Caveat],
    [Solar system], [$epsilon_(\"SS\") = a_3\/a_N$], [$< 10^(-6)$ (requires $cal(C)_dot.o << 1$)], [Constraint satisfied], [Imposed coherence suppression; not independent confirmation],
    [Galaxy ($< 50$ kpc)], [Flat $v_c$], [$v_c^2 = B$, BTFR], [Consistent], [Standard flat-rotation regime],
    [Galaxy ($50$--$300$ kpc)], [Outer $v_c$ decline], [$v_c^2 approx B(1 - r\/r_(3 4))$], [Constraining], [Mass-dependent; stacks may wash signal],
    [Weak lensing ($< 1$ Mpc)], [$Delta Sigma(R)$ profile], [Flat $+$ constant, mass-dependent], [Open], [Requires $eta$ derivation],
    [Long-range], [$R_5^+$ force law], [$|a_5| tilde r ln r$ (Section 6)], [Derived], [Physical coupling and sign open],
  ),
  caption: [Observational status. Two predictions are consistent with existing data (solar-system bounds, flat rotation). One is constraining but not yet decisive (outer decline). One requires further theoretical development (gravitational slip for lensing). The $R_5^+$ force law is derived but its physical coupling is open.]
) <tab:observations>

=== Falsification conditions

The model makes several falsifiable predictions:

1. *Solar-system coherence.* If a persistent $1\/r$ anomalous acceleration exceeding $approx 10^(-6) a_N$ is detected at $approx 1$ AU, either the coherence functional $cal(C)[rho_(\"compact\")]$ is larger than required or the model is wrong.

2. *Mass-dependent outer decline.* Under the universal-$D$, galaxy-sourced, GR-like-lensing branch ($q_4 = 1$), the three-sector formula $V^2\/B = 1 + r_(2 3)\/r - r\/r_(3 4)$ predicts at $r = 300$ kpc (with illustrative $r_(2 3) approx 9$ kpc, $r_(3 4) approx 1$ Mpc): $V^2\/B approx 0.73$, hence $V\/V_(\"flat\") approx 0.85$, roughly a 15% velocity decline. This is qualitatively pressured by the nearly flat lensing-inferred profiles of Mistele et al.; a quantitative significance assessment requires their covariance matrix and a POT-specific lensing map ($eta_alpha$). If mass-binned stacks at $200$--$300$ kpc show no mass-dependent signal ($a_4\/a_3 prop M^(-1\/2)$), then either $D$ is not universal, the coherence functional modifies the prediction, or $R_4^+$ is not galaxy-sourced.

3. *Gravitational slip.* If combined dynamical and lensing observations yield $eta_3 != 1$ or $eta_4$ values inconsistent with the observed lensing profile, the sector-metric structure requires modification.

4. *Second critical sector.* The $R_5^+$ static shadow is $r^2 ln(r\/r_0) + beta r^2$ (Section 6), not a pure $r^2$. The physical coupling, sign, and long-range dynamics of this sector are the subject of separate analysis. The relationship to $Lambda$ and large-scale expansion is an open problem.

5. *Cancellation-radius prediction.* For a galaxy with known $A = G M$ and measured $D$, the radius at which $v_c^2 = B$ exactly is $r_(\"cancel\") = sqrt(A\/D)$. This is the radius where the rotation curve crosses the $R_3^+$ floor, and is testable from rotation-curve shape.")


// =============================================================================
// Section 10: Discussion and Limitations
// =============================================================================

define sec_discussion = ArxivSection("Discussion",
"=== Relationship to existing modified-gravity theories

*Fractional-Laplacian and Riesz-potential gravity.* Critical or fractional constructions producing logarithmic gravitational potentials and flat-rotation phenomenology predate the present work. Giusti (2020) derives MOND-like behavior from a fractional Poisson operator near its critical order $s = 3\/2$, explicitly connecting the resulting $ln r$ potential with the Tully-Fisher relation. Varieschi (2018) discusses gravitational Riesz-type potentials in a fractional-dimension framework, noting their relevance to flat rotation curves. Calcagni and Varieschi (2022) obtain logarithmic large-radius potentials in a multi-fractional theory and apply them to SPARC galaxy rotation data. The novelty claimed here is therefore *not* the isolated logarithmic potential or its $1\/r$ force law, which are already present in the fractional-gravity literature.

