spec: small edits to typos in protocol docs

docs/protocol/src/concepts/batching_flows.md

@@ -8,7 +8,7 @@ public chain state, and the desire to shield value amounts in individual
 transactions.
 
 To address this tension, Penumbra provides a mechanism to aggregate value flows
-across a batch of transactions, revealing the only the total amount and not the
+across a batch of transactions, revealing only the total amount and not the
 value contributed by each individual transaction.  This mechanism is built using
 an integer-valued homomorphic encryption scheme that supports threshold
 decryption, so that the network's validators can jointly control a decryption

docs/protocol/src/crypto/decaf377-rdsa.md

@@ -93,6 +93,7 @@ Let’s say we have two actions in a transaction: one spend (indicated with subs
 The balance commitments for those actions are:
 
 $cv_o = [v_o] G_v + [rcv_o] G_{rcv}$
+
 $cv_s = [v_s] G_v + [rcv_s] G_{rcv}$
 
 where

docs/protocol/src/protocol/action_descriptions/delegator_vote.md

@@ -48,9 +48,9 @@ using the above witnessed values and where `ds` is a constant domain separator:
 
 The zk-SNARK certifies that the public input balance commitment $cv$ was derived from the witnessed values as:
 
-$cv = [v] G_v + [\tilde v] G_{\tilde v}$
+$cv = [v] G_v + [\widetilde{v}] G_{\widetilde{v}}$
 
-where $G_{\tilde v}$ is a constant generator and $G_v$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md). For delegator votes, $[\tilde v] = 0$.
+where $G_{\widetilde{v}}$ is a constant generator and $G_v$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md). For delegator votes, $[\widetilde{v}] = 0$.
 
 ### Nullifier Integrity
 

docs/protocol/src/protocol/action_descriptions/outputs.md

@@ -36,9 +36,9 @@ using the above witnessed values and where `ds` is a constant domain separator:
 
 The zk-SNARK certifies that the public input balance commitment $cv$ was derived from the witnessed values as:
 
-$cv = [v] G_v + [\tilde v] G_{\tilde v}$
+$cv = [v] G_v + [\widetilde{v}] G_{\widetilde{v}}$
 
-where $G_{\tilde v}$ is a constant generator and $G_v$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md).
+where $G_{\widetilde{v}}$ is a constant generator and $G_v$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md).
 
 ### Diversified Base is not Identity
 

docs/protocol/src/protocol/action_descriptions/spend.md

@@ -42,9 +42,9 @@ using the above witnessed values and where `ds` is a constant domain separator:
 
 The zk-SNARK certifies that for non-zero values $v \ne 0$, the public input balance commitment $cv$ was derived from the witnessed values as:
 
-$cv = [v] G_v + [\tilde v] G_{\tilde v}$
+$cv = [v] G_v + [\widetilde{v}] G_{\widetilde{v}}$
 
-where $G_{\tilde v}$ is a constant generator and $G_v$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md).
+where $G_{\widetilde{v}}$ is a constant generator and $G_v$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md).
 
 ### Nullifier Integrity
 

docs/protocol/src/protocol/action_descriptions/swap.md

@@ -40,9 +40,9 @@ using the above witnessed values and where `ds` is a constant domain separator:
 
 The zk-SNARK certifies that the public input fee commitment $cv_f$ was derived from the witnessed values as:
 
-$cv_f = [v_f] G_{v_f} + [\tilde v_f] G_{\tilde v}$
+$cv_f = [-v_f] G_{v_f} + [\widetilde{v_f}] G_{\widetilde{v}}$
 
-where $G_{\tilde v}$ is a constant generator and $G_{v_f}$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md).
+where $G_{\widetilde{v}}$ is a constant generator and $G_{v_f}$ is an asset-specific generator point derived as described in [Value Commitments](../../protocol/value_commitments.md).
 
 ### Balance Commitment Integrity
 

docs/protocol/src/protocol/action_descriptions/undelegate_claim.md

@@ -19,9 +19,9 @@ And the corresponding public inputs:
 
 The zk-SNARK certifies that the public input balance commitment $cv$ was derived from the witnessed values as:
 
-$cv = v_e + [\tilde v] G_{\tilde v}$
+$cv = v_e + [\widetilde{v}] G_{\widetilde{v}}$
 
-where $G_{\tilde v}$ is a constant generator.
+where $G_{\widetilde{v}}$ is a constant generator.
 
 $v_e$ is the expected balance computed from the penalty $p$, the unbonding
 amount $v_u$ and the unbonding asset ID $G_u$:

docs/protocol/src/stake/example.md

@@ -5,7 +5,7 @@ delegators, Alice, Bob, and Charlie, and two validators, Victoria and
 William.  Tendermint consensus requires at least four validators and no one
 party controlling more than $1/3$ of the stake, but this example uses only a few parties just to illustrate the dynamics.
 
-For simplicity, the the base reward rates and commission rates
+For simplicity, the base reward rates and commission rates
 are fixed over all epochs at $r = 0.0006$ and $c_v = 0$, $c_w = 0.1$.
 The `PEN` and `dPEN` holdings of participant $a, b, c, \ldots$ are
 denoted by $x_a$, $y_a$, etc., respectively.

docs/protocol/src/stake/tokens.md

@@ -31,7 +31,7 @@ adjusted by governance.
 
 Each validator declares a set of funding streams, which comprise both the
 destinations of their commission and the total commission rate $c_{v,e} \in
-[0,1]$. $c{v,e}$ is subtracted from the base reward rate to get a
+[0,1]$. $c_{v,e}$ is subtracted from the base reward rate to get a
 validator-specific reward rate $$r_{v,e} = (1 - c_{v,e})r_e.$$.
 
 The base exchange rate between `PEN` and `dPEN` is given by the function

docs/protocol/src/stake/validator-rewards.md

@@ -1,9 +1,9 @@
 # Validator Rewards and Fees
 
 Validators declare a set of funding streams that comprise the destination of
-all of their staking rewards. Each funding stream contains a rate $ri \in
-[0,1]$ and a destination address $ai$.  The validator's total commission rate
-is defined as $c_{v,e} = \sum ri$, the sum of the rate of each funding stream.
+all of their staking rewards. Each funding stream contains a rate $r_i \in
+[0,1]$ and a destination address $a_i$.  The validator's total commission rate
+is defined as $c_{v,e} = \sum r_i$, the sum of the rate of each funding stream.
 $c_{v,e}$ cannot exceed 1.
 
 The spread between the base reward rate $r_e$ and the reward rate for their