igneum/docs/analysis/difficulty-2026-10-03.md

Difficulty controller for Igneum: data, simulation, implementation, test network

3 October 2026, consensus-engineer. Everything measured here is reproducible from sim/difficulty/ (record, simulator, raw outputs) and the difficulty branch of vendor/igneum-node.

1. The question

Igneum's effective hash rate steps every hour when the mining program changes (35 to 48 Mhash/s across seeds on the M5 Max, 228 vs 185 Mhash/s for 104 vs 128 loads on the RTX 5090, docs/bench-log.md), and GPU miners hop in and out fast. Kaspa's sampled DAA (KIP-4) averages a 44-minute window and holds genesis bits for 600 blocks. The devnet of 3 October showed what that costs. This document measures five controllers on synthetic steps and on the devnet record, tunes the Igneum candidate on the synthetic set only, then validates it on the record and on a live 3-node test network.

2. Data: the overnight devnet record

sim/difficulty/devnet-2026-10-03.csv: 3,682 headers pulled through the observer node's wRPC JSON endpoint (getBlocks from genesis, read only), columns hash, DAA score, blue score, timestamp, bits, difficulty (2^255 / target, the node's own convention; expected hashes per block is twice that), log2 target, epoch, chain flag, parent count, selected parent. Genesis bits 0x1d100000 (difficulty 134,217,727, 2^28 expected hashes per block), sized for the RTX 5090 plus the M5 Max.

Timeline (block 1 at 19:07:49 UTC = 20:07 BST):

Minutes from block 1	Blocks	Blocks/s	Bits at end	Difficulty at end	DAA score at end	What was mining
0 to 5	36	0.12	0x1d100000	134,217,727	35	Mac Metal worker, 30 MH/s
5 to 10	43	0.14	0x1d100000	134,217,727	78	same
10 to 15	24	0.08	0x1d100000	134,217,727	102	same
15 to 20	34	0.11	0x1d100000	134,217,727	136	same, stopped at 19:28 UTC
20 to 25	3	0.01	0x1d100000	134,217,727	139	about 1.5 MH/s (three isolated blocks)
25 to 30	1	0.00	0x1d100000	134,217,727	140	same
30 to 35	9	0.03	0x1d100000	134,217,727	149	PC joins at 19:42 UTC
35 to 40	172	0.57	0x1d100000	134,217,727	321	PC, 116 MH/s estimated from the blocks (coordinator: 154 MH/s plus 4 MH/s AMD)
40 to 45	187	0.62	0x1d100000	134,217,727	507	same
45 to 50	421	1.40	0x1e00b6a3	11,758,225	925	first retarget at DAA 600 (19:56:12 UTC): bits 0x1d4caca1, then easing every block
50 to 55	1,633	5.44	0x1e00d950	9,882,007	2,559	difficulty bottoms at 8,552,118 (15.7x below genesis); peak 355 blocks in one minute
55 to 60	639	2.13	0x1d36b2ef	39,260,045	3,201	the window fills with fast blocks, difficulty climbs past the correct level
60 to 65	400	1.33	0x1d3a4442	36,856,148	3,599	still above the correct level (about 58 million for 116 MH/s at 1 block/s)
65 to 70	79	0.26	0x1d44da0d	31,189,959	3,680	epoch boundary at DAA 3,600; the PC's launcher rebuilds its kernel (miner exit 42) and the chain idles

What Kaspa's rule did, as the coordinator saw it live and the record confirms: genesis difficulty held through block 600 while the real rate was 0.6 blocks/s; the first retarget saw a window whose 600 blocks spanned 48 minutes (the Metal period, the idle gap, the PC period) and eased 4.8x at once, then kept easing to 15.7x as more of the slow history entered the average; the chain ran at 5.4 blocks/s for five minutes (1,633 blocks) and 355 blocks in its peak minute; it then overshot the other way to 1.5x too hard. The DAG widened accordingly: 3,681 blocks against 1,656 chain blocks.