What the present paper contributes is the organization of *consecutive* causal Riesz sectors ($R_2^+ arrow.r R_3^+ arrow.r R_4^+ arrow.r R_5^+$) into a single static hierarchy, including: the even/odd critical pattern (even sectors yield clean powers, odd sectors acquire logarithmic factors); the multi-sector kernel with algebraically determined transition scales $r_(2 3) = A\/B$, $r_(3 4) = B\/D$; the second critical sector at $alpha = 5$ with its $(r_0, beta)$ covariance; source-dependent sector admissibility through the coherence framework; and a machine-verified projection algebra (phantom densities, ESD, lensing identities). None of these structural features appear in the fractional-Laplacian treatments, which typically study a single fractional order rather than the full integer-indexed hierarchy.

*Conformal gravity.* The three-sector kernel $Phi = -A\/r + B ln r - D r$ has structural similarities to the Mannheim conformal gravity potential, which includes $-A\/r$ and $gamma r$ terms. The present construction differs in that the logarithmic sector $R_3^+$ (which conformal gravity skips) is the source of flat rotation rather than a linear potential, and the sector hierarchy has a mathematical origin in the Riesz operator family. The hierarchy's deeper roots in projective characteristic geometry and the Bernstein--Sato structure of the complexified quadric are established in (Atik 2026e).

*MOND.* The constant-acceleration $R_4^+$ sector resembles the MOND transition, and the critical acceleration $a_0$ may be related to $D$, but the present framework does not derive this identification.

*Lorentzian existence versus physical admissibility.* All sectors $R_alpha^+$ belong to the causal Lorentzian Riesz family (BGP convention), with $hat(R)_alpha^+|_(omega = 0) prop |bold(k)|^(-alpha)$ and the recursion $square.stroked R_(alpha+2)^+ = R_alpha^+$ connecting them. The companion paper (Atik 2026a) establishes for $R_2^+$ and $R_3^+$ a detailed physical admissibility package: retarded support, wavefront-set analysis, characteristic variety, source interpretation, and coupling normalization. For $R_4^+$ and $R_5^+$, the Lorentzian *existence* is immediate from the Riesz family, and the static shadows are derived here, but the corresponding admissibility analysis — retarded support verification, microlocal structure, and physical source coupling — has not been performed. Lorentzian existence is necessary but not sufficient for physical-sector admissibility. The analytic structure of the full causal Riesz family — including its exceptional sectors, Bernstein--Sato structure, tube-domain representation, and relation to Petrovsky cycles — is developed separately (Atik 2026e).

=== Possible large-scale interpretation

The motivation for extending the hierarchy beyond $R_3^+$ is not purely formal. Since the adjacent $R_2^+$ and $R_3^+$ sectors generate the Newtonian and logarithmic radial structures relevant to two distinct gravitational regimes, it is natural to ask whether subsequent sectors contribute to still larger-scale phenomena. The present analysis does not identify $R_4^+$ or $R_5^+$ with cosmological expansion. Indeed, the second critical static shadow $R_5^+ tilde r^2 ln r$ rules out the simplest identification with a pure quadratic de Sitter potential. More generally, however, the existence of a sector in the underlying Riesz hierarchy need not imply that it appears with the same weight in every physical or observational projection. Determining which sectors are excited by a source, and which sectors are accessible to a particular observable, requires the still-unknown coherence and projection functionals. Thus the possible relation between higher Riesz sectors and large-scale cosmological observations remains a question posed by the hierarchy rather than a conclusion of this paper.

=== No-escape constraint and sector transitions

The companion multi-center analysis (Atik 2026b) establishes that the isolated $R_3^+$ logarithmic interaction cannot remain globally valid to arbitrarily large separation: $U prop ln r arrow +infinity$ implies no finite-energy asymptotic escape, which conflicts with the observed existence of gravitationally unbound systems. The present hierarchy supplies a possible mathematical mechanism for terminating $R_3^+$ dominance: the subsequent $R_4^+$ sector, whose repulsive acceleration $+D$ competes with the attractive $R_3^+$ force $-B\/r$ and drives $v_c^2 arrow 0$ near $r_(3 4) = B\/D$. Whether this sector transition provides the physically required global completion — rather than merely a mathematical cancellation in $v_c^2$ — depends on the source dependence and domain of validity of $D$, which remain open. The structural suggestion is that the no-escape theorem and the sector hierarchy may be related: $R_3^+$ cannot persist globally, and the hierarchy provides the next sector that can terminate it.

