Normalization

This phase deliberately ignores the design from ERModel (P1) and RelationalDesign (P2) as a starting point. Instead it starts over from a single flat relation containing every attribute of the model, derives the functional dependencies that hold on it, and decomposes it formally, step by step. The final section compares what falls out of that process with the P2 design.

De-normalized database form

Building one relation out of the whole model

The ER model has ten entity/relationship sets carrying attributes (see ERModel): Users, Cryptos, Markets, Orders, Transactions, MarketTrades, MarketCandles, Watchlists, and the two attributed relationships Holds and Contains. Eight more relationships (QuotedOn, PlacedOn, Places, Records, Settles, Fills, Aggregates, Owns) carry no attributes of their own — in Chen notation they need none, because the diagram expresses the link itself as a relationship, not a column. A single flat relation has no such device: the only way to keep one entity's rows pointed at another's is a plain attribute holding the referenced key, which is exactly what P2's ER-to-relational transformation already introduces for each of those eight relationships (markets.crypto_id, orders.market_id, orders.user_id, transactions.user_id, transactions.related_order, market_trades.market_id, market_candles.market_id, watchlists.user_id). Those linking attributes are included below for that reason — not because they were copied from P2's design, but because a "single table with everything in it" cannot represent the model at all without them.

Every attribute name is prefixed by a two-or-three-letter code for the entity/relationship it came from, because several names repeat across the model (id, created_at, quantity, type, name, price, side all appear more than once) and the de-normalized relation may not contain duplicate names.

Prefix	Origin (P1 entity / relationship)	Attributes
U_	Users	U_ID, U_USERNAME, U_EMAIL, U_FULL_NAME, U_PASSWORD_HASH, U_AVAILABLE_BALANCE, U_INVESTED_BALANCE, U_CREATED_AT, U_UPDATED_AT
C_	Cryptos	C_ID, C_SYMBOL, C_NAME, C_CREATED_AT
M_	Markets (+ QuotedOn)	M_ID, M_CRYPTO_ID, M_QUOTE_CURRENCY, M_IS_ACTIVE, M_CREATED_AT
H_	Holds (+ surrogate key)	H_ID, H_USER_ID, H_CRYPTO_ID, H_QUANTITY, H_RESERVED_QUANTITY, H_AVG_PRICE, H_CREATED_AT, H_UPDATED_AT
O_	Orders (+ PlacedOn, Places)	O_ID, O_USER_ID, O_MARKET_ID, O_SIDE, O_TYPE, O_STATUS, O_QUANTITY, O_PRICE, O_PLACED_AT, O_EXECUTED_AT
T_	Transactions (+ Records, Settles)	T_ID, T_USER_ID, T_TYPE, T_AMOUNT, T_CURRENCY, T_RELATED_ORDER, T_CREATED_AT, T_DESCRIPTION
MT_	MarketTrades (+ Fills)	MT_ID, MT_MARKET_ID, MT_EXECUTED_AT, MT_PRICE, MT_QUANTITY, MT_SIDE, MT_SOURCE
MC_	MarketCandles (+ Aggregates)	MC_ID, MC_MARKET_ID, MC_TIMEFRAME, MC_OPEN, MC_HIGH, MC_LOW, MC_CLOSE, MC_VOLUME, MC_CANDLE_TIME
W_	Watchlists (+ Owns)	W_ID, W_USER_ID, W_NAME, W_CREATED_AT
WI_	Contains (+ surrogate key)	WI_ID, WI_WATCHLIST_ID, WI_CRYPTO_ID, WI_ADDED_AT

H_ID and WI_ID exist for the same reason they exist in P2: Holds and Contains are M:N relationships with their own attributes, and giving each its own surrogate key (rather than relying solely on the {user,crypto} / {watchlist,crypto} pair) is the same design choice already justified in RelationalDesign (section "Descriptive representation of the relational schema").

This gives one relation, R_EDUBERZA, of 68 attributes:

R_EDUBERZA(
  U_ID, U_USERNAME, U_EMAIL, U_FULL_NAME, U_PASSWORD_HASH, U_AVAILABLE_BALANCE,
  U_INVESTED_BALANCE, U_CREATED_AT, U_UPDATED_AT,
  C_ID, C_SYMBOL, C_NAME, C_CREATED_AT,
  M_ID, M_CRYPTO_ID, M_QUOTE_CURRENCY, M_IS_ACTIVE, M_CREATED_AT,
  H_ID, H_USER_ID, H_CRYPTO_ID, H_QUANTITY, H_RESERVED_QUANTITY, H_AVG_PRICE,
  H_CREATED_AT, H_UPDATED_AT,
  O_ID, O_USER_ID, O_MARKET_ID, O_SIDE, O_TYPE, O_STATUS, O_QUANTITY, O_PRICE,
  O_PLACED_AT, O_EXECUTED_AT,
  T_ID, T_USER_ID, T_TYPE, T_AMOUNT, T_CURRENCY, T_RELATED_ORDER, T_CREATED_AT,
  T_DESCRIPTION,
  MT_ID, MT_MARKET_ID, MT_EXECUTED_AT, MT_PRICE, MT_QUANTITY, MT_SIDE, MT_SOURCE,
  MC_ID, MC_MARKET_ID, MC_TIMEFRAME, MC_OPEN, MC_HIGH, MC_LOW, MC_CLOSE, MC_VOLUME,
  MC_CANDLE_TIME,
  W_ID, W_USER_ID, W_NAME, W_CREATED_AT,
  WI_ID, WI_WATCHLIST_ID, WI_CRYPTO_ID, WI_ADDED_AT
)

Every attribute is single-valued and atomic (a balance, a timestamp, a symbol, an amount — nothing here is a list or a nested record), so R_EDUBERZA satisfies 1NF as soon as it is written down. Whether it satisfies anything beyond that is exactly what the rest of this page checks.

