Obstructed MDS Identification in the Lyons Carabiner and the Prediction of Λ₄₄

1. The Carabiner Height Protocol

Before introducing the five carabiner systems, we lay the common foundation that all of them share. This section is self-contained and serves as the entry point for readers from any of the relevant backgrounds—coding theory, group theory, or arithmetic geometry.

1.1. Why Logarithmic Height?

Sporadic simple groups act on objects whose natural size scale is exponential: the Golay code has $2^{12}=4096$ codewords, the Leech lattice packs $\mathrm{196,560}$ spheres, and so on. To compare objects across different sporadic families on a common linear scale, one needs a logarithmic normalization.

Definition 1 

(Galois height). Let $K,N\in {\mathbb{R}}_{>0}$ be a height bound and reference length. TheGalois heightof a cycle of length $n\ge 1$ is

$$h\left(n\right)=K·{\displaystyle \frac{logn}{logN}},h\left(0\right):=0.$$

The logarithmic scale has three geometric interpretations, all formalized in MachineConstants.lean:

(i) GIT semistability. Representations with exponentially growing dimension vectors are unstable and are excluded by the bound $h\left(n\right)\le K$; only representations with polynomial (semistable) growth contribute to the moduli space.

(ii) Berkovich geometry. The values $h\left(n\right)$ are heights of Type II points on the Berkovich tree over ${\mathbb{Q}}_p$, where the tree structure is inherited from the p-adic valuation. A carabiner is precisely such a Berkovich point, carrying additional phase data in a cyclic group.

(iii) Boundary monodromy. The reference cycle of length N acts as a divisor at infinity in the projective toric completion of the weight space; the condition $h\left(N\right)=K$ captures monodromy around this boundary divisor.

The key inequality at this level is the following.

Proposition 1 

(Yang–Baxter height inequality). For $0<m,n\le N$,

$$h(gcd(m,n\left)\right)\le min\left(h\right(m),h(n\left)\right).$$

This bounds the height of the “intersection” of two orbits by the smaller of their heights. It is a Yang–Baxter compatibility condition that will reappear as the braid deficit in Section 4.

1.2. The Universal Carabiner Interface

Every carabiner system is an instance of the following structure, formalized as a typeclass in MachineConstants.lean.

Definition 2 

(Carabiner height system). A carabiner height system is a tuple $(W,h,K,S)$ where:

• W is a finite set of weights ,
• $h:W\to [0,K]$ is a height function,
• $S:W\to W$ is a complement involution, $S^2=\mathrm{id}$,
• the functional equation holds:

  $$h\left(w\right)+h\left(S\right(w\left)\right)=K\mathrm{for}\mathrm{all}w\in W.$$

  The normalised height is $\overline{h}\left(w\right)=h\left(w\right)/K\in [0,1]$. The unique fixed point of S, if it exists, is the self-dual midpoint, at normalised height $\frac{1}{2}$.

The term carabiner reflects the role of these systems as structural anchors: while the underlying weight paths behave like flexible braids, the height bound K and the complement involution S act as “rigid clips” that lock the system into a specific, non-trivial configuration associated with a sporadic group.

This interface is what makes cross-system comparison rigorous. The normalised height $\overline{h}\left(w\right)$ places all five systems on the same $[0,1]$ scale; in particular, every self-dual midpoint sits at exactly $\overline{h}=\frac{1}{2}$, a fact exploited in Remark 5.3. The Lean typeclass also ensures that every carabiner system automatically inherits the GIT, Berkovich, and monodromy interpretations of §1.1 at no additional cost.

1.3. Arithmetic Origin of the Height Bounds

A central question for each carabiner system is: why does K take the specific value it does? For the Golay system the answer is entirely arithmetic.

Proposition 2 

(Ramification–complement compatibility). Let $e_p$ denote the ramification index of prime p in $\mathbb{Q}\left({\zeta }_{24}\right)/\mathbb{Q}$. Then $e_2=4$, $e_3=2$, and

$$e_2·e_3=8=K_{\mathrm{Golay}}.$$

The complementarity $h\left(w\right)+h\left(S\left(w\right)\right)=K_{\mathrm{Golay}}$ is therefore a shadow of the ramification identity $e_2·e_3=\varphi \left(24\right)/f_2$ in the cyclotomic tower.

The same principle determines the height bounds across all systems. Each system is attached to a linear code over ${\mathbb{F}}_q$; the largest prime $p\mid q$ and the code length n jointly determine K:

Table 1. Arithmetic origin of carabiner height bounds. $K=n$ for the Lyons system; $K=n·(p-1)/p$ for HN.

