some changes to PROP-001.md text

PROP-001.md

@@ -19,17 +19,17 @@ directory:
 - `balances.json` https://atomone.fra1.digitaloceanspaces.com/cosmoshub-4/prop848/balances.json 
 - `auth_genesis.json` https://atomone.fra1.digitaloceanspaces.com/cosmoshub-4/prop848/auth_genesis.json
 
-The way those files were extracted from the snaphots is available
+The way these files were extracted from proposal 848 snaphot is available
 [here](SNAPSHOT-EXTRACT.md). If you prefer you can start from a Gaia blockchain
 node and extract all the data, everything is detailed in the link above.
 
 ## Verify tally
 
 A good way to check the correctness of the data is to compute the prop 848
-tally from the data and compare it with the proposal `TallyResult` field stored
-in the blockchain's store.
+tally using it and compare the results with the `TallyResult` field of the
+proposal object stored on-chain.
 
-You can achieve that by running this command:
+You can achieve this by running the following command:
 ```
 $ go run . tally data/prop848/
 173,165 votes
@@ -48,15 +48,16 @@ Yes percent: 0.517062127500689774
 +-----------+------------+------------+------------+------------+-------------+
 ```
 
-As printed in the output of the command, the diff is always 0, meaning there's
+As printed in the output of the command, the diff is 0, meaning there's
 no difference between the tally computed from the data and the `TallyResult`
 field of the proposal.
 
 ## Consolidate accounts
 
 The program allows you to create a `data/prop848/accounts.json` file that
-consolidates all the interesting data from an account. This file will be used
-by the following command to compute other things.
+consolidates all the relevant data for accounts. This file will be the starting
+point for the following steps, and will be fed to this very same program
+but using a different command.
 
 ```
 $ go run . accounts data/prop848/
@@ -107,8 +108,8 @@ For example, here is the representation of an account in this file:
 }
 ```
 
-It gives access to the liquid and staked amount, the vote, the delegations and
-their relative vote.
+It gives access to the unbonded and bonded amounts (here *liquid* and *staked*
+amounts), the direct vote if present, the delegations and their relative vote.
 
 > [!NOTE]
 > `ModuleAccount` and `InterchainAccount` are skipped.
@@ -153,10 +154,10 @@ unbonded $ATOM is calculated to result in them having less than 1/3 of the
 final $ATONE supply.
 
 Let's define the following variables:
-- `C` the multiplier
-- `t` the target percent (known, 33%)
-- `X` a supply in $ATOM (known)
-- `Y` a supply in $ATONE
+- `C` is the multiplier we want to compute to be applied to non-voter cathegories, i.e. *Not Staked*, *Abstain* and *Did Not Vote*
+- `t` the target relative percentage of $ATONE supply distributed to non-voter cathegories we want to achieve (known, 33%)
+- `X` is the supply in $ATOM (known)
+- `Y` is the supply in $ATONE
 - both `X` and `Y` will have an annotation indicating the portion of the supply:
     - `Y` voted Yes
     - `A` voted Abstain
@@ -167,42 +168,35 @@ Let's define the following variables:
 
   For example, $X_{A}$ is the number of $ATOM that has votes ABSTAIN.
 
-Intuitively, we can start by writing this formula, which expresses our need:
+Using the above notation, we can express our what we want to achieve mathemacally as:
 ```math
-\begin{flalign}
-& \frac{Y_{A} + Y_{DNV} + Y_{U}}{Y_{A} + Y_{DNV} + Y_{U} + Y_{Y} + Y_{N} + Y_{NWV}} <= t &
-\end{flalign}
+\frac{Y_{A} + Y_{DNV} + Y_{U}}{Y_{A} + Y_{DNV} + Y_{U} + Y_{Y} + Y_{N} + Y_{NWV}} \leq t
 ```
-
-Which can be translated by the number of abstainers, non-voters and unbonded
-$ATONE divided by the total number of $ATONE must be less or equal to `t`, thus
-33%.
-
-Now let's replace the `Y`s, which are unknown at this point, with the `X`s,
-using the multipliers we know and the multiplier we are looking for `C`:
+It is know from the specifications of the distribution mechanism, disregarding
+any additional bonus or malus:
 ```math
-\begin{flalign}
-& Y_{Y} = X_{Y} &\\
-& Y_{N} = 4 \cdot X_{N} & \\
-& Y_{NWN} = 4 \cdot X_{NWV} & \\
-& Y_{A} + Y_{DNV} + Y_{U} = C \cdot (X_{A} + X_{DNV} + X_{U}) &
-\end{flalign}
+\left\{
+\begin{aligned}
+& Y_{Y} = X_{Y} \\
+& Y_{N} = 4 \cdot X_{N} \\
+& Y_{NWV} = 4 \cdot X_{NWV} \\
+& Y_{A} + Y_{DNV} + Y_{U} = C \cdot X_{A} + C \cdot X_{DNV} + C \cdot X_{U} = C \cdot (X_{A} + X_{DNV} + X_{U})
+\end{aligned}
+\right.
 ```
 
-Which, with respect to the first equation, gives:
+Which if plugged in the above equation gives:
 ```math
-\begin{flalign}
-& \frac{C \cdot (X_{A} + X_{DNV} + X_{U})}{C \cdot (X_{A} + X_{DNV} + X_{U}) + X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV}} <= t &
-\end{flalign}
+\frac{C \cdot (X_{A} + X_{DNV} + X_{U})}{C \cdot (X_{A} + X_{DNV} + X_{U}) + X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV}} \leq t
 ```
 
-Now let's isolate `C`:
+Finally, let's isolate `C`:
 ```math
-\begin{flalign}
-& C \cdot (X_{A} + X_{DNV} + X_{U}) <= t \cdot C \cdot (X_{A} + X_{DNV} + X_{U}) + t \cdot (X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV}) &\\
-& (1 - t) \cdot C \cdot (X_{A} + X_{DNV} + X_{U}) <=  t \cdot (X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV}) &\\
-& C  <=  \frac{t}{1-t} \cdot \frac{(X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV})}{(X_{A} + X_{DNV} + X_{U})} &\\
-\end{flalign}
+\begin{align}
+C \cdot (X_{A} + X_{DNV} + X_{U}) &\leq t \cdot C \cdot (X_{A} + X_{DNV} + X_{U}) + t \cdot (X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV}) \\[10pt]
+(1 - t) \cdot C \cdot (X_{A} + X_{DNV} + X_{U}) &\leq  t \cdot (X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV}) \\[10pt]
+C  &\leq  \frac{t}{1-t} \cdot \frac{(X_{Y} + 4 \cdot X_{N} + 4 \cdot X_{NWV})}{(X_{A} + X_{DNV} + X_{U})}
+\end{align}
 ```
 Which gives the final formula described in the [proposal 001][001].