Hash-rate profile derived from the record (expected hashes of every block over wall time, between the two visible events, Metal stop 19:28:05 UTC and PC start 19:42:40 UTC):

From s	To s	Blocks/s at genesis difficulty	MH/s
0	1,215	0.114	30.7
1,215	2,090	0.006	1.5
2,090	4,064	0.433	116.2

Exact replay check: the record's own timestamps and bits replayed through rusty-kaspa's integer arithmetic (sim.py kaspa_bits_exact, same sampling rule (parent_daa + 1) mod 4 = 0, 661 samples, earliest sample dropped from the average, compact-bits rounding) reproduce the chain's bits exactly for the 244 retargets from DAA 600 to 844 and diverge from DAA 845, when the chain had passed 1 block/s and the sampled window began to include merged blocks that a chain-only replay cannot see. The simulator's Kaspa model is therefore exact on a chain and approximate on a wide DAG.

3. Simulator

sim/difficulty/sim.py. One selected chain, DAA score = height, 1 block per second target, solve times exponential with mean (expected hashes) / (hash rate); a hash-rate profile is piecewise constant in time (miners) or keyed to the 3,600-block epoch (programs); timestamps are the true times, optionally with uniform jitter. The difficulty unit is 2^28 hashes (the devnet genesis), so every synthetic run starts at a realistic target far from MAX_DIFFICULTY_TARGET.

Metrics. "First within 10%": the first time after a step when the centred 121-block mean of the expected block rate (hash rate over difficulty) is within 10% of target. "Settled": the first time it then stays within 10% for 100 blocks. "Overshoot": how far past target the mean goes on the far side before the next step. "Blocks above 2x / below 0.5x": blocks produced while the expected rate was outside that band. "Worst gap": longest inter-block time. Steady state: standard deviation of the expected rate ratio over a steady hour ("steady std"), and the coefficient of variation of blocks per minute, which is 0.129 for a Poisson process at 60 per minute (the controller adds to it only when it is noisy).

Controllers.

Name	Rule	Source
kaspa	KIP-4 sampled DAA as the fork runs it	consensus/src/processes/difficulty.rs, constants.rs (661 samples, every 4th block, min 150 samples)
monero	720 blocks, sorted timestamps, 60 cut each side, lag 15	Monero next_difficulty, approximate from memory of the reference
lwma60, lwma120	Zawy LWMA-1: linear weights, average target, solvetimes in [-132 s, 6 T]	Zawy's 2018 reference, approximate from memory
igneum	the rule of spec 2.3	this document
igneum-literal	the brief's trigger: short lane engaged while the short-window rate is more than 25% off target	kept to measure the chatter

Profiles. up50 (x50 at 3 h), down50 (/50 at 3 h), epoch30 (x1.3 or /1.3 drawn at each epoch boundary from epoch 2, six boundaries), walk10 (10% per hour log random walk), hop3 and hop10 (a 3x or 10x pool in and out every 15 minutes, twelve steps), polluted (the devnet case: 21 minutes at 0.11 blocks/s, 13 minutes idle, then 0.6 blocks/s at genesis difficulty), warmup-hard (genesis 10x too hard), warmup-easy (genesis 10x too easy), real (the record's profile above).

4. Designing the Igneum candidate

Two findings changed the brief's sketch.

Zawy's LWMA estimator is biased while targets ramp. LWMA computes average target times average solvetime. While the clamp drags the target down 3% per block after a 50x step, the short window holds blocks mined at targets from 1x to 15x apart; average target times average solvetime then over-estimates the target (Jensen), and the fast lane stalled at about 15x of the 50x step for a full window length (trajectory dump, first implementation). Replacing every lane's estimator with work over time (the sum of blue-work increments over the sum of clamped solvetimes, linear weights for the short lane) removed the stall: the 50x step-up settles in 62 s instead of 224 s. This is the estimator behind Kaspa's estimateNetworkHashesPerSecond, so the lineage stays Kaspa's.

The brief's trigger chatters. "Fast lane takes over when the short-window rate departs from target by more than 25%" releases the fast lane as soon as the short window is back on target, while the long window is still polluted by the pre-step blocks for 44 minutes; the long lane then pulls the target the wrong way until the short window departs again. Measured as igneum-literal: polluted case settles at 1,910 s against 72 s, hop3 fails to settle on 8 of 12 steps against 0, hop10 on 10 against 0. The trigger adopted compares the two lanes: the short lane rules while it disagrees with the reference lane by more than 25%, which stays true exactly as long as the long window is wrong.