=== Open problems

Three theoretical problems remain open:

1. *Gravitational slip.* Deriving $eta_alpha = Psi_alpha \/ Phi_alpha$ from the projected metric structure. This determines whether the Layer 1 lensing projections (Section 7) translate directly into physical lensing predictions.

2. *Coherence functional and compositionality.* Deriving $cal(C)[rho]$ from first principles. The solar-system constraint ($cal(C)_dot.o << 1$) and the galactic condition ($cal(C)_(\"gal\") approx 1$) are boundary conditions on this unknown functional. Its form would determine whether the sector hierarchy is a property of the gravitational field itself or of the projected observation. A related open question is compositionality: the BTFR scaling $B prop sqrt(M)$ is nonlinear in mass and therefore not additive under source superposition. The theory must establish whether the Riesz operator's linearity survives when $B[rho]$ depends nonlinearly on the source configuration.

3. *Source dependence and determination of $D$.* The present analysis treats $D$ as a universal coefficient and uses an illustrative numerical value. Weak-lensing data (Mistele et al. 2024, finding flat lensing profiles to $approx 1$ Mpc) may constrain $D$ but also may suggest that $D$ is not universally galaxy-sourced. Determining whether $D$ depends on source class, projection geometry, or cosmological boundary conditions is necessary for the mass-dependent predictions of Section 8.

=== What has been established

The first three static shadows and their force laws are mathematically established within the construction. The Riesz hierarchy determines *radial dependence*; physical *amplitude and sign* are set by phenomenological matching. Within the present hierarchy construction, the $R_3^+$ sector supplies the radial functional form for flat rotation ($v_c^2 = B$); determination of the coupling $B$ and BTFR normalization lies outside the sector hierarchy itself and is addressed in the companion POT treatment (Atik 2026c). The physical role and source dependence of $R_4^+$ (constant acceleration) remain conditional predictions of the universal-$D$ branch. The $alpha = 5$ sector produces a static shadow $r^2 ln(r\/r_0) - 5 r^2\/6$ with a reference-scale covariance (Section 6); its physical coupling is an open problem. The model generates conditional predictions (mass-dependent outer decline under universal $D$; lensing projections under assumed $eta_alpha$) and has a machine-verified algebraic skeleton.

What has not been established is: the physical coupling signs and amplitudes from first principles; the coherence functional $cal(C)[rho]$ and its compositionality; the gravitational slip $eta_alpha$; the source dependence of $D$; or whether this hierarchy is the unique gravitational kernel. These distinguish the mathematical sector construction from a physical theory of gravity.

=== The critical-exponent mechanism beyond gravity

The pole-times-tangent mechanism producing the logarithmic static shadow at $alpha = 3$ is not unique to the spatial Riesz reduction. In companion work on one-loop amplitudes [Atik2026g], critical Riesz homogeneity ($alpha_(\"tot\") = d$) similarly converts the first-order tangent of a complex power into logarithmic kinematic dependence:

$ Delta^(-epsilon) = 1 - epsilon ln Delta + O(epsilon^2). $

The projections and physical interpretations are different --- $Q_(\"static\")$ produces $ln(r\/r_0)$ while $Q_(\"obs\")$ produces $ln(Delta\/mu^2)$ --- but the underlying analytic resonance mechanism is the same. This suggests that the critical-exponent structure is a property of complex-power homogeneity itself, not a peculiarity of three-dimensional Fourier inversion. No gravitational claims of the present paper are affected.")


// =============================================================================
// Section 11: Conclusions
// =============================================================================

define sec_conclusions = ArxivSection("Conclusions",
"The Euclidean Riesz kernel hierarchy $I_alpha (r) prop r^(alpha - 3)$, evaluated in three spatial dimensions, produces the potential sequence $1\/r arrow.r ln(r\/r_0) arrow.r r$ at $alpha = 2, 3, 4$ with force laws $r^(-2) arrow.r r^(-1) arrow.r r^0$. The logarithmic member arises at the first critical exponent $alpha = 3$ as a finite-part extraction, not as a phenomenological insertion. At the second critical exponent $alpha = 5$, the same finite-part mechanism produces a static shadow $r^2 ln(r\/r_0) + beta r^2$, with a reference-scale covariance $beta' = beta + 1\/6 ln(r_0'\/r_0)$.

