Technical report — not peer-reviewed. Every MEASURED result carries a reproduction command, and every proof cited as machine-checked names its certificate or Lean theorem; claims of scope, attribution and interpretation are argued, not verified. One caveat is structural, and it frames all the rest: the same author wrote the claims, the software that checks them, and this report that grades the check. Verification here is independent in mechanism, never in authorship; no independent party has yet audited or reproduced any of it (METHODS.md §"Authorship independence").
Methods, environment pinning, statistics conventions, and artifact access: see METHODS.md.
What if the sequence is a circle — the last hexagram wrapping around to the first? Several scholars, notably Terence McKenna, read it that way. This report re-derives the mathematics under the circular reading. Two results stand out. First, the wrap-around step is forced to be odd (proved formally), which makes McKenna's observed 3-to-1 ratio of even-to-odd transitions a necessity, not a choice. Second, a surprise: the sequence's missing distance-5 transition is a genuine extra rule in the circular reading — orderings that wrap at distance 5 make up 17.4% of the valid space, yet not one appears among 10.5 billion enumerated records. That gap between the full space and the enumerated slice is a stark demonstration of why bounded search results need independent measurement — and why we decided this rule, though real, stays documented rather than adopted.
McKenna & McKenna (1975) read the King Wen sequence as a cycle — position 64 wrapping to position 1 —
and their published counts (64 transitions, "three even integers to each odd integer") depend on that
closure. We work out exactly what the ROAE constraint system says under the circular reading. Three
theorems and one SAT decision result: (i) the wrap-around Hamming distance d(s₆₃, s₀) is odd for every
C4+C5-valid ordering — now machine-checked in Lean 4 at full generality (wrap_parity_general, structural
induction, not finite enumeration); (ii) McKenna's exact 3:1 even:odd transition ratio is a forced
consequence of C4 + C5 plus the XOR parity identity — a regularity he read as a design feature that turns
out to be a theorem, not a choice; (iii) every valid circular reading has exactly 16 parity-class
alternations, and the first and last hexagrams of any valid linear ordering lie in opposite
popcount-parity classes. Finally, the circular form of C2 ("no 5-line transition anywhere on the cycle")
is a genuine extra constraint: valid linear orderings with a 5-line wrap exist (SAT-decided, explicit
witness) even though exactly zero appear among the 10,525,271,997 records of the deepest canonical slice;
the full-space wrap-distance masses are measured at d=1: 17.5%, d=3: 65.2%, d=5: 17.4% (2×10¹⁰
weighted-Knuth probes; independently reproduced by the archived v2.0 r6 run —
evidence/r6/rc1c_primary.out: 17.45 / 65.18 / 17.37% — agreeing
within 0.05 percentage points per class; per-class ± figures were not emitted by the instrument, so
the two-run agreement is the published uncertainty statement). The operator's
documented decision: circular C2 is not promoted into the constraint system — the circular reading is
McKenna's interpretive frame, not an attested property of the received artifact.
- The circular frame and its provenance. McKenna & McKenna (1975, The Invisible Landscape, Part Two, Ch. 9) constructed their difference wave over 64 transitions including the wrap s₆₃ → s₀ (KW's wrap has Hamming distance 3) — the circular reading is theirs, with full attribution (CITATIONS.md, MCKENNA.md). What closure does not touch: C1, C3, C4 are position/pair properties, unaffected. What it touches: the transition multiset (C5) gains a 64th member, and C2 acquires a 64th application — the wrap itself.
- The wrap-parity theorem, three ways. For any sequence satisfying C4 and C5, the wrap distance
d(s₆₃, s₀) is odd — proven via the XOR parity identity (popcount(a⊕b) ≡ popcount(a)+popcount(b) mod 2).
KW's wrap is d = 3. Verification stack: (a) the prose proof (SPECIFICATION.md); (b) the Lean 4
kernel-checked general form
wrap_parity_general— verified for EVERY C4+C5 sequence of 6-bit values by structural induction (telescoping transition-parity lemma + sum-parity/odd-count machinery), upgrading the formal core from "finite facts checked" to "sequence-level theorem proven"; (c) empirical corroboration at the d3 560T canonical (10,525,271,997 records, sha 9a968fa2…): 100.000000% odd wrap — necessarily, sincesolve --verifyenforces the C4+C5 hypotheses; the theorem holds deductively, the enumeration validates the implementation. - McKenna's 3:1 is forced, not designed. His "perfect ratio of three to one" (16 odd of 64 circular transitions, 25.00% exact) is the circular reading of the wrap-parity theorem plus C5's 16-odd-of-64 count. Every C4+C5-valid ordering has it. McKenna discovered the ratio empirically before its proof was articulated here — one of his most accurate quantitative claims, and stronger than he may have realized: forced given C4 + C5, and hence not an independent design choice within that constraint system — though C5 is itself a regularity read off King Wen, so "forced" here is relative to KW-derived constraints, not to an unconstrained arranger.