The epoch windows. At an epoch boundary the program changes and the hash rate steps; blocks of the old program carry no information about the new one. Every lane is therefore restricted to the block's own epoch: the short lane is the newest 120 blocks of the epoch, the reference lane is the epoch window (blended with the parent's implied rate as k : 16 for small k) until the epoch has 600 blocks and Kaspa's sampled window restricted to the epoch after that. The parent's target is held for the first 8 blocks of an epoch. Epoch 0 starts at genesis, so this is also the warm-up rule: the dead zone is 8 blocks, not 600.

Asymmetric clamps. A step down is the painful case for any past-only controller: after a 50x drop blocks arrive every 50 s and each one is the only evidence. With a symmetric 3% clamp the 50x step-down took 1,786 s with a 211 s worst gap. The clamps are not what bounds timestamp manipulation (the symmetric solvetime cap and the cancellation of a forged timestamp by the next honest block are), so the ease direction can be faster: 10% per block gives 1,164 s and a 65 s worst gap with no measurable change anywhere else, and widening the cap from 6 to 20 block times (section 5) brings it to 657 s and 35 s.

5. Tuning on the synthetic set

One parameter at a time, two sweeps. Columns per profile: settled s ("rec"), first within 10% s ("first"), steady std; down50 adds the worst gap; polluted adds the peak rate. clamp_pct in the first sweep moves both clamps together (the ease clamp did not exist yet).

First sweep, around the brief's symmetric 3% clamp and the 6 T cap (ns 120, trigger 25%, clamp 3% both ways, prior 16):

Tune ns

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
ns=60	72.8	72.8	0.069	1843.1	1843.1	0.069	114.2	127.0	80.2	0.067	none	none	0.089	0/0	301.6	251.0	0.069	56.4	56.4	0.083	4.4
ns=120	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
ns=180	63.3	63.3	0.035	2131.4	2131.4	0.035	125.1	156.3	99.0	0.017	none	none	0.056	0/0	356.1	356.1	0.035	76.5	76.5	0.042	5.6

Tune trig

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
trig=0.15	66.7	66.7	0.077	1591.5	1591.5	0.077	98.7	137.3	107.3	0.070	none	none	0.082	0/0	312.9	261.8	0.077	58.1	58.1	0.075	5.8
trig=0.25	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
trig=0.35	62.9	62.9	0.034	1835.4	1835.4	0.034	126.4	156.3	99.0	0.016	none	none	0.054	0/0	328.5	279.2	0.034	78.4	78.4	0.042	5.5

Tune clamp_pct

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
clamp_pct=0.02	84.0	84.0	0.039	2588.2	2588.2	0.039	216.8	156.3	99.4	0.031	none	none	0.060	0/0	373.6	336.9	0.039	80.1	80.1	0.046	5.4
clamp_pct=0.03	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
clamp_pct=0.05	62.2	62.2	0.040	1312.5	1312.5	0.040	173.8	155.8	98.9	0.034	none	none	0.062	0/0	284.6	258.8	0.040	59.9	59.9	0.049	5.4

Tune ease_pct

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
ease_pct=0.03	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
ease_pct=0.06	62.7	62.7	0.039	1282.5	1282.5	0.039	176.0	155.6	98.8	0.031	none	none	0.061	0/0	299.5	265.0	0.039	58.5	58.5	0.048	5.8
ease_pct=0.1	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2

Tune k0

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
k0=8	63.0	63.0	0.040	1786.6	1786.6	0.040	211.1	156.0	99.0	0.033	none	none	0.061	0/0	311.5	280.9	0.040	67.1	67.1	0.048	5.5
k0=16	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
k0=32	63.0	63.0	0.040	1786.9	1786.9	0.040	211.9	156.0	99.1	0.033	none	none	0.061	0/0	290.0	263.0	0.040	71.4	71.4	0.047	5.7

Tune cap

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
cap=6	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
cap=20	63.8	63.8	0.039	1703.7	1703.7	0.039	165.8	143.7	86.8	0.033	none	none	0.061	0/0	274.9	257.2	0.039	62.9	62.9	0.048	5.8
cap=132	63.8	63.8	0.039	1649.6	1649.6	0.039	161.0	143.7	86.8	0.033	none	none	0.061	0/0	263.5	238.2	0.039	75.8	75.8	0.048	13.1

