The Shapley Supercluster fits Geier's programme well - A first look by Stefan Geier et al., Gerhart-Hauptmann-Straße 6, 83071 Haidholzen

The Shapley Supercluster fits Geier's programme well - A first look

by Stefan Geier et al., Gerhart-Hauptmann-Straße 6, 83071 Haidholzen

The Shapley Supercluster displays a moderate, coherent, and falsifiable exploratory concordance with the Fibonacci–Lucas-Φ-coordinate family of Geier’s Programme.

Please, compare with: The Great Attractor fits Geier's Programme very well with more than 99% by Stefan Geier et al., ISTS, Gerhart-Hauptmann-Straße 6, 83071 Haidholzen, https://humanistischebetrachtungen1.blogspot.com/2026/08/the-great-attractor-fits-geiers.html 

20.08.2026 (to be improved):

The Shapley Supercluster is both a real high-density structure and a scale-dependent feature of reconstructed cosmic velocity fields. Its central concentration is collapsing, whereas a much wider 50-Mpc region has been modelled as gravitationally unbound in an accelerating universe. This paper asks whether published Shapley data follow the restricted Φ-Fibonacci-Lucas coordinate family of the GEIER programme and places the result within Geier’s related Penrose, Kerr, BABAR, glueball, Yang–Mills and Great Attractor works. We conducted a primary-source audit of cluster dynamics, density, peculiar-velocity, catalogue and magnetic-field studies; retained dimensionless quantities as the primary evidence; calculated bounded proportional proximity P(a,b)=100 min(|a|,|b|)/max(|a|,|b|), logarithmic residuals and standardized differences; and audited dependence, target density and representation invariance. The modeled Shapley contribution to the Local Group peculiar velocity, f=0.090±0.021, is close to Φ⁻⁵=0.09016994 (P=99.81%; standardized difference -0.008σ). The mean density ratio 1+δ=1.76±0.17 is close to L₄/L₃=7/4=1.75 (P=99.43%; +0.059σ). Equivalently, δ=0.76±0.17 lies near 3/4; this is the same datum and not independent. The three successive ratios of rounded POSSUM core masses are 2.227, 2.200 and 4.000, compared with √5, √5 and L₃=4, but ranking and rounding prevent a dynamical interpretation. Multi-scale counts 4, 5, 11, 21, 25 and 45 include several exact sequence targets; however, changing catalogue definitions and a 27.45-35.29% exact integer target coverage over 1-51 substantially reduce their evidential weight. The expanded cross-domain comparison reveals an evidential ladder: the Penrose paper supplies a structuralist separation of theorem core from bridge claims; Kerr, Great Attractor and Shapley supply astrophysical coordinate tests; BABAR adds an amplitude-square and mechanical-analogue network; glueballs offer a spectrum-wide recurrence test; and q-deformed SU(2)₃ Yang–Mills supplies an exact but local operator realization of the Fibonacci-Lucas-Φ kernel. This coherence is scientifically fertile but is not independent replication because authorship, target grammar and programme provenance overlap. The strongest Shapley conclusion remains a coherent, dimensionless and falsifiable exploratory concordance, not physical confirmation. No Shapley observable has yet been shown to obey rₙ₊₁=1+1/rₙ, the projective contraction zₙ₊₁/zₙ=-Φ⁻², a common basin of attraction, or a derived e-α-2ħ bridge. We formulate a preregistered shell-trajectory and constrained-simulation test that can decide the stronger claim on held-out data.

Keywords: Shapley Supercluster; Shapley Attractor; Great Attractor; GEIER programme; golden ratio Φ; Fibonacci numbers; Lucas numbers; Penrose programme; Kerr black holes; BABAR; glueballs; Yang–Mills theory; cosmic flows; density contrast; projective dynamics; theory-nets; target density; preregistration; falsification

Highlights

Primary numerical correspondence audit.

Observable	Published value	Target	Target value	P	Dlog	Z	Evidence status
Local Group velocity fraction	0.090000 ± 0.021000	Φ⁻⁵	0.090169944	99.8115%	0.001886	-0.0081σ	Primary, dimensionless; linked source
Mean density ratio, 1 + δ	1.760000 ± 0.170000	L₄/L₃ = 7/4	1.750000000	99.4318%	0.005698	+0.0588σ	Primary, dimensionless; linked source
Fractional overdensity δ	0.760000 ± 0.170000	3/4	0.750000000	98.6842%	0.013245	+0.0588σ	Same datum as density ratio; do not double count

Multi-scale count hierarchy.

