Authority note (legacy narrative). This page predates the 2026-07 technical-report suite and is retained as the project's plain-language narrative. Where this page and the technical reports (or CRITIQUE.md) disagree, the reports win — the discrepancy is a bug in this page; please report it (the same convention as CLAIMS_DECIDED.md).
A plain-language introduction to what solve.py and solve.c compute. Several of the core observations (the pair structure C1, the no-5-line-transition property C2) have been noted in prior literature — see CITATIONS.md for credits. ROAE's specific contribution is threefold: budget-exact enumeration of solutions under the conjoined constraint system (each canonical count is exact at its node budget; the space itself is far too large to exhaust at any budget — see SEARCH_SPACE_SIZE.md), partition-invariant reproducibility of the canonical counts (the same byte-identical results on any hardware), and a seven-family null-model framework testing how the King Wen structure compares to structured and unstructured permutation families.
For deeper material referenced throughout this article:
- Canonical enumeration counts and their reproducibility shas: CLAUDE.md §Canonical shas and HISTORY.md.
- Partition invariance theorem and the cross-path validation grid: PARTITION_INVARIANCE.md.
- Joint-observable distributional analysis (where King Wen sits in the 3.43B-ordering density): DISTRIBUTIONAL_ANALYSIS.md.
- Methodological critique, including the null-model caveat on the "N boundaries uniquely determine King Wen" finding: CRITIQUE.md.
- Formal constraint definitions and theorems: SPECIFICATION.md.
- Command-line reference: SOLVE_C_CLI.md (
solve.csubcommands, env vars, exit codes), ROAE_PY_CLI.md (roae.pyanalysis sections).
For HOW the enumeration actually works — what a branch / sub-branch / node means, the difference between all-branch and single-branch enumeration, and the open research questions — see BRANCHES_EXPLAINED.md. It's the step-by-step companion to this summary.
Long ago — traditionally about 3,000 years, though modern scholarship debates when the ordering was fixed — someone in ancient China, or successive generations of practitioners, arranged 64 symbols (called hexagrams) in a specific order. This ordering is called the King Wen sequence. There are more possible arrangements of 64 things than there are atoms in the universe — roughly 10^89 of them (a 1 followed by 89 zeros; the observable universe holds about 10^80 atoms). But somehow, whoever arranged them picked one specific arrangement:
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
We wanted to know: what rules did they follow? And can we figure out those rules just by studying the result?
Each hexagram is a stack of 6 lines. Each line is either solid (⚊ yang) or broken (⚋ yin) — like a 6-digit binary number with only 1s and 0s. With 6 positions and 2 choices each, there are exactly 2^6 = 64 possible hexagrams. The King Wen sequence puts all 64 in a specific order.
| ䷀ The Creative #1 | ䷄ Waiting #5 | ䷁ The Receptive #2 | |
|---|---|---|---|
| Line 6 (top) | ⚊ 1 | ⚋ 0 | ⚋ 0 |
| Line 5 | ⚊ 1 | ⚊ 1 | ⚋ 0 |
| Line 4 | ⚊ 1 | ⚋ 0 | ⚋ 0 |
| Line 3 | ⚊ 1 | ⚊ 1 | ⚋ 0 |
| Line 2 | ⚊ 1 | ⚊ 1 | ⚋ 0 |
| Line 1 (bottom) | ⚊ 1 | ⚊ 1 | ⚋ 0 |
| Binary | 111111 | 010111 | 000000 |
To get the binary code, read the 1s and 0s from the top of the table downward. For example, ䷀ The Creative #1 is all solid lines: 111111. ䷁ The Receptive #2 is all broken lines: 000000. ䷄ Waiting #5 reads 0, 1, 0, 1, 1, 1 from top to bottom, giving 010111 — a mix of solid and broken.
We found five rules (C1–C5 in SPECIFICATION.md; four are formally independent — C2 is implied by C5), each one eliminating more of the possible arrangements. Rules 1 and 2 are known in prior literature (Rule 1 from I Ching scholarship / Cook 2006; Rule 2 from Terence & Dennis McKenna's The Invisible Landscape, 1975); Rules 3–5 are formalized and quantified here. See CITATIONS.md.
(A sixth "XOR rule" was identified during discovery but later proven mathematically redundant — it follows automatically from Rule 1; see SOLVE.md Theorem 2. It is therefore not listed below.)
The 64 hexagrams are grouped into 32 consecutive pairs. Each hexagram is paired with the one you get by flipping it upside down. (For 4 symmetric hexagrams that look the same upside down — ䷀䷁ ䷚䷛ ䷜䷝ ䷼䷽ — the partner is the one with every line toggled instead.)
What this does: Cuts the possibilities from 10^89 down to about 10^45. Still enormous, but 44 zeros gone in one step.
Flip (28 pairs, partner highlighted):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 3↔4, 5↔6, 7↔8, 9↔10, 11↔12, 13↔14, 15↔16, 17↔18, 19↔20, 21↔22, 23↔24, 25↔26, 31↔32, 33↔34, 35↔36, 37↔38, 39↔40, 41↔42, 43↔44, 45↔46, 47↔48, 49↔50, 51↔52, 53↔54, 55↔56, 57↔58, 59↔60, 63↔64
Inverse (4 pairs, partner highlighted):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 1↔2, 27↔28, 29↔30, 61↔62
When you move from one hexagram to the next, some number of lines change (1 through 6). This is called the Hamming distance. In the King Wen sequence, the number 5 never appears — no two consecutive hexagrams differ by exactly 5 lines.