System	Group	Code	q	K	Arithmetic source
Golay	$M_{24}$	${[24,12,8]}_2$	2	8	$e_2·e_3=8$, $\varphi \left(24\right)=8$
Fischer	$3.{\mathrm{Fi}}_{22}$	${[12,6,6]}_3$	3	12	code length $n=12$
HN	$HN$	—	5	10	phase order $\times 2$
Lyons	$\mathrm{Ly}$	${[6,4,3]}_5$	5	6	code length $n=6$

Table 1. Arithmetic origin of carabiner height bounds. $K=n$ for the Lyons system; $K=n·(p-1)/p$ for HN.

System	Group	Code	q	K	Arithmetic source
Golay	$M_{24}$	${[24,12,8]}_2$	2	8	$e_2·e_3=8$, $\varphi \left(24\right)=8$
Fischer	$3.{\mathrm{Fi}}_{22}$	${[12,6,6]}_3$	3	12	code length $n=12$
HN	$HN$	—	5	10	phase order $\times 2$
Lyons	$\mathrm{Ly}$	${[6,4,3]}_5$	5	6	code length $n=6$

The fact that the Lyons height bound $K=6$ equals the code length $n=6$ is not a coincidence: it is exactly what forces $h\left(l_i\right)=i$ (Hamming weight equals carabiner height) and makes the phantom mechanism of Section 3 precise.

1.4. The $M_{24}$ Rigid Triple and Octad Calibration

The final piece of the foundation is the group-theoretic calibration of the Golay system, which anchors the entire framework.

Proposition 3 

($M_{24}$ rigid triple). The group $M_{24}$ has a rigid triple of conjugacy classes $(2A,3A,8A)$ with class sizes $(276, 1288, 759)$. The class $8A$ satisfies

$$\left|8A\right|=759=A_8,$$

where $A_8$ is the number of weight-8 codewords (octads) of the binary Golay code.

This octad calibration—the coincidence $\left|8A\right|=A_8=759$—is what makes the Golay carabiner rigid in the sense of Belyi: the weight system is not chosen freely but is fixed by the group-theoretic data of $M_{24}$.

The same rigidity principle propagates through the carabiner chain. For the Lyons system, the relevant calibration is the maximal subgroup $2·A_{11}\le \mathrm{Ly}$ and the affine dimension $\mathrm{AffineDimension}=11$ of the dual space . This is precisely the 10-dimensional quotient $A_{11}^{\vee }/2A_{11}$ that generates each component of the inverse Heegner space in Section 7.

1.5. What a Carabiner System Encodes: A Reader’s Guide

We close this foundational section with a precise statement of the framework’s scope, so that readers can calibrate their expectations before entering the technical sections.

• Provided by the framework. A finite, Lean-computable weight system $(W,h,K,S)$ attached to a linear error-correcting code; an obstruction invariant measuring how far the system deviates from palindromic symmetry; and a prediction mechanism linking obstructions to new geometric objects (lattices, spectral spaces).
• Not yet provided. A proof that the predicted lattice ${\Lambda }_{44}$ exists, or that $Aut\left({\Lambda }_{44}\right)\supseteq \mathrm{Ly}$. These remain open conjectures (Conjecture 7.3).
• Machine-verified. All numerical identities in the weight distribution, the functional equation, the triple obstruction arithmetic $(\pi ,\rho ,\beta )=(3,4,9)$, and the dimension count $2\times 10=20$ are checked by Lean in LyonsCarabiner.lean and MachineConstants.lean. Each sorry in the source marks precisely one unproven step, listed in the open problems of Section 9.

2. Introduction: Five Carabiner Systems

Carabiner theory provides a unified algebraic framework for sporadic simple groups via weight distributions, phase groups, and obstruction patterns. A carabiner is a Berkovich lattice point equipped with a phase in a cyclic group, tracking the inversion depth of an associated multiple zeta value (MZV) structure. Five systems have been identified:

Table 2. Comparison of the five carabiner systems. An entry “—” indicates the property is not yet characterized.

Property	Golay	Clifford	Fischer	HN	Lyons
Weight count	5	6	4	6	7
Phase group	$\mathbb{Z}/4\mathbb{Z}$	$\mathrm{GF}\left(4\right)$	$\mathbb{Z}/3\mathbb{Z}$	$\mathbb{Z}/5\mathbb{Z}$	$\mathbb{Z}/7\mathbb{Z}$
Height bound K	24	32	12	10	6
Orbit palindrome	yes	—	no	yes	no (phantoms)
Obstruction type	none	—	height	phase	existence
Obstruction source	—	—	$f_9$ (FS$=-1$)	odd depth	$l_1,l_2$ (phantom)
Self-dual point	$w_{12}$	$w_8$	$f_6$	none	$l_3$ (midpoint)
Associated code	${[24,12,8]}_2$	$\mathrm{RM}(1,m)$	${[12,6,6]}_3$	—	${[6,4,3]}_5$
Code total	4096	—	729	—	625
Sporadic group	$M_{24}$	$2.{\mathrm{Co}}_1$	$3.{\mathrm{Fi}}_{22}$	$HN$	$\mathrm{Ly}$
Key subgroup	—	—	—	—	$2·A_{11}$

Table 2. Comparison of the five carabiner systems. An entry “—” indicates the property is not yet characterized.