Functional dependencies

Canonical cover

Read directly off the model: each entity's/relationship's own key determines its own attributes, nothing more. This is already minimal — no functional dependency below has an extraneous attribute on its left side, and no dependent attribute is repeated on the right side of more than one dependency, which is what "canonical cover" requires.

#	Functional dependency	Source
FD1	U_ID → U_USERNAME, U_EMAIL, U_FULL_NAME, U_PASSWORD_HASH, U_AVAILABLE_BALANCE, U_INVESTED_BALANCE, U_CREATED_AT, U_UPDATED_AT	Users
FD2	U_USERNAME → U_ID	Users (UNIQUE(username))
FD3	U_EMAIL → U_ID	Users (UNIQUE(email))
FD4	C_ID → C_SYMBOL, C_NAME, C_CREATED_AT	Cryptos
FD5	C_SYMBOL → C_ID	Cryptos (UNIQUE(symbol))
FD6	M_ID → M_CRYPTO_ID, M_QUOTE_CURRENCY, M_IS_ACTIVE, M_CREATED_AT	Markets
FD7	M_CRYPTO_ID, M_QUOTE_CURRENCY → M_ID	Markets (UNIQUE(crypto_id, quote_currency))
FD8	H_ID → H_USER_ID, H_CRYPTO_ID, H_QUANTITY, H_RESERVED_QUANTITY, H_AVG_PRICE, H_CREATED_AT, H_UPDATED_AT	Holds
FD9	H_USER_ID, H_CRYPTO_ID → H_ID	Holds (UNIQUE(user_id, crypto_id))
FD10	O_ID → O_USER_ID, O_MARKET_ID, O_SIDE, O_TYPE, O_STATUS, O_QUANTITY, O_PRICE, O_PLACED_AT, O_EXECUTED_AT	Orders
FD11	T_ID → T_USER_ID, T_TYPE, T_AMOUNT, T_CURRENCY, T_RELATED_ORDER, T_CREATED_AT, T_DESCRIPTION	Transactions
FD12	MT_ID → MT_MARKET_ID, MT_EXECUTED_AT, MT_PRICE, MT_QUANTITY, MT_SIDE, MT_SOURCE	MarketTrades
FD13	MC_ID → MC_MARKET_ID, MC_TIMEFRAME, MC_OPEN, MC_HIGH, MC_LOW, MC_CLOSE, MC_VOLUME, MC_CANDLE_TIME	MarketCandles
FD14	MC_MARKET_ID, MC_TIMEFRAME, MC_CANDLE_TIME → MC_ID	MarketCandles (UNIQUE(market_id, timeframe, candle_time))
FD15	W_ID → W_USER_ID, W_NAME, W_CREATED_AT	Watchlists
FD16	WI_ID → WI_WATCHLIST_ID, WI_CRYPTO_ID, WI_ADDED_AT	Contains
FD17	WI_WATCHLIST_ID, WI_CRYPTO_ID → WI_ID	Contains (UNIQUE(watchlist_id, crypto_id))

Minimality, checked by example (Markets): could FD7 drop an attribute from its left side? M_CRYPTO_ID alone does not determine M_ID — many markets can reference the same crypto in different quote currencies (that is the entire point of the market entity), so two rows can share M_CRYPTO_ID and disagree on M_ID. M_QUOTE_CURRENCY alone fails the same way in the other direction. Neither attribute is extraneous, so the left side of FD7 cannot shrink. The same check applies to FD9, FD14 and FD17, whose composite left sides come directly from the UNIQUE constraints already justified per-relation in RelationalDesign; none of those constraints holds on a proper subset of its columns either.

No redundant dependency: each of FD1–FD17 has a right side that is not implied by any other dependency in the set — for instance, nothing outside FD1 mentions U_AVAILABLE_BALANCE, so FD1 cannot be derived from the rest and cannot be dropped. This set is the canonical cover.

Dependencies carried by foreign keys

Six attributes above are foreign keys: M_CRYPTO_ID, H_USER_ID, H_CRYPTO_ID, O_USER_ID, O_MARKET_ID, T_USER_ID, T_RELATED_ORDER, MT_MARKET_ID, MC_MARKET_ID, W_USER_ID, WI_WATCHLIST_ID, WI_CRYPTO_ID — each one draws its values from the same domain as some other attribute's key. Because of that, every dependency that holds on the referenced key also holds, by substitution, on the referencing attribute:

Foreign key	References	Therefore also determines
M_CRYPTO_ID	C_ID	C_SYMBOL, C_NAME, C_CREATED_AT
H_USER_ID	U_ID	all of U_*
H_CRYPTO_ID	C_ID	all of C_*
O_USER_ID	U_ID	all of U_*
O_MARKET_ID	M_ID	all of M_*, and transitively all of C_*
T_USER_ID	U_ID	all of U_*
T_RELATED_ORDER	O_ID	all of O_*, and transitively U_*, M_*, C_* (when not null)
MT_MARKET_ID	M_ID	all of M_*, transitively C_*
MC_MARKET_ID	M_ID	all of M_*, transitively C_*
W_USER_ID	U_ID	all of U_*
WI_WATCHLIST_ID	W_ID	all of W_*, transitively U_*
WI_CRYPTO_ID	C_ID	all of C_*

None of these is added to the canonical cover — each is derivable from FD1–FD17 by transitivity plus the foreign-key identity, which is exactly why a canonical cover excludes them. They matter anyway: they are precisely the transitive dependencies the 3NF check below has to rule out.

Candidate keys and primary key

Orders, Transactions, MarketTrades, MarketCandles, Holds, Watchlists and Contains are, with respect to each other, independent record types: nothing about an order's id says anything about which market-candle row, or which unrelated transaction, or which watchlist item is in the same tuple of R_EDUBERZA — a user can exist with zero of any of them, and having one order says nothing about how many holdings, trades or candles exist alongside it. (The one FK that crosses between two of these — T_RELATED_ORDER — is nullable, so it cannot be relied on to always connect a transaction row back to an order.) That means no proper subset of attributes can functionally determine all 68 attributes of R_EDUBERZA: the only way to pin down a H_* value, an O_* value, a T_* value, an MT_* value, an MC_* value, a W_* value and a WI_* value at once is to state one identifying attribute from each cluster explicitly.

Chosen primary key (closure shown below):

{ U_ID, C_ID, M_ID, H_ID, O_ID, T_ID, MT_ID, MC_ID, W_ID, WI_ID }

Closure check, applying FD1–FD17 in turn to this set:

Step	Attributes added	Dependency used
start	U_ID, C_ID, M_ID, H_ID, O_ID, T_ID, MT_ID, MC_ID, W_ID, WI_ID	—
1	U_USERNAME, U_EMAIL, U_FULL_NAME, U_PASSWORD_HASH, U_AVAILABLE_BALANCE, U_INVESTED_BALANCE, U_CREATED_AT, U_UPDATED_AT	FD1 (U_ID → …)
2	C_SYMBOL, C_NAME, C_CREATED_AT	FD4
3	M_CRYPTO_ID, M_QUOTE_CURRENCY, M_IS_ACTIVE, M_CREATED_AT	FD6
4	H_USER_ID, H_CRYPTO_ID, H_QUANTITY, H_RESERVED_QUANTITY, H_AVG_PRICE, H_CREATED_AT, H_UPDATED_AT	FD8
5	O_USER_ID, O_MARKET_ID, O_SIDE, O_TYPE, O_STATUS, O_QUANTITY, O_PRICE, O_PLACED_AT, O_EXECUTED_AT	FD10
6	T_USER_ID, T_TYPE, T_AMOUNT, T_CURRENCY, T_RELATED_ORDER, T_CREATED_AT, T_DESCRIPTION	FD11
7	MT_MARKET_ID, MT_EXECUTED_AT, MT_PRICE, MT_QUANTITY, MT_SIDE, MT_SOURCE	FD12
8	MC_MARKET_ID, MC_TIMEFRAME, MC_OPEN, MC_HIGH, MC_LOW, MC_CLOSE, MC_VOLUME, MC_CANDLE_TIME	FD13
9	W_USER_ID, W_NAME, W_CREATED_AT	FD15
10	WI_WATCHLIST_ID, WI_CRYPTO_ID, WI_ADDED_AT	FD16

The closure now contains all 68 attributes, so the set is a superkey; removing any one of its ten attributes drops an entire cluster that nothing else in the set can reach (e.g. drop T_ID and no remaining attribute determines any T_* value), so it is minimal — a candidate key.

It is not the only one. Any attribute that is itself a determinant of a whole cluster can stand in for that cluster's id — U_USERNAME or U_EMAIL for U_ID (FD2/FD3), C_SYMBOL for C_ID (FD5), {M_CRYPTO_ID, M_QUOTE_CURRENCY} for M_ID (FD7), {H_USER_ID, H_CRYPTO_ID} for H_ID (FD9), {MC_MARKET_ID, MC_TIMEFRAME, MC_CANDLE_TIME} for MC_ID (FD14), {WI_WATCHLIST_ID, WI_CRYPTO_ID} for WI_ID (FD17) — giving 3 × 2 × 2 × 2 × 1 × 1 × 1 × 2 × 1 × 2 = 96 candidate keys in total. The all-surrogate-id combination above is chosen as primary key for the same reason id was chosen over username/email/symbol/etc. per entity in ERModel: it is opaque, and none of its parts are things a user would ever legitimately change.

Normal form of R_EDUBERZA before decomposition: 1NF only, and barely that — see 2NF below. It cannot be in 2NF, 3NF or BCNF, since each of those requires 2NF as a precondition.

1NF decomposition

No decomposition happens at this step. 1NF requires atomic, single-valued attributes and no repeating groups; R_EDUBERZA was built that way from the start (every column above is a single scalar), so the relation already satisfies 1NF as written in De-normalized database form. The real work starts at 2NF.