Count	Source-defined meaning	Nearest restricted target	Proximity	Caution
4	Two clusters + two groups in POSSUM core [6]	L₃=4	100%	Different core definition
5	Three Abell + two poor clusters in optical core [4]	F₅=5	100%	Different object taxonomy
11	Central collapse / ShaSS clusters [1,4]	L₅=11	100%	Repeated partly overlapping core description
21	X-ray luminous cluster sample [2]	F₈=21	100%	Analysis sample, not intrinsic state
≥25	Broad Abell membership [4]	(F₈+L₇)/2=25	Exact at 25	Lower-bound wording; boundary dependent
45	eROSITA FoF members [5]	L₈=47	95.745%	Algorithm and linking-length dependent

Relations deliberately not counted as physical evidence.

Apparent relation	Why it is representation- or selection-sensitive	Decision
55 km s⁻¹ = F₁₀; 612 km s⁻¹ ≈ F₁₅=610	Velocity numerals change to 55,000 and 612,000 in m s⁻¹.	Excluded; only 55/612 is admissible.
8 h⁻¹ Mpc = F₆	Distance coefficient changes with unit and h convention.	Excluded as sequence evidence.
51 Mpc = (L₈+F₁₀)/2	51 Mpc is a selected analysis boundary.	Recorded as a design choice, not a natural constant.
2.58×10¹⁶ M⊙ ≈ Φ²×10¹⁶ M⊙	Coefficient changes if the mass unit is 10¹⁵ M⊙.	Excluded without a unit-bearing bridge.
169 detected H I galaxies = 13²	Survey sample size depends on selection and completeness.	Excluded as an intrinsic Shapley state.

Final evidence grading.

Claim	Assessment
Shapley is a gravitational overdensity and dynamical structure	Supported.
Shapley can be a velocity-flow attractor under specified reconstruction and smoothing	Supported, but scale-dependent.
fShapley≈Φ⁻⁵	Strong descriptive dimensionless correspondence; retrospective and model-derived.
1+δ≈L₄/L₃=7/4	Strong descriptive dimensionless correspondence; broad uncertainty; same source model.
Core masses form a √5/Lucas hierarchy	Suggestive secondary pattern; rounded, rank-ordered, not dynamical.
Catalogue counts form a Fibonacci orbit	Not supported; definitions differ and no recurrence is obeyed.
Shapley displays q=−Φ⁻² contraction	Not tested.
Shapley corroborates an e-α-2ħ cosmological mechanism	Not demonstrated.
Requested Geier corpus independently validates Shapley	No. It provides hypothesis provenance, methodological coherence and one exact model identity, but shares authorship and target grammar and supplies no Shapley transport law.
Overall classification	Moderate, coherent and falsifiable exploratory compatibility with the restricted GEIER coordinate family.

Conclusion:

The Shapley Supercluster follows the restricted GEIER programme in several non-trivial descriptive instances. Its modeled contribution to the Local Group peculiar velocity is

f(Shapley)   = 0.090 ± 0.021 ≈ Φ⁻⁵ = 0.09016994,

and its mean 50-Mpc density ratio is

1 + δ = 1.76 ± 0.17 ≈ 7/4 = L₄/L₃.

These correspondences are representation-invariant, low-complexity and jointly coherent enough to justify a serious prospective Shapley-GEIER hypothesis. The secondary √5 mass hierarchy and multi-scale count pattern add context, but their evidential value is reduced by rounding, ranking, catalogue boundaries and target density.

The stronger conclusion does not yet follow. No published Shapley trajectory obeys the Fibonacci update law, no projective residual contraction q=−Φ⁻² has been measured, no common GEIER basin has been demonstrated, and no e-α-2ħ mechanism has been derived from cosmological dynamics. The truth-oriented formulation is therefore: the Shapley Supercluster displays a moderate, coherent and falsifiable exploratory concordance with the Φ-Fibonacci-Lucas coordinate family of the GEIER programme, but it has not yet been shown to instantiate that programme as a causal physical attractor or universal law.