Second sweep, around the adopted defaults (ns 120, trigger 25%, harden 3%, ease 10%, prior 16, cap 6 T at the time of the sweep; the cap row decided the final value):

Tune ns

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
ns=60	57.5	57.5	0.080	595.9	595.9	0.080	87.5	120.2	92.6	0.076	none	none	0.093	0/0	201.2	152.1	0.080	60.8	60.8	0.090	5.2
ns=120	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2
ns=180	63.5	63.5	0.034	1742.3	1742.3	0.034	93.4	155.8	98.8	0.017	none	none	0.056	0/0	354.2	344.3	0.034	77.3	77.3	0.042	5.7

Tune trig

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
trig=0.15	63.2	63.2	0.078	1098.2	1098.2	0.078	75.1	136.1	106.1	0.072	none	none	0.083	0/0	244.8	234.9	0.078	62.1	62.1	0.075	5.2
trig=0.25	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2
trig=0.35	63.5	63.5	0.034	1238.4	1238.4	0.034	75.5	155.8	98.8	0.016	none	none	0.054	0/0	276.3	263.9	0.034	63.1	63.1	0.042	5.8

Tune clamp_pct

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
clamp_pct=0.02	97.1	97.1	0.037	1091.3	1091.3	0.037	71.6	155.3	98.7	0.027	none	none	0.061	0/0	269.7	258.5	0.037	71.9	71.9	0.048	6.2
clamp_pct=0.03	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2
clamp_pct=0.05	60.1	60.1	0.040	1371.6	1371.6	0.040	175.9	155.5	98.6	0.033	none	none	0.063	0/0	295.1	262.7	0.040	71.6	71.6	0.049	6.1

Tune ease_pct

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
ease_pct=0.03	63.0	63.0	0.040	1786.2	1786.2	0.040	211.2	155.9	99.0	0.033	none	none	0.061	0/0	275.6	275.6	0.040	69.3	69.3	0.048	5.8
ease_pct=0.06	62.7	62.7	0.039	1282.5	1282.5	0.039	176.0	155.6	98.8	0.031	none	none	0.061	0/0	299.5	265.0	0.039	58.5	58.5	0.048	5.8
ease_pct=0.1	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2
ease_pct=0.15	61.6	61.6	0.038	1111.4	1111.4	0.038	65.4	155.2	98.4	0.030	none	none	0.062	0/0	261.6	250.5	0.038	67.1	67.1	0.048	6.4

Tune k0

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
k0=8	62.2	62.2	0.038	1144.7	1144.7	0.038	65.3	155.5	98.7	0.030	none	none	0.062	0/0	247.7	247.7	0.038	70.2	70.2	0.048	6.2
k0=16	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2
k0=32	62.2	62.2	0.038	1169.9	1169.9	0.038	65.3	155.7	98.9	0.030	none	none	0.061	0/0	287.5	266.4	0.038	71.4	71.4	0.048	6.3

Tune cap

ctrl	up50 rec	up50 first	up50 std	down50 rec	down50 first	down50 std	down50 gap	epoch30 rec	epoch30 first	epoch30 std	walk10 rec	walk10 first	walk10 std	walk10 >2x/<0.5x	hop10 rec	hop10 first	hop10 std	polluted rec	polluted first	polluted std	polluted peak
cap=6	62.2	62.2	0.039	1164.4	1164.4	0.039	65.3	155.5	98.8	0.030	none	none	0.062	0/0	253.4	253.4	0.039	71.7	71.7	0.048	6.2
cap=20	61.7	61.7	0.038	657.4	657.4	0.038	35.3	143.7	86.5	0.030	none	none	0.062	0/0	211.9	208.1	0.038	70.0	70.0	0.049	8.5
cap=132	61.7	61.7	0.038	629.5	629.5	0.038	32.4	143.7	86.5	0.030	none	none	0.062	0/0	212.9	209.1	0.038	63.8	63.8	0.049	16.4

Decisions: ns 120 (60 doubles the steady noise, 180 is slower on hops); trigger 25% (15% engages on noise and doubles the steady noise, 35% slower on hops); harden 3% (2% slows the step-up to 97 s, 5% buys nothing on the up step and worsens the down step's gap); ease 10% (3% leaves the step-down at 1,786 s, 15% buys 50 s more at no cost but was not adopted to keep a margin); prior 16 (no effect across 8 to 32); cap 20 T (6 T leaves the step-down at 1,164 s, 20 T gives 657 s and a 35 s worst gap and also helps the epoch steps and hopping; the 132 s cap, Kaspa's future tolerance, gains nothing more and lets the idle gap of the polluted case over-ease the target to a 16x peak against 8.5x at 20 T and 6.2x at 6 T). The cap is not the manipulation bound: under the symmetric cap a forged timestamp's contribution is cancelled by the next honest block whatever the cap, so the cap only decides how much a genuine slow block may say. Nothing was tuned on the record.

