Fixed point arithmetic represents values as integers with an implicit scale, commonly a power-of-two binary point. It delivers predictable, compact, energy-efficient arithmetic in DSPs, FPGAs, microcontrollers, control loops, and quantized AI inference. In a signed Q format, selected bits represent the integer range and remaining bits represent fractional resolution; a stored integer q corresponds to q times a scale such as two raised to minus the fractional-bit count. A professional performance claim defines workload, useful work, input and output shapes, numerical format, batch and concurrency, warmup and measurement interval, hardware and software versions, power state, correctness tolerance, and aggregation method. Peak specifications are ceilings under particular conditions; delivered behavior includes utilization, data movement, synchronization, control overhead, and tail effects. A contract states total bits, signedness, integer and fraction allocation, scale and zero point, rounding, saturation or wraparound, accumulator width, rescaling, and overflow policy.

Architecture, quantitative model, and operating behavior. Addition needs aligned scales; multiplication combines scales and doubles potential width; accumulation requires guard bits; shifting changes binary scale efficiently. Symmetric AI quantization often uses zero-centered signed integers, while asymmetric quantization adds a zero point. INT8 or INT4 inference maps weights and activations using per-tensor, per-channel, or per-group scales, computes integer dot products into wider accumulators, and requantizes outputs. This is fixed-point-like arithmetic with learned or calibrated scaling metadata. Qm.n binary formats, block floating point, affine integer quantization, logarithmic number systems, saturating DSP arithmetic, and mixed-width accumulators balance range, precision, and implementation cost. Useful analysis separates arithmetic, memory hierarchy, interconnect, storage, control, and queuing. It counts operations and bytes at each boundary, identifies dependencies and reuse, estimates ideal ceilings, and then uses counters and traces to explain the gap between the model and measurement. Ratios without a clearly named numerator and denominator invite invalid comparisons. Report useful throughput together with latency distribution, utilization, arithmetic intensity, achieved bandwidth, cache hit rate, occupancy, communication time, memory capacity, power, energy per result, quality, and cost. Include median and tail behavior, sustained rather than burst operation, repeated trials, and uncertainty. A faster approximation is not equivalent unless it meets the same accuracy and service constraints.

Implementation, hardware mapping, and bottlenecks. Derive ranges from analysis and calibration, allocate guard bits, choose unbiased rounding where useful, saturate safety-critical paths, propagate scales through graphs, test worst-case sums, and make conversions explicit at module boundaries. Integer adders and multipliers are smaller and often lower power than floating units; FPGA DSP blocks and ASIC MAC arrays exploit fixed widths. Narrow operands reduce SRAM, interconnect, and memory energy, often more than arithmetic energy. Overflow wraparound, quantization dead zones, biased truncation, scale mismatch, insufficient accumulator width, double requantization, per-channel metadata errors, and inputs outside calibration range can cause abrupt errors. Begin with a correct reference and representative shapes. Profile end to end, classify the dominant resource, inspect kernel and system timelines, change one bottleneck at a time, and remeasure because optimization moves pressure elsewhere. Tiling, fusion, batching, vectorization, layout, precision, compression, overlap, prefetch, sharding, and algorithm choice are useful only when they reduce the limiting resource. The execution path spans registers, local SRAM and caches, HBM or GDDR, host DRAM, PCIe or coherent links, scale-up fabric, network, and storage. Compute units consume tensors only when compilers and kernels issue enough independent work and the hierarchy supplies operands. Package wiring, memory stacks, clocks, voltage, thermal headroom, and power delivery determine sustained limits. Frequent mistakes include quoting peak instead of achieved rates, omitting data conversion and transfer, measuring a cached toy input, timing asynchronous work without synchronization, mixing decimal and binary units, ignoring warmup or throttling, changing precision or quality, averaging away tails, and optimizing a component that is not on the critical path.

Measurement, validation, and engineering controls. Use exhaustive tests for small widths, boundary values, worst-case accumulation, reference rational arithmetic, random distributions, saturation counts, signal-to-quantization noise, and end-to-end quality. Least significant step, representable range, saturation rate, quantization error, SQNR, accumulator headroom, area, frequency, power, memory, throughput, and task quality matter. Log integer codes and scales together, decode intermediates into real units, inject extrema, and use bit-accurate golden models shared by RTL, compiler, and software. Verification combines analytical bounds, microbenchmarks, hardware counters, kernel timelines, end-to-end traces, scaling sweeps, sensitivity to batch and shape, cold and warm runs, long-duration thermal tests, correctness comparisons, fault and congestion tests, and independent reproduction. Roofline and queueing models guide diagnosis but must be calibrated against the deployed machine. Benchmark code, datasets, model and compiler artifacts, drivers, firmware, topology, clock and power settings, environment, commands, raw samples, counter traces, and analysis notebooks remain versioned. Continuous tests detect regressions in quality, latency, throughput, bandwidth, memory, power, and cost, with thresholds chosen from variance rather than a single run. Published comparisons disclose configuration, exclusions, tuning effort, measurement boundary, quality criteria, and uncertainty. Energy and carbon claims distinguish chip, IT, and facility boundaries and avoid extrapolating one benchmark to all workloads. Owners review regressions and retain evidence sufficient to reproduce decisions.

