Appendix F: Notation Quick Reference

00-START-HERE · 01-READING-ROUTES · A5_annotated_reading_list · A7_glossary This appendix provides a compact, calculation-oriented reference for the source’s symbols, overload rules, and conventions while leaving the extended discussion to the front matter.

TL;DR

Preserve case, primes, subscripts, function arguments and units.
This reference writes $m$ for PQ subquantizers and $M$ for the HNSW neighbor parameter. Libraries may reuse uppercase $M$ for PQ; map the API’s meaning explicitly.
Keep $R@k$ , $P@k$ and $P_k$ distinct: recall, precision and segmentation error.
Write the logarithm base. Use natural logs for nats/log loss and the stated BM25 form, and $\log_2$ for DCG and code-bit counts.
Decimal and binary storage units are different. Vectors are columns unless a derivation explicitly uses row-vector stacking.
The $d=768$ , nominal $110$ -million-parameter encoder is an illustrative comparison setup, not a universal architecture requirement.

The story

Think of the notation sheet as the control board in a rail yard. Each symbol is a track label. $x$ brings user input into the yard. $q=\Phi(x)$ sends a query toward retrieval. $c_1,\ldots,c_k$ carries contexts onward, and $y$ marks the generated answer at the end of the route. Subscripts and primes are track markers. $k$ and $k^{\prime}$ do not stop at the same platform. Uppercase $D$ , lowercase $d$ , and primed $D^{\prime}$ also lead to different destinations. The overload table is the switch chart. It tells the operator whether $d$ means a document or dimensionality, whether $k$ counts contexts or PQ centroids, and whether $N$ counts vectors or model parameters. The switch chart chooses $m$ for PQ here and $M$ for HNSW. That is a local notation policy, not a claim about every paper or library. Faiss, for example, also uses uppercase $M$ as its PQ subquantizer parameter.[1] The conventions state units, vector orientation and the recurring setup. Logarithm bases must follow the quantity: bits and DCG use base $2$ , while nats, natural-log BM25 and natural-log losses use base $e$ . A change of base changes reported numeric values and units. The board works only when the operator reads the local route before interpreting a reused label.

Decoder table

The three blocks below preserve the source layout and all 57 primary symbol-meaning entries.

Block 1: System, retrieval, and index notation

Symbol	Meaning	Why it matters
$\theta$	Generator parameters	Locates the trainable or frozen generator state
$D$	Datastore	Names the corpus available to retrieval
$D_1,\ldots,D_m$	Component stores	Separates federated or multimodal stores
$x$	User input	Marks the pipeline input before encoding
$q=\Phi(x)$	Query issued to the retriever	Connects user input to retrieval geometry
$g(x)$	Retrieval gate with values $\{0,1\}$	Decides whether retrieval runs
$\mathop{\mathrm{RET}}\nolimits$	Retriever	Names the evidence-selection component
$\mathop{\mathrm{GEN}}\nolimits_{\theta}$	Generator	Names the answer-producing component
$c_1,\ldots,c_k$	Retrieved contexts	Identifies evidence passed downstream
$y$	Generated answer	Marks the pipeline output
$k$	Contexts reaching the generator	Sets the final context depth
$k^{\prime}$	Retrieval depth before reranking	Separates candidate depth from final depth
$N$	Vectors in the index	Sets index scale
$d$	Embedding dimension	Sets vector width and memory cost
$M$	HNSW neighbor parameter in this reference	A configured slot/connectivity parameter, not every node’s realized out-degree
$M_0$	Base-layer neighbor-slot capacity, often $2M$	A construction/allocation convention, not exact realized degree
$\mathrm{ef}$	HNSW search candidate-width parameter	Not necessarily graph hops, examined nodes or hardware parallelism
$n_{\mathrm{probe}}$	Inverted File (IVF) cells probed	Trades search breadth for latency
$m$	PQ sub-quantizers	Splits a vector for compression
$k$	PQ centroids per subspace	Sets the local codebook size
$b$	Bits per sub-code when $k=2^b$ ; total PQ bits are $mb$	Declare total packed bytes $\lceil mb/8\rceil$ and padding separately
$R@k$	Recall at a declared retrieval cutoff	Distinguish judged-relevance recall, ANN-neighbor recall and full evidence-set coverage

