the implementation — markowitz

§07 / the implementation

the implementation

the implementation of markowitz is a single rust program of roughly nine hundred lines, deployed as a token-2022 transfer hook against the markowitz mint on solana mainnet. the program does four things on every transfer: it reads the active price from the bonding curve, updates the cohort assignment of the wallets involved in the transfer, recomputes the covariance matrix across all cohorts, and applies a surcharge proportional to the squared distance between the current holder distribution and the nearest point on the efficient frontier. all four operations execute atomically inside the same transaction as the transfer that triggered them, and all four complete within the program's compute unit budget of two hundred thousand units.

the data layout

the program maintains three program derived addresses, each with a fixed account layout. the first pda, seeded at ["cohorts", mint_pubkey], holds a packed table of cohort assignments. each entry is a single byte representing a wallet's current cohort, with four bits for the balance bucket index and four bits for the tenure window index. the table is sized to hold up to sixty four thousand wallet entries inside a single account well under the ten kilobyte limit. the second pda, seeded at ["covariance", mint_pubkey], holds the covariance matrix as a thirty six by thirty six array of i128 fixed point values, totaling roughly twenty kilobytes. the third pda, seeded at ["vault", mint_pubkey], holds the accumulated surcharge tokens and a small distribution state struct tracking the most recent payout slot.

// account layouts for the markowitz transfer hook program
const COHORT_GRID_SIZE: usize = 36;          // 6 balance buckets × 6 tenure windows
const MAX_WALLETS: usize = 65_536;            // capacity of the cohort table
const FP_SCALE: i128 = 1_000_000_000_000;    // 1e12 fixed-point

#[account]
pub struct CohortTable {
    pub mint: Pubkey,
    pub wallet_count: u32,
    pub last_update_slot: u64,
    pub assignments: [u8; MAX_WALLETS],       // packed cohort index per wallet
    pub balance_log: [u64; MAX_WALLETS],      // last known balance (for cohort transitions)
    pub first_seen_slot: [u64; MAX_WALLETS],  // for tenure window calculation
}

#[account]
pub struct CovarianceMatrix {
    pub mint: Pubkey,
    pub last_update_slot: u64,
    pub cohort_means: [i128; COHORT_GRID_SIZE],
    pub cohort_variances: [i128; COHORT_GRID_SIZE],
    pub covariance: [[i128; COHORT_GRID_SIZE]; COHORT_GRID_SIZE],
}

#[account]
pub struct VaultState {
    pub mint: Pubkey,
    pub accumulated: u64,
    pub last_distribution_slot: u64,
    pub total_distributed: u128,
    pub bump: u8,
}

the transfer hook

the transfer hook is the entry point. it is invoked by the token-2022 program every time the markowitz mint participates in a transfer, regardless of which application initiated the transfer. the hook reads the active bonding curve price, looks up the source and destination wallets in the cohort table, transitions their cohort assignments if their balance or tenure has crossed a bucket boundary, updates the running cohort statistics, and applies the surcharge calculated against the efficient frontier. the return value is the lamports to be withheld from the input amount before the transfer settles.

// the transfer hook entry point — invoked on every transfer
pub fn execute(ctx: Context<Execute>, amount: u64) -> Result<u64> {
    let clock = Clock::get()?;
    let cohorts = &mut ctx.accounts.cohort_table;
    let cov = &mut ctx.accounts.covariance_matrix;
    let vault = &mut ctx.accounts.vault;

    // 1. read active price from the bonding curve
    let price_fp = read_bonding_curve_price(&ctx.accounts.curve_state)?;

    // 2. update source and destination cohort assignments
    let src_idx = lookup_or_insert(cohorts, &ctx.accounts.source.owner, clock.slot)?;
    let dst_idx = lookup_or_insert(cohorts, &ctx.accounts.destination.owner, clock.slot)?;

    let src_old_cohort = cohorts.assignments[src_idx];
    let dst_old_cohort = cohorts.assignments[dst_idx];

    let src_new_balance = ctx.accounts.source.amount.saturating_sub(amount);
    let dst_new_balance = ctx.accounts.destination.amount.saturating_add(amount);

    let src_new_cohort = compute_cohort(
        src_new_balance,
        clock.slot.saturating_sub(cohorts.first_seen_slot[src_idx]),
    );
    let dst_new_cohort = compute_cohort(
        dst_new_balance,
        clock.slot.saturating_sub(cohorts.first_seen_slot[dst_idx]),
    );

    cohorts.assignments[src_idx] = src_new_cohort;
    cohorts.assignments[dst_idx] = dst_new_cohort;
    cohorts.balance_log[src_idx] = src_new_balance;
    cohorts.balance_log[dst_idx] = dst_new_balance;
    cohorts.last_update_slot = clock.slot;

    // 3. update covariance matrix incrementally
    incremental_covariance_update(
        cov,
        src_old_cohort, src_new_cohort,
        dst_old_cohort, dst_new_cohort,
        price_fp,
    )?;

    // 4. compute distance to efficient frontier and surcharge
    let distance_fp = frontier_distance(cov);
    let dist_sq_fp = (distance_fp * distance_fp) / FP_SCALE;
    let surcharge_rate_fp = (SURCHARGE_K_FP * dist_sq_fp) / FP_SCALE;
    let surcharge = ((amount as i128 * surcharge_rate_fp) / FP_SCALE)
        .max(0)
        .min(amount as i128) as u64;

    vault.accumulated = vault.accumulated.saturating_add(surcharge);
    Ok(surcharge)
}

the covariance update

the covariance matrix update is the most expensive operation in the hook, and the part that required the most engineering. a naive implementation would recompute the full thirty six by thirty six matrix on every transfer, which would require thirteen hundred entry recalculations and exceed the compute budget by an order of magnitude. the implementation used here is incremental. when a transfer moves a wallet from one cohort to another, only the rows and columns of the matrix that involve those two cohorts need to be updated, and within those rows the update can be computed in constant time using welford's online algorithm for variance and covariance. the total cost is bounded at four cohort updates and one hundred forty four matrix cell updates, regardless of how many wallets the protocol is tracking.

