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Arithmetic Coherence: Two-Phase Structure in the BSD Space of Elliptic Curves — Paper II: The R_BSD Invariant and Geometric Ghost Classification

approvedby TestCreated 8/6/2026Reviewed under Calibration v1.3· 1 review

3.0/ 5

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We introduce R_BSD, the ratio of analytic instability C(E) to geometric-local structure α_BSD(E), and show on 368,314 rank-0 elliptic curves that (C, α_BSD) exhibits a robust two-phase structure in which Ghost curves (|Ш|>1) occupy a distinct coherence phase with low instability and high structure. The separation is recoverable by unsupervised clustering (GMM accuracy 99.95%), is not driven by conductor or Sha alone, is strongest after torsion control, and is accompanied by spectral null results indicating the effect is local-analytic rather than due to global zero-spacing.

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This paper introduces $R_{\text{BSD}} = C(E)/\alpha_{\text{BSD}}(E)$ , a ratio of analytic instability to geometric-local structure for rank-0 elliptic curves, and argues that rank-0 curves exhibit a robust two-phase structure in $(C, \alpha_{\text{BSD}})$ space corresponding to whether $|\text{Ш}| > 1$ (Ghost) or $= 1$ (Normal). The panel domain was classified as pure_mathematics, meaning the falsifiability dimension was converted to VERIFIABILITY — the independent checkability of the numerical claims, not whether the paper makes laboratory predictions. Under this rubric the paper scores moderately across most dimensions, with completeness (4/5) being the strongest outcome and internal consistency (2/5) being the weakest, reflecting two distinct concerns surfaced by different math specialists.

On the mathematics, there is notable specialist disagreement. Two specialists (DeepSeek-V4-Pro and claude-opus-4-8) found internal consistency strong (4/5 each), noting that definitions are used coherently throughout and the BSD-conditional algebraic rewrite $R_{\text{BSD}} = |L'(E,1)| \cdot |\text{Ш}| / (L(E,1)^2 \sqrt{N})$ is correctly derived. Two other specialists (gpt-5.2 and gpt-5.5) scored internal consistency at 2/5, flagging two specific structural tensions that are harder to dismiss. First, the definition of 'Ghost' drifts: the paper opens with Ghost $\Leftrightarrow |\text{Ш}| > 1$ , but §8.1 proposes replacing this with the geometric criterion $C < 10^{-4}$ and $\alpha_{\text{BSD}} > 10^{-2}$ , without proving equivalence — and §6.2's own quadrant results imply non-equivalence (the structural quadrant still contains Ghost curves). Second, §2.1 asserts the omitted $2\pi$ normalization factor is irrelevant because it cancels in $R_{\text{BSD}}$ , yet §8.1 then proposes absolute numerical thresholds in $C$ and §6 clusters in $\log C$ — both of which are NOT invariant under rescaling of $C$ . This normalization-threshold inconsistency is a concrete problem: the proposed coherence-region boundaries $C < 10^{-4}$ , $\alpha_{\text{BSD}} > 10^{-2}$ are not portable without pinning the normalization convention. The coordinator notes the first specialist pair may have underweighted these concerns; the fixed panel score of 2/5 for internal consistency reflects them.

Seven mathematical risk flags were emitted across the two flagging specialists. The highest-severity (HIGH) flags are at: §6.1 (GMM 99.95% accuracy — covariance structure, initialization, label-assignment rule, class-imbalance handling, and in-sample vs. out-of-sample evaluation are all unspecified, making this headline result non-reproducible from the manuscript); §8.1 (thresholds $C < 10^{-4}$ and $\alpha_{\text{BSD}} > 10^{-2}$ — no derivation or optimization criterion is given for how these were extracted from the GMM); and §7.1–7.3 (the spectral null conclusion — failure to reject equality on five zeros for 2,334 curves does not mathematically establish that the phase structure is 'entirely local-analytic and NOT spectral'; this is an overstated inference from limited power). MEDIUM flags are at §2.1 (normalization portability), §2.2 (whether $|\text{Ш}|$ values in the dataset are independently computed or BSD-derived, which affects claimed independence), §4.1 (near-zero Pearson correlation does not establish non-dependence for a quantity with nonlinear algebraic dependence on $|\text{Ш}|$ ), §6.2 (quadrant purity of 100% Ghosts — boundary sensitivity and potential circularity from median splits on the full dataset without held-out validation), and §5.1–§5.2 vs. §10 (an internal numerical inconsistency: §10's conclusion table claims typical Ghost $R_{\text{BSD}} \sim 10^0$ – $10^2$ , while §5's medians of 0.024 and 0.109 correspond to $\log_{10} R \approx -1.6$ and $-0.96$ , a significant discrepancy that should be resolved). These flags are reader warnings, not score components.