2NF decomposition

Relation analyzed: R_EDUBERZA, all 68 attributes, primary key {U_ID, C_ID, M_ID, H_ID, O_ID, T_ID, MT_ID, MC_ID, W_ID, WI_ID} (10 attributes), FD1–FD17 in force.

Current normal form: 1NF only (previous section).

Violations: 2NF forbids a non-prime attribute from depending on part of a candidate key. Every single functional dependency in the canonical cover (FD1–FD17) has a left side that is a proper subset of the ten-attribute primary key — U_ID alone, C_ID alone, …, down to the two-attribute {WI_WATCHLIST_ID, WI_CRYPTO_ID}. There is no non-prime attribute in R_EDUBERZA that depends on the whole ten-attribute key and nothing smaller. In other words, every non-prime attribute violates 2NF at once — the violation is not a handful of stray columns to peel off, it is the entire relation, because gluing ten independent record types together under one artificial composite key was never going to satisfy 2NF to begin with.

Decomposition. This uses 3NF/BCNF synthesis (Bernstein's algorithm) rather than the binary decomposition algorithm: since the canonical cover is already in hand (as the phase instructions recommend building first), synthesis creates one relation per left-hand side in the cover directly, instead of hunting for one offending dependency at a time and splitting in two repeatedly. Grouping FD1–FD17 by determinant produces ten relations:

New relation	Attributes	Key(s)	Source FDs
R_USERS	U_ID, U_USERNAME, U_EMAIL, U_FULL_NAME, U_PASSWORD_HASH, U_AVAILABLE_BALANCE, U_INVESTED_BALANCE, U_CREATED_AT, U_UPDATED_AT	U_ID, U_USERNAME, U_EMAIL	FD1, FD2, FD3
R_CRYPTO	C_ID, C_SYMBOL, C_NAME, C_CREATED_AT	C_ID, C_SYMBOL	FD4, FD5
R_MARKETS	M_ID, M_CRYPTO_ID, M_QUOTE_CURRENCY, M_IS_ACTIVE, M_CREATED_AT	M_ID, {M_CRYPTO_ID, M_QUOTE_CURRENCY}	FD6, FD7
R_HOLDINGS	H_ID, H_USER_ID, H_CRYPTO_ID, H_QUANTITY, H_RESERVED_QUANTITY, H_AVG_PRICE, H_CREATED_AT, H_UPDATED_AT	H_ID, {H_USER_ID, H_CRYPTO_ID}	FD8, FD9
R_ORDERS	O_ID, O_USER_ID, O_MARKET_ID, O_SIDE, O_TYPE, O_STATUS, O_QUANTITY, O_PRICE, O_PLACED_AT, O_EXECUTED_AT	O_ID	FD10
R_TRANSACTIONS	T_ID, T_USER_ID, T_TYPE, T_AMOUNT, T_CURRENCY, T_RELATED_ORDER, T_CREATED_AT, T_DESCRIPTION	T_ID	FD11
R_MARKET_TRADES	MT_ID, MT_MARKET_ID, MT_EXECUTED_AT, MT_PRICE, MT_QUANTITY, MT_SIDE, MT_SOURCE	MT_ID	FD12
R_MARKET_CANDLES	MC_ID, MC_MARKET_ID, MC_TIMEFRAME, MC_OPEN, MC_HIGH, MC_LOW, MC_CLOSE, MC_VOLUME, MC_CANDLE_TIME	MC_ID, {MC_MARKET_ID, MC_TIMEFRAME, MC_CANDLE_TIME}	FD13, FD14
R_WATCHLISTS	W_ID, W_USER_ID, W_NAME, W_CREATED_AT	W_ID	FD15
R_WATCHLIST_ITEMS	WI_ID, WI_WATCHLIST_ID, WI_CRYPTO_ID, WI_ADDED_AT	WI_ID, {WI_WATCHLIST_ID, WI_CRYPTO_ID}	FD16, FD17

Every one of these ten relations now has all of its non-prime attributes depending on its whole key (in every case there is only one non-composite or one designated key doing the determining, so 2NF holds trivially in each).

Dependency preservation. FD1–FD17 is the canonical cover of R_EDUBERZA. Each FD's determinant and every one of its dependent attributes land inside exactly one of the ten new relations (see the "Source FDs" column above — no FD is split across two relations). The union of the FDs that hold on R_USERS, …, R_WATCHLIST_ITEMS is therefore exactly FD1–FD17 again: nothing was lost.

Lossless join — chase test.

Note: the chase algorithm is not part of the course material. I was curious about a stricter way to test lossless join than the usual "the common attributes are a key of one side" argument, so I applied it here.

The chase decides whether a decomposition R = R1 ∪ … ∪ Rn is lossless under a set of functional dependencies. Build a tableau with one column per attribute of R and one row per relation Ri. In row i, put a distinguished symbol a in every column of Ri and a unique symbol b_i in every other column. Then repeat, until nothing changes: for each FD X → Y, whenever two rows agree on all of X, make them agree on Y. If they disagree, an a wins, otherwise one b replaces the other. The decomposition is lossless exactly when some row ends up with a in every column.