Block 2: Scoring and evaluation notation

Symbol	Meaning	Why it matters
$s(q,d)$	Relevance score	Orders a document for a query
$\mathop{\mathrm{rank}}\nolimits(d)$	Position in a ranked list	Connects a score to rank-based metrics
$f$	Term frequency	Controls lexical evidence strength
$k_1,b$	Best Matching 25 (BM25) saturation and length normalization	Control repetition and document-length effects
$\mathop{\mathrm{avgdl}}\nolimits$	Mean document length	Anchors BM25 length normalization
$E_Q,E_P$	Query and passage encoders	Map both sides into retrieval space
$L_q,L_p$	Query and passage token lengths	Set encoder and interaction cost
$\mathop{\mathrm{MaxSim}}\nolimits$	Late-interaction operator	Aggregates token-level similarity
$\mathop{\mathrm{MRR}}\nolimits$	Mean reciprocal rank of the first relevant hit	Permits multiple relevant items; zero when none appears within cutoff
$\mathop{\mathrm{DCG}}\nolimits$	Discounted Cumulative Gain	Discounts useful results by rank
$\mathop{\mathrm{nDCG}}\nolimits @k$	Normalized discounted cumulative gain at a declared cutoff	State gain, judgment universe, tie and zero-IDCG policies
$P@k$	Precision at $k$	Measures useful-result density
$P_k$	Segmentation error	Measures chunk-boundary error rather than precision
$\mathop{\mathrm{IoU}}\nolimits$	Intersection over Union for chunk-span overlap	Measures overlap between predicted and reference spans
$F_1$	$2PR/(P+R)$ when $P+R\gt 0$	State zero-denominator and macro/micro averaging policies

Block 3: Training, serving, and graph notation

Symbol	Meaning	Why it matters
$\mathcal{L}$	Training loss	Defines the optimization target
$\tau$	Positive softmax temperature $\tau\gt 0$	Relative to fixed scores: smaller sharpens, larger flattens
$D^{\prime}$	Retrieved subset in training	Separates training evidence from the full store
$\lambda(d,x)$	Normalized document weight for a stated ensemble	Define candidate support and normalization; not every mixture uses this same convention
$P_R$	Retriever distribution	Describes retrieval preferences
$Q_{\mathrm{LM}}$	Generator preference distribution	Describes generator preferences
$\gamma,\beta$	Retriever and generator temperatures	Calibrate the two distributions
$r$	Low-Rank Adaptation (LoRA) rank	Sets adapter capacity
$A,B$	For $W\in\mathbb R^{d_{\mathrm{out}}\times d_{\mathrm{in}}}$ , $A\in\mathbb R^{r\times d_{\mathrm{in}}}$ , $B\in\mathbb R^{d_{\mathrm{out}}\times r}$	$W+(\alpha/r)BA$ is common; factor order follows the chosen orientation
$E_q,E_d$	Shared-space embeddings	Enable cross-modal or shared-space scoring
$t_i$	Stage latency, distinct from attributable device service time	Parallel stages need a critical path; resource cost needs service/capacity allocation
$C_q$	Cost per query	Connects design choices to serving economics
$B$	Serving batch size	Controls throughput and memory use
$s$	Tokens in context or answer	Sets sequence-dependent serving cost
$L$	Transformer layers	Sets model depth and repeated compute
$G$	Knowledge graph	Names the graph used for grounded retrieval
$\lvert V\rvert,\lvert E\rvert$	Entities and edges	Quantify graph size
$\omega$	Graph grounding operator	Maps a query into graph-grounded form
$h$	Traversal depth in hops	Bounds graph search and reasoning depth
$\theta_m$	Example modality-specific admission threshold	Validate modality/score comparability; no theorem forbids a calibrated global policy

Core mechanics

Purpose and use

The full notation chapter carries the conventions and extended overload discussion. This appendix is the compact version. It gives symbols and meanings for use during a derivation rather than during continuous reading. Read the exact glyph before assigning meaning. Case distinguishes $D$ from $d$ . A prime distinguishes $D^{\prime}$ from $D$ and $k^{\prime}$ from $k$ . Subscripts distinguish $M_0$ , $P_R$ , $Q_{\mathrm{LM}}$ , $\theta_m$ , and related symbols from their base letters. Function arguments also establish scope. $s(q,d)$ is a relevance score, while standalone $s$ counts tokens in a context or answer. The calligraphic symbol $\mathcal{L}$ is training loss. Plain $L$ counts Transformer layers.