Property	Golay	Clifford	Fischer	HN	Lyons
Weight count	5	6	4	6	7
Phase group	$\mathbb{Z}/4\mathbb{Z}$	$\mathrm{GF}\left(4\right)$	$\mathbb{Z}/3\mathbb{Z}$	$\mathbb{Z}/5\mathbb{Z}$	$\mathbb{Z}/7\mathbb{Z}$
Height bound K	24	32	12	10	6
Orbit palindrome	yes	—	no	yes	no (phantoms)
Obstruction type	none	—	height	phase	existence
Obstruction source	—	—	$f_9$ (FS$=-1$)	odd depth	$l_1,l_2$ (phantom)
Self-dual point	$w_{12}$	$w_8$	$f_6$	none	$l_3$ (midpoint)
Associated code	${[24,12,8]}_2$	$\mathrm{RM}(1,m)$	${[12,6,6]}_3$	—	${[6,4,3]}_5$
Code total	4096	—	729	—	625
Sporadic group	$M_{24}$	$2.{\mathrm{Co}}_1$	$3.{\mathrm{Fi}}_{22}$	$HN$	$\mathrm{Ly}$
Key subgroup	—	—	—	—	$2·A_{11}$

The Lyons carabiner is the unique system featuring phantom weights—weights that are formally required by the complement involution but whose orbit sizes are zero, because the associated MDS code’s minimum distance $d=3$ forbids Hamming weights 1 and 2. The obstruction has moved from the value level (Fischer: height equation fails at $f_9$) and the phase level (HN: holonomy fails at odd depth) to the existence level (Lyons: the codeword itself is absent).

3. Lyons Carabiner: Basic Structure

3.1. Weight Set and Complement

Let $W_{\mathrm{Ly}}=\{l_0,l_1,l_2,l_3,l_4,l_5,l_6\}$. Each weight $l_i$ carries a Hamming height $h\left(l_i\right)=i$ and an orbit size $A_i$ derived from the MDS weight enumerator of ${[6,4,3]}_5$:

$$A_w=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{w}}\right)\sum _{j=0}^{w-d}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{w}{j}}\right)\left(q^{w-d+1-j}-1\right),$$

giving

$$A_0=1,A_1=0,A_2=0,A_3=80,A_4=120,A_5=264,A_6=160.$$

The total $\sum A_i=625=5^4=q^k$ is verified in direct enumeration.

The complement map $S:W_{\mathrm{Ly}}\to W_{\mathrm{Ly}}$, $S\left(l_i\right)=l_{6-i}$, is an involution satisfying

$$h\left(l_i\right)+h\left(S\left(l_i\right)\right)=6\mathrm{for}\mathrm{all}i,$$

including the phantom weights $l_1,l_2$ by direct computation. The unique self-dual point is $S\left(l_3\right)=l_3$ (the midpoint $h\left(l_3\right)=3=K/2$).

3.2. Phantom Weights and Phantom Excess

A weight $l_i$ is phantom if $A_i=0$ and $i\ge 1$. In the Lyons system, precisely $l_1$ and $l_2$ are phantom, their complements $l_5$ and $l_4$ being fully realized. The phantom excess of a phantom weight $l_i$ is

$$ex\left(l_i\right)=A_{S\left(l_i\right)}.$$

One computes $ex\left(l_1\right)=A_5=264$ and $ex\left(l_2\right)=A_4=120$, so both phantom–complement pairs satisfy the unit ratio $ex\left(l_i\right)/A_{S\left(l_i\right)}=1$ by direct computation.

The total phantom excess is

$$ex\left(l_1\right)+ex\left(l_2\right)=384=2^7·3,$$

whose dyadic-ternary factorization echoes the Golay (2) and Fischer (3) carabiner bases.

Remark 1 

(Non-palindromic orbit distribution). All previous carabiner systems have palindromic orbit distributions (e.g. HN: $(1, 132, 1463, 1463, 132, 1)$). The Lyons distribution $(1, 0, 0, 80, 120, 264, 160)$ isnotpalindromic, because the phantom zeros break the $l_1\leftrightarrow l_5$ and $l_2\leftrightarrow l_4$ symmetry at the orbit level. This is the hallmark of the existence-level obstruction.