All attributes of one cluster (U_*, C_*, M_*, …) always appear together, and FD1–FD17 never mix clusters. So each cluster is one column group below: a means every column of the group holds a distinguished symbol, and b means none of them does. The foreign-key attributes (H_USER_ID, O_MARKET_ID, …) belong to their own cluster (H_*, O_*, …), not to the cluster they reference.

Step 1 — the ten relations from the table above.

              U*  C*  M*  H*  O*  T*  MT* MC* W*  WI*
R_USERS        a   b   b   b   b   b   b   b   b   b
R_CRYPTO       b   a   b   b   b   b   b   b   b   b
R_MARKETS      b   b   a   b   b   b   b   b   b   b
R_HOLDINGS     b   b   b   a   b   b   b   b   b   b
R_ORDERS       b   b   b   b   a   b   b   b   b   b
R_TRANSACTIONS b   b   b   b   b   a   b   b   b   b
R_MARKET_TR.   b   b   b   b   b   b   a   b   b   b
R_MARKET_CA.   b   b   b   b   b   b   b   a   b   b
R_WATCHLISTS   b   b   b   b   b   b   b   b   a   b
R_WATCHLIST_I. b   b   b   b   b   b   b   b   b   a

Every FD has its left side inside one cluster, for example U_ID → U_* or H_USER_ID, H_CRYPTO_ID → H_ID. For such an FD to fire, two rows would have to agree on that left side. But only one row has as in that cluster, and the bs of different rows are all different, so no two rows ever agree on any left side. *The chase changes nothing, and no row becomes all a.' Under FD1–FD17 alone, the ten relations are not guaranteed to join back to R_EDUBERZA. This is not an accident of this model. It is exactly why Bernstein's synthesis algorithm has a final step: *if no synthesised relation contains a candidate key of R, add one that does. None of the ten contains the ten-attribute key.

Step 2 — add the key relation `R_KEY(U_ID, C_ID, M_ID, H_ID, O_ID, T_ID, MT_ID, MC_ID, W_ID, WI_ID). Its row has a only in the ten ID columns, written a· for "a` in the ID, b in the rest of the group":

              U*  C*  M*  H*  O*  T*  MT* MC* W*  WI*
R_KEY          a·  a·  a·  a·  a·  a·  a·  a·  a·  a·
(the ten rows of step 1 unchanged)

Now FD1 U_ID → U_* fires: row R_KEY and row R_USERS both have a in U_ID, so they must agree on the rest of U_*, and R_USERS has a there. R_KEY becomes a in the whole U* group. The same happens with FD4 (C*), FD6 (M*), FD8 (H*), FD10 (O*), FD11 (T*), FD12 (MT*), FD13 (MC*), FD15 (W*) and FD16 (WI*):

              U*  C*  M*  H*  O*  T*  MT* MC* W*  WI*
R_KEY          a   a   a   a   a   a   a   a   a   a     <- all distinguished

Row R_KEY is all a, so the decomposition into the ten relations plus R_KEY is lossless.

Why R_KEY is not kept in the final schema. An instance of R_KEY would only record which ID of one cluster appears together with which ID of every other cluster. As shown under Candidate keys and primary key, the ten clusters are independent record types, and R_EDUBERZA pairs every row of one with every row of the others. So R_KEY would be just the cross product of the ten ID sets and would carry no information. The same independence means the join dependency ⋈[R_USERS, …, R_WATCHLIST_ITEMS] holds on R_EDUBERZA by construction. Under that dependency the ten relations alone already reconstruct it: their natural join, with no common attributes, is exactly that cross product. The chase makes this reasoning explicit. FDs by themselves cannot prove the join lossless; you need either the key relation or the independence of the clusters. That was hidden in the earlier "foreign key equals primary key" argument, which described the equi-joins the application runs, not the natural join the lossless-join property is about.

3NF decomposition

Relations analyzed: each of the ten relations produced above, individually.

For each relation, 3NF asks whether any non-prime attribute is transitively dependent on a key — i.e. determined by another non-prime attribute rather than directly by the key. This is exactly where the foreign-key-carried dependencies from Dependencies carried by foreign keys have to be checked, because that table is precisely the list of "dependency that would cause a problem at the next higher normal form" the phase template asks for.

Worked example — R_MARKETS. Its key M_ID determines M_CRYPTO_ID, and M_CRYPTO_ID → C_SYMBOL, C_NAME, C_CREATED_AT also holds (M_CRYPTO_ID draws its values from C_ID's domain). If C_SYMBOL, C_NAME and C_CREATED_AT were still columns of R_MARKETS, this would be exactly the transitive dependency M_ID → M_CRYPTO_ID → C_SYMBOL that violates 3NF. They are not: the 2NF step above already put them in R_CRYPTO, keyed directly by C_ID (FD4), because FD4 — not the derived M_CRYPTO_ID → C_SYMBOL — is what the canonical cover actually contains. R_MARKETS itself has no attribute that determines another non-prime attribute of R_MARKETS; the transitive dependency is real, but it points out of the relation, not within it.