6. Results on the synthetic set

Seed 7, honest timestamps, final parameters. Full output in sim/difficulty/results.md.

up50: hash rate x50 at 3 h

controller	first within 10% s	settled s	recovery blocks	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	1542.1	1542.1	4967	0.009	0.012	0.126	9.5	3760	0	21511	none
monero	94.0	94.0	713	0.089	0.037	0.144	10.3	657	0	18490	none
lwma60	105.0	105.0	182	0.208	0.131	0.088	11.3	78	0	17977	none
lwma120	231.2	231.2	388	0.189	0.092	0.096	10.5	164	0	18078	none
igneum	61.7	61.7	181	0.140	0.038	0.135	11.3	125	0	17927	377
igneum-literal	63.7	63.7	186	0.217	0.034	0.130	11.3	127	0	17956	230

down50: hash rate /50 at 3 h

controller	first within 10% s	settled s	recovery blocks	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	12295.7	12295.7	1569	5.102	0.012	0.126	178.5	1648	1256	16185	none
monero	6432.8	6432.8	701	0.039	0.037	0.144	187.3	0	567	19356	none
lwma60	577.9	577.9	111	0.154	0.131	0.088	118.7	0	55	24588	none
lwma120	1073.6	1073.6	177	0.129	0.092	0.096	64.9	0	97	24205	none
igneum	657.4	657.4	147	0.062	0.038	0.135	35.3	0	90	24548	474
igneum-literal	967.0	967.0	146	0.062	0.034	0.130	121.1	0	86	24276	267

epoch30: x1.3 or /1.3 at each epoch boundary from epoch 2

controller	first within 10% s	settled s	worst recovery s	not recovered	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	1583.0	1583.0	1974.6	0	0.043	0.010	0.130	12.3	0	0	29403	none
monero	456.2	456.2	603.3	0	0.116	0.038	0.141	16.0	0	0	28857	none
lwma60	91.1	156.9	310.6	0	0.291	0.132	0.090	11.6	0	0	28658	none
lwma120	136.9	153.1	240.8	0	0.229	0.094	0.097	11.5	0	0	28719	none
igneum	86.5	143.7	260.8	0	0.126	0.030	0.133	11.3	0	0	28735	351
igneum-literal	86.4	143.8	260.8	0	0.050	0.018	0.130	11.3	0	0	28775	149

walk10: 10% per hour log random walk

controller	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	0.049	0.144	11.1	0	0	21728	none
monero	0.049	0.143	10.6	0	0	21541	none
lwma60	0.132	0.088	11.1	0	0	21486	none
lwma120	0.093	0.105	10.5	0	0	21518	none
igneum	0.062	0.130	11.2	0	0	21521	519
igneum-literal	0.055	0.136	11.1	0	0	21523	184

hop3: 3x pool hops in and out every 15 min from 3 h

controller	first within 10% s	settled s	worst recovery s	not recovered	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	none	none	none	12	0	0.012	0.126	21.0	1208	1036	22128	none
monero	336.7	336.7	362.4	6	0.062	0.037	0.144	18.6	2438	1437	20382	none
lwma60	109.5	166.9	373.4	0	0.270	0.131	0.088	14.7	111	42	21408	none
lwma120	161.8	184.3	292.1	0	0.201	0.092	0.096	15.0	215	129	21364	none
igneum	124.0	189.5	410.3	0	0.229	0.038	0.135	11.8	213	182	21228	8054
igneum-literal	429.3	372.9	493.1	8	0.202	0.034	0.130	16.1	86	529	20913	4294