Examples from the King Wen sequence:
| Lines changed | Transition | Example |
|---|---|---|
| 1 | ䷳ Keeping Still #52 → ䷴ Development #53 | 100100 → 110100 |
| 2 | ䷁ The Receptive #2 → ䷂ Difficulty at the Beginning #3 | 000000 → 010001 |
| 3 | ䷅ Conflict #6 → ䷆ The Army #7 | 111010 → 000010 |
| 4 | ䷂ Difficulty at the Beginning #3 → ䷃ Youthful Folly #4 | 010001 → 100010 |
| 5 | (never occurs in King Wen) | |
| 6 | ䷀ The Creative #1 → ䷁ The Receptive #2 | 111111 → 000000 |
The difference wave — each character represents one transition, height proportional to lines changed (1-6):
█▂▅▅▅▃▂▅▂▅█▂▂▅▂▂█▃▅▃▂▂▂▃▅▂█▂█▃▂▃▅▅▅▂▅█▅▃▂▅▂▃▅▃▂▃▅▅▅▁█▂▂▃▅▃▂▁█▃█
The tallest blocks (█) are distance 6. The shortest (▁) are distance 1. No block reaches height 5 (▅) — that transition never occurs. See all 63 transitions in the appendix.
What this does: Eliminates about 96% of the remaining arrangements.
Every hexagram has a complement (its "opposite") — the one where every line is flipped, solid↔broken. In King Wen, complements sit strikingly near each other: 9 of the 32 complement pairs are directly adjacent, and the average separation is only 12.1 positions (random arrangements average about 22). Summed over all 64 hexagrams, King Wen's total complement-distance is exactly 776.
One honest subtlety, learned from the large enumerations: 776 is a ceiling King Wen sits at, not a minimum it achieves. Among the billions of valid orderings, many have their complements even closer — the minimum seen is 424 at the 100T canonical and 392 at the deeper 560T canonical — and roughly one in ten (10.11% at the 560T canonical — re-measured 2026-07-08, 1,063,580,364 of 10,525,271,997 records; was 9.91% at 100T) ties King Wen at exactly 776. So the rule we can actually state is "total complement-distance stays at or below 776" — a bound taken from King Wen's own value. (An earlier version of this page framed Rule 3 as active minimization; the enumeration corrected that. Scope note added 2026-07-04: the minimum was previously quoted unscoped as 424 — that is the 100T value; the 560T §[22] range is [392, 776].)
A reference baseline for the "one in ten" figure (exact, added 2026-07-22): under the bare pair-slot
null — all pair orderings with the Creative/Receptive start, with no other constraint applied and no
enumeration budget — the exact distribution (verify.py --check-null-g) gives P(total ≤ 776) = 8.106231%
and a ceiling-tie share P(total = 776 | total ≤ 776) = 7.86%. That null population is not like-for-like
with the enumerated canonical sets (which also enforce the no-5 and difference-wave rules and are
budget-truncated), so the baseline neither confirms nor refutes the ~10% figure. What it shows is that a
null with no transition structure at all already produces a ceiling-tie share close to the observed ~10% —
so the tie fraction should be read as largely generic to the pair-slot geometry, not as King-Wen-specific
structure.
What this does: Eliminates orderings whose complements drift farther apart than King Wen's — but leaves an enormous family, King Wen among them.
Closest complements highlighted (distance 1 — adjacent in the sequence):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 1↔2, 11↔12, 17↔18, 27↔28, 29↔30, 38↔39, 53↔54, 61↔62, 63↔64
9 of 32 complement pairs sit directly next to each other. The farthest apart are ䷂ #3 and ䷱ #50 (distance 47) — but the average is only 12.1. See all 32 complement pairs in the appendix.
Farthest complements highlighted (䷂ #3 and ䷱ #50, 47 positions apart):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 3↔50
The sequence begins with the two most extreme hexagrams: all solid lines ䷀ Qian (The Creative) #1, representing Heaven, followed by all broken lines ䷁ Kun (The Receptive) #2, representing Earth.
What this does: Eliminates another 98%.
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 1↔2
The "jumps" between consecutive hexagrams follow a specific recipe — called the difference wave: exactly 2 jumps of size 1, 20 jumps of size 2, 13 jumps of size 3, 19 jumps of size 4, and 9 jumps of size 6. No jumps of size 0 or 5.
What this does: After all previous rules, a backtracking enumeration (solve.c, 10 trillion nodes partitioned across parallel threads) finds hundreds of millions of valid orderings — an enormous reduction from 10^89, but far more than the "near-unique" result suggested by earlier Monte Carlo sampling. Canonical counts (d3 re-established 2026-05-13 on post-resume-fix code; see CANONICAL_HASHES.md): 706,427,594 at the depth-3 partition, 286,357,503 at depth-2. The count depends on which partition strategy samples the search space; under true exhaustive enumeration both would converge.
Canonical enumerations using solve.c at the 10T node budget find hundreds of millions of unique orderings satisfying Rules 1-5. At the depth-3 partition (158,364 sub-branches): 706,427,594. At depth-2 (3,030 sub-branches): 286,357,503. Both enumerations are partial in the sense that each sub-branch hits its per-sub-branch node budget rather than completing naturally — so the true count under exhaustive enumeration is unknown and likely larger; an unbiased Monte-Carlo estimate (Knuth random-probe) now puts the total at ≈10³⁸ (≈3×10³⁷ distinct-canonical) — see SEARCH_SPACE_SIZE.md. Only Position 1 (Creative/Receptive) is universally locked — the same pair appears in every valid ordering. The remaining 31 positions show a gradient of constraint:
| Positions | Pairs observed | KW match rate | Character |
|---|---|---|---|
| 1 | 1 | 100% | Fully determined |
| 3-18 | at least 2 each | 87-99% | Highly constrained |
| 19-20 | at least 2-4 | ~50% | Moderately constrained |
| 21-32 | at least 7-16 each | 10-22% | Progressively free |
| 2 | at least 16 | 0.2% | Branch-dependent |
Caveats on these rates: The match rates are computed from a partial, non-uniform sample. Each search branch was explored to an unknown depth (none completed), so shallower solutions are overrepresented. The "at least N pairs" counts are lower bounds — a complete enumeration could reveal additional pairs at each position. The rates should be treated as indicative, not definitive.
