Voting Rights: Anomaly Free Boards

The following are the only anomaly free Board structures possible for board sizes 1, 4, 7, 11 and 13.

Note that Corporate Governance research says that anomaly free boards are in 4D and 11D+ and anything in between doesn't work (because of the obstruction from exotic S7 which can embed only in R11 (flat) and R12 torsioned). We fix that by taking the 7 member board and mapping it to 11D. [See the discussion on Stable Homotopy Groups of Spheres in the appendix].

 Please double check the case of 13 (only 12 people must vote on any issue, 1 must be the observer and the special resolution pass threshold is 9/12 and the budget pass threshold is less than 5/12 unified no-votes and 7/12 yes votes, abstains don't sink the budget vote. [There's a more nuanced variation using the Binary Golay Code voting at the bottom of this document; the ultimate bottom of this document is an attempt by Gemini to create something that works, I'm not sure how well it did, but it does maintain some semblance of Entire Fairness]. 

Voting Rights Agreement

Theorem #1: No 1D voting system is anomaly free. Use 3D voting.

The Green Schwarz anomaly freeness criteria requires a 2D field to resolve the anomaly. 

Consequence: Nettle only uses 2D voting rules. Our recommended voting rule is called Qualified Double Majority for AGM votes and a weighted voting rule for Director votes. 

AGM Voting: Qualified Double Majority (2D voting + 1D minority protection rule)

Qualified Double Majority: It’s a supranational voting rule adopted by the EU to vote at its highest governing body as part of the Lisbon Treaty. It has been looked over extensively for fairness and unbiasedness. It works well when the underlying populations are heavily skewed and avoids the tyranny of the majority. 

Details of the voting rule can be found here: https://www.consilium.europa.eu/en/council-eu/voting-system/qualified-majority/ 

In this case, replace people/population with Shares and members with Shareholders. 80% of EU legislative votes happen according to this rule. If it’s fair for a supranational body, it’s fair for us. [Some advantages: the voting rule is built in such a way that quorums etc. are not needed, the voting rule itself incorporates a quorum and minority blocking rights].

The proposed text of the rule is in the appendix.

Director Voting: Entire Fairness & the MFW rule (3D voting)

Since the Directors are fewer in number, it helps to be specific about what we’re trying to achieve. We’re trying to ensure that any decision of the Board is entirely fair to all parties. Anomalous decisions are defined as those where the outcome of a vote doesn’t reflect the will of the shareholders (ie. one shareholder was able to overpower the others).

In this case, we actually require a 3D voting rule (a tri-linear vertex is used to resolve a hexagonal anomaly). [It is intuitive if you see that 3-part House, Senate, Executive structures are successful in the real-world]. 

The Entire Fairness / MFW standard requires a majority vote of the interested party, an uncoerced vote of the independents and a majority vote of the minority party. 

The Entire Fairness standard is encoded into the Director voting rules by weights:

A 4 person board with 3-way voting: (approve-abstain-disapprove):  

Pass threshold 7/12 (simple majority), 8/12 (special majority),  4/12 (minority blocking), 10/12 (drag along), max weight: 6

Director Class	Seats	Weight	Representative
Common Stock	1	3	Independent
Preferred Stock-1	1	2	Investor Representatives
Preferred Stock-2	1	2	Investor Representatives
Founder Stock	1	5	Founders
Total Seats/Weights	4	12

For a 7 person board with 3-way voting (approve-abstain-disapprove): pass threshold 6/11 (simple majority) and 7/11 (special majority), minority block threshold 4/11, drag along: 9/11, max-weight: 5

Director Class	Seats	Weight	Representative
Common Stock	1	2	Independent
Minority Preferred	1	1	Minority Preferred w/o Liq. Pref.
Preferred Stock-A	1	1	Investor Representatives
Preferred Stock-A	1	1	Investor Representatives
Preferred Stock-B	1	1	Investor Representatives
Preferred Stock-B	1	1	Investor Representatives
Founder Stock	1	4	Founders
Totals	7	11

To avoid quorum issues, we use 7/11 and 4/11 as the thresholds and so if 4 votes are against instead of abstain, the motion does not pass. 

This ratio gives Founders consent rights, Preferred the ability to override Founder’s Consent rights and Independents to balance the board. 

This is again exactly the same as the Qualified Double Majority Rule for Shareholder voting and happily is the same as the MFW standard. 

The above voting structure is only reasonable when economic rights and ownership rights are closely aligned. If economic rights and control rights differ, then we need to pick a larger board and a voting structure suited to the purpose.

For technical mathematical reasons, there are no “board-structures” in 9D and 10D. The next “board structure” is available at 11D which we will compact onto 10D, 12D and 13D. 12D and 13D voting is enormously stressful for the company because the underlying mathematical structures indicate that the company will be going through a period of change and transformation. 

For a 11 person board with 3-way voting (approve-abstain-disapprove): pass threshold 6/11 (simple majority) and 9/11 (special majority), minority block threshold 4/11, drag along 9/11, max-weight: 5

Director Class	Seats	Weight	Representative
Common Stock	1	1	Independent
Employee Options	1	1	Employee representative
Minority Preferred Stock	1	1	Minority Preferred
Preferred Stock	5	5	Investor Representatives
Founder Stock	3	3	Founders (up to 3 seats vs votes - must vote as a bloc)
Totals	11	11

To avoid quorum issues, we use 7/11 and 4/11 as the thresholds and so if 4 votes are against instead of abstain, the motion does not pass. Rank = 8, so 3 people (Founder group) must vote as a block. [Ternary Golay Code]

The last board structure is available in 13D. In this board structure, only up-down voting is possible. 