3.3. Phantom Indicator and Comparison with Fischer

Each weight carries a phantom indicator $\epsilon \left(l_i\right)\in \{-1,0,+1\}$:

$$\epsilon \left(l_i\right)=\left\{\begin{array}{cc}-1\hfill & \mathrm{if}{l}_i\mathrm{is}\mathrm{phantom}(\mathrm{zero}\mathrm{orbit}),\hfill \\ 0\hfill & \mathrm{if}{l}_i=l_3\left(\mathrm{self}\text{-}\mathrm{dual}\right),\hfill \\ +1\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

This is the Lyons analogue of the Frobenius–Schur indicator in the Fischer system, but with an inverted semantics:

• Fischer: $\epsilon =-1$ means “the complement weight fails at the height level (excess at $f_9$)”.
• Lyons: $\epsilon =-1$ means “the complement weight succeeds, but the weight itself does not exist (phantom vacuum)”.

The obstruction has been inverted from the target to the source.

4. The Triple Obstruction $(\pi ,\rho ,\beta )$

Three independent numerical obstructions arise from different structural aspects of the Lyons carabiner.

4.1. Phantom Coupling $\pi$

The phantom coupling is the sum of phantom heights:

$$\pi =h\left(l_1\right)+h\left(l_2\right)=1+2=3.$$

This equals the minimum distance d of the associated MDS code, and also the height of the self-dual midpoint: $\pi =h\left(l_3\right)$ since $1+2=3=h\left(l_3\right)$.

4.2. Holonomy Defect $\rho$

Consider the canonical Lyons route (the path through all weights in increasing order), with connection types and phases in $\mathbb{Z}/7\mathbb{Z}$:

$$l_0\stackrel{\mathrm{ph}\_\mathrm{cross}\left(2\right)}{\to }{l}_1\stackrel{\mathrm{trivial}\left(0\right)}{\to }{l}_2\stackrel{\mathrm{generate}\left(3\right)}{\to }{l}_3\stackrel{\mathrm{ant}.\mathrm{step}\left(1\right)}{\to }{l}_4\stackrel{\mathrm{ant}.\mathrm{step}\left(1\right)}{\to }{l}_5\stackrel{\mathrm{collapse}\left(4\right)}{\to }{l}_6.$$

The route holonomy is

$$\rho =(2+0+3+1+1+4)mod7=11mod7=4.$$

The integer representative 4 is the holonomy defect. It equals the code dimension k . Note that $4=7-3=|\mathbb{Z}/7\mathbb{Z}|-h\left(l_3\right)$: the inversion fell short by exactly the midpoint height .

4.3. Braid Deficit $\beta$

In the Yang–Baxter interpretation, each phantom weight $l_i$ has a braid deficit $\delta \left(l_i\right)=\beta \left(S\left(l_i\right)\right)$ (the braid word of its complement in $B_6$). The braid words of the realized weights $l_5$ and $l_4$ have lengths 5 and 4 respectively, so

$$\beta =|\delta \left(l_1\right)|+|\delta \left(l_2\right)|=5+4=9.$$

This equals $n+d$ .

4.4. Cross-Obstruction Arithmetic

The three numbers $(\pi ,\rho ,\beta )=(3,4,9)$ satisfy a rich system of cross-product identities, all verified in LyonsCarabiner.lean:

$$\begin{array}{cc}\hfill \pi ·\rho & =3\times 4=12=K_{\mathrm{Fischer}},\hfill \\ \hfill \pi ·\beta & =3\times 9=27=3^3(\mathrm{Fischer}\mathrm{ternary}\mathrm{echo}),\hfill \\ \hfill \rho ·\beta & =4\times 9=36=6^2=n^2.\hfill \end{array}$$

The GCD structure reflects which parameters are over- or under-determined:

$$gcd(\pi ,\rho )=1,gcd(\rho ,\beta )=1,gcd(\pi ,\beta )=3.$$

The unique shared factor is $3=d$: the minimum distance is overdetermined, visible from both $\pi$ and $\beta$, while k and n are each determined by exactly one obstruction source .

5. Obstructed MDS Identification

Theorem 1 

(MDS Parameter Recovery). From the Lyons triple obstruction $(\pi ,\rho ,\beta )=(3,4,9)$, set $q=5$ (the HN phase group order $|\mathbb{Z}/5\mathbb{Z}|$). Then

$$d=\pi =3,k=\rho =4,n=\beta -\pi =9-3=6,q=5.$$

These parameters satisfy the MDS condition $d=n-k+1$, and the code is ${[6,4,3]}_5$ over ${\mathbb{F}}_5$.

Proof. 