Earlier analysis based on a partial sample (438 solutions from a single search branch) claimed 23 of 32 positions were locked and that two adjacency constraints sufficed for uniqueness. The larger enumeration reveals this was an artifact of undersampling. The constraint gradient is real but much more nuanced.
Among 6 billion C3-valid solutions (including orientation variants), only 0.0018% satisfy both legacy adjacency constraints (C6+C7). Note: this rate mixes orientation variants (~297 per unique ordering) with unique orderings, so the per-ordering rate would differ. These constraints significantly narrow the space, but their sufficiency for uniqueness is refuted, not open: TR-4 §4 measured it on 2026-07-02 and found ≈5.21×10³¹ orderings surviving C1–C7. That refutation is a headline result of this project, not an outstanding item.
What makes King Wen unique among the astronomically many valid orderings is an open question. The C1–C5 space is estimated at ≈10³⁸ (Knuth random-probe estimate, see SEARCH_SPACE_SIZE.md), so King Wen is one of astronomically many valid orderings — and its early appearance during enumeration is a search-setup artifact (order-invariant to the findings), not evidence of scarcity. It is equally possible that additional mathematical rules exist to be discovered, or that King Wen is simply one choice among many with no further mathematical distinction.
The millions of alternative arrangements satisfying Rules 1-5 are not random — they share strong structural similarities with King Wen. The closest non-King-Wen solutions differ by only 2 pair positions — and those distance-2 twins occur at the front (positions 2/3 and 4/5) and middle (positions 22-25) of the sequence as well as the back, at every canonical scale checked (d3 10T, 100T, 560T §[24]). One front twin — KW with the position-2/3 pair blocks interchanged, rec#330177707 — is exactly the 560T record that survives four of the five greedy minimum boundaries (see BOUNDARY_MINIMUM.md). (Corrected 2026-07-04: an earlier version claimed the twins were "always in the last third (positions 26-32)" — a pre-canonical-era claim that no canonical dataset supports.) This means:
- Position 1 is fixed by C4's definition. The Creative/Receptive pair always comes first (the pair choice is classically attested centuries before any enumeration; see the orientation bullet below for what is and is not forced).
- Position 2 partially constrains positions 3-19, but not deterministically. Earlier docs claimed "for 16 of 31 branches, positions 3-19 are fully locked" (based on
--prove-cascade). That result is correct only within the shift-pattern subspace (where every position is restricted to KW's pair or the previous pair) — and the canonical analyses show this subspace is a shrinking minority of valid orderings: 2.69% at d2, 0.062% at d3. In the full canonical datasets, every reachable first-level branch admits multiple distinct pair sequences at positions 3-19; none are uniquely determined. The cascade region is heavily constrained (per-position entropy 0.3-1.9 bits at d3, well below the 5-bit maximum) but not deterministic. - Freedom is concentrated in the back half but spread across the cascade too. The canonical analyses show position 3 has the highest freedom (d3: 4.52 bits, d2: 4.05 bits, 28-31 distinct pairs observed), positions 24-31 carry 3.41-3.54 bits each at d3 (14 distinct pairs; positions 22-23 sit lower, at 1.74 and 3.15 bits — range corrected 2026-07-04 from "22-31 / 3.40-3.54"), and the "cascade region" (5-20) carries 0.48-1.85 bits each at d3 — heavily constrained but not zero. The traditional Xugua commentary explaining why specific hexagrams follow each other in the back half retrospectively rationalizes steps that the enumeration shows were genuine selections among available alternatives, not mathematical necessity.
- King Wen's complement distance is 776 (mean 12.1) — small vs random permutations (~22), and low among orderings satisfying every other constraint (C1+C2+C4+C5) — the ledger puts that at ≈12%, and under the bare C1&C4 null the exact tail is 8.1% (
verify.py --check-null-g). (Amended 2026-08-01, lens sweep: this sentence previously read "at the 3.9th percentile (sampled) of orderings satisfying every other constraint". That figure is flagged — not supported by the population it is labelled with; see SOLVE.md §Rule 3.) But within the full C1–C5 canonical set, KW is at the ceiling, not the minimum — at the 560T canonical (10.5B orderings, re-measured 2026-07-08) 1,063,580,364 (10.11%) tie with KW at 776 and all are ≤776 (the ceiling); was 340M/9.91% at 100T. Rule 3 is a ceiling constraint, not a minimization. See §Rule 3 for the updated framing. - The starting orientation is definitional, not forced (corrected 2026-07-26; an earlier version claimed it was forced — that claim is retracted). ䷀ The Creative comes before ䷁ The Receptive by C4's definition, classically attested (the Xugua opens Heaven-then-Earth). The other constraints do not force it: complementing every hexagram maps any valid arrangement to one opening (0, 63) that still satisfies C1, C2, C3 (= 776), and C5 — machine-checked in lean/KingWen.lean.
- Within-pair orientation follows no simple rule, but it is not free (corrected — an earlier version of this line called it "a free choice at each pair"; that gloss is retracted). Which hexagram comes first within each pair follows no consistent single pattern — not yang count, not binary value, not trigram weight. But TR-1 §7 measured the freedom and found it coupled, not independent: ≈20.7 bits, not 31, with only 9 of the 31 pairs flippable individually.