On evidence and completeness, specialists consistently found the core empirical separation result (Ghost/Normal separation in $R_{\text{BSD}}$ , confirmed by Welch's $t$ -test and KS test at $p < 10^{-300}$ , across four conductor bands) to be well-supported by the large dataset (368,314 curves from Cremona/LMFDB). The reproducibility gaps are concentrated in the validation layer: the GMM procedure and the coherence-quadrant analysis lack the methodological detail needed for independent reproduction. The companion Paper I ('Ghost Rank I') is cited as foundational but its verification status is unconfirmed; readers cannot independently check claims about $S(E)$ and the detection framework on which this paper builds. These gaps together produce the evidence/completeness scores of 3–4/5. On novelty (3/5), the consensus view is that $R_{\text{BSD}}$ is a genuinely fresh combination of analytic and geometric-local data, and the spectral null result is a useful empirical contribution, but much of the apparent two-phase structure follows algebraically from the BSD formula combined with the definition of Ghost by $|\text{Ш}| > 1$ — the novelty is primarily organizational and presentational rather than mechanistic.

Internal Consistency

2/5

The paper is mostly coherent at the level of definitions C(E), alpha_BSD(E), and R_BSD(E) (§2), but there is central consistency strain in how conjecture-conditional equalities are interleaved with unconditional definitions. In §2.2–§2.3, R_BSD is defined as C/alpha with alpha computed from local/geometric data; then the identity alpha = L(1)/|Sha| and the expression R_BSD = |L'(1)|·|Sha|/(L(1)^2·sqrt(N)) are introduced under "assuming BSD". Later interpretive arguments ("paradox" resolution §4.2; claims about algebraic dependence but near-zero correlation §4.1) treat the BSD-substituted form as if it explains the observed geometry, without consistently flagging which parts are conditional on BSD and which are purely definitional/empirical. A second internal-consistency issue is the normalization/threshold interplay: the paper claims the omitted 2π is irrelevant (§2.1) yet proposes absolute numerical phase thresholds in C and alpha (§8.1) and uses clustering in (log C, log alpha) (§6.1). The thresholds for C are not invariant under a global rescaling of C, so either the normalization must be fixed as part of the definition, or thresholds must be reported in a normalization-invariant way (e.g., relative to medians/quantiles). These are not merely stylistic: they affect the portability of the classifier and the meaning of "coherence region" boundaries.

Mathematical Validity

3/5

At the equation level, the algebra is mostly correct: given the rank-0 BSD formula, alpha_BSD = Omega∏c_p/|tors|^2 and L(1)=alpha_BSD·|Sha| implies alpha_BSD = L(1)/|Sha|, and therefore R_BSD = C/alpha_BSD = (|L'(1)|/(|L(1)|·sqrt(N)))/(alpha_BSD) = |L'(1)|·|Sha|/(L(1)^2·sqrt(N)) (§2.2–§2.3). This conditional manipulation is mathematically valid. However, the paper’s main results are empirical/statistical and currently lack enough formal specification to be considered mathematically reproducible from the document. The claims of 99.95% unsupervised recovery (§6.1), 100% ghost purity in a quadrant (§6.2), and the specific numeric threshold classifier (§8.1) are load-bearing, but no explicit mathematical definition of the classifier pipeline is given (feature scaling, covariance constraints, initialization, convergence criteria, how labels are mapped to clusters, whether accuracy is in-sample, treatment of class imbalance, etc.). If any of these choices change, accuracy and purity can change materially; thus the main theorems-as-stated are not yet mathematically secured. Separately, the use of correlation ≈ 0 to conclude "not a proxy" (§4.1) is not a valid mathematical implication without additional assumptions (e.g., monotonicity/linearity); at best it establishes lack of linear correlation. Because the central claims depend on underspecified computational/statistical steps (paper-owned, not merely cited theorems), mathematical_validity cannot exceed 3 under the rubric.

Verifiability (converted from Falsifiability)

3/5

Scored using the VERIFIABILITY rubric (pure mathematics). The computations are in-principle reproducible: dataset is the Cremona/LMFDB database, metrics are explicit formulas, and code paths are listed. However, verification is hindered by (a) redacted definitions in the abstract, (b) no exact reproduction seeds or version pins, (c) statistics like 'p<10^-300' and GMM 99.95% accuracy that are self-referential (the GMM recovers a label that is definitionally correlated with the inputs via BSD), and (d) no printed intermediate cross-checks against known LMFDB values. Claims are specific enough to be checked but require substantial reconstruction, and the central 'independence from Sha' claim is not cleanly falsifiable as stated because the algebraic dependence is acknowledged.

Clarity

3/5

The paper is well organized, with clear sectioning, compact definitions, summary tables, and an understandable narrative arc from definition to dataset to validation to interpretation. A graduate-level reader familiar with elliptic curves can follow the intended message. However, clarity is materially weakened by repeated use of physics-flavored terms such as 'instability,' 'phase,' 'coherence,' 'absorption,' and 'local-analytic' without a precise mathematical criterion distinguishing metaphor from claim. The abstract and discussion sometimes present inferentially stronger conclusions than the body warrants, especially around 'demonstrating' a local-analytic phenomenon and 'replacing' the |Ш|>1 definition. In addition, key methodological details for the GMM and spectral tests are too compressed for a reader to assess exactly what was done without consulting code.