For a 13 person board with 2-way voting (approve-disapprove): pass threshold 17/24 (simple majority) and 18/24 (special majority), 19/24 (drag along), minority block threshold 13/24, max-weight: 8 for any class and no more than 1 member absent. Voting will proceed in 2 rounds: one for economic ownership and one for control ownership.

Director Class	Seats	Economic Weight	Control Weight	Representative
Preferred Stock	7	7	5 (2 investors 0)	Investor Representatives
Minority Preferred Stock	1	1	1	Must agree to vote as a bloc minority with 2 other investors
Common Stock	1	1	1
Employee Options	1	0 or 1 Indep	1
Founder Stock	3	2	4	Founders (must vote as a bloc with common stock, employee options and minority preferred)
Totals	13	11 or 12	12

This structure is based on the Binary Golay Code.

[23, 12, 7] – for Ordinary Resolutions (guaranteed to never block)

[24, 12, 8] – for Special Resolutions (can block and has highest error detection capability)

13-person Board

12 person rotation board structure of the 13 people. 1 person must be an observer.

The vote must be held twice - once on economic weights and once on control weights. 

A resolution must:

1. Launch: Achieve the base vote threshold (4/12 or 8/12).

2. Stay on Course: Avoid being shot down by a blocking coalition (5/12 "No" votes).

3. Achieve Orbit: Secure final approval from the Keyholder supermajority (4/5 consent).

This is the ultimate balance of power. It allows for operational agility on minor issues but demands an interlocking, undeniable consensus between the Founders, key Investors, and the Independent Director for any strategic move. It is a system built to last, ensuring that no matter how late the night or how high the pressure, the company's foundational interests are always protected.

1. The Keyholder Group: A 5-person "Keyholder Committee" is designated, representing a cross-section of the board's core interests:

• The 2 Founder Directors

• The Independent Director

• 2 Lead Preferred Stock Directors (e.g., the representatives from the two largest funds)

The rules above encode the following “exceptional” error correction codes:

1. The Binary Golay Code and the Extended Binary Golay Code. 

2. The Ternary Golay Code and the Extended Ternary Golay Code

These are the only ones that have the right structure to be used for anomaly free processing. All other voting rules are anomalous or inefficient. 

Appendix: Qualified Double Majority Voting

The rule: 

A qualified majority requires 55% of voting EU member states, representing at least 65% of the population of voting members for a European Commission proposal to be approved. This increases to 72% of voting members states, representing at least 65% of the EU population of voting members should the proposal originate from a member state. Proposals can be blocked should a qualified majority of at least four Council members representing more than 35% of the EU population be formed.

Appendix: Error Correcting Codes 

The Ternary Golay Code is used for boards of size 4, 7 and 11.

The Binary Golay Code is used for boards of size 13. 

Ternary Golay Code

[Total voting weights, number of Directors, max weight to any Director and pass threshold]

[11, 6, 5] - for Ordinary Resolutions (guaranteed to never block)

[12, 6, 6] - for Special Resolutions (can block and has highest error detection capability)

Binary Golay Code

[23, 12, 7] – for Ordinary Resolutions (guaranteed to never block)

[24, 12, 8] – for Special Resolutions (can block and has highest error detection capability)

These error correction codes have been called: “the single best published page in coding theory”. Their automorphism groups are the correct ones for a startup to work well (the Matthew Groups). Deviating even slightly causes the dimension of the system to change and that dimension doesn’t have the right board structure.

Appendix: Homotopy Groups of Spheres

The Homotopy Groups of Spheres are used to decide which dimensions have exotic Spheres. Only those dimensions that have exotic spheres support resolution of anomalies that a board has to face. Other dimensions don’t have the board structure available. 

Only the following dimensions have exotic spheres available for the use of board-like structures:

S7, S8, S11, S13, S15, S19. All 3D voting rules fit within a path length of 20.

For Board Sizes 11 and where the critical pass threshold is 7.

S7 - R11 (all flat embeddings of day to day workings)

S7 - R12 (all torsioned embeddings of day to day workings)

For Board Sizes 13 and where the critical pass threshold is 9

S8 - R9 (critical threshold for special resolutions, as we need S8 to exist for smooth operations).

S11 - R13 (critical threshold for dissent and blocking vote in the 24D Leech lattice).

S13 - R16 (minimal pass thresholds), R17-R19 (metastable ranges), R20 (probably anomaly free), R26 (embedding dimension) – used in the construction of the Binary Golay Code voting rules. = powers the bosonic string theory (activist investors). 

S15 - R16 (immersion possible, embedding not possible), R30 (embedding upper bound)

S19 - R20 (immersion possible, embedding not possible), R38 (embedding upper bound).

Summary: 16 to 20 is the critical threshold range for voting when the board size is 13. 

Appendix: Uncommon Voting Rules

For an 8 person board with 3-way voting (approve-abstain-disapprove): pass threshold 7/12 (simple majority), 8/12 (special majority),  4/12 (minority blocking), 9/12 (drag along), max weight: 6

Director Class	Seats	Weight	Representative
Common Stock	1	1	Independent
Preferred Stock	6	1	Investor Representatives
Founder Stock	1	6	Founders
Total Weights		12

This configuration is not common, but it is possible. This configuration has been compacted to the 4 person board situation.