Five constraints, discoverable through analysis, narrow 10^89 possible arrangements to hundreds of millions (d3 canonical: 706M; d2 canonical: 286M). Position 1 is fully determined. The cascade region (positions 3-20) is heavily constrained (per-position entropy 0.3-1.9 bits at d3 — far below the 5-bit maximum) but not deterministic: every first-level branch admits multiple distinct pair sequences across positions 3-19. Positions 24-31 have substantially more freedom (entropy 3.4-3.5 bits at d3; positions 22-23 are transitional at 1.7 and 3.1 bits). The rules were extracted from King Wen (confirmatory analysis, not independent prediction), but the constraint structure they reveal is genuine.
Someone — or successive generations of practitioners — in ancient China (traditionally ~3,000 years ago; the dating of the ordering's fixation is debated) arranged 64 symbols into an ordering that satisfies a set of interlocking mathematical constraints admitting an estimated ≈3×10³⁷ valid arrangements of the 10^89 possible (SEARCH_SPACE_SIZE.md) — a reduction of more than 50 orders of magnitude. Within every enumerated dataset, a handful of boundary constraints (specifying which pairs must be adjacent at specific positions) then single out King Wen exactly: 4 boundaries at the 10T datasets, 5 boundaries at canonical depth (100T and 560T — the identical set {1, 4, 21, 25, 27} at both, isolating KW among 10.5 billion orderings) (corrected 2026-07-04: an earlier version said "4 at 560T" — a survivor-counting error; see BOUNDARY_MINIMUM.md); whether that holds over the full space is open (extrapolation suggests ~13–20 boundaries would be required). The minimums are proven within the enumerated datasets — exhaustive 3-subset testing confirms no triple suffices anywhere, and exhaustive 4-subset testing confirms no 4-set suffices at 100T/560T.
A key partition-scope finding (2026-04-19): the specific boundaries that work are partition-dependent. What IS stable across d2 and d3: boundaries {25, 27} are mandatory in every working 4-set. What is NOT stable: the other 2 boundaries.
- d2 10T (286M): 4 working 4-subsets —
{25, 27} ∪ one-of-{2, 3} ∪ one-of-{21, 22}. - d3 10T (706M): 8 working 4-subsets —
{25, 27}plus pairs drawn from{1..6}(specifically {1,3}, {1,4}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {3,6}).
Only {25, 27} can be cited as the stable "mandatory boundaries" result. The broader phrasing about "one-of-{2,3} ∪ one-of-{21,22}" is a d2-specific statement — at deeper partition sampling, different early-zone boundaries become interchangeable. This is exactly the kind of partition-depth sensitivity the null-model caveat (see CRITIQUE.md) anticipates: structural claims that look specific at one partition may look different at another.
| Step | Rule | Arrangements remaining |
|---|---|---|
| 0 | All possible orderings | 10^89 |
| 1 | Pair structure | 10^45 |
| 2 | No 5-line jumps | ~4% of step 1 |
| 3 | Complements ceiling (total distance ≤776) | ~0.3% of step 1 |
| 4 | Start with Heaven/Earth | ~0.005% of step 1 |
| 5 | Specific transition counts | 706,427,594 (d3 10T canonical); 286,357,503 (d2 10T canonical) — budgeted 10T slices, not the layer size; the true C1–C5 layer is estimated ≈1.3×10³⁸, see SEARCH_SPACE_SIZE.md |
| 6 | 4 boundary constraints (5 at canonical depth) | 1 (King Wen) — within the enumerated slice; see SEARCH_SPACE_SIZE.md |
Counting units in this table are mixed by construction (legacy presentation, flagged 2026-08-06; the
numbers themselves are unchanged and correct). Steps 0–1 count raw orientation-explicit arrangements
(steps 2–4 are fractions of step 1). Step 5's budgeted counts are orientation-deduplicated canonical
records — one record per canonical pair-sequence, orientation variants collapsed
(SOLUTIONS_FORMAT.md §Deduplication) — while its ≈1.3×10³⁸ layer estimate is
raw (orientation-explicit; ≈3.3×10³⁷ after ~4× orientation-dedup —
SEARCH_SPACE_SIZE.md), so the two conventions sit in one cell and should not be
divided into each other without converting (VERIFY.md --fiber-sweep is the exact
converter).