hop10: 10x pool hops in and out every 15 min from 3 h

controller	first within 10% s	settled s	worst recovery s	not recovered	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	none	none	none	12	0	0.012	0.126	52.1	5700	1035	22951	none
monero	283.5	283.5	312.9	6	0.085	0.037	0.144	83.2	3551	569	19203	none
lwma60	165.5	263.9	542.0	0	0.242	0.131	0.088	34.3	315	198	21008	none
lwma120	282.8	325.5	789.0	0	0.199	0.092	0.096	30.2	576	417	20676	none
igneum	208.1	211.9	382.9	0	0.230	0.038	0.135	25.0	422	321	20631	8445
igneum-literal	466.6	505.8	698.0	6	0.237	0.034	0.130	35.7	816	666	20683	5432

polluted: window polluted by a pre-step slow period (21 min at 0.11 blocks/s, 13 min idle, then 0.6 blocks/s at genesis difficulty)

controller	first within 10% s	settled s	recovery blocks	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks	peak rate	trough rate after 10 min
kaspa	650.7	2748.3	5163	6.942	0.010	0.134	11.2	2320	0	18592	none	7.9	0.6
monero	124.0	124.0	735	0.092	0.032	0.142	10.4	675	0	17641	none	15.0	0.9
lwma60	66.1	66.1	113	0.183	0.132	0.089	11.3	36	0	17138	none	5.7	0.6
lwma120	110.3	110.3	201	0.147	0.091	0.108	10.5	74	0	17190	none	6.0	0.7
igneum	70.0	70.0	136	0.180	0.049	0.122	11.3	72	0	17005	2585	8.5	0.6
igneum-literal	817.0	1876.4	2546	0.197	0.040	0.141	11.3	571	0	17618	1986	8.4	0.6

warmup-hard: genesis difficulty 10x too hard

controller	first within 10% s	settled s	recovery blocks	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	5790.7	none	none	92.876	32.727	3.002	67.2	4860	599	6444	none
monero	286.6	286.6	159	0.047	0.038	0.135	17.5	0	33	7011	none
lwma60	155.2	155.2	118	0.147	0.132	0.078	10.5	0	18	7128	none
lwma120	187.9	187.9	152	0.111	0.095	0.102	10.5	0	18	7128	none
igneum	321.9	321.9	199	0.024	0.031	0.126	10.5	0	23	6953	138
igneum-literal	351.4	702.7	568	0.024	0.019	0.134	10.5	0	23	6979	179

warmup-easy: genesis difficulty 10x too easy

controller	first within 10% s	settled s	recovery blocks	overshoot	steady std of rate	steady blocks/min CV	worst gap s	blocks above 2x	blocks below 0.5x	blocks	fast-lane blocks
kaspa	1554.8	1554.8	3168	0.011	0.008	0.137	9.8	599	0	8869	none
monero	95.4	95.4	76	0.091	0.038	0.154	16.1	16	16	7145	none
lwma60	57.9	57.9	87	0.183	0.133	0.073	9.2	8	0	7192	none
lwma120	88.3	88.3	136	0.147	0.096	0.099	9.4	8	0	7225	none
igneum	58.7	58.7	114	0.082	0.029	0.133	10.3	61	0	7192	120
igneum-literal	58.4	58.4	114	0.086	0.015	0.129	10.3	61	0	7209	191

With +-500 ms timestamp jitter (half a block time, a harsh clock model), the ordering holds: up50 settled Kaspa 1,535 s, LWMA60 159 s, LWMA120 140 s, Igneum 169 s; down50 Kaspa 12,523 s, LWMA60 847 s, LWMA120 966 s, Igneum 1,047 s (first within 10% at 896 s, worst gap 49 s against 78 and 81); epoch30 Kaspa 1,535 s, LWMA60 63 s, LWMA120 124 s, Igneum 55 s; hop10 Kaspa never, LWMA60 210 s, LWMA120 318 s, Igneum 228 s; polluted Kaspa 2,722 s (peak 8.5x), LWMA60 89 s, LWMA120 174 s, Igneum 71 s; steady std Kaspa 0.010, LWMA60 0.125, LWMA120 0.085, Igneum 0.046. The jitter costs the Igneum short lane on the 50x cases because 20 ms solve times are drowned by 500 ms clock noise; it still beats Kaspa by 9x and 12x there.