Overloaded symbols

Symbol	Meaning A	Meaning B or scope condition
$d$	A document or chunk	Embedding or hidden dimensionality
$k$	Retrieved contexts	PQ centroids per sub-quantizer
$m$	PQ sub-quantizers	Datastore count or modality index
$M$	HNSW neighbor parameter in this reference	Other notations, including Faiss PQ, reuse $M$ ; never translate by case alone
$b$	BM25 length-normalization parameter	Sub-code bits, total code bits and bytes/parameter must be distinguished by scope
$N$	Vectors in the index	Model parameter count
$P$	Model parameter count	Instance price per hour
$G$	Knowledge graph	Use $\mathop{\mathrm{schema}}\nolimits(G)$ for its schema where confusion is possible

The overload table is a scope rule. It does not authorize swapping meanings within one derivation. Use the surrounding topic to select the intended meaning.

Conventions

Convention	Rule	Scope condition
Logarithms	Write $\ln$ or $\log_2$ as appropriate.	Nats/log loss and displayed BM25 use natural logs; DCG and bit counts use base $2$ .
Storage	Use decimal units. $1\ \text{gigabyte (GB)}=10^9\ \text{bytes}$ .	This is the default storage convention.
Quoted storage source	Preserve binary units.	Apply this only when the source is quoted in binary units and the text says so.
Latency	Measure in milliseconds.	Applies when no different unit is written.
Throughput	Measure in queries per second.	Applies when no different unit is written.
Cost	Measure in United States dollars per query.	Applies when no different unit is written.
Vectors and transpose	Column vectors by default; $\mathsf T$ is transpose.	A stated row-stacked matrix convention overrides the default; check dimensions.
Recurring default	Illustrative $d=768$ and nominal $110\times10^6$ parameters.	Model parameter count does not by itself determine embedding dimension or FLOPs.

Scope-sensitive reading sequence

Match exact glyphs, arguments and units.
Define overloaded symbols locally and map any library parameter explicitly.
Check dimensions and vector orientation before multiplying matrices.
State the logarithm base and whether the quantity is in bits or nats.
Separate document count, indexed vector count and model parameters.
Treat $d=768$ and $110\times10^6$ parameters as illustrative inputs, not universal facts.

Distinctions that must survive transcription

Pair or family	Required distinction
$D$ , $d$ , $D^{\prime}$	Datastore, document or dimension by scope, and retrieved training subset
$k$ , $k^{\prime}$ , $k_1$	Context count or PQ centroid count, pre-rerank depth, and BM25 saturation
$M$ , $M_0$ , $m$	HNSW upper/base slot parameters and this reference’s PQ subquantizer count
$b$	Distinguish BM25 normalization, bits per sub-code, total code bits and bytes per parameter
$P@k$ , $P_k$ , $P_R$ , $P$	Precision at $k$ , segmentation error, retriever distribution, and an overloaded parameter or price symbol
$R@k$ , $\mathop{\mathrm{MRR}}\nolimits$ , $\mathop{\mathrm{DCG}}\nolimits$ , $\mathop{\mathrm{nDCG}}\nolimits @k$ , $F_1$	Distinct evaluation quantities
$s(q,d)$ , $s$	Relevance score and token count
$\mathcal{L}$ , $L$ , $L_q$ , $L_p$	Training loss, Transformer layer count, and query or passage token lengths
$E_Q$ , $E_P$ , $E_q$ , $E_d$	Query and passage encoders versus shared-space embeddings
$G$ , $\lvert V\rvert$ , $\lvert E\rvert$ , $\omega$ , $h$	Graph, entity count, edge count, grounding operator, and hop depth

Diagrams

The manifest records 0 numbered figures and 0 numbered tables. The source contains two unnumbered reference layouts. The main symbol layout is recreated in the three Decoder table blocks. The overloaded-symbol layout is recreated under Core mechanics. No source number or caption is invented.

Source layout accounting

Main notation layout: 57 symbol-meaning entries across three horizontal blocks.
Overloaded symbols layout: 8 symbols with two meanings or a scope restriction.
Conventions text: 8 rules covering bases, units, orientation, and the recurring default.

Whiteboard pack

What to draw

Draw system flow $x\to q=\Phi(x)\to\mathop{\mathrm{RET}}\nolimits\to c_{1:k}\to\mathop{\mathrm{GEN}}\nolimits_\theta\to y$ .
Separate index, evaluation, training, serving and graph symbols.
Mark local overloads and library-specific parameter names.
Distinguish PQ sub-code bits from total bits and HNSW slots from realized degree.
Add explicit log bases, storage units, latency versus service time and vector orientation.
State the example dimension and parameter count without claiming a universal setup.

Spoken script

Read the notation sheet by scope, not by a familiar letter alone. Trace input, query transformation, retrieval, contexts and generation. Then separate index sizes from model parameters, first-hit metrics from set coverage, and stage latency from attributable resource time. In this reference, PQ uses $m$ and HNSW uses $M$ , but a library can reuse uppercase $M$ for PQ. Distinguish bits per sub-code from total code bits, and use dimension checks for matrix products. Finish with explicit logarithm bases, decimal or binary units, and the illustrative encoder setup.