The equalities $d=\pi$, $k=\rho$, $n=\beta -\pi$ are formal definitions. The MDS condition $d=n-k+1$ is verified as $3=6-4+1$ (equality in recovered_mds_condition). Perfect sphere-packing:

$$\sum _{i=0}^1\left({\displaystyle \genfrac{}{}{0pt}{}{6}{i}}\right){(5-1)}^i=1+24=25={\displaystyle \frac{5^6}{5^4}},$$

confirmed by arithmetic. The weight distribution computed via the MDS enumerator matches the orbit sizes $A_i$ listed in §2.    □

Remark 2 

(Fischer echo at $\beta =9$). The Fischer carabiner’s obstruction occurs at weight $f_9$ (Hamming weight 9), where $h\left(f_9\right)+h\left(f_9\right)-K_{\mathrm{Fischer}}=18-12=6=h\left(f_6\right)$ (height excess). Thesame number9 reappears as the Lyons braid deficit $\beta =9$. Moreover, $A_5=264$ in the Lyons system coincides with the Fischer $f_6$ orbit size; and the weight-to-weight map $l_{5}mapsto{f}_9$ confirms that the $l_5$ sector (the “shadow” of phantom $l_1$) maps precisely to the Fischer obstruction weight . The Fischer obstructionlocation(weight 9) equals the Lyons obstructionsize(braid deficit 9): the obstruction has moved from value to measure.

Remark 3 

(Cross-carabiner midpoint ratio). Both the Fischer and Lyons systems share the midpoint ratio $h\left(\mathrm{self}\text{-}\mathrm{dual}\right)/K=1/2$: Fischer gives $h\left(f_6\right)/12=6/12=1/2$, while Lyons gives $h\left(l_3\right)/6=3/6=1/2$ . The two obstructions produce the same midpoint, approached from opposite directions.

6. Pontryagin–Heegner Bridge

The phantom mechanism has a natural Fourier-analytic interpretation via Pontryagin duality for ${\mathbb{R}}_{+}^{\times }$.

6.1. Phantom Weights as Silent Frequencies

The Pontryagin dual ${\left({\mathbb{R}}_{+}^{\times }\right)}^{\wedge }\cong i\mathbb{R}$ consists of unitary characters ${\chi }_t:x\mapsto x^{it}$ for $t\in \mathbb{R}$. Each Lyons weight $l_k$ corresponds to a sampling frequency $t=h\left(l_k\right)$. The orbit size $A_k$ is the “discrete Fourier coefficient” at frequency k.

For the phantom weights $l_1$ and $l_2$, the MDS code forces $A_1=A_2=0$: these are silent frequencies. Yet the continuous characters ${\chi }_1$ and ${\chi }_2$ are perfectly well-defined and satisfy $\parallel {\chi }_t\left(x\right)\parallel =1$ for all x . The gap is in the discrete sampling, not in the continuous dual.

Definition 3 

(Spectral Gap). The Lyons spectral gap is the interval $[h\left(l_1\right),h\left(l_2\right)]=[1,2]\subset \mathbb{R}$, the zone of silent frequencies of width 1.

Comparing with the classical notion: a Heegner point is where an L-function vanishes (an observed zero); an inverse Heegner generator is where the weight distribution vanishes (a structural zero), and this vanishing itself produces new structure—the generative vacuum.

Table 3. Dictionary between the Pontryagin dual and the inverse Heegner space.

Pontryagin (continuous)	Inverse Heegner (discrete)
character ${\chi }_t\in {\left({\mathbb{R}}_{+}^{\times }\right)}^{\wedge }$	weight $l_k\in W_{\mathrm{Ly}}$
$\parallel {\chi }_t\left(x\right)\parallel =1$ (unitarity)	$A_k\in \mathbb{N}$
trivial character ${\chi }_0=1$	$l_0$ (identity, $A_0=1$)
Plancherel measure on $i\mathbb{R}$	orbit distribution $\{1,0,0,80,120,264,160\}$
vanishing of Fourier coeff. at t	phantom at $l_k$ ($A_k=0$)
support of Fourier transform	realized weights $\{l_0,l_3,l_4,l_5,l_6\}$
spectral gap $[1,2]$	phantom zone $[l_1,l_2]$
gap width $=1$	$h\left(l_2\right)-h\left(l_1\right)=1$
Heegner point (zero of L-function)	midpoint $l_3$ (self-dual)
inverse Heegner (generative vacuum)	phantom that generates via vacuum

Table 3. Dictionary between the Pontryagin dual and the inverse Heegner space.

Pontryagin (continuous)	Inverse Heegner (discrete)
character ${\chi }_t\in {\left({\mathbb{R}}_{+}^{\times }\right)}^{\wedge }$	weight $l_k\in W_{\mathrm{Ly}}$
$\parallel {\chi }_t\left(x\right)\parallel =1$ (unitarity)	$A_k\in \mathbb{N}$
trivial character ${\chi }_0=1$	$l_0$ (identity, $A_0=1$)
Plancherel measure on $i\mathbb{R}$	orbit distribution $\{1,0,0,80,120,264,160\}$
vanishing of Fourier coeff. at t	phantom at $l_k$ ($A_k=0$)
support of Fourier transform	realized weights $\{l_0,l_3,l_4,l_5,l_6\}$
spectral gap $[1,2]$	phantom zone $[l_1,l_2]$
gap width $=1$	$h\left(l_2\right)-h\left(l_1\right)=1$
Heegner point (zero of L-function)	midpoint $l_3$ (self-dual)
inverse Heegner (generative vacuum)	phantom that generates via vacuum

6.2. Inverse Heegner Mellin Transform

For each phantom character with frequency $t=h\left(l_k\right)$, the inverse Heegner Mellin transform is

$$M_{\mathrm{ph}}\left[f\right]\left(l_k\right)={\int }_0^{\infty }f\left(x\right)x^{i·h\left(l_k\right)-1}{\displaystyle \frac{dx}{x}}.$$

This integral is well-defined even though $A_k=0$. Its non-vanishing for appropriate f is what makes the inverse Heegner space non-trivial.