Reading. Kaspa's rule has the lowest steady-state noise (0.012) and the worst response to every step: 26 minutes for a 50x up, 3.4 hours for a 50x down with a 5x overshoot, never settling under pool hopping, 46 minutes and a 7.9x peak on the devnet case, and a 600-block dead zone that turns a 10x-too-hard genesis into 97 minutes of near-silence. LWMA 60 is the fastest on step-downs and the noisiest (0.132, 13% block-rate swings all day). The Igneum rule keeps the steady noise at 0.038 (three times Kaspa's, less than half of LWMA120's), is the fastest on the 50x step-up (62 s), the epoch steps (144 s settled, 87 s first within 10%, against Kaspa's 1,583 s), the devnet case (70 s), the 3x and 10x hopping (190 s and 212 s) and the too-easy genesis (59 s), second to LWMA60 on the 50x step-down (657 s against 578 s, with a 35 s worst gap against 119 s) and on the too-hard genesis (322 s against 155 s, the price of the 10% ease clamp). On the polluted case every past-only controller shows a peak when the PC arrives, because the difficulty was correctly set for the Metal miner; the number that matters is how long the peak lasts: Igneum 70 s, Kaspa 46 minutes with a second, larger peak of its own making.

7. Validation on the record

The record's hash-rate profile (section 2) replayed through every controller, not tuned on it:

Controller	First within 10% s	Settled s	Overshoot	Blocks above 2x	Blocks within 10% of target (of about 3,700)	Fast-lane blocks
kaspa	262	never	6.24 (peak 7.6x)	2,340	0
monero	136	136	0.09	672	2,822
lwma60	75	75	0.18	87	2,175
lwma120	157	157	0.15	174	2,390
igneum	79	79	0.18	132	2,608	2,484
igneum-literal	482	1,516	0.01	515	2,170	1,959

The simulated Kaspa trajectory reproduces the live one: a late first retarget, a multi-x peak, thousands of blocks above 2x, no settling within the record. The Igneum rule settles 79 s after the PC arrives; it runs on the short lane for most of the remaining 33 minutes (2,484 blocks) because the long window stays polluted by the 75x step for 44 minutes, which is the designed behaviour.

8. Implementation

Branch difficulty of vendor/igneum-node (worktree vendor/igneum-node-diff).

File	Change
consensus/core/src/igneum.rs	DifficultyRule { KaspaSampled, IgneumDual } (serde kebab-case) and the difficulty constants module (SHORT_WINDOW 120, EPOCH_HOLD 8, PRIOR_BLOCKS 16, LONG_MIN 600, TRIGGER_PERCENT 25, HARDEN_PERCENT 3, EASE_PERCENT 10, CAP_BLOCKS 20)
consensus/core/src/config/params.rs	Params.difficulty_rule (IgneumDual on all four networks), OverrideParams.difficulty_rule and OverrideParams.genesis_bits (test networks: the genesis hash is recomputed from the new bits)
consensus/src/processes/difficulty.rs	calculate_difficulty_bits(window, ghostdag, daa_score) dispatches on the rule; kaspa_difficulty_bits is the old body unchanged; igneum_difficulty_bits gathers the chain steps (blue-work increment and clamped solvetime per selected-chain block of the epoch, at most 600 or 120) and the in-epoch samples of the window, then calls the pure igneum_target (Uint320 arithmetic, rates as unreduced fractions, no floating point)
consensus/src/processes/window.rs, consensus/src/consensus/services.rs	the rule and the epoch length threaded to the manager; the DAA score of the block passed to the retarget
testing/integration/src/consensus_integration_tests.rs	difficulty_test pins KaspaSampled (it asserts Kaspa's behaviour on mainnet parameters)

Unit tests (cargo test -p kaspa-consensus --lib difficulty, 9 tests): hold for the first 8 blocks, steady state keeps the target within 0.1%, a 50x step up hardens by exactly 3% per block, a 50x step down eases by exactly 10% per block, the short lane takes over only beyond 25% disagreement with the long lane (on target and 20% fast stay on the long lane, 2x fast binds the 3% clamp), the epoch lane's shrinkage (8 blocks at 10% slower move the target 3.1%, 8 blocks at half speed bind the 10% clamp), the maximum target is never exceeded, plus Kaspa's two level-work tests. cargo test -p kaspa-consensus-core --lib params: 4 tests including the override-file parsing.