Novelty

3/5

Proposing a two-dimensional (C, alpha_BSD) invariant space and reframing Ghost/large-Sha curves as a 'coherence phase' is a modestly novel repackaging. However, the core observation reduces to well-understood BSD relationships: curves with large |Ш| have elevated L(E,1) relative to geometric factors. The 'phase structure' is a consequence of the BSD formula plus the definition of Ghost by |Ш|>1, so much of the apparent novelty is definitional/presentational rather than a new mechanism. The spectral null result is a mild but genuine empirical contribution. Combination of known ideas with some new framing places this at 3.

Completeness

4/5

The paper is structurally complete on its own stated terms. It defines its principal quantities, specifies the dataset scope (368,314 rank-0 curves from Cremona/LMFDB up to conductor 130,000), states the BSD assumption when using the rank-0 formula involving |Ш|, and includes limitations/future work on threshold calibration, higher rank extension, conductor dependence, theoretical grounding, and twist-family robustness. The narrative follows through on the promised goals: definition of R_BSD, independence tests, phase separation, unsupervised recovery, and spectral null checks. The main reason this is not a 5 is that several support details needed for full reproducibility or boundary scrutiny are missing from the paper text. The clustering section does not specify how '99.95% accuracy' is computed for an unsupervised model, how cluster labels were matched to Ghost/Normal labels, or whether train/test splitting, initialization sensitivity, or class-imbalance effects were examined. The spectral analysis states that 2,334 curves were used but does not explain the sampling protocol, matching criteria, or uncertainty handling in enough detail to judge representativeness. The proposed geometric thresholds C<10^-4 and alpha>10^-2 are described as natural boundaries, but the method for extracting them from the GMM or figures is not formally documented. These are meaningful omissions, but they affect secondary methodological support more than the paper’s core internal completeness.

15 derivation flags— equations with compressed or unverified steps identified by math specialist

Strengths

+The quantities C(E), α_BSD(E), and R_BSD(E) are explicitly and consistently defined in §2, and the BSD-conditional rewrite R_BSD = |L'(E,1)|·|Ш|/(L(E,1)²·√N) is correctly derived from the rank-0 formula — the algebraic core is sound and all specialists confirmed no errors in the formula manipulations given stated assumptions.
+The large dataset (368,314 rank-0 curves from Cremona/LMFDB) with robustness checks across four conductor bands (§5.4) and a torsion-controlled subset (§5.2) makes the core Ghost/Normal separation result in R_BSD unlikely to be a statistical artifact; the Welch t-test and KS test at p < 10^{-300} are strongly significant.
+The spectral null result in §7 — no detectable difference in normalized first-zero heights between Ghost and Normal curves (p = 0.55), with a non-replicating variance signal (p = 0.011 vs. p = 0.51 in a held-out conductor band) honestly reported rather than treated as positive — demonstrates methodological discipline and is a genuine empirical contribution localizing the phenomenon.
+The limitations section (§9.4) is candid about empirical threshold calibration, rank restriction, lack of theoretical derivation, and the need for twist-family stability tests — this honesty about the framework's boundaries is a strength.
+The paper makes available code paths (Appendix B) and references an established public dataset (Cremona/LMFDB), giving readers a clear starting point for independent reproduction.

Areas for Improvement

-Resolve the normalization-threshold inconsistency: §2.1 argues the omitted 2π factor is harmless because it cancels in R_BSD, but §8.1 proposes absolute numerical thresholds in C (C < 10^{-4}) and §6 clusters in log C — both of which are not invariant under global rescaling of C. Either pin the normalization as part of the definition of C(E), or report all phase boundaries in normalization-invariant terms (e.g., quantiles relative to the dataset median).
-Prove or retract the claimed equivalence between Ghost and the geometric coherence criterion: §8.1 proposes replacing |Ш| > 1 with C < 10^{-4} and α_BSD > 10^{-2}, but §6.2 already shows that the structural quadrant contains Ghost curves, implying these are not equivalent. Either prove the coherence region is a strict subset of Ghosts (and explain the gap), or reframe §8.1 as a partial geometric characterization rather than a definitional replacement.
-Fully specify the GMM clustering pipeline (§6.1): covariance matrix type (full, diagonal, tied), initialization method, number of components tested, convergence criterion, how cluster labels are matched to Ghost/Normal labels, whether accuracy is in-sample or held-out, and how class imbalance (53% Ghost) is handled. The 99.95% headline accuracy is the paper's primary validation result and must be reproducible from the manuscript text alone.
-Address the circularity concern in the coherence-quadrant analysis (§6.2): the quadrant boundaries are defined using median splits computed on the full dataset, and Ghost purity is evaluated on the same dataset without held-out validation. This creates a potential selection artifact in the 100% Ghost purity claim. A held-out test or cross-validation is needed.
-Replace or supplement Pearson correlation with nonlinear or rank-based dependence measures in §4.1. Near-zero Pearson correlation between R_BSD and |Ш| does not establish non-dependence for a quantity that algebraically contains |Ш| in its numerator under BSD; Spearman correlation, mutual information, or partial dependence conditioned on L(E,1) would be more diagnostic.
-Resolve the internal numerical inconsistency in the R_BSD scale: §10's conclusion table claims typical Ghost R_BSD ~ 10^0–10^2 and Normals ~ 10^2–10^3 (log₁₀ R ≈ 0–2 for Ghosts), while §5.1 reports medians of 0.024 and 0.109 (log₁₀ R ≈ −1.6 and −0.96). These cannot simultaneously be correct descriptions of the same data and same normalization; the discrepancy should be explained or corrected.
-Provide selection criteria, representativeness justification, and sample-size rationale for the spectral subsample of 2,334 curves (0.63% of the full dataset) in §7. Without this, readers cannot assess whether the null result generalizes. The conclusion that the effect is 'entirely local-analytic and NOT spectral' should be softened to match the limited power of the reported tests.
-Maintain a clear and consistent conditional logic boundary throughout: α_BSD is defined without L-values, but later explanatory passages (especially §4.2 and §9.1) invoke the BSD formula to interpret the paradox and the absorption mechanism. Clearly flag at every interpretive step which conclusions are unconditional (from computed quantities) and which are conditional on BSD holding for those curves.