The rules above were only the beginning. Three further instruments — a machine-checked theorem prover, a logic solver that can settle "does any ordering with property X exist?" questions definitively, and a measurement technique that samples the full space of valid orderings without enumerating it — produced a second wave of results, each with a full technical report:
- Every valid ordering has 23 mathematical "twins" (counted at the record level) — relabelings that the rules cannot tell apart. King Wen is not special in this respect: the twins are a property of the rules themselves. [TR-5]
- Some "design choices" turn out to be forced. Eight rules — stating seven distinct regularities, since r3 and p1c4 turn out to be the same fact under two scholars' names (TR-1 §3(2)) — that scholars across eight centuries attributed to the arranger's intent are mathematical consequences of the rules — any valid ordering has them, chosen or not. And every valid ordering has exactly 15 alternations between even-balanced and odd-balanced pairs — provably, always. [TR-1, TR-6]
- The classical rules contradict each other. A central result: the four strongest design rules proposed in the literature — two by Steve Moore, two traceable through Larry Schulz back to a 13th-century commentator — cannot all be satisfied by any C1∩C2∩C4∩C5-valid ordering. King Wen keeps one of them perfectly and misses the other three by the smallest measured margins (2 each). Its famous "irregularities," puzzled over for centuries, are the visible seam of a forced trade-off — not damage to a once-perfect-under-all-four original (none could exist), and not sloppiness. (Whether they instead reflect damage to the three-rule-perfect precursor that does exist is a separate, open question — TR-2's pre-registered two-model comparison initially favored that corruption reading over a soft-preference arranger — but as of 2026-08-04 those figures are not calibrated in the pooled sense: the M_tend self-recovery arm FAILED at 68/100 against a frozen bar of 70, so the calibration gate vetoed, and as of 2026-08-07 the Bayes factor and posterior are withdrawn as claimed results — retained as the as-computed record only. The question is weighed, not settled, and no verdict is asserted. See CORRECTIONS CX-25 and CX-26.) [TR-2]
- In information terms, half the sequence is explained — nearly all of it by the pairing rule, which turns out to be mathematically optimal among pairing rules of its kind (complement/reversal matchings; Radisic 2026). The other half is explained by nothing anyone has found yet. [TR-9]
Each of these is reported in full: every MEASURED result carries a reproduction command, and every proof cited as machine-checked names its certificate or Lean theorem. That is not the same as saying all four are machine-verifiable, and one of them is not. The bit-ledger above is the suite's most judgment-dependent result — its accounting conventions are chosen, not derived, which is why TR-9 states a range (about 105–139 bits unexplained) rather than a number and publishes its conventions so a skeptic can re-price everything under their own. What a machine checks there is the arithmetic under a stated convention, never the convention.
Applying the same methodology to random pair-constrained sequences — extract their diff distribution, complement distance, and starting pair, then test for uniqueness — also produces apparent uniqueness in 9 out of 10 cases. The constraint extraction approach makes almost any sequence appear uniquely determined. This means the three properties that test extracts — complement distance, starting pair and transition counts (Rules 3, 4 and 5 above) — are not individually remarkable: any sequence's specific properties would similarly narrow the search space.
What IS genuinely special about King Wen is Rules 1 and 2: the perfect pair structure and the no-5-line-transition property. Only ~4% of pair-constrained orderings avoid 5-line transitions (ROAE's 10^9-sample pair-constrained null gives 4.29% precisely). The pair structure itself is vanishingly unlikely by chance — a random 64-permutation has probability ~10^-44 of satisfying it. Both observations are prior knowledge — Rule 1 is classical (Yi Zhuan commentary, Cook 2006) and Rule 2 is from McKenna & McKenna 1975 (The Invisible Landscape). ROAE's contribution is independent computational verification at scale, plus null-model testing across seven structured permutation families showing no family other than KW simultaneously satisfies C1+C2+C3. See CITATIONS.md and CRITIQUE.md §Missing analyses.
The analysis shows the King Wen sequence satisfies a set of interlocking mathematical constraints. It cannot show whether whoever arranged it understood those constraints explicitly or arrived at them through centuries of refinement. A simple practice — "pair each hexagram with its mirror, keep complements nearby, avoid jarring transitions" — applied consistently over generations could produce the same result as deliberate mathematical design. The sequence is the same either way; only the history differs, and the history is outside the reach of computation.
ROAE's contribution to the study of the King Wen sequence — distinct from what was previously known — is primarily about quantifying how specific and KW-distinctive the combined constraint system is, and about reproducibility of that quantification.
What was already known (CITATIONS.md):
- The pair structure (Rule 1) is ancient, discussed in Yi Zhuan commentary (traditionally dated 5th–3rd c. BCE, though modern dating is later), formalized in modern combinatorial terms in Cook 2006 (Classical Chinese Combinatorics).
- The no-5-line-transition property (our Rule 2 = C2) is documented in McKenna & McKenna 1975 (The Invisible Landscape, Chapter 9 "Order in the I Ching and Order in the World"). In the same chapter McKenna also identifies two additional design rules: a secondary "minimize transitions of value 1 except where doing so would force a value 5" rule and a "three to one ratio of even to odd transitions" rule. The 3:1 ratio is provably forced by C4+C5 per the Theorem in SPECIFICATION.md — derived from first principles, not a separate constraint. The value-1 rule was empirically measured (
solve --verify-rule2) and found to be a real KW-specific regularity: 80.03% of C1–C5 records violate it at the d3 560T canonical (2026-06-15,9a968fa2…, 10,525,271,997 records — KW is in the 19.97% minority that obeys it). (Figure updated 2026-08-01: this page previously quoted 83.77% from the 2026-05-19 run on the v2 11.2T canonical — a lineage the project has since closed. Both figures are correct at their own depth and lineage; the 560T number is the current one, and is what CITATIONS.md carries.) but NOT promoted to a formal C-rule, because it would be reverse-engineered from KW's specific value-1 placements; see MCKENNA.md for the full peer-review-defensibility analysis. - Statistical properties of King Wen vs random permutations are documented in Chan 2026 (arXiv:2604.09234) — Monte Carlo permutation analysis against 100,000 random baselines, reporting KW's higher-than-random mean Hamming distance (3.35 at 98.2nd percentile), negative lag-1 autocorrelation of Hamming distances (3.7th percentile), within-pair vs between-pair distance asymmetry (99.2nd percentile), yang-balanced groups of four (99.8th percentile), and surprise-distribution variance. Chan's research predates ROAE; where ROAE's findings overlap with Chan's (specifically: the mean-Hamming, lag-1 autocorrelation, and within/between-pair asymmetry observations) Chan's prior art is acknowledged.
