Figure: a worked 16-token example of RoPE geometry, and the block-movement arithmetic that survives it. The scenario is a 16-token span, a GPU cache of 8 KV block slots, a distance band of W = 3 meaning plus or minus 3 positions, and a tier fetch for a 5.00 mebibyte block of 16 tokens at 320 kibibytes per token. RoPE gives an upper-bound envelope, the magnitude of the inner product of q_m and k_n is at most B of the absolute distance m minus n with B non-increasing; it is never proportional to cosine of (m minus n) times theta, and because RoPE is a rotation applied to Q and K at compute time it carries no information about which blocks are needed next, so it cannot drive prefetch. Section 2 shows an assumed attention distribution around a query at position 8, in which positions 5 to 11, within plus or minus 3, are assumed to hold about 72 percent of the attention mass; that concentration is an input assumption, model-, layer-, head- and workload-dependent, not a measurement, and it is not something RoPE proves. Section 3 walks the movement itself: a demand miss on the block covering position 8, a tier fetch that completes in 164, 82 or 43 microseconds at an assumed 32, 64 or 121 gigabytes per second, attention then reading positions 6, 7, 9 and 10 out of the resident block, and finally an eviction in which the block's identity key and score travel with it and are restored on re-admission rather than reset. Section 4 gives the comparison that survives: moving a 5.00 mebibyte block costs 164, 82 or 43 microseconds across the assumed bandwidth range, while regenerating the same 16 tokens by forward compute costs 9, 11 or 18 milliseconds at 50, 40 or 25 percent model FLOPs utilisation. Moving is two to three orders of magnitude cheaper than regenerating, and no precise multiple is claimed, because the two quantities do not measure the same operation; regenerating an arbitrary interior block is at least that expensive and potentially far higher, since interior KV is not independently reproducible and hidden states depend recursively on all preceding context. Retracted from this figure and not replaced: the 97 percent hit rate, the 72 to 80 to 87 to 93 to 97 ladder, the plus 4 percentage point RoPE-prefetch rung, the 8.0 times 400 to 50 microsecond prefetch speedup, the 25 microsecond flash round-trip, the 200 nanosecond CXL access and the 100 nanosecond HBM access. No CXL hardware was measured. No single hit rate exists: the simulated range is 8.5 to 91.6 percent across budgets, policies and workloads. The largest effect in the study is not on this page and is not a scoring or prefetch mechanism at all: at a 32 gibibyte budget, letting identity-keyed policy state survive HBM eviction is worth plus 34.15 percentage points to LFU, from 15.56 to 49.71 percent, and plus 24.85 to EMA alpha 0.01, from 31.54 to 56.39 percent, while EMA alpha 0.15 is unchanged at minus 0.21 percentage points — larger than the whole spread between policies, and a protocol result testable with no CXL hardware. Simulated figures are provisional, because a single data-structure defect in Revision 1 inverted four of five findings.
RoPE Geometry: Worked Example
What the distance bound says, and what moving a block actually costs
RoPE is a rotation applied to Q/K at compute time: a function of position alone, known in
advance for every position, identical whether a block is resident or evicted. It carries no information
about which blocks are needed next. Distance decay is an upper bound,
|⟨q_m, k_n⟩| ≤ B(|m − n|) — never ∝ cos((m − n)·θ).
Withdrawn from this figure: the 97% hit rate and the 72 → 80 → 87 → 93 → 97 ladder (never run as an ablation; increments assumed additive and independent), the +4 pts RoPE-prefetch rung, the 8.0× “400 µs → 50 µs” prefetch speedup, and the 25 µs / 200 ns / 100 ns tier latencies. No CXL hardware was measured. What remains on this page is the geometry, and the move-versus-regenerate arithmetic in §4.
Assumption, not measurement. This walkthrough assumes attention concentrates
around the query position, so that positions 5–11 (within ±3 of query P=8) capture
~72% of total attention mass. The concentration is
model-, layer-, head- and workload-dependent and is an input to the model, not a result of it.
What RoPE itself provides is an upper-bound envelope on the inner-product magnitude,
|⟨q_m, k_n⟩| ≤ B(|m − n|) with B non-increasing — the inner product is a sum over
d/2 frequency pairs and does not decay monotonically.
PLACE scores the admission against state restored with the block, not reset.
The evicted blocks keep their identity key and score — restored on re-admission, not reset. That is the mechanism worth +34.15 pts to LFU at 32 GiB, and it is a protocol property, testable with no CXL hardware. Simulated (provisional)
Both columns are an Analytical model over stated inputs. The move column divides 5.00 MiB by three assumed tier bandwidths; no CXL hardware was measured anywhere in this package, and the 200 ns CXL access figure that once appeared here is demoted to an unmeasured assumption inside the 32–121 GB/s range. The regenerate column is the linear term of forward compute for 16 tokens at three MFU assumptions.
Claim the order of magnitude, not a multiple. The two quantities do not measure the same operation, so a ratio between them is not a speedup. The 218× figure that can be formed from one pair (17.9 ms ÷ 82 µs) is one point in the range and must always be quoted with its inputs. The honest statement is: moving a block is two to three orders of magnitude cheaper than regenerating it.
Interior blocks are worse than the table suggests. Regenerating an arbitrary interior block costs at least the figures above and potentially far more: interior KV is not independently reproducible, because hidden states depend recursively on all preceding context. Avoiding regeneration is what the tier is for.
Retracted, and not replaced. The former "16 serial round-trips at 25 µs = 400 µs versus
2 batched round-trips = 50 µs, an 8.0× reduction" is withdrawn: it rested on a 25 µs flash round-trip and on
a prefetcher that RoPE cannot supply, and batching does not survive a bandwidth-limited transfer model
anyway. The +4 pts / 93% → 97% rung and the 97% hit rate go with it — no single hit rate
exists, and the corrected simulation gives 8.5–91.6% across budgets, policies and
workloads. Simulated (provisional)
See Chapter 7, and ARCHITECTURE-v6.md for where
MOVE sits in the control plane.
The tier's value is avoiding regeneration — and the biggest win is a protocol, not a predictor
RoPE supplies only a non-increasing upper bound on inner-product magnitude with distance. It predicts nothing about residency, and the attention concentration assumed in §2 is an input, not a proof. What the tier actually buys is that a block already computed need not be computed again. And the largest effect measured anywhere in this study is neither the scoring function nor any prefetcher: it is letting a block's identity-keyed policy state survive its eviction from HBM — LFU 15.56 → 49.71 (+34.15 pts) and EMA α=0.01 31.54 → 56.39 (+24.85 pts) at 32 GiB, with EMA α=0.15 unchanged (−0.21 pts). Larger than the spread between policies, and testable with no CXL hardware. Simulated (provisional)
Illustrative walkthrough over 16 positions — never a benchmark. The attention concentration in §2 is an assumption; the RoPE bound in §1 is an Analytical model; the transfer and compute costs in §4 are an Analytical model over assumed bandwidths and MFU — no CXL hardware was measured. Hit-rate figures are Simulated (provisional): one data-structure defect in Revision 1 inverted four of five findings, so Revision 2 is provisional, not settled. Retracted here: 97%, the 72 → 80 → 87 → 93 → 97 ladder, the +4 pts prefetch rung, the 8.0× 400 µs → 50 µs speedup, and the 25 µs / 200 ns / 100 ns latencies. Work built on, cited not competed with: H2O, SnapKV, Scissorhands, PyramidKV, Quest, Ada-KV, HeadKV, InfiniGen, LayerKV, KVPR, PagedAttention. Canonical numbers v6.0.