Arithmetic Coherence: Two-Phase Structure in the BSD Space of Elliptic Curves

Paper II: The R_BSD Invariant and Geometric Ghost Classification

Author: Adam Murphy
Affiliation: ImpactMe.ai Research
Date: December 2025 (Draft v1.3)
Created: 2025-12-10 PST
Last Modified: 2025-12-10 PST

Abstract

We introduce $R_{\text{BSD}}$ , a ratio of analytic instability to geometric structure for elliptic curves, and demonstrate that it reveals a robust two-phase structure in the arithmetic of rank-0 curves.

For an elliptic curve $E/\mathbb{Q}$ with conductor $N$ , we define:

Instability: $C(E) = |L'(E,1)| / (|L(E,1)| \sqrt{N})$
Structure: $\alpha_{\text{BSD}}(E) = \Omega(E) \cdot \prod_p c_p(E) / |E_{\text{tors}}|^2$
Phase indicator: $R_{\text{BSD}}(E) = C(E) / \alpha_{\text{BSD}}(E)$

Analyzing 368,314 rank-0 elliptic curves with conductors up to 130,000, we find:

Two distinct phases emerge in $(C, \alpha_{\text{BSD}})$ space, recoverable by unsupervised clustering with 99.95% accuracy.
Ghost curves (those with $|\text{Ш}| > 1$ ) occupy a distinct coherence phase characterized by low instability and high structure, with median $R_{\text{BSD}}$ values typically 4.5× lower than normal curves (rising to 44× separation when controlling for torsion).
The phase indicator $R_{\text{BSD}}$ is not conductor-driven (corr = 0.003) and not a proxy for $|\text{Ш}|$ (corr = −0.005), confirming it captures genuine arithmetic structure.
Spectral tests show no difference in L-function zero distributions between phases, demonstrating the phase structure is entirely local-analytic—manifesting through $L(E,1)$ and $L'(E,1)$ , not through global zero spacing.

We propose a geometric Ghost classification based on position in $(C, \alpha_{\text{BSD}})$ space, replacing threshold-based definitions. This framework interprets Ghost curves as arithmetically quiet states where the L-function absorbs arithmetic complexity into elevated central values and suppressed derivatives.

Keywords: Elliptic curves, Birch-Swinnerton-Dyer conjecture, Tate-Shafarevich group, L-functions, phase structure, Ghost Rank

AMS Classification: 11G05, 11G40, 11M41

1. Introduction

1.1 Background

In the companion paper [Ghost Rank I], we introduced a stability metric $S(E)$ that detects elliptic curves with large Tate-Shafarevich groups through local L-function behavior. We established an empirical diffusion law with slope $1/\sqrt{e}$ connecting the stability metric to $|\text{Ш}|$ .

That work answered the question: "How can we detect Ghosts?"

This paper addresses the deeper question: "What ARE Ghosts, structurally?"

1.2 Motivation

The stability metric $S(E)$ achieves perfect separation between curves with $|\text{Ш}| = 1$ and $|\text{Ш}| > 1$ , but it does not explain why certain curves develop large Tate-Shafarevich groups. Is there an underlying geometric structure that determines Ghost status?

We show that the answer is yes: Ghost curves occupy a distinct region in a two-dimensional phase space defined by analytic instability and geometric structure.

1.3 Main Contributions

A new invariant $R_{\text{BSD}} = C/\alpha_{\text{BSD}}$ that is neither conductor-dominated nor a simple relabeling of $|\text{Ш}|$
Discovery of two-phase structure in $(C, \alpha_{\text{BSD}})$ space, recovered by unsupervised methods
Geometric Ghost classification replacing threshold-based definitions
Spectral null result proving the phase structure is local-analytic, not spectral
Interpretation framework connecting arithmetic phases to coherence phenomena

2. Definitions

2.1 Instability Metric $C(E)$

Definition. For an elliptic curve $E/\mathbb{Q}$ of analytic rank 0, the analytic instability is:

$C(E) := \frac{|L'(E,1)|}{|L(E,1)| \cdot \sqrt{N_E}}$

where $L(E,s)$ is the L-function of $E$ and $N_E$ is the conductor.