The three-sector kernel $Phi(r) = -A\/r + B ln(r\/r_0) - D r$ is the principal result:

- The $R_2^+$ sector reproduces Newtonian gravity with the standard $G$.
- The $R_3^+$ sector supplies the radial functional form for flat circular velocity ($v_c^2 = B$) and the isothermal effective density ($rho prop r^(-2)$). The coupling $B$ is empirical within this construction; the Riesz sector hierarchy alone does not determine the BTFR normalization or zero point.
- The $R_4^+$ sector introduces a constant acceleration; its physical role and source dependence are predictions to be tested, not observationally established facts.

The $R_5^+$ second critical sector has force $|a_5| tilde r ln r$ asymptotically. The physical coupling sign and the resulting long-range dynamics are open questions requiring separate analysis. The pair $(r_0, beta)$ has a reparameterization covariance that is structurally distinct from the Poisson homogeneous freedom $(C_1\/r + C_2)$; the invariant content of the $alpha = 5$ static shadow is the $r^2 ln r$ term.

Two of the first three coefficients ($A$, $B$) are independently anchored. Transition scales are algebraically determined. Under the assumption of universal $D$, the model predicts mass-dependent outer-rotation decline ($a_4\/a_3 prop M^(-1\/2)$). Lensing projections are conditional on the unknown gravitational slip $eta_alpha$; until $Phi + Psi$ is derived from POT, the ESD calculations are effective-GR phantom mappings, not the theory's photon-deflection law.

The central unresolved questions are: the physical coupling and dynamics of the $R_5^+$ second critical sector; the coherence functional $cal(C)[rho]$ (why different source classes admit different sectors); the gravitational slip $eta_alpha$ (whether lensing mass equals dynamical mass in each sector); and whether $D$ is cosmological or source-dependent.")


// =============================================================================
// Appendix A: Machine-Verified Identities
// =============================================================================

define app_verification = ArxivAppendix("A", "Machine-Verified Algebraic Identities",
"All algebraic identities in this paper were verified with Z3 through the Kleis language (https://kleis.io). The companion theory file `pot_riesz_sector_signs.kleis` contains 80 verified examples.

#figure(
  table(
    columns: (auto, auto, auto),
    align: (left, left, left),
    [Paper result], [Kleis proposition], [Type],
    [Reference-scale independence], [RS-4d], [Identity],
    [Flat rotation $v_c^2 = B$], [RS-4d3], [Identity],
    [$nabla^2(ln r) = 1\/r^2$], [RS-4d4], [Laplacian],
    [Force equality at $r_(2 3)$], [RS-9b], [Identity],
    [Force equality at $r_(3 4)$], [RS-9c], [Identity],
    [Solar-system $B_dot.o$ bound], [RS-10b], [Inequality],
    [Coherence suppression], [RS-10c], [Limit],
    [Contamination $a_4\/a_3 = r\/r_(3 4)$], [RS-11a], [Identity],
    [Rotation decline], [RS-11e], [Identity],
    [Isothermal density], [RS-12a], [Laplacian],
    [Phantom mass consistency], [RS-12h], [Identity],
    [$Delta Sigma_3 = B\/(4 G R)$], [RS-13b], [Projection],
    [$R_4^+$ ESD cancellation], [RS-13d], [Projection],
    [Exact finite-cutoff ESD], [RS-13d2], [Inequality],
    [$q_4 = 0$ removes $R_4^+$ lensing], [RS-14a], [Identity],
    [$V^2$ in dimensionless form], [RS-14e], [Identity],
    [Cancellation radius], [RS-14g], [Identity],
    [$r_(\"cancel\") = sqrt(r_(2 3) r_(3 4))$], [RS-14g2], [Identity],
    [Exact $q_4$ bound], [RS-14k4a], [Inequality],
    [Exact bound $<$ linearized], [RS-14k4a], [Inequality],
    [$a_4\/a_3 prop M^(-1\/2)$], [RS-14l2], [Scaling],
    [$r_(3 4) prop sqrt(M)$], [RS-14l3], [Scaling],
  ),
  caption: [Mapping between paper results and Z3-verified propositions. Z3 verifies the algebraic structure; physical assumptions (coherence, gravitational slip, coefficient values) are not claimed as machine-verified.]
) <tab:verification>

The verification establishes internal algebraic consistency of the sector hierarchy. It does not verify the physical assumptions (choice of sign convention, universality of $D$, coherence functional form) or observational inferences.")