Interview traps

1. Are $m$ and $M$ interchangeable in product quantization?

Not within one locally defined derivation. This reference uses $m$ for PQ and $M$ for HNSW, but this is not universal: Faiss PQ also names its subquantizer parameter $M$ . Read the API and map it explicitly; HNSW parameters are not exact realized degrees.[1]

2. Does $k$ always mean the same retrieval depth?

No. $k$ can mean contexts reaching the generator or PQ centroids per sub-quantizer, while $k^{\prime}$ is retrieval depth before reranking. $k_1$ is the BM25 saturation parameter, so the exact mark and local scope choose the meaning.

3. Which metric symbols are easy to collapse incorrectly?

$R@k$ is Recall at $k$ , while $P@k$ is Precision at $k$ . $P_k$ is segmentation error. $\mathop{\mathrm{MRR}}\nolimits$ , $\mathop{\mathrm{DCG}}\nolimits$ , $\mathop{\mathrm{nDCG}}\nolimits @k$ , $\mathop{\mathrm{IoU}}\nolimits$ , and $F_1$ also retain separate meanings, so similar letters do not make them interchangeable.

4. What defaults apply when a derivation omits units?

Do not repair a missing definition by guessing. State $\ln$ for natural-log quantities such as nats and the displayed BM25 IDF, and $\log_2$ for DCG and bit counts. Use explicitly decimal $\mathrm{GB}$ or binary $\mathrm{GiB}$ . Milliseconds, QPS and dollars/query are convenient defaults only when dimensionally appropriate; stage wall time is not automatically billable device time.

5. Is $d=768$ a universal requirement?

No. It is the book’s recurring default with a $110$ million-parameter encoder, and its purpose is to make worked examples comparable without repeating the setup. A written alternative overrides a convention default.

Key numbers

Item	Value	Meaning
Primary reference entries	57	All symbol-meaning entries in the main layout
Primary source blocks	3	System and index, evaluation, and training or serving or graph notation
Explicit overloaded symbols	8	$d$ , $k$ , $m$ , $M$ , $b$ , $N$ , $P$ , and $G$
Convention rules	8	Base, storage, unit, orientation, and recurring-default rules
Retrieval gate values	$2$	$g(x)$ takes values $\{0,1\}$
Default logarithm base	$2$	Applies unless another base is written
Decimal gigabyte	$10^9$ bytes	Binary units require an explicit quoted-source exception
Default embedding dimension	$768$	Recurring setup for comparable worked examples
Default encoder size	$110$ million parameters	Paired with $d=768$ in the recurring setup
Manifest numbered visuals	0 figures, 0 tables	The source layouts are unnumbered

00-START-HERE · 01-READING-ROUTES · A5_annotated_reading_list · A7_glossary

Appendix F: Notation Quick Reference

TL;DR

The story

Decoder table

Block 1: System, retrieval, and index notation

Block 2: Scoring and evaluation notation

Block 3: Training, serving, and graph notation

Core mechanics

Purpose and use

Overloaded symbols

Conventions

Scope-sensitive reading sequence

Distinctions that must survive transcription

Diagrams

Source layout accounting

Whiteboard pack

What to draw

Spoken script

Interview traps

1. Are $m$ and $M$ interchangeable in product quantization?

2. Does $k$ always mean the same retrieval depth?

3. Which metric symbols are easy to collapse incorrectly?

4. What defaults apply when a derivation omits units?

5. Is $d=768$ a universal requirement?

Key numbers

Contents

Search the book

TL;DR

The story

Decoder table

Block 1: System, retrieval, and index notation

Block 2: Scoring and evaluation notation

Block 3: Training, serving, and graph notation

Core mechanics

Purpose and use

Overloaded symbols

Conventions

Scope-sensitive reading sequence

Distinctions that must survive transcription

Diagrams

Source layout accounting

Whiteboard pack

What to draw

Spoken script

Interview traps

1. Are mmm and MMM interchangeable in product quantization?

2. Does kkk always mean the same retrieval depth?

3. Which metric symbols are easy to collapse incorrectly?

4. What defaults apply when a derivation omits units?

5. Is d=768d=768d=768 a universal requirement?

Key numbers

1. Are $m$ and $M$ interchangeable in product quantization?

2. Does $k$ always mean the same retrieval depth?

5. Is $d=768$ a universal requirement?