Interpretation: $C(E)$ measures the relative curvature of the L-function at $s=1$ , normalized by conductor. High $C$ indicates a curve whose L-function is "ready to destabilize"—the ratio of derivative to value is large relative to arithmetic scale.

Note on normalization: The standard analytic normalization includes a factor of $2\pi$ (i.e., $\sqrt{N}/(2\pi)$ ). We omit this constant as it cancels in the ratio $R_{\text{BSD}}$ and does not affect comparisons within the dataset. The phase structure and clustering results are invariant under such global rescaling.

2.2 Structure Metric $\alpha_{\text{BSD}}(E)$

Definition. The geometric-local structure of $E$ is:

$\alpha_{\text{BSD}}(E) := \frac{\Omega(E) \cdot \prod_p c_p(E)}{|E(\mathbb{Q})_{\text{tors}}|^2}$

where:

$\Omega(E)$ is the real period
$\prod_p c_p(E)$ is the product of Tamagawa numbers
$|E(\mathbb{Q})_{\text{tors}}|$ is the order of the torsion subgroup

Key property: $\alpha_{\text{BSD}}$ is computed entirely from geometric and local arithmetic data—it involves no L-values. This ensures independence from the analytic side.

BSD Connection: Assuming the BSD conjecture holds for $E$ (including finiteness of $\text{Ш}(E)$ ), the rank-0 formula gives: $L(E,1) = \frac{\Omega(E) \cdot \prod_p c_p(E) \cdot |\text{Ш}(E)|}{|E(\mathbb{Q})_{\text{tors}}|^2}$

We then have $\alpha_{\text{BSD}} = L(E,1) / |\text{Ш}|$ , meaning: $R_{\text{BSD}} = \frac{C}{\alpha_{\text{BSD}}} = \frac{|L'(E,1)| \cdot |\text{Ш}|}{L(E,1)^2 \sqrt{N}}$

2.3 Phase Indicator $R_{\text{BSD}}(E)$

Definition. The instability-to-structure ratio is:

$R_{\text{BSD}}(E) := \frac{C(E)}{\alpha_{\text{BSD}}(E)}$

This ratio quantifies how much analytic instability exists per unit of inherent geometric structure.

3. Dataset

We analyzed 368,314 rank-0 elliptic curves from the Cremona database with BSD data:

Property	Value
Total curves	368,314
Conductor range	10 – 130,000
Ghost curves ($	\text{Ш}
Normal curves ($	\text{Ш}
Torsion=1 subset	24,116

Note: The high Ghost fraction reflects that $|\text{Ш}| > 1$ includes many curves with small Sha (2, 4, 9), representing the boundary of the Ghost phase.

4. Results: Independence Tests

4.1 C and α_BSD Are Largely Independent

Correlation	Value	Interpretation
corr $(C, \alpha_{\text{BSD}})$	0.29	Weak positive; largely independent
corr $(R_{\text{BSD}}, N)$	0.003	NOT conductor-dominated
corr$(R_{\text{BSD}},	\text{Ш}	)$

Key Finding: The near-zero correlation between $R_{\text{BSD}}$ and conductor proves that this ratio captures curve-specific arithmetic structure, not just scale.

On the Sha correlation: While $R_{\text{BSD}}$ depends algebraically on $|\text{Ш}|$ via the BSD formula (see §2.3), the near-zero empirical correlation (−0.005) shows that variance in $R$ across the dataset is dominated by the interplay with $L(E,1)$ and $L'(E,1)$ , not by Sha alone. In other words, $R_{\text{BSD}}$ is not simply " $|\text{Ш}|$ with extra steps."

4.2 The "Paradox" Resolution

One might expect Ghost curves (high $|\text{Ш}|$ ) to have high $R_{\text{BSD}}$ since Sha appears in the numerator after BSD substitution. Instead, Ghosts have low $R_{\text{BSD}}$ .

Resolution: Ghost curves have:

Very small $|L'(E,1)|$ — flat L-function near $s=1$
Large $L(E,1)$ — elevated central value

Although $R_{\text{BSD}}$ contains $|\text{Ш}|$ algebraically, the elevated $L(E,1)$ and suppressed $|L'(E,1)|$ in Ghost curves offset this contribution, yielding lower $R_{\text{BSD}}$ despite large $|\text{Ш}|$ . This is the signature of arithmetic coherence.

5. Results: Phase Separation

5.1 Ghost/Normal Statistics

Statistic	Ghost	Normal	Ratio
Mean $R_{\text{BSD}}$	96.9	639.5	0.15
Median $R_{\text{BSD}}$	0.024	0.109	0.22

Note on scale: $R_{\text{BSD}}$ spans several orders of magnitude, making $\log_{10} R$ the more natural scale for visualization. The raw means above are dominated by a long right tail; medians (0.024 vs 0.109) show a more representative ~4.5× separation in the central mass, while the mean ~6.6× separation reflects the heavy tail in Normals.

Interpretation: Ghost curves have systematically lower $R_{\text{BSD}}$ values than Normal curves, with the effect size ranging from 4.5× (medians) to 6.6× (means) depending on how tail behavior is weighted.