// =============================================================================
// References
// =============================================================================

define ref_atik_pot = ArxivReference("Atik2026a", "Atik, E. (2026). Adjacent Causal Riesz Sectors and Galactic Gravity. Preprint, kleis.io/papers.")

define ref_atik_interaction = ArxivReference("Atik2026b", "Atik, E. (2026). Multi-Center Dynamics of Adjacent Causal Riesz Sectors. Preprint, kleis.io/papers.")

define ref_atik_flat = ArxivReference("Atik2026c", "Atik, E. (2026). Flat Galactic Rotation Curves from Projected Ontology. Preprint, kleis.io/papers.")

define ref_atik_lensing = ArxivReference("Atik2026d", "Atik, E. (2026). POT vs GR: Gravitational Lensing Predictions. Preprint, kleis.io/papers.")

define ref_atik_analytic = ArxivReference("Atik2026e", "Atik, E. (2026). The Analytic Geometry of the Causal Riesz Hierarchy: Bernstein--Sato Singularities, Tube Boundary Values, and Petrovsky Cycles. Preprint, kleis.io/papers.")

define ref_mistele = ArxivReference("Mistele2024", "Mistele, T., McGaugh, S., & Lelli, F. (2024). Indefinitely flat circular velocities and the baryonic Tully-Fisher relation from weak lensing. Astrophys. J. Lett., 969, L3.")

define ref_mcgaugh_btfr = ArxivReference("McGaugh2000", "McGaugh, S. S., Schombert, J. M., Bothun, G. D., & de Blok, W. J. G. (2000). The baryonic Tully-Fisher relation. Astrophys. J., 533, L99.")

define ref_mannheim = ArxivReference("Mannheim2012", "Mannheim, P. D. & O'Brien, J. G. (2012). Fitting galactic rotation curves with conformal gravity and a global quadratic potential. Phys. Rev. D, 85, 124020.")

define ref_milgrom = ArxivReference("Milgrom1983", "Milgrom, M. (1983). A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J., 270, 365.")

define ref_giusti = ArxivReference("Giusti2020", "Giusti, A. (2020). MOND-like fractional Laplacian theory. Phys. Rev. D, 101, 124029. arXiv:2002.07133.")

define ref_varieschi = ArxivReference("Varieschi2018", "Varieschi, G. U. (2018). Newtonian fractional-dimension gravity and MOND. Found. Phys., 48, 1526--1548. arXiv:1805.05950.")

define ref_calcagni_varieschi = ArxivReference("CalcagniVarieschi2022", "Calcagni, G. & Varieschi, G. U. (2022). Gravitational potential and galaxy rotation curves in multi-fractional spacetimes. J. High Energy Phys., 2022, 24. arXiv:2106.15430.")

define ref_oneloop = ArxivReference("Atik2026g", "Atik, E. (2026). Riesz Homogeneity, Logarithmic Resonance, and the K--Q Structure of One-Loop Amplitudes. Preprint, kleis.io/papers.")


// =============================================================================
// Paper Assembly
// =============================================================================

define riesz_paper = arxiv_paper(
    paper_title,
    paper_authors,
    paper_affiliations,
    paper_abstract,
    paper_keywords,
    [
        sec_intro,
        sec_riesz_hierarchy,
        fig_critical,
        fig_hierarchy,
        sec_kernel,
        sec_coherence,
        sec_galactic,
        fig_rotation,
        sec_cosmological,
        sec_lensing,
        fig_esd,
        sec_predictions,
        fig_mass,
        fig_q4_lensing,
        fig_q4_tightening,
        sec_observations,
        sec_discussion,
        sec_conclusions,
        app_verification,
        ref_atik_pot,
        ref_atik_interaction,
        ref_mistele,
        ref_mcgaugh_btfr,
        ref_mannheim,
        ref_milgrom,
        ref_atik_flat,
        ref_atik_lensing,
        ref_atik_analytic,
        ref_giusti,
        ref_varieschi,
        ref_calcagni_varieschi,
        ref_oneloop
    ]
)


// =============================================================================
// Compilation
// =============================================================================

example "compile" {
    let typst_output = compile_arxiv_paper(riesz_paper) in
    out(typst_raw(typst_output))
}