The expanded corpus changes the interpretation, not the Shapley arithmetic. It shows that the GEIER programme contains both exact mathematical or model-specific identities and exploratory empirical specializations. The Penrose theory-net analysis strengthens the methodological separation of these levels; the Kerr, BABAR, glueball and Great Attractor papers demonstrate a progressively constrained family of applications; and the Yang–Mills paper provides an exact local operator anchor. Shapley remains in the empirical-specialization class until a cosmological operator and transport law are derived and successfully predicted on held-out trajectories.

The next advance should not be another retrospective identity. It should be a preregistered, complete-trajectory prediction on constrained simulations and held-out observations, with fixed variables, target grammar, covariance, rivals and failure criteria. A particularly strong test would transport one operator-level specification - inspired, but not assumed, by the exact projective and Yang–Mills kernels - from a training ensemble to independent Shapley products without retuning. Shapley is especially suitable for this transition because its nested scales supply both the attraction language that motivates the hypothesis and the rich dynamical data needed to refute or strengthen it.

References ... :

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Discussion welcome  |  Comments welcome  |  Critique welcome  |  Please check the calculations

19.08.2026:
In several interesting instances the Shapley Supercluster is compatible with our Geier’s Programme at the level of exploratory Φ/Fibonacci–Lucas coordinates. However, it has not yet been shown to follow the Programme as a causal, generative, or dynamical law.

The Shapley case is arguably at least as suggestive numerically as the Great Attractor case, particularly because its two strongest correspondences are dimensionless:

[
f_{\mathrm{Shapley}}\approx \Phi^{-5},
\qquad
1+\delta\approx \frac{L_4}{L_3}=\frac74.
]

Nevertheless, these are retrospective correspondences. Our stricter GEIER–TRANSFORM formulation requires fixed dimensionless observables, a finite target grammar, full target-density accounting, an independently justified transformation, comparison with matched rivals, and prediction on new data. It explicitly shifts the evidential unit from a recurring number to a generator–orbit–invariant–regulation system.

1. The established astrophysical situation

The Shapley structure is genuinely dynamical, but it is not one simple equilibrium object. Its central concentration is collapsing and extends at least (8h^{-1}) Mpc around Abell 3558, enclosing 11 Abell clusters, with inferred infall velocities approaching (2{,}000\ \mathrm{km,s^{-1}}). On a much larger scale, however, a roughly 50–51 Mpc Shapley region was estimated to have mass ((4.4\pm0.44)\times10^{16}M_\odot), density ratio (1+\delta=1.76\pm0.17), and to remain gravitationally unbound; in the spherical-collapse calculation it never reaches turnaround because of cosmic acceleration. Thus the core may collapse while the wider region continues expanding. (arXiv)

Even the ordinary astrophysical expression “Shapley attractor” is model- and scale-dependent. A 2017 Cosmicflows reconstruction described the local flow as dominated by a Shapley-associated attractor together with a dipole repeller. A 2026 digital-twin preprint instead finds that streamline convergence shifts from Virgo to Hydra–Centaurus and then Shapley as the smoothing scale grows, while no single structure dominates the Local Group’s complete velocity budget. (arXiv)

This distinction is important:

• Cosmological attractor: a mass overdensity or reconstructed velocity-flow basin.

• Geier Φ-attractor: an invariant state generated by a specified transformation, ideally with a measurable basin and contraction law.

The first is scientifically supported for Shapley in a qualified, scale-dependent sense. The second is not yet established.

2. The strongest Geier-family correspondences

For transparent comparison, define the bounded proportional concordance

[
A(x,t)
=100\exp!\left[-\left|\ln\frac{x}{t}\right|\right]
=100\frac{\min(x,t)}{\max(x,t)}.
]

This percentage describes numerical proximity only. It is not a probability that the hypothesis is true, a confidence level, or a statistical significance.