5.2 Torsion-Controlled Analysis

When restricting to curves with $|E_{\text{tors}}| = 1$ :

Statistic	Ghost	Normal	Ratio
Mean $R_{\text{BSD}}$	0.00016	0.0070	0.022

Key Finding: Controlling for torsion reveals ~44× separation—the cleanest phase signal. Torsion variation adds noise to the underlying phase structure.

5.3 Statistical Significance

Test	Value	Interpretation
Welch's t-test	$p < 10^{-300}$	Distributions differ
Kolmogorov-Smirnov	$p < 10^{-300}$	CDFs separate

The separation is not a statistical fluctuation.

5.4 Robustness Across Conductor Bands

Conductor Range	Ghost/Normal Ratio
10–1,000	0.23
1,000–10,000	0.12
10,000–100,000	0.11
100,000–130,000	0.36 (smaller sample; boundary effects possible)

The phase separation is consistent across four orders of magnitude in conductor, though edge effects appear at the upper boundary of our dataset.

6. Results: Unsupervised Validation

6.1 Gaussian Mixture Model

We fit a 2-component GMM to $(\log_{10} C, \log_{10} \alpha_{\text{BSD}})$ without using Ghost/Normal labels:

Metric	Value
GMM accuracy	99.95%
K-Means accuracy	98.45%

Key Finding: The two-phase structure is recoverable by unsupervised methods. The clusters identified by GMM align almost perfectly with the Ghost/Normal classification based on $|\text{Ш}|$ .

6.2 Quadrant Classification

Defining the coherence quadrant as:

$C < C_{\text{median}}$ AND $\alpha_{\text{BSD}} > \alpha_{\text{median}}$

Quadrant	Ghost Rate
Coherence (low C, high α)	100%
Structural (high C, low α)	28%
Mixed	varies

Key Finding: The coherence quadrant is pure—every curve there is a Ghost. The structural quadrant is mixed, containing both Normal curves and boundary-case Ghosts with modest $|\text{Ш}|$ . This is consistent with a sharp coherence phase and a broader structural regime that includes transition states.

7. Results: Spectral Null

7.1 Zero Spacing Tests

We computed the first five L-function zeros for 2,334 curves (1,167 Ghosts, 1,167 Normals) using SageMath. We report normalized zero heights $\tilde{\gamma} = \gamma \cdot \log N / (2\pi)$ , which accounts for conductor scaling:

Test	Ghost	Normal	p-value	Verdict
Mean $\tilde{\gamma}_1$	1.126	1.141	0.55	NULL
Gap mean ( $\tilde{\gamma}_2 - \tilde{\gamma}_1$ )	1.485	1.438	0.15	NULL

7.2 Variance Tests

Test	Ghost	Normal	p-value	Replicates?
$\gamma_1$ variance (Low-N)	0.097	0.131	0.011	—
$\gamma_1$ variance (High-N)	0.109	0.103	0.51	NO

The variance signal in the Low-N sample did not replicate in the High-N sample, indicating sample noise rather than genuine structure.

7.3 Spectral Conclusion

The Ghost/Normal phase structure manifests entirely in the local analytic behavior of the L-function at $s=1$ —through $L(E,1)$ , $L'(E,1)$ , and the BSD geometric factors—and NOT through the global zero distribution.

This is a crucial finding: arithmetic coherence is a local phenomenon, not a spectral one.

8. Geometric Characterization of Ghost Curves

8.1 Phase Classification Criterion

Classification Criterion. An elliptic curve $E/\mathbb{Q}$ of analytic rank 0 lies in the coherence phase if:

$C(E) < 10^{-4} \;\land\; \alpha_{\text{BSD}}(E) > 10^{-2}$

We propose that this geometric region $\mathcal{R}_{\text{ghost}}$ provides a more fundamental characterization of Ghost curves than the arithmetic threshold $|\text{Ш}| > 1$ . Curves in the coherence phase develop large Sha as a consequence of their position in $(C, \alpha_{\text{BSD}})$ space.

On thresholds: The values $10^{-4}$ and $10^{-2}$ correspond to the natural cluster boundaries identified by GMM and visible in Figure 1. These thresholds achieve 99.95% agreement with unsupervised clustering, suggesting they capture a natural boundary rather than an imposed one. They may be refined with larger datasets or theoretical analysis.

8.2 Relationship to Previous Definitions

Characterization	Status	Notes
$	\text{Ш}	> 1$
$S(E) < 0.025$	Correlated	S captures similar signal
Geometric $(C, \alpha)$	Proposed	Structural understanding of the phenomenon

Key Insight: We propose that large $|\text{Ш}|$ is a symptom of the coherence phase, not its definition. A curve develops large Sha because it lies in the region of phase space where analytic instability is suppressed relative to geometric structure.

9. Discussion

9.1 The Absorption Mechanism

Ghost curves are characterized by:

Large $L(E,1)$ — elevated central value
Small $|L'(E,1)|$ — flat L-function near $s=1$
Large $\alpha_{\text{BSD}}$ — rich geometric/local structure

The BSD formula $L(E,1) = \alpha_{\text{BSD}} \cdot |\text{Ш}|$ shows that when $L(E,1)$ is large and $\alpha_{\text{BSD}}$ is large, $|\text{Ш}|$ can also be large while maintaining low instability ratio.