The same reasoning applies to every other foreign key in the list: H_USER_ID/H_CRYPTO_ID, O_USER_ID/O_MARKET_ID, T_USER_ID/T_RELATED_ORDER, MT_MARKET_ID, MC_MARKET_ID, W_USER_ID, WI_WATCHLIST_ID/WI_CRYPTO_ID are all foreign keys sitting alongside a non-key attribute set that depends only on their own relation's key, never on the foreign key itself. None of R_USERS, R_CRYPTO, R_HOLDINGS, R_ORDERS, R_TRANSACTIONS, R_MARKET_TRADES, R_MARKET_CANDLES, R_WATCHLISTS, R_WATCHLIST_ITEMS has a non-prime attribute that another non-prime attribute of the same relation determines.

Conclusion: synthesising directly from the canonical cover in the 2NF step already avoided every transitive dependency — there is nothing left to decompose for 3NF. All ten relations from the previous section satisfy 3NF unchanged.

BCNF if possible

Relations analyzed: the same ten relations, checked against the stricter BCNF rule: every determinant of every functional dependency that holds on the relation must be a candidate key of that relation (3NF allows an exception when the dependent side is prime; BCNF does not).

Relation	Functional dependencies in force	Determinant	Is it a candidate key?
R_USERS	FD1, FD2, FD3	U_ID, U_USERNAME, U_EMAIL	Yes — all three are candidate keys
R_CRYPTO	FD4, FD5	C_ID, C_SYMBOL	Yes — both candidate keys
R_MARKETS	FD6, FD7	M_ID, {M_CRYPTO_ID, M_QUOTE_CURRENCY}	Yes — both candidate keys
R_HOLDINGS	FD8, FD9	H_ID, {H_USER_ID, H_CRYPTO_ID}	Yes — both candidate keys
R_ORDERS	FD10	O_ID	Yes — the only candidate key
R_TRANSACTIONS	FD11	T_ID	Yes — the only candidate key
R_MARKET_TRADES	FD12	MT_ID	Yes — the only candidate key
R_MARKET_CANDLES	FD13, FD14	MC_ID, {MC_MARKET_ID, MC_TIMEFRAME, MC_CANDLE_TIME}	Yes — both candidate keys
R_WATCHLISTS	FD15	W_ID	Yes — the only candidate key
R_WATCHLIST_ITEMS	FD16, FD17	WI_ID, {WI_WATCHLIST_ID, WI_CRYPTO_ID}	Yes — both candidate keys

Every determinant in every relation is one of that relation's own candidate keys. All ten relations are already in BCNF — the highest of the four normal forms this phase asks for, reached in the same step that fixed 2NF. This is not a coincidence: it happens because the canonical cover already grouped each relation's own key directly against its own attributes with no attribute appearing on the right side of two different relations' dependencies, which is exactly what synthesis from a canonical cover guarantees when, as here, none of the per-cluster functional dependencies overlap.

No further decomposition is possible or necessary; splitting any of the ten relations further would only separate attributes that already depend on the whole key of a BCNF relation, which cannot fix anything and only costs a join.

Final result and discussion

Normalized relational model

R_USERS          (U_ID, U_USERNAME, U_EMAIL, U_FULL_NAME, U_PASSWORD_HASH,
                   U_AVAILABLE_BALANCE, U_INVESTED_BALANCE, U_CREATED_AT, U_UPDATED_AT)
R_CRYPTO         (C_ID, C_SYMBOL, C_NAME, C_CREATED_AT)
R_MARKETS        (M_ID, M_CRYPTO_ID → R_CRYPTO, M_QUOTE_CURRENCY, M_IS_ACTIVE, M_CREATED_AT)
R_HOLDINGS       (H_ID, H_USER_ID → R_USERS, H_CRYPTO_ID → R_CRYPTO, H_QUANTITY,
                   H_RESERVED_QUANTITY, H_AVG_PRICE, H_CREATED_AT, H_UPDATED_AT)
R_ORDERS         (O_ID, O_USER_ID → R_USERS, O_MARKET_ID → R_MARKETS, O_SIDE, O_TYPE,
                   O_STATUS, O_QUANTITY, O_PRICE, O_PLACED_AT, O_EXECUTED_AT)
R_TRANSACTIONS   (T_ID, T_USER_ID → R_USERS, T_TYPE, T_AMOUNT, T_CURRENCY,
                   T_RELATED_ORDER → R_ORDERS, T_CREATED_AT, T_DESCRIPTION)
R_MARKET_TRADES  (MT_ID, MT_MARKET_ID → R_MARKETS, MT_EXECUTED_AT, MT_PRICE, MT_QUANTITY,
                   MT_SIDE, MT_SOURCE)
R_MARKET_CANDLES (MC_ID, MC_MARKET_ID → R_MARKETS, MC_TIMEFRAME, MC_OPEN, MC_HIGH, MC_LOW,
                   MC_CLOSE, MC_VOLUME, MC_CANDLE_TIME)
R_WATCHLISTS     (W_ID, W_USER_ID → R_USERS, W_NAME, W_CREATED_AT)
R_WATCHLIST_ITEMS(WI_ID, WI_WATCHLIST_ID → R_WATCHLISTS, WI_CRYPTO_ID → R_CRYPTO, WI_ADDED_AT)

Ten relations, every one in BCNF, connected by the eleven foreign keys spelled out above.