Shapley observable	Geier-family target	Numerical result	Assessment
Fraction of the Local Group peculiar velocity: (55/612=0.089869); reported as (0.090\pm0.021)	(\Phi^{-5}=0.09016994)	(A=99.6666%); deviation (=-0.014\sigma). Using the rounded 9.0% gives (A=99.8115%)	Strongest dimensionless correspondence, but model-derived and retrospective
Mean density ratio (1+\delta=1.76\pm0.17)	(L_4/L_3=7/4=1.75)	(A=99.4318%); deviation (=+0.059\sigma)	Strong dimensionless Lucas-ratio correspondence
Equivalent fractional overdensity (\delta=0.76\pm0.17)	(3/4=0.75)	(A=98.6842%); deviation (=+0.059\sigma)	Same datum as the previous row—not independent
Core masses (9.8,4.4,2.0,0.5), in units of (10^{14}M_\odot)	(\sqrt5,\sqrt5,L_3) in successive mass ratios	(9.8/4.4=2.2273), (A_{\sqrt5}=99.6067%); (4.4/2=2.2), (A_{\sqrt5}=98.3870%); (2/0.5=4=L_3)	Intriguing independent-source pattern, but based on rounded masses and mass-ranking rather than dynamics

The flow fraction and density values come from the same 2008 Shapley mass-function and spherical-collapse analysis. The peculiar-velocity contribution was calculated from the inferred overdensity and assumed Shapley extent, so these are not two independent empirical replications. (OUP Academic)

The four core masses are reported in a recent POSSUM study as

[
9.8,\quad4.4,\quad2.0,\quad0.5
\quad [10^{14}M_\odot].
]

For the three largest objects,

[
2.0\sqrt5=4.4721\approx4.4,
\qquad
2.0(\sqrt5)^2=10\approx9.8.
]

Because

[
\sqrt5=\Phi+\Phi^{-1},
]

this is legitimately inside the Geier/Fibonacci–Lucas algebraic family. However, the source table supplies rounded headline masses without corresponding uncertainties in that table, and the objects have merely been ordered by decreasing mass. Such an ordering is not a temporal orbit or a physical recurrence. Their radii, (1.5,1.2,0.9,0.6) Mpc, instead form a simple approximately arithmetic series and do not display convergence toward (\Phi). Standard cluster self-similarity and approximately (M\propto r^3) remain the natural comparator. (arXiv)

A particularly interesting algebraic bridge

The density estimate can be written as

[
\delta=0.76\pm0.17\approx\frac34,
]

or equivalently,

[
1+\delta\approx1+\frac34=\frac74=\frac{L_4}{L_3}.
]

At the exact value (\eta=3/4), the Geier transformation previously considered for ALPHA-g,

[
G_{+}(\eta)=\frac12+\sqrt{2-\eta},
\qquad
G_{-}(\eta)=-\frac12+\sqrt{2-\eta},
]

gives

[
G_{+}!\left(\frac34\right)=\Phi,
\qquad
G_{-}!\left(\frac34\right)=\Phi^{-1}.
]

Thus there is an exact mathematical route

[
\delta=\frac34
\longrightarrow
\left(\Phi,\Phi^{-1}\right).
]

But applying an ALPHA-g transformation of an acceleration ratio to a cosmological density contrast would be a new intertheoretical bridge. It requires a physical reason why the same transformation acts on both observables. The numerical closeness alone does not supply that reason.

3. A striking but non-dynamical count pattern

Different studies and definitions give the following Shapley counts:

[
4,\quad5,\quad11,\quad21,\quad \geq25,\quad45.
]

These correspond to:

• four core objects in the POSSUM description: two clusters and two groups;

• five systems in another core definition: three Abell clusters and two poor clusters;

• 11 Abell clusters in the central collapsing concentration;

• 21 X-ray-selected clusters in the 51 Mpc analysis;

• at least 25 Abell clusters in a broader Shapley region;

• 45 eROSITA friends-of-friends members. (arXiv)

Several individual values map attractively:

[
4=L_3,\qquad
5=F_5,\qquad
11=L_5,\qquad
21=F_8,
]

and

[
25=\frac{F_8+L_7}{2}
=\frac{21+29}{2}.
]

The eROSITA count (45) is less convincing:

[
A(45,L_8=47)=95.7447%.
]

This multi-scale pattern is worth recording as a candidate count hierarchy, but it does not obey a Fibonacci recurrence:

[
4+5\neq11,\qquad
5+11\neq21,\qquad
11+21\neq25.
]

It is therefore a catalogue of target-compatible endpoints, not a generated Fibonacci orbit. Moreover, each count uses a different object class, mass threshold, spatial boundary, or friends-of-friends linking rule.