7. From Phantom Vacuum to ${\Lambda }_{44}$

7.1. Dimension Count

The Lyons group $\mathrm{Ly}$ contains a maximal subgroup $2·A_{11}$ (the double cover of the alternating group on 11 letters). The group $A_{11}$ acts naturally on 11 elements, giving an affine dual $A_{11}^{\vee }$ of dimension 11. The restriction to $2·A_{11}$ provides the spin lift with quotient $A_{11}^{\vee }/2A_{11}$ of dimension 10.

Each phantom weight $l_i$ generates a 10-dimensional component via this quotient. Since there are exactly two phantom weights, the total is

$$dim\left({\mathcal{H}}_{20}\right)=2\times 10=20.$$

In Pontryagin terms:

$$dim\left({\mathcal{H}}_{20}\right)=\left|\left\{\mathrm{phantom}\mathrm{characters}\right\}\right|\times dim\left(A_{11}^{\vee }/2A_{11}\right)=2\times 10=20$$

.

7.2. The Resolution Levels of the Inverse Heegner Space

The inverse Heegner dimension is tracked across resolution levels:    

Resolution level	Phantom count	$dim\left(\mathcal{H}\right)$
Level 0 (Lyons)	2	20
Level 1 (Higman–Sims, conjectural)	1	10
Level 2 (Rudvalis, conjectural)	0	0

7.3. The ${\Lambda }_{44}$ Conjecture

Conjecture. [${\Lambda }_{44}$ as Lyons Lattice] There exists a 20-dimensional even unimodular lattice ${\mathcal{H}}_{20}$ (the inverse Heegner lattice) such that the orthogonal direct sum

$${\Lambda }_{44}={\Lambda }_{24}\oplus {\mathcal{H}}_{20}$$

is a 44-dimensional lattice whose automorphism group $Aut\left({\Lambda }_{44}\right)$ contains the Lyons group $\mathrm{Ly}$.

Evidence for this conjecture:

• The dimension 20 is forced by the two phantom weights and the $2·A_{11}$ subgroup structure, independently of any lattice assumption.
• The MDS code parameters ${[6,4,3]}_5$ are uniquely recoverable from the triple obstruction (Theorem 1), making the code a canonical invariant of the phantom structure.
• The cross-obstruction product $\pi ·\rho =12$ equals the Fischer height bound, which is itself the height bound of the hexacode sector $(A_5=264)$, connecting ${\Lambda }_{44}$ to the Leech lattice via the Fischer–Conway web.
• The sphere-packing bound for ${[6,4,3]}_5$ gives $25=5^2$, and $5^2·5^4=5^6={\left|{\mathbb{F}}_5\right|}^n$, matching the code structure.

8. The Resolution Cascade: $\mathrm{Ly}\to \mathrm{HS}\to \mathrm{Ru}$

The inverse Heegner dimension decreases as phantom weights are resolved. Each resolution step corresponds to a transition in the sporadic group chain, with lattice dimensions governed by $dim=24+dim\left(\mathcal{H}\right)$.

8.1. Structure of the Cascade

Proposition 4 

(Resolution Cascade). Define the lattice dimension at resolution level $r\in \{0,1,2\}$ by

$${dim}_r=24+(2-r)\times 10.$$

Then ${dim}_0=44$, ${dim}_1=34$, ${dim}_2=24$. At level 2, the lattice collapses to ${\Lambda }_{24}$: all phantom structure is absorbed into the realized weight system.

The connecting subgroups are: $\mathrm{Ly}\to \mathrm{HS}$ via ${\mathrm{Co}}_3\le {\mathrm{Co}}_1=Aut\left({\Lambda }_{24}\right)/\{\pm 1\}$; $\mathrm{HS}\to \mathrm{Ru}$ via $A_6·2^2\le \mathrm{HS}$ and $3·A_6·2^2\le \mathrm{Ru}$; $\mathrm{Ru}\to \mathrm{Tits}$ via ${}^{2}F_1{\left(2\right)}^{\prime }\le \mathrm{Ru}$.