Discussion

This is the P2 design. Strip the U_/C_/M_/… prefixes back to plain column names and R_USERS, R_CRYPTO, R_MARKETS, R_HOLDINGS, R_ORDERS, R_TRANSACTIONS, R_MARKET_TRADES, R_MARKET_CANDLES, R_WATCHLISTS, R_WATCHLIST_ITEMS are, attribute for attribute and key for key, users, crypto, markets, holdings, orders, transactions, market_trades, market_candles, watchlists, watchlist_items from RelationalDesign. Every foreign key matches, every candidate key matches (including the less obvious composite ones — {user_id, crypto_id} on holdings, {crypto_id, quote_currency} on markets, {market_id, timeframe, candle_time} on market_candles), and the normal form matches (P2 already claimed 3NF; this phase shows the stronger result that the design is actually in BCNF).

That is not a coincidence of two people happening to agree — it is what should happen when a design is derived correctly twice by two different methods from the same underlying model: P2 got here by applying the standard ER-to-relational transformation rules (each entity becomes a table on its own key, each attributed M:N relationship becomes a table on the combined key, each attributeless 1:N relationship becomes a foreign key on the "many" side). This phase got here by ignoring that transformation entirely, writing down only the attributes and the functional dependencies they obey, and mechanically applying 2NF/3NF/BCNF synthesis. Landing on the same ten relations either means the P2 transformation rules are sound for this particular model (which they are, for exactly the reason RelationalDesign (Normalisation section) already argued: single-column UUID primary keys everywhere rule out partial dependencies by construction, and no non-key attribute references another non-key attribute anywhere in the model, which rules out transitive dependencies too), or it is a coincidence spanning ten independently-checked relations and dozens of functional dependencies — the first explanation is the only credible one.

The one substantive difference is holdings.avg_price, which P2 documents as a derived attribute — the running weighted-average buy price, recomputable from the buy rows in transactions — kept as a stored column anyway for read performance (RelationalDesign (Normalisation section) calls this out explicitly as an accepted denormalisation). Nothing in this phase's functional-dependency analysis can see that H_AVG_PRICE is derivable from T_* rows rather than stored independently — FD8 (H_ID → H_AVG_PRICE) is a perfectly ordinary functional dependency either way, because derivability from a different relation's rows is a property of the data and the application logic that maintains it (see UseCase0004's ON CONFLICT … DO UPDATE), not something that shows up as a violation of any single-relation normal form. Formal normalization and "no column is a cached computation of other columns" are related but different concerns; this phase only checked the first one.

Which design is used going forward: P2's, unchanged. Since the two designs coincide exactly, "restructuring the database objects" means confirming there is nothing to change rather than writing new DDL. server/db/schema_creation.sql already matches R_USERS…R_WATCHLIST_ITEMS column-for-column (including holdings.reserved_quantity, added between P2 and this phase — see RelationalDesignAIUsage (section "Session 3 — 2026-09-16") — which is H_RESERVED_QUANTITY above, correctly grouped under R_HOLDINGS's key alongside H_QUANTITY and not treated as needing a relation of its own). P4's prototype (server/trade.go, server/portfolio.go) keeps working against the same schema without change. RelationalDesign has been updated with a short note pointing here as the formal validation of its normal-form claim.

The table definitions in server/db/schema_creation.sql:

CREATE TABLE project.users (
    id                uuid            PRIMARY KEY DEFAULT gen_random_uuid(),
    username          varchar(50)     NOT NULL UNIQUE,
    email             varchar(255)    NOT NULL UNIQUE,
    full_name         varchar(200),
    password_hash     varchar(255)    NOT NULL,
    available_balance numeric(18,4)   NOT NULL DEFAULT 0 CHECK (available_balance >= 0),
    invested_balance  numeric(18,4)   NOT NULL DEFAULT 0 CHECK (invested_balance  >= 0),
    created_at        timestamptz     NOT NULL DEFAULT now(),
    updated_at        timestamptz
);

-- ============================================================================
-- CRYPTO
-- Catalog of crypto assets available on the platform.
-- ============================================================================
CREATE TABLE project.crypto (
    id         uuid         PRIMARY KEY DEFAULT gen_random_uuid(),
    symbol     varchar(20)  NOT NULL UNIQUE,
    name       varchar(255) NOT NULL,
    created_at timestamptz  NOT NULL DEFAULT now()
);

-- ============================================================================
-- MARKETS
-- A market is a (crypto, quote_currency) pair, e.g. BTC/USD.
-- ============================================================================
CREATE TABLE project.markets (
    id             uuid        PRIMARY KEY DEFAULT gen_random_uuid(),
    crypto_id      uuid        NOT NULL REFERENCES project.crypto(id),
    quote_currency char(3)     NOT NULL DEFAULT 'USD',
    is_active      boolean     NOT NULL DEFAULT true,
    created_at     timestamptz NOT NULL DEFAULT now(),
    CONSTRAINT uq_markets UNIQUE (crypto_id, quote_currency)
);