What ROAE adds:
-
Enumeration at scale, exact at its budget. Under the conjoined C1–C5 constraint system, ROAE's deepest published canonical counts 10,525,271,997 distinct orderings at a 560 trillion-node partial enumeration (d3 560T; sha
9a968fa2…— see CANONICAL_HASHES.md); the 10T d3 canonical (706,427,594,b85c8871…) remains a smaller-scale reference anchor. Each count is exact at its budget and reproducible byte-identically across hardware and region — and, to our knowledge, enumeration at this scale had not previously been carried out, only estimated or approximated; corrections welcome. (Corrected 2026-08-01: this bullet stated the flagship contribution at the 10T figure — understating the project's own deepest result ~15× — labelled the population "C1 + C2 + C3" (legacy shorthand for C1–C5; see METHODS.md), and asserted the prior-art claim with no hedge while the three bullets around it were hedged.) -
Seven-family null-model framework. Measures how other structured permutation families compare to KW's structural properties. Main finding: zero of 1.86 billion permutations across six unconditional families satisfy C1 (consistent with the theoretical rate of ~10⁻⁴⁴ for random permutations). For the de Bruijn and Gray code families, 0% is also proved analytically, not just observed. To our knowledge this is the first systematic null-model test of this scope for the KW structural constraints; corrections welcome.
-
C1 does most of the structural work. A 10⁹-sample pair-constrained null (C1 guaranteed by construction) shows that given C1, the conditional C2 hit rate jumps from 0.18% (random) to 4.29% (a ~24× multiplier; the precise ratio is computed from the unrounded bases, 23.5×, and does not reproduce from the two rounded figures shown), and C3 from 0.003% to 6.42% (a ~2,100–2,300× multiplier; the precise 2,264× is computed from the unrounded bases and does not reproduce from the rounded figures shown; the C3|C1 rate is now exact — 6.4211367496%,
verify.py --check-null-g --unpinned, MC-consistent). This quantifies what was previously intuitive: the pair structure is doing the heavy lifting; C2 and C3 are relatively modest additional filters. -
C3 (complement-distance ceiling of 776) as a specifically quantified constraint. Believed ROAE-original; if prior work exists, please see CITATIONS.md.
-
Mawangdui and Jing Fang 8 Palaces comparison (corrected 2026-07-05). Of four tested comparison orderings (Fu Xi, King Wen, Mawangdui, Jing Fang), King Wen and Jing Fang's 8 Palaces (arrangement traditionally attributed to Jing Fang, 77–37 BCE; scored as linearized in Nielsen 2003's printed palace order) satisfy C2 (zero 5-line transitions) despite Jing Fang failing C1 and C3. The authentic Mawangdui silk-text ordering (168 BCE) does not: it has exactly one 5-line transition, at the octet seam where its trigram-block construction resets (#48 Jing → #51 Zhen). An earlier version of this item reported that Mawangdui also satisfied C2 and inferred a "shared classical Chinese design principle" — that was computed on an erroneous Mawangdui array and is withdrawn (see CITATIONS.md errata; corrected array per Shaughnessy 2022, Table 11.2).
-
Latin-square C2-rate decomposition. The 8! × 8! Latin-square row × column traversal family has a surprisingly high C2 rate (57.96%). ROAE provides an analytic decomposition that reproduces this rate exactly from first principles: only 7 of 63 transitions in a Latin-square traversal can be 5-line, and the rate is controlled by the row-permutation's Hamming profile (Hamiltonian path in the 3-cube) combined with the column-permutation's first-last XOR. See CRITIQUE.md §Latin-square C2-rate decomposition.
-
Partition Invariance theorem. A formal guarantee (PARTITION_INVARIANCE.md) that the canonical solutions.bin sha256 is byte-identical across any choice of hardware, thread count, region, merge algorithm, or enumeration path — for fixed solver + input parameters. Cross-validated across the "4 corners": {Zen 4 F64 westus2, Zen 5 D128 westus3} × {external merge, in-memory heap-sort}.
Open questions this work surfaces: see CRITIQUE.md §Open questions. In brief: Costas arrays at order 64 remain untested (infeasible at 64! candidates); Latin-square's 57.96% C2 rate invites parallel analysis of whether KW's own adjacency geometry has an analogous within/between decomposition; whether any published order-64 Costas arrays accidentally satisfy C1+C2+C3 is a concrete follow-up.
For full technical details, methodology, and reproducible commands, see SOLVE.md and CITATIONS.md.
Visualization of Rule 3 (Complements stay close). Each line shows the King Wen sequence with one complement pair highlighted. Sorted from closest (distance 1 — adjacent) to farthest (distance 47). The mean across all 32 pairs is 12.1.
Close complement pairs (distance 1-7) are always in the same half of the sequence. Far complement pairs (distance 19+) always cross the midpoint. No exceptions. (Note: some distance groups show highlights on both sides — e.g., distance 6 has four pairs, two in the first half and two in the second. Each individual pair stays in its own half.)
Multi-pair distances always come in groups of 2 (e.g., 5↔35 AND 6↔36). This is because complementation maps King Wen pairs to King Wen pairs: if A and reverse(A) form a pair, then complement(A) and complement(reverse(A)) also form a pair. So both members of one pair always point to members of the same other pair, at the same distance.