Cost: up to 600 compact-header and ghostdag reads per header for the first 600 blocks of an epoch, 120 afterwards, all from in-memory caches at 1 BPS. Re-measure before the 4 BPS step.

9. Test network

Three igneumd nodes from the difficulty branch (gRPC 26800, 26810, 26820; P2P 26801, 26811, 26821; node 2 and node 3 --connect to node 1; appdir /tmp/igneum-diff-test/igneum; override file {"difficulty_rule": "igneum-dual", "genesis_bits": 505413632}, i.e. bits 0x1e200000, 2^19 expected hashes, genesis hash f682b78a...e099), three igneum-miner mine --engine igneum-pow CPU miners of 4 threads each with their own payout identities, on the shared Mac at load average 50 to 90 (two other agents were building). Schedule: miner A on node 1 for 1,200 s from block 1; miners B and C on nodes 2 and 3 from 359 s to 779 s. 1,133 blocks, 0 rejected (814 + 159 + 160 found), one sink at the end. Record: sim/difficulty/testnet-igneum-2026-10-03.csv.

What the miners delivered (expected hashes over wall time, the same estimator the simulator's record profile uses): 0.0653 MH/s with A alone, 0.0988 MH/s with A, B and C (1.51x, not 3x: three 4-thread miners on a loaded 18-core Mac share the cores, B and C each reported 0.032 MH/s against A's 0.054), 0.0686 MH/s after B and C stopped (0.69x). Genesis difficulty 262,144 was 4.0x too hard for A alone (the correct value is about 65,000).

Event	Measured, first within 10% (61-block mean)	Simulator prediction on the same profile, 5 seeds	Kaspa's rule, simulator
Warm-up from a 4x too hard genesis	132 s (held 8 blocks, then eased 10% per block; difficulty 190,283 at 60 s, 62,248 at 90 s)	3, 92, 263, 278, 309 s (median 263)	never within 20 min (no retarget before block 600; 337 to 378 blocks in 20 min)
Miners B and C join, x1.51	214 s (within 10% at 23 s, then a 1.9 blocks/s burst at 360 to 390 s as the DAA score jumped with the merged blocks)	49, 54, 61, 87, 180 s (median 61)	never
Miners B and C leave, /1.45	85 s	110, 231, 300 s and twice not within the remaining 413 s	never
Worst gap after the warm-up	7.3 s	4.5 to 13.0 s	13 to 21 s

Reading: the live network recovered faster than the simulator on the warm-up and the step-down and slower on the step-up. The differences have a cause the simulator does not model: the measured hash rate of CPU miners on a loaded machine swings by 2x between 30-second bins (0.03 to 0.15 MH/s in the table above), so a 1.5x step is of the same size as the noise, and the 61-block mean crosses the 10% band on noise as often as on control. The warm-up, the one clean signal (4x), came in at 132 s against a simulator median of 263 s, both an order of magnitude inside Kaspa's 600-block dead zone (which at 0.25 blocks/s would have lasted 40 minutes). Kaspa's rule, live, on the same network and genesis (sim/difficulty/testnet-kaspa-2026-10-03.csv, override {"difficulty_rule": "kaspa-sampled", "genesis_bits": 505413632}, 10 minutes, A for 600 s, B and C from 200 s to 500 s): 70 blocks, bits 0x1e200000 on all 70 (difficulty 262,144 never moved), 0.08 blocks/s, worst gap 63 s, delivered hash rate 0.042 MH/s (the other agents' builds were heavier during this run). Never within 10% of target at any point: the 600-block dead zone would have ended after about 2 hours at that rate. The Igneum run on the same genesis was within 10% at 132 s.

Per-30-second table (blocks, mean difficulty, work over time) in sim/difficulty/results.md.

10. Limits and what is still open

The simulator has no DAG: blocks off the selected chain enter the implementation only through blue work, and the record replay is chain-only (exact for 244 retargets, approximate after). No network latency, no template refresh delay. Monero and LWMA are reproduced from memory of their reference code and labelled approximate. The step-down remains the slowest case (657 s for 50x); real-time targeting would cut it further and is left out on purpose (spec 2.3). Red blocks' work is ignored by every lane, as by Kaspa's estimator. The 4 BPS and 10 BPS steps need the sampled window to replace the chain walk for the epoch lane and a re-derivation of the block-count constants.