-- ============================================================================
-- HOLDINGS
-- Per-user crypto position with running weighted average entry price.
-- ============================================================================
CREATE TABLE project.holdings (
    id                uuid           PRIMARY KEY DEFAULT gen_random_uuid(),
    user_id           uuid           NOT NULL REFERENCES project.users(id)  ON DELETE CASCADE,
    crypto_id         uuid           NOT NULL REFERENCES project.crypto(id),
    quantity          numeric(20,4)  NOT NULL CHECK (quantity >= 0),
    -- Committed to the user's own open sell orders, not yet removed from the
    -- position. quantity - reserved_quantity is what is actually free to
    -- sell — the crypto-side equivalent of users.available_balance.
    reserved_quantity numeric(20,4)  NOT NULL DEFAULT 0
                                      CHECK (reserved_quantity >= 0 AND reserved_quantity <= quantity),
    -- Weighted-average entry price. NOT NULL so that the P/L arithmetic in
    -- v_portfolio can never silently produce NULL for an existing position.
    avg_price         numeric(18,6)  NOT NULL DEFAULT 0 CHECK (avg_price >= 0),
    created_at        timestamptz    NOT NULL DEFAULT now(),
    updated_at        timestamptz,
    CONSTRAINT uq_holdings_user_crypto UNIQUE (user_id, crypto_id)
);

-- ============================================================================
-- ORDERS
-- Orders placed by users on a market.
-- ============================================================================
CREATE TABLE project.orders (
    id          uuid           PRIMARY KEY DEFAULT gen_random_uuid(),
    user_id     uuid           NOT NULL REFERENCES project.users(id)   ON DELETE CASCADE,
    market_id   uuid           NOT NULL REFERENCES project.markets(id),
    side        varchar(4)     NOT NULL CHECK (side   IN ('buy', 'sell')),
    type        varchar(20)    NOT NULL CHECK (type   IN ('market', 'limit')),
    status      varchar(20)    NOT NULL CHECK (status IN ('open', 'executed', 'cancelled')),
    quantity    numeric(20,4)  NOT NULL CHECK (quantity > 0),
    price       numeric(18,6),
    placed_at   timestamptz    NOT NULL DEFAULT now(),
    executed_at timestamptz
);

CREATE INDEX idx_orders_user      ON project.orders(user_id);
CREATE INDEX idx_orders_market    ON project.orders(market_id);
CREATE INDEX idx_orders_status    ON project.orders(status);

-- ============================================================================
-- TRANSACTIONS
-- Financial ledger: deposits, buys, sells, fees.
-- ============================================================================
CREATE TABLE project.transactions (
    id            uuid           PRIMARY KEY DEFAULT gen_random_uuid(),
    user_id       uuid           NOT NULL REFERENCES project.users(id) ON DELETE CASCADE,
    type          varchar(50)    NOT NULL CHECK (type IN ('deposit', 'buy', 'sell', 'fee')),
    amount        numeric(18,4)  NOT NULL,
    currency      char(3)        NOT NULL DEFAULT 'USD',
    related_order uuid           REFERENCES project.orders(id),
    created_at    timestamptz    NOT NULL DEFAULT now(),
    description   text
);

CREATE INDEX idx_transactions_user ON project.transactions(user_id, created_at DESC);

-- ============================================================================
-- MARKET TRADES
-- Raw executed trades on a market. Source of truth for current price.
-- ============================================================================
CREATE TABLE project.market_trades (
    id          bigserial      PRIMARY KEY,
    market_id   uuid           NOT NULL REFERENCES project.markets(id),
    executed_at timestamptz    NOT NULL,
    price       numeric(18,6)  NOT NULL CHECK (price    > 0),
    quantity    numeric(20,6)  NOT NULL CHECK (quantity > 0),
    side        varchar(4)     CHECK (side IN ('buy', 'sell')),
    source      varchar(50)    NOT NULL DEFAULT 'simulation'
);

CREATE INDEX idx_market_trades_market_time ON project.market_trades(market_id, executed_at DESC);

-- ============================================================================
-- MARKET CANDLES
-- OHLCV aggregates over standard timeframes.
-- ============================================================================
CREATE TABLE project.market_candles (
    id          bigserial      PRIMARY KEY,
    market_id   uuid           NOT NULL REFERENCES project.markets(id),
    timeframe   varchar(5)     NOT NULL CHECK (timeframe IN ('1m', '5m', '1h', '1d')),
    open        numeric(18,6)  NOT NULL,
    high        numeric(18,6)  NOT NULL,
    low         numeric(18,6)  NOT NULL,
    close       numeric(18,6)  NOT NULL,
    volume      numeric(20,6)  NOT NULL,
    candle_time timestamptz    NOT NULL,
    CONSTRAINT uq_candle UNIQUE (market_id, timeframe, candle_time)
);

CREATE INDEX idx_market_candles_market_tf_time ON project.market_candles(market_id, timeframe, candle_time DESC);

-- ============================================================================
-- WATCHLISTS
-- ============================================================================
CREATE TABLE project.watchlists (
    id         uuid         PRIMARY KEY DEFAULT gen_random_uuid(),
    user_id    uuid         NOT NULL REFERENCES project.users(id) ON DELETE CASCADE,
    name       varchar(100) NOT NULL,
    created_at timestamptz  NOT NULL DEFAULT now()
);

CREATE TABLE project.watchlist_items (
    id           uuid        PRIMARY KEY DEFAULT gen_random_uuid(),
    watchlist_id uuid        NOT NULL REFERENCES project.watchlists(id) ON DELETE CASCADE,
    crypto_id    uuid        NOT NULL REFERENCES project.crypto(id),
    added_at     timestamptz NOT NULL DEFAULT now(),
    CONSTRAINT uq_watchlist_crypto UNIQUE (watchlist_id, crypto_id)
);