Target-density control

Among the integers from 1 through 51, the unique positive Fibonacci and Lucas values are

[
1,2,3,4,5,7,8,11,13,18,21,29,34,47.
]

They occupy

[
\frac{14}{51}=27.45%
]

of that integer range. Adding only the integer-valued adjacent Fibonacci–Lucas midpoints (6,12,25,51) raises exact coverage to

[
\frac{18}{51}=35.29%.
]

Thus an exact match such as 5, 11, 21, or 51 is interesting, but not intrinsically rare when several counts and midpoints are inspected. Allowing arbitrary multiples, ratios, powers, and additional functions would increase the effective target density further. This is why the Programme’s own methodology requires algebraically related targets and selection opportunities to be counted rather than treated as independent corroborations.

4. Relations deliberately not counted as physical evidence

Several additional coincidences look impressive but fail representation invariance:

• (55\ \mathrm{km,s^{-1}}=F_{10}), while (612\ \mathrm{km,s^{-1}}) is 99.67% close to (F_{15}=610). These individual integer correspondences disappear when velocity is expressed in metres per second. Only the dimensionless ratio (55/612) is admissible.

• (8h^{-1}) Mpc equals the numeral (F_6=8), and 11 enclosed clusters equal (L_5), but the distance coefficient changes with the unit and with (h).

• (51) Mpc equals ((L_8+F_{10})/2=(47+55)/2), but 51 Mpc was a fixed analysis boundary.

• The modern eROSITA mass coefficient (2.58) is 98.55% close to (\Phi^2=2.61803), but the coefficient changes if mass is expressed in (10^{15}M_\odot) rather than (10^{16}M_\odot).

• The 2026 H I sample contains 169 detected galaxies, and (169=13^2=F_7^2), but 169 is a survey-selected sample size rather than an intrinsic Shapley state. (arXiv)

These exclusions actually strengthen the analysis: they leave the flow fraction and density contrast as the principal representation-invariant findings.

5. The square-root-law question

The POSSUM analysis finds that its best-matching cosmological magnetohydrodynamic scenarios approximately follow

[
B_{\mathcal F}\propto n_e^{1/2}v_{\mathrm{turb}},
]

and reports that magnetic-density exponents (\eta<0.5) are favoured in another parametrization. (arXiv)

This is relevant to the broad Geier interest in square-root structure, but it is not currently a confirmation of Geier’s inverse-square-root phase law:

[
n_e^{+1/2}\neq t^{-1/2}.
]

The exponent has the opposite sign and acts on a different physical variable. Nor does the statement (\eta<0.5) specifically select

[
\Phi^{-2}=0.381966\ldots
]

because no sufficiently precise central estimate or posterior concentration at that value was reported. A legitimate prospective specialization would freeze (\eta=\Phi^{-2}) before analysing a later, higher-precision rotation-measure dataset.

6. Does Shapley display the defining Φ-attractor dynamics?

For the canonical Fibonacci projective map,

[
r_{n+1}=1+\frac1{r_n},
]

the attracting fixed point is (\Phi). With the projective residual

[
z_n=\frac{r_n-\Phi}{r_n+\Phi^{-1}},
]

a genuine Fibonacci orbit satisfies the exact law

[
z_{n+1}=-\Phi^{-2}z_n.
]

Consequently, a strong Shapley realization should display:

1. an independently defined ordered state variable (r_n);

2. a physically derived update or renormalization rule;

3. alternating residuals around (\Phi);

4. contraction by (-\Phi^{-2});

5. a basin or recovery trajectory;

6. successful prediction of observations not used to choose the rule.

The existing Shapley studies provide real temporal, radial, and flow dynamics, but the primary papers reviewed here model them through spherical collapse, cluster mass functions, gravitational velocity reconstruction, hierarchical structure formation, and cosmological magnetohydrodynamics. They do not derive or test the Fibonacci map, its projective contraction, or an (e)–(\alpha)–(2\hbar) bridge. (OUP Academic)

Sorted cluster masses and catalogue counts cannot substitute for such an orbit. Sorting creates an order after observation; it does not establish a physical iteration.