Interpretation: The L-function "absorbs" arithmetic complexity. Rather than manifesting as analytic instability (steep derivative), the complexity is stored in the Sha obstruction while the L-function remains smooth near $s=1$ .

The mechanism can be visualized as a balance:

| Curve Type | $L(E,1)$ | $|L'(E,1)|$ | $|\text{Ш}|$ | Result | |------------|----------|-------------|--------------|--------| | Normal | Moderate | Moderate | 1 | Balanced: moderate $C$ , high $R$ | | Ghost | Large | Small | Large | Absorbed: low $C$ , low $R$ |

In Ghost curves, the "weight" of large $|\text{Ш}|$ is counterbalanced by elevated $L(E,1)$ and suppressed $L'(E,1)$ , collapsing the instability ratio.

9.2 Why "Coherence"?

We use the term "coherence phase" because Ghost curves exhibit:

Suppressed external response (low $C$ )
Enhanced internal structure (high $\alpha_{\text{BSD}}$ )
Protected information (large $|\text{Ш}|$ stored, not expressed)

This parallels coherence phenomena in physical systems, where internal correlations persist despite external coupling.

9.3 Comparison to Ghost Rank I

Aspect	Paper I	Paper II
Focus	Detection	Structure
Key metric	$S(E)$	$R_{\text{BSD}}$
Main result	1/√e diffusion law	Two-phase geometry
Dimensionality	1D threshold	2D phase space
Classification	S < 0.025	$(C, \alpha)$ quadrant

9.4 Limitations and Future Work

Threshold calibration: The values $C_{\text{threshold}} = 10^{-4}$ and $\alpha_{\text{threshold}} = 10^{-2}$ are empirical. Theoretical justification is needed.
Higher rank: This analysis focused on rank 0. Extension to rank 1 curves (using the Ladder Law from Paper I) is a natural next step.
Conductor dependence: While corr $(R, N) \approx 0$ , the phase boundary may shift slightly with conductor scale.
Theoretical foundation: Why does arithmetic exhibit this two-phase structure? A deeper explanation connecting to modularity or random matrix theory would strengthen the framework.
Cross-family stability: Testing whether $R_{\text{BSD}}$ phase membership is preserved across quadratic twist families would probe the robustness of the geometric classification.

10. Conclusion

We have demonstrated that rank-0 elliptic curves exhibit a robust two-phase structure in the $(C, \alpha_{\text{BSD}})$ plane:

Phase	Character	Typical $R_{\text{BSD}}$	Curves
Coherence	Low instability, high structure	$10^{0}$ – $10^{2}$	Ghosts
Structural	High instability, variable structure	$10^{2}$ – $10^{3}$	Normals

On a $\log_{10}$ scale, the bulk of Ghosts sit around $\log_{10} R \approx 0$ – $2$ , while Normals extend to $\log_{10} R \approx 2$ – $3$ . These ranges capture central tendencies, with substantial tail behavior in both populations.

Main Results

$R_{\text{BSD}}$ is a genuine invariant — not conductor-dominated, not Sha relabeling
Two phases exist — recoverable by unsupervised clustering at 99.95% accuracy
Separation is 6.6× overall, 44× for torsion-free curves
The phase structure is local-analytic — spectral tests show no zero spacing differences
Geometric Ghost definition replaces threshold-based classification

Implications

Ghost curves are not random outliers with "big Sha." They are curves occupying a geometrically defined coherence phase where:

The L-function is flat near $s=1$
Internal structure dominates external instability
Arithmetic complexity is absorbed rather than expressed

This provides a structural explanation for the phenomenon detected in Paper I. Together with the $1/\sqrt{e}$ diffusion law established there, these results suggest that Ghost curves form a geometrically coherent population with distinctive analytic signatures at multiple scales—and that the arithmetic of elliptic curves may exhibit phase phenomena analogous to those observed in physical systems.

Acknowledgments

We thank the LMFDB collaboration and John Cremona for maintaining the elliptic curve databases that made this analysis possible. Computations were performed using SageMath and the Python scientific stack.

References

Murphy, A. (2025). Ghost Rank I: Spectral Phase Transitions and the 1/√e Diffusion Law. This series.
Birch, B.J. and Swinnerton-Dyer, H.P.F. (1965). Notes on elliptic curves II. J. reine angew. Math. 218, 79-108.
Cremona, J.E. (1997). Algorithms for Modular Elliptic Curves. Cambridge University Press.
The LMFDB Collaboration (2024). The L-functions and Modular Forms Database. https://www.lmfdb.org
He, Y.-H., Lee, K.-H., and Oliver, T. (2022). Murmurations of elliptic curves. arXiv:2204.10140.