Distance 1 (9 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 1↔2, 11↔12, 17↔18, 27↔28, 29↔30, 38↔39, 53↔54, 61↔62, 63↔64
Distance 3 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 37↔40
Distance 4 (2 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 55↔59, 56↔60
Distance 5 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 10↔15
Distance 6 (4 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 7↔13, 8↔14, 51↔57, 52↔58
Distance 7 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 9↔16
Distance 10 (2 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 31↔41, 32↔42
Distance 14 (2 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 19↔33, 20↔34
Distance 19 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 26↔45
Distance 20 (2 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 23↔43, 24↔44
Distance 21 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 25↔46
Distance 25 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 22↔47
Distance 27 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 21↔48
Distance 30 (2 pairs):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pairs: 5↔35, 6↔36
Distance 45 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 4↔49
Distance 47 (1 pair):
䷀䷁ ䷂䷃ ䷄䷅ ䷆䷇ ䷈䷉ ䷊䷋ ䷌䷍ ䷎䷏ ䷐䷑ ䷒䷓ ䷔䷕ ䷖䷗ ䷘䷙ ䷚䷛ ䷜䷝ ䷞䷟ ䷠䷡ ䷢䷣ ䷤䷥ ䷦䷧ ䷨䷩ ䷪䷫ ䷬䷭ ䷮䷯ ䷰䷱ ䷲䷳ ䷴䷵ ䷶䷷ ䷸䷹ ䷺䷻ ䷼䷽ ䷾䷿
Pair: 3↔50
Every consecutive hexagram transition, showing the number of lines that change (Hamming distance). A distance of 5 never occurs — this is Rule 2.
The difference wave as a sparkline (each character = one transition, height = lines changed):
█▂▅▅▅▃▂▅▂▅█▂▂▅▂▂█▃▅▃▂▂▂▃▅▂█▂█▃▂▃▅▅▅▂▅█▅▃▂▅▂▃▅▃▂▃▅▅▅▁█▂▂▃▅▃▂▁█▃█
The difference wave — the sequence of Hamming distances read left to right:
6 2 4 4 4 3 2 4 2 4 6 2 2 4 2 2 6 3 4 3 2 2 2 3 4 2 6 2 6 3 2 3 4 4 4 2 4 6 4 3 2 4 2 3 4 3 2 3 4 4 4 1 6 2 2 3 4 3 2 1 6 3 6
Transition counts: 1: 2, 2: 20, 3: 13, 4: 19, 5: 0 (never), 6: 9
| # | From | To | Distance |
|---|---|---|---|
| 1 | ䷀ #1 The Creative | ䷁ #2 The Receptive | 6 |
| 2 | ䷁ #2 The Receptive | ䷂ #3 Difficulty | 2 |
| 3 | ䷂ #3 Difficulty | ䷃ #4 Youthful Folly | 4 |
| 4 | ䷃ #4 Youthful Folly | ䷄ #5 Waiting | 4 |
| 5 | ䷄ #5 Waiting | ䷅ #6 Conflict | 4 |
| 6 | ䷅ #6 Conflict | ䷆ #7 The Army | 3 |
| 7 | ䷆ #7 The Army | ䷇ #8 Holding Together | 2 |
| 8 | ䷇ #8 Holding Together | ䷈ #9 Small Taming | 4 |
| 9 | ䷈ #9 Small Taming | ䷉ #10 Treading | 2 |
| 10 | ䷉ #10 Treading | ䷊ #11 Peace | 4 |
| 11 | ䷊ #11 Peace | ䷋ #12 Standstill | 6 |
| 12 | ䷋ #12 Standstill | ䷌ #13 Fellowship | 2 |
| 13 | ䷌ #13 Fellowship | ䷍ #14 Great Possession | 2 |
| 14 | ䷍ #14 Great Possession | ䷎ #15 Modesty | 4 |
| 15 | ䷎ #15 Modesty | ䷏ #16 Enthusiasm | 2 |
| 16 | ䷏ #16 Enthusiasm | ䷐ #17 Following | 2 |
| 17 | ䷐ #17 Following | ䷑ #18 Decay | 6 |
| 18 | ䷑ #18 Decay | ䷒ #19 Approach | 3 |
| 19 | ䷒ #19 Approach | ䷓ #20 Contemplation | 4 |
| 20 | ䷓ #20 Contemplation | ䷔ #21 Biting Through | 3 |
| 21 | ䷔ #21 Biting Through | ䷕ #22 Grace | 2 |
| 22 | ䷕ #22 Grace | ䷖ #23 Splitting Apart | 2 |
| 23 | ䷖ #23 Splitting Apart | ䷗ #24 Return | 2 |
| 24 | ䷗ #24 Return | ䷘ #25 Innocence | 3 |
| 25 | ䷘ #25 Innocence | ䷙ #26 Great Taming | 4 |
| 26 | ䷙ #26 Great Taming | ䷚ #27 Nourishment | 2 |
| 27 | ䷚ #27 Nourishment | ䷛ #28 Preponderance of Great | 6 |
| 28 | ䷛ #28 Preponderance of Great | ䷜ #29 The Abysmal | 2 |
| 29 | ䷜ #29 The Abysmal | ䷝ #30 The Clinging | 6 |
| 30 | ䷝ #30 The Clinging | ䷞ #31 Influence | 3 |
| 31 | ䷞ #31 Influence | ䷟ #32 Duration | 2 |
| 32 | ䷟ #32 Duration | ䷠ #33 Retreat | 3 |
| 33 | ䷠ #33 Retreat | ䷡ #34 Great Power | 4 |
| 34 | ䷡ #34 Great Power | ䷢ #35 Progress | 4 |
| 35 | ䷢ #35 Progress | ䷣ #36 Darkening | 4 |
| 36 | ䷣ #36 Darkening | ䷤ #37 The Family | 2 |
| 37 | ䷤ #37 The Family | ䷥ #38 Opposition | 4 |
| 38 | ䷥ #38 Opposition | ䷦ #39 Obstruction | 6 |
| 39 | ䷦ #39 Obstruction | ䷧ #40 Deliverance | 4 |
| 40 | ䷧ #40 Deliverance | ䷨ #41 Decrease | 3 |
| 41 | ䷨ #41 Decrease | ䷩ #42 Increase | 2 |
| 42 | ䷩ #42 Increase | ䷪ #43 Breakthrough | 4 |
| 43 | ䷪ #43 Breakthrough | ䷫ #44 Coming to Meet | 2 |
| 44 | ䷫ #44 Coming to Meet | ䷬ #45 Gathering | 3 |
| 45 | ䷬ #45 Gathering | ䷭ #46 Pushing Upward | 4 |
| 46 | ䷭ #46 Pushing Upward | ䷮ #47 Oppression | 3 |
| 47 | ䷮ #47 Oppression | ䷯ #48 The Well | 2 |
| 48 | ䷯ #48 The Well | ䷰ #49 Revolution | 3 |
| 49 | ䷰ #49 Revolution | ䷱ #50 The Cauldron | 4 |
| 50 | ䷱ #50 The Cauldron | ䷲ #51 The Arousing | 4 |