// incremental covariance update using welford's online algorithm
fn incremental_covariance_update(
    cov: &mut CovarianceMatrix,
    src_old: u8, src_new: u8,
    dst_old: u8, dst_new: u8,
    price_fp: i128,
) -> Result<()> {
    let affected_cohorts: [u8; 4] = [src_old, src_new, dst_old, dst_new];

    for &cohort_i in affected_cohorts.iter() {
        let i = cohort_i as usize;

        // update the mean and variance for cohort i using Welford's method
        let n = (cov.cohort_means[i].abs() / FP_SCALE).max(1);
        let delta = price_fp - cov.cohort_means[i];
        cov.cohort_means[i] += delta / n;
        let delta2 = price_fp - cov.cohort_means[i];
        cov.cohort_variances[i] += (delta * delta2) / FP_SCALE;

        // update the row and column of the covariance matrix
        for j in 0..COHORT_GRID_SIZE {
            if j == i { continue; }
            let cross = ((price_fp - cov.cohort_means[i]) * (price_fp - cov.cohort_means[j]))
                / FP_SCALE;
            cov.covariance[i][j] = (cov.covariance[i][j] * (n - 1) + cross) / n;
            cov.covariance[j][i] = cov.covariance[i][j];  // symmetric
        }

        // diagonal entries are the cohort's own variance
        cov.covariance[i][i] = cov.cohort_variances[i];
    }

    cov.last_update_slot = Clock::get()?.slot;
    Ok(())
}

the frontier distance

the distance between the current holder distribution and the efficient frontier is computed in closed form. the standard approach to portfolio optimization requires inverting the covariance matrix, which is an expensive operation. the implementation here uses a banded approximation that exploits a property of the cohort grid: cohorts that are far apart on the balance-tenure grid are nearly uncorrelated, so the off-diagonal entries of the covariance matrix that lie outside a band around the diagonal can be approximated as zero. inverting a banded matrix is linear in the matrix size rather than cubic, which brings the operation into the compute budget. the result is a single i128 fixed point value representing the distance from the holder set to the frontier, in mean-variance units.

// frontier distance via banded matrix inversion
const BAND_WIDTH: usize = 5;

fn frontier_distance(cov: &CovarianceMatrix) -> i128 {
    let mut banded_inv = [[0i128; COHORT_GRID_SIZE]; COHORT_GRID_SIZE];

    // populate the banded approximation
    for i in 0..COHORT_GRID_SIZE {
        for j in 0..COHORT_GRID_SIZE {
            if (i as isize - j as isize).abs() as usize <= BAND_WIDTH {
                banded_inv[i][j] = cov.covariance[i][j];
            }
        }
    }

    // tridiagonal inversion in linear time using the Thomas algorithm
    // (full implementation omitted for brevity, ~120 lines)
    let inverted = invert_banded(&banded_inv);

    // weight vector is the current cohort distribution
    let weights = compute_cohort_weights(cov);

    // portfolio variance = w^T · Σ^-1 · w
    let mut portfolio_var: i128 = 0;
    for i in 0..COHORT_GRID_SIZE {
        for j in 0..COHORT_GRID_SIZE {
            portfolio_var += (weights[i] * inverted[i][j] * weights[j])
                / (FP_SCALE * FP_SCALE);
        }
    }

    // distance to frontier = √(portfolio_var - frontier_var(target_return))
    let frontier_var = frontier_variance_at_return(target_return_fp());
    let diff = portfolio_var.saturating_sub(frontier_var);
    isqrt_fp(diff.max(0))
}

the seal

once the program was compiled and tested against simulated holder distributions, it was deployed to solana mainnet through the bpf upgradeable loader. the upgrade authority was set to none in the same session as the deployment, which permanently sealed the bytecode at the program's address. the mint was created with the token-2022 standard and the transfer hook extension configured at creation to point at the program. the extra account metas list was published in the same transaction, specifying the three pdas (cohorts, covariance, vault) that the hook would read and write on every invocation. from that moment, the protocol has been running without intervention. the bytecode is the bytecode. the math is the math. the chain is whatever the chain produces.

# the deployment sequence — one shot, no rollback

# 1. build and verify
anchor build --verifiable
sha256sum target/deploy/markowitz.so > deployment.hash

# 2. deploy program to mainnet
solana program deploy \
    --url https://api.mainnet-beta.solana.com \
    --keypair ./deployer.json \
    target/deploy/markowitz.so

# 3. revoke upgrade authority in the same session
solana program set-upgrade-authority \
    <PROGRAM_ID> \
    --new-upgrade-authority none \
    --skip-new-upgrade-authority-signer-check

# 4. create the mint with the transfer hook extension
spl-token --program-2022 create-token \
    --transfer-hook <PROGRAM_ID>

# 5. publish extra account metas for the hook
markowitz-cli initialize-extra-metas \
    --mint <MINT_ADDRESS> \
    --cohort-pda <COHORT_PDA> \
    --covariance-pda <COVARIANCE_PDA> \
    --vault-pda <VAULT_PDA>

# 6. close the deployer keypair file
shred -u ./deployer.json

the steps above were executed once. there is no version two, no patch, no configuration change, no parameter that can be adjusted by anyone. the program is what runs. the math is what produces the surcharge. the chain is what decides whether the holder set sits on the frontier or off it. nothing else is in the path.