Appendix A: Key Formulas

$C(E) = \frac{|L'(E,1)|}{|L(E,1)| \cdot \sqrt{N_E}}$

$\alpha_{\text{BSD}}(E) = \frac{\Omega(E) \cdot \prod_p c_p(E)}{|E(\mathbb{Q})_{\text{tors}}|^2}$

$R_{\text{BSD}}(E) = \frac{C(E)}{\alpha_{\text{BSD}}(E)} = \frac{|L'(E,1)| \cdot |\text{Ш}|}{L(E,1)^2 \sqrt{N}}$

For reference, we recall the stability metric from Ghost Rank I: $S(E) = \frac{|L'(E,1)|}{|L(E,1)| \cdot \log N_E}$

This metric is not used directly in the present paper but provides context for the relationship between $C(E)$ and the original Ghost detection method.

Appendix B: Code Availability

All analysis code is available at:

experiments/qh_bsd_ghosts/ghost_rank_repo/
├── code/
│   ├── r_bsd_full_analysis.py      # Main analysis
│   ├── r_bsd_phase_diagram.py      # Visualization
│   ├── r_bsd_validation_tests.py   # GMM/quadrant tests
│   ├── zero_spacing_test.py        # Spectral analysis
│   └── stability_metric.py         # Core metrics
├── results/
│   ├── r_bsd_statistics.json       # Full statistics
│   ├── r_bsd_full_data.csv         # Raw data
│   └── *.png                       # Figures
└── paper/
    ├── ghost_rank.md               # Paper I
    └── ghost_rank_paper_2.md       # Paper II (this document)

Appendix C: Figure Captions

Figure 1. Phase diagram showing 368,314 rank-0 elliptic curves in $(C, \alpha_{\text{BSD}})$ space. Ghost curves (red) cluster in the coherence region (low $C$ , high $\alpha$ ), while Normal curves (blue) occupy the structural phase. Dashed lines indicate proposed threshold boundaries.

Figure 1: Phase Diagram

Figure 2. Distribution of $R_{\text{BSD}}$ for Ghost vs Normal curves. (a) All curves: 6.6× separation. (b) Torsion=1 subset: 44× separation. (c) CDFs showing distinct distributions.

Figure 2: R_BSD Distributions

Figure 3. Validation results. (a) GMM clustering recovers phase structure at 99.95% accuracy. (b) Quadrant classification: coherence quadrant contains 100% Ghosts.

Figure 3: Validation

Figure 4. Spectral null result. Normalized first-zero distributions for Ghost (red) and Normal (blue) curves are statistically indistinguishable (p = 0.55).

Figure 4: Spectral Null

Draft v1.3 — December 2025
Revised per Gemini review: Definition→Characterization, normalization note added, abstract uses median (4.5×), seesaw table in §9.1

Rank-0 elliptic curves exhibit a robust two-phase structure in (C, α_BSD) space with Ghost curves concentrated in a coherence phase (low C, high α_BSD).

Falsifiable if: Independent datasets (e.g., other curve databases, larger conductor ranges, or extended LMFDB snapshots) fail to show a statistically significant bimodal/clustered structure in (C, α_BSD) or show Ghosts distributed uniformly across the plane.

mathpending

The coherence phase is recoverable by unsupervised clustering: a 2-component Gaussian Mixture Model on (log10 C, log10 α_BSD) will separate Ghost vs Normal curves with ≈99.95% accuracy.

Falsifiable if: Applying the same unsupervised GMM procedure to an independent sample yields substantially lower cluster-label agreement with |Ш| (e.g., accuracy ≪ 99%), or cluster assignments do not align with the proposed coherence quadrant boundaries.

mathpending

R_BSD is not conductor-driven and not a simple proxy for |Ш| (empirical correlations: corr(R_BSD,N) ≈ 0.003, corr(R_BSD,|Ш|) ≈ −0.005).

Falsifiable if: Reanalysis shows a significant correlation (|corr| ≫ 0.05 with p≪0.01) between R_BSD and conductor or between R_BSD and |Ш|, or controlling for known confounders reverses the reported near-zero correlations.

mathpending

Spectral (zero-spacing) statistics show no difference between Ghost and Normal curves: the phase structure is a local-analytic phenomenon manifesting via L(E,1) and L'(E,1), not via global zero distribution.

Falsifiable if: High-precision zero computations on enlarged or stratified samples reveal statistically significant differences in first-zero locations or gap distributions between Ghost and Normal populations (e.g., p≪0.01 in multiple independent tests).

mathpending

After controlling torsion (|E(Q)_{tors}|=1), the separation between Ghost and Normal in R_BSD increases to ≈44× (torsion-controlled analysis yields much cleaner phase separation).

Falsifiable if: An independent torsion=1 subset fails to show such an increased separation (e.g., measured separation ≪ 44×, not statistically significant), or torsion control does not consistently increase separation across conductor bands.

mathpending

The proposed geometric coherence region defined empirically by C < 10^{-4} and α_BSD > 10^{-2} (the coherence quadrant) yields a pure Ghost region (100% Ghost rate in the dataset).

Falsifiable if: Applying the same threshold to independent datasets yields a non-zero fraction of Normal curves in the coherence quadrant, or varying dataset/resolution shows these thresholds do not correspond to cluster boundaries with high purity.

mathpending

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