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  <text x="380" y="31" fill="#e6edf3" font-size="21" font-weight="700" text-anchor="middle">Fixed-Point Arithmetic — The Binary Point Is Implied</text>
  <text x="380" y="53" fill="#8b98a5" font-size="12.5" text-anchor="middle">integers carry a scale factor; every operation must preserve range, precision, and binary-point position</text>

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    <text x="165" y="-10" fill="#93c5fd" font-size="10.5" font-weight="700" text-anchor="middle">SIGNED Q4.4 ENCODING · SCALE = 2⁻⁴</text>
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        <text x="17" y="23">0</text><text x="55" y="23">0</text><text x="93" y="23">1</text><text x="131" y="23">0</text>
        <text x="176" y="23">1</text><text x="214" y="23">1</text><text x="252" y="23">0</text><text x="290" y="23">0</text>
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      <text x="35" y="82" fill="#fca5a5">sign</text>
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    <text x="165" y="106" fill="#e6edf3" font-size="13" font-weight="700" text-anchor="middle">raw 44 × 2⁻⁴ = 2.75</text>
    <text x="165" y="120" fill="#8b98a5" font-size="8.2" text-anchor="middle">the point is metadata—not a stored bit</text>
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      <text x="25" y="13" fill="#93c5fd" font-size="11" font-weight="700">0010.1100</text>
      <text x="126" y="13" fill="#e6edf3" font-size="9">= +2.75</text>
      <text x="0" y="36" fill="#8b98a5" font-size="9">B</text>
      <text x="25" y="36" fill="#c4b5fd" font-size="11" font-weight="700">1111.1000</text>
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      <text x="229" y="70" fill="#e6edf3" font-size="9">Q8.8</text>
      <text x="25" y="91" fill="#8b98a5" font-size="9">raw 44 × (−8) = −352</text>
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    <text x="333" y="-10" fill="#c4b5fd" font-size="10.5" font-weight="700" text-anchor="middle">QUANTIZATION: CONTINUOUS VALUES SNAP TO A DISCRETE GRID</text>
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      <text x="110" y="78">2.6250</text><text x="181" y="78">2.6875</text><text x="252" y="78">2.7500</text>
      <text x="323" y="78">2.8125</text><text x="394" y="78">2.8750</text><text x="465" y="78">2.9375</text><text x="536" y="78">3.0000</text>
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    <text x="365" y="31" fill="#6ee7b7" font-size="8.5">stored 2.8125</text>
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    <text x="145" y="94" fill="#fbbf24" font-size="8" text-anchor="middle">Δ = 2⁻⁴ = 0.0625</text>
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    <text x="76" y="-8" fill="#fca5a5" font-size="10.5" font-weight="700" text-anchor="middle">OVERFLOW EXAMPLE</text>
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    <text x="102.5" y="17" fill="#fca5a5" font-size="9.5" font-weight="700" text-anchor="middle">WRAPAROUND · MODULO 2⁸</text>
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    <text x="123" y="17" fill="#6ee7b7" font-size="9.5" font-weight="700" text-anchor="middle">SATURATION · CLAMP TO RAIL</text>
    <text x="123" y="37" fill="#e6edf3" font-size="11" font-family="ui-monospace,SFMono-Regular,Menlo,monospace" text-anchor="middle">0111.1111 → +7.9375</text>
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  <text x="380" y="456" fill="#6b7684" font-size="11" text-anchor="middle">Fixed-point design is an explicit contract among numerical error, dynamic range, word width, area, power, and latency.</text>
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Selection and system-level application. Use fixed point where range is controlled and efficiency matters, floating point where dynamic range is unpredictable, and mixed schemes when sensitive operations need range while bulk MACs benefit from integers. Quantized neural networks, audio and wireless DSP, motor control, sensor interfaces, image pipelines, FPGA accelerators, and embedded inference use fixed-point arithmetic. Number format is co-designed with calibration, compiler quantization, MAC and accumulator RTL, SRAM width, interfaces, verification, safety limits, and model quality. Optimization is a system exercise across algorithms, precision, kernels, compiler, runtime, accelerator, memory, interconnect, scheduler, serving policy, cooling, and facility limits. Removing one ceiling often exposes another, so architecture decisions should optimize time and energy to a useful result rather than an isolated metric. A professional performance claim defines workload, useful work, input and output shapes, numerical format, batch and concurrency, warmup and measurement interval, hardware and software versions, power state, correctness tolerance, and aggregation method. Peak specifications are ceilings under particular conditions; delivered behavior includes utilization, data movement, synchronization, control overhead, and tail effects. Report useful throughput together with latency distribution, utilization, arithmetic intensity, achieved bandwidth, cache hit rate, occupancy, communication time, memory capacity, power, energy per result, quality, and cost. Include median and tail behavior, sustained rather than burst operation, repeated trials, and uncertainty. A faster approximation is not equivalent unless it meets the same accuracy and service constraints. CFS connects this topic to semiconductor architecture, implementation, verification, manufacturing, packaging, test, and deployed AI-system tradeoffs across the platform.