| 51 | ䷲ #51 The Arousing | ䷳ #52 Keeping Still | 4 |
| 52 | ䷳ #52 Keeping Still | ䷴ #53 Development | 1 |
| 53 | ䷴ #53 Development | ䷵ #54 Marrying Maiden | 6 |
| 54 | ䷵ #54 Marrying Maiden | ䷶ #55 Abundance | 2 |
| 55 | ䷶ #55 Abundance | ䷷ #56 The Wanderer | 2 |
| 56 | ䷷ #56 The Wanderer | ䷸ #57 The Gentle | 3 |
| 57 | ䷸ #57 The Gentle | ䷹ #58 The Joyous | 4 |
| 58 | ䷹ #58 The Joyous | ䷺ #59 Dispersion | 3 |
| 59 | ䷺ #59 Dispersion | ䷻ #60 Limitation | 2 |
| 60 | ䷻ #60 Limitation | ䷼ #61 Inner Truth | 1 |
| 61 | ䷼ #61 Inner Truth | ䷽ #62 Small Preponderance | 6 |
| 62 | ䷽ #62 Small Preponderance | ䷾ #63 After Completion | 3 |
| 63 | ䷾ #63 After Completion | ䷿ #64 Before Completion | 6 |
Revision 2026-08-01 (lens-sweep adjudication, area q-methods-tr4): three wording corrections, no number or sha changed. (i) The page title read "How the King Wen Sequence Was Built" — a historical-process claim this page's own §"What we can and cannot say" explicitly disowns, and the most-liftable string on the page; it now reads "Is Structured". (ii) The opening contribution sentence claimed "exhaustive enumeration", refuted 138 lines below by "both enumerations are partial … the true count under exhaustive enumeration is unknown"; it now reads "budget-exact enumeration", and the §"What ROAE adds" bullet heading is retitled to match. (iii) §"An important caveat" concluded that "Rules 3-7 are not individually remarkable" while this page defines only Rules 1-5 (C6/C7 are named once, never numbered or defined here); it now names the three properties that test actually extracts — complement distance, starting pair, transition counts (Rules 3, 4, 5).
Revision 2026-07-22 (C3 scope-consistency sweep): the 3.9th-percentile figure is now labeled at its measured scope — orderings satisfying every constraint except C3 itself (C1+C2+C4+C5; earlier text said "C1-only") — and §Rule 3 gained an exact reference baseline for the ~10% ceiling-tie figure (verify.py --check-null-g: tie share 7.86% under the bare pair-slot null; a baseline, not a refutation — the populations are not like-for-like). No counts or shas changed.
Revision 2026-08-01 (lens sweep — C3 percentile flag): the 3.9th-percentile complement-distance figure is flagged and withdrawn from citation. It is a statistic of the 13,296-ordering solve.py differential slice, whose stated range [11.75, 14.5] cannot be the range of C1+C2+C4+C5 — the strictly smaller C1–C5 canonical contains orderings at cd = 6.125 — and the suite's own ledger gives 1.3287×10³⁸ / 1.097051×10³⁹ ≈ 12% at that scope. The 2026-07-22 scope correction above fixed the figure's label, not the figure. Authoritative statement of the flag, and the measurement that would settle it: SOLVE.md §Rule 3. No canonical count, sha, or theorem changed.
Revision 2026-08-02 (the last surviving blanket-verifiability over-claim; registry key RP-a823340f): §"The story continued" closed on a one-line assertion that all four of its bullets were machine-verifiable. One of them is TR-9's bit-ledger — the result reports/README.md itself describes as "judgment-dependent by construction" — so the sentence was false in exactly the way the eleven report covers' banner was before 14d8751 retired it. It was stated in different words and so was invisible to the gate written for the covers; a repo-wide sweep confirms this was the last instance of the class. Replaced with the covers' own formulation — measured results carry a reproduction command, machine-checked proofs name their certificate — plus the explicit exception for the bit-ledger and TR-9's stated range. The retracted wording is now in RETRACTED_PHRASES.tsv so it cannot return; it is cited here by key rather than quoted, per that registry's convention. No number, count, sha or theorem changed.
Revision 2026-08-06 (counting-unit label pass, UNASKED-2 batch): §"The numbers at a glance" gained an explicit units note — the funnel table mixes raw orientation-explicit arrangement counts (steps 0–1), fractions (steps 2–4), and orientation-deduplicated canonical-record counts (step 5's budgeted figures) against a raw layer estimate in the same cell. The note flags the legacy presentation and points at SOLUTIONS_FORMAT.md §Deduplication, SEARCH_SPACE_SIZE.md, and the exact record↔sequence converter (VERIFY.md --fiber-sweep). No number changed — the defect was unstated units, not wrong values.