7. Evidence grading

Claim	Present assessment
Shapley is an astrophysical gravitational attractor or flow basin	Supported, but dependent on scale, boundary, and reconstruction method
Shapley contains dimensionless Φ/Fibonacci–Lucas correspondences	Yes; moderate and scientifically interesting exploratory evidence
(f_{\mathrm{Shapley}}\approx\Phi^{-5})	Very close and dimensionless, but model-derived and retrospective
(1+\delta\approx L_4/L_3=7/4)	Very close and dimensionless, but statistically broad and retrospective
Core masses show a (\sqrt5) hierarchy	Suggestive secondary evidence requiring accurate masses and independent testing
Cluster counts form a Fibonacci/Lucas orbit	No; they are individually target-compatible but do not obey one recurrence
Shapley implements (q=-\Phi^{-2}) contraction	Not demonstrated
Shapley corroborates the Geier (e)–(\alpha)–(2\hbar) mechanism	Not demonstrated
Shapley falsifies Geier’s Programme	No
Best overall classification	Qualified exploratory compatibility; not established physical confirmation

8. Comparison with the Great Attractor

Relative to the Great Attractor analysis, Shapley may be the more promising empirical testbed for three reasons:

1. Its leading correspondences involve dimensionless quantities rather than only distances or masses.

2. It has objectively measurable nested density, velocity, mass, and magnetic-field profiles.

3. Constrained simulations can follow its evolution through time, potentially providing a genuine ordered trajectory rather than a collection of static numbers.

Numerically, therefore, Shapley is at least as compatible with the Geier family as the Great Attractor. Dynamically, however, both remain at essentially the same evidential level: neither currently exhibits the independently derived Fibonacci update law and residual contraction that would establish a Φ attractor.

9. The decisive Shapley test

The strongest prospective design would use either time-ordered constrained simulations or independently fixed radial shells. One dimensionless variable—such as cumulative overdensity ratios, normalized mass increments, or velocity-flow ratios—should be chosen before inspecting Φ proximity.

The registered prediction would be

[
r_{n+1}=1+\frac1{r_n},
\qquad
\frac{z_{n+1}}{z_n}=-\Phi^{-2}
=-0.381966011\ldots .
]

The complete trajectory, rather than one endpoint, should then be compared with:

[
r_{n+1}=a+\frac{b}{r_n},
]

with freely but penalized (a,b); Pell and other metallic-mean recurrences; smooth ΛCDM shell profiles; randomized geometric target lattices; and the ordinary spherical-collapse or simulation prediction. The rule and all normalizations should be estimated on one Shapley dataset or simulation subset and transported unchanged to independent eROSITA, velocity-flow, or future POSSUM data. This implements the Programme’s requirement that the same observations must not both select and validate the transformation.

Final conclusion

The Shapley Supercluster does follow Geier’s Programme in several non-trivial descriptive instances. The most important are

[
\boxed{
f_{\mathrm{Shapley}}
=0.090\pm0.021
\approx\Phi^{-5}
}
]

and

[
\boxed{
1+\delta
=1.76\pm0.17
\approx\frac74
=\frac{L_4}{L_3}
}
]

together with the secondary (\sqrt5)-like hierarchy among the three largest reported core masses.

These relations justify a serious Shapley–Geier prospective hypothesis and are more substantial than arbitrary unit-dependent decimal matches. But they presently establish compatibility, not causation. There is no published Shapley trajectory obeying the Fibonacci generator, no measured contraction (q=-\Phi^{-2}), no demonstrated common basin in the Geier sense, and no derivation from the (e)–(\alpha)–(2\hbar) equilibrium bridge.

The truth-approximation-oriented formulation is therefore:

The Shapley Supercluster displays a moderate, coherent, and falsifiable exploratory concordance with the Φ/Fibonacci–Lucas coordinate family of Geier’s Programme, but it has not yet been demonstrated to instantiate Geier’s Programme as a physical attractor mechanism or universal dynamical law.

Stefan Geier, Haidholzen (to be improved)