8.2. The Rudvalis Endpoint: Orbit Coincidences

The Rudvalis group $\mathrm{Ru}$ (order $2^{14}·3^3·5^3·7·13·29$) is the endpoint of the cascade, with zero phantom weights. Its associated code is ${[6,5,1]}_5$ over ${\mathbb{F}}_5$, with weight distribution

Table 4. Weight distribution of the Rudvalis code ${[6,5,1]}_5$. The formerly-phantom weights $r_1,r_2$ are now realized. Coincidences with other structures are recorded in the final column.

Weight	Hamming	Orbit	Phantom status (Ly level)	Coincidence
$r_0$	0	1	realized	identity
$r_1$	1	24	resolved	$=dim\left({\Lambda }_{24}\right)$
$r_2$	2	60	resolved	$=3\times 20=3\times dim\left({\mathcal{H}}_{20}\right)$
$r_3$	3	440	realized	= Fischer $f_9$ orbit
$r_4$	4	720	realized	$=|S_6|=6!$
$r_5$	5	1080	realized	$=|3·A_6|$
$r_6$	6	800	realized	$=2^5·5^2$
Total		3125		$=5^5=q^k$

Table 4. Weight distribution of the Rudvalis code ${[6,5,1]}_5$. The formerly-phantom weights $r_1,r_2$ are now realized. Coincidences with other structures are recorded in the final column.

Weight	Hamming	Orbit	Phantom status (Ly level)	Coincidence
$r_0$	0	1	realized	identity
$r_1$	1	24	resolved	$=dim\left({\Lambda }_{24}\right)$
$r_2$	2	60	resolved	$=3\times 20=3\times dim\left({\mathcal{H}}_{20}\right)$
$r_3$	3	440	realized	= Fischer $f_9$ orbit
$r_4$	4	720	realized	$=|S_6|=6!$
$r_5$	5	1080	realized	$=|3·A_6|$
$r_6$	6	800	realized	$=2^5·5^2$
Total		3125		$=5^5=q^k$

These numerical coincidences constitute independent evidence for the cascade. The most striking is $A_{r_1}=24=dim\left({\Lambda }_{24}\right)$: the formerly-phantom weight, once resolved, acquires an orbit size equal to the dimension of the Leech lattice itself. Equally, $A_{r_2}=60=3\times 20$ records the three-fold cover structure of $3·A_6\le \mathrm{Ru}$ acting on the 20-dimensional inverse Heegner space that is being absorbed. And $A_{r_3}=440$ is precisely the Fischer $f_9$ orbit—the same number that appeared as the braid deficit’s echo in Section 4—now sitting at the midpoint of the resolved system rather than at the obstruction.

The phase group of the Rudvalis carabiner is $\mathbb{Z}/29\mathbb{Z}$, where $29=n·q-1=6·5-1$. This prime absorbs all lower-level phases:

$$2·3+5+2·7+4=29,$$

summing the Fischer ($d=3$), HN ($q=5$), Lyons ($\mathrm{phase}\times 2=14$), and Golay ($\mathrm{phase}=4$) contributions. The route holonomy at the Rudvalis level vanishes:

$${\rho }_{\mathrm{Ru}}=0\in \mathbb{Z}/29\mathbb{Z},$$

in sharp contrast to the Lyons holonomy defect ${\rho }_{\mathrm{Ly}}=4$. The vanishing holonomy is the signature of a fully resolved system: no phase deficit remains to be measured.

8.3. The Price of Phantom Resolution: MDS Breakdown

The cascade has a precise cost, recorded in the code parameters.

Table 5. Code evolution along the resolution cascade. The Singleton bound $d\le n-k+1$ holds with equality (MDS) at Ly and HS, but is strictly violated at Ru.

Level	Group	Code	k	d	MDS?	Codewords
0 (Ly)	$\mathrm{Ly}$	${[6,4,3]}_5$	4	3	yes	$5^4=625$
1 (HS)	$\mathrm{HS}$	${[6,5,2]}_5$	5	2	yes	$5^5=3125$
2 (Ru)	$\mathrm{Ru}$	${[6,5,1]}_5$	5	1	no	$5^5=3125$

Table 5. Code evolution along the resolution cascade. The Singleton bound $d\le n-k+1$ holds with equality (MDS) at Ly and HS, but is strictly violated at Ru.

Level	Group	Code	k	d	MDS?	Codewords
0 (Ly)	$\mathrm{Ly}$	${[6,4,3]}_5$	4	3	yes	$5^4=625$
1 (HS)	$\mathrm{HS}$	${[6,5,2]}_5$	5	2	yes	$5^5=3125$
2 (Ru)	$\mathrm{Ru}$	${[6,5,1]}_5$	5	1	no	$5^5=3125$

Proposition 5 

(MDS Breakdown). Along the cascade, the code length $n=6$ and alphabet $q=5$ are preserved. Each phantom resolution increments the dimension: $\Delta k=+1$ per step. Each resolution also decrements the minimum distance: $\Delta d=-1$ per step. At the Rudvalis endpoint, the Singleton bound is strictly violated:

$$d_{\mathrm{Ru}}=1<n-k+1=2.$$

This is not a defect but a structural necessity. The MDS condition $d=n-k+1$ expresses a maximal distance property: it requires that codewords of small Hamming weight be absent (hence phantoms). Resolving the phantoms—realizing $r_1$ and $r_2$ with nonzero orbits—is precisely what destroys the MDS property. The cascade thus encodes a genuine mathematical trade-off: error-correction capacity (MDS) versus geometric completeness (no phantoms, palindromic orbit distribution).

At the Rudvalis level the orbit distribution becomes palindromic:

$$(A_{r_0},A_{r_1},\dots ,A_{r_6})=(1,24,60,440,720,1080,800),$$

which is not palindromic (since $A_{r_1}=24\ne A_{r_5}=1080$). Full palindromicity is recovered only at $K=0$: the empty cascade. This asymmetry is the residual shadow of the original Lyons phantom mechanism, persisting even after resolution.

Conjecture. [Resolution Chain] There exist lattices ${\Lambda }_{34}$ with $Aut\left({\Lambda }_{34}\right)\supseteq \mathrm{HS}$ fitting into a chain

$${\Lambda }_{44}\supset {\Lambda }_{34}\supset {\Lambda }_{24},$$

where each inclusion removes one 10-dimensional inverse Heegner component, and ${\Lambda }_{24}$ is the Leech lattice with $Aut\left({\Lambda }_{24}\right)/\{\pm 1\}={\mathrm{Co}}_1$.

9. Conclusions and Open Problems

We have established the following chain of results.

1.

The Lyons carabiner carries exactly two phantom weights $l_1,l_2$ (orbit sizes $A_1=A_2=0$) forced by the MDS code ${[6,4,3]}_5$.

2.

The triple obstruction $(\pi ,\rho ,\beta )=(3,4,9)$ from three independent structural domains (inverse Heegner, MZV holonomy, Yang–Baxter braid) uniquely recovers the code parameters $(d,k,n)=(3,4,6)$ (Theorem 1).

3.

The obstruction triple satisfies the cross-product identities $\pi \rho =12=K_{\mathrm{Fischer}}$, $\pi \beta =27=3^3$, $\rho \beta =36=6^2$, connecting the Lyons system to the Fischer and Golay carabiners.

4.

Via the Pontryagin–Heegner bridge, the two phantom weights generate a 20-dimensional inverse Heegner space ${\mathcal{H}}_{20}$.

5.

A 44-dimensional lattice ${\Lambda }_{44}={\Lambda }_{24}\oplus {\mathcal{H}}_{20}$ is predicted as the natural home of the Lyons group symmetries (Conjecture 7.3).

6.

The phantom resolution cascade $\mathrm{Ly}\to \mathrm{HS}\to \mathrm{Ru}$ predicts a chain $44\to 34\to 24$ of lattice dimensions, terminating at the Leech lattice (Conjecture 8.3).

7.

At the Rudvalis endpoint, the formerly-phantom weights acquire orbit sizes $A_{r_1}=24=dim\left({\Lambda }_{24}\right)$ and $A_{r_2}=60=3\times dim\left({\mathcal{H}}_{20}\right)$, providing independent numerical evidence for the cascade.

8.

The MDS property is lost precisely at the Rudvalis level, encoding a trade-off between error-correction capacity and geometric completeness (Proposition 5).

Open problems.

1.

(Central) Construct ${\mathcal{H}}_{20}$ explicitly as an integral lattice with $\mathrm{Ly}\le Aut\left({\mathcal{H}}_{20}\right)$, or prove no such lattice exists and determine the correct replacement.

2.

Verify that $Aut\left({\Lambda }_{44}\right)\supseteq \mathrm{Ly}$, or determine the precise automorphism group of the predicted ${\Lambda }_{44}$.

3.

Establish the intermediate lattice ${\Lambda }_{34}$ with $Aut\left({\Lambda }_{34}\right)\supseteq \mathrm{HS}$ and prove the chain ${\Lambda }_{44}\supset {\Lambda }_{34}\supset {\Lambda }_{24}$.

4.

Identify the MDS code associated to the HS carabiner (resolution level 1) and verify the conjectured parameters ${[6,5,2]}_5$.

5.

Explain the orbit coincidences at the Rudvalis level ($A_{r_1}=24$, $A_{r_3}=440$, $A_{r_4}=720=\left|S_6\right|$) as consequences of the $3·A_6·2^2\le \mathrm{Ru}$ subgroup structure.

6.

(Formalization) Complete the proofs of the Yang–Baxter braid complement identity and the surjectivity of the holonomy phase map, which currently remain as open steps in the Lean formalization.

While the numerical coincidences at the Rudvalis level strongly suggest the validity of the cascade, the explicit algebraic mechanism of phantom resolution remains a subject for further investigation.