- What closure changes: circular C5 and the 16-alternation corollary. Under closure KW's transition
multiset becomes {1:2, 2:20, 3:14, 4:19, 6:9} (the d=3 count rises 13→14); orderings with d=1 wraps
read {1:3, …, 3:13, …} instead. The parity-alternation theorem (PARITY_ALTERNATION.md; Lean
alternations_15_general) forces exactly 15 alternations linearly; on the cycle the count must be even and the wrap boundary is forced to alternate (equivalent to wrap parity — two routes to one fact). Corollary: every valid circular reading has exactly 16 alternations, and the first and last hexagrams of any valid linear ordering lie in opposite popcount-parity classes (KW: 63 even → 42 odd ✓). - Circular C2 is a genuine extra constraint — the SAT decision. The wrap-parity theorem restricts the
wrap to d ∈ {1, 3, 5}. At the 560T canonical the wrap is d=3 in 91.83% of records, d=1 in 8.17%, and
d=5 in exactly zero of 10,525,271,997. Nevertheless, valid linear orderings with a 5-line wrap
EXIST — SAT-decided (2026-07-03) with an explicit C1–C5-valid witness (final pair (32, 1); wrap
d(1, 63) = 5; complement-distance sum 752). So the circular reading is not free: it excludes real
members of the linear solution set. Per the twins lesson (SYMMETRY_SEARCH.md), budgeted-slice absence
does not measure full-space rarity — and the full-space wrap-distance masses are now MEASURED (2×10¹⁰
weighted-Knuth probes, 2026-07-03; estimator per METHODS.md; mass ratios are
heavy-tail dominated — small probe budgets will not resolve them):
d=1: 17.5%, d=3: 65.2%, d=5: 17.4%. Uncertainty, stated from archived artifacts (2026-07-26):
the instrument prints point masses without per-class CIs, so no ± figure is quoted; instead the
independent 2×10¹⁰-probe v2.0 r6 run (
evidence/r6/rc1c_primary.out, 2026-07-10) re-measured the same three masses at 17.45 / 65.18 / 17.37% — two independent draws agreeing within 0.05 percentage points per class, which bounds the run-to-run scatter at the precision every figure here is quoted to. The 5-wrap orderings that no budgeted slice has ever contained are between a fifth and a sixth of the full space; circular C2 would cut the space by ×1.21. - The non-promotion decision, on the record. Operator decision 2026-07-03: circular C2 is documented, NOT promoted, and not implemented in solve.c in any form. Rationale (consistent with the R-series non-promotion discipline): the circular reading is McKenna's interpretive frame, not an attested property of the received artifact; enforcing it would add a reverse-engineered constraint. The implementation analysis, for the record: as a pure leaf-emission filter it would be byte-identical to the current lineage at every published canonical scale (zero 5-wrap records exist in any slice — divergence begins only in territory no budget has reached, as the SAT witness proves); as a prune it would change node consumption and open a new sha lineage. Neither is warranted. Closure also invites a larger symmetry question — without C4, a circular system would be invariant under the 32 pair-slot rotations as well as the B₃ relabelings — but under the actual system (C4 kept) the circular reading changes nothing about the symmetry group.
- Wrap-parity theorem, statement + proof: documentation/SPECIFICATION.md §Theorem (Wrap-around parity is odd)
- Lean general form:
lean lean/KingWen.lean(Lean 4, tested 4.31.0; silence = all theorems check) —wrap_parity_general, supporting lemmastransitions_sum_parity,sum_parity_odd_count,odd_count_partition; see lean/README.md §Tier 2 - 560T wrap measurement (91.83% d3 / 8.17% d1 / zero d5):
./solve --verify-wrap-parityagainst the d3 560T canonical (sha registry: documentation/CANONICAL_HASHES.md). Note: this mode's printed theorem line formerly claimed "C2 forbids 5 → d ∈ {1,3}", contradicting §5's SAT result; corrected in public commit0c24637(2026-07-03) to state d ∈ {1,3,5} with d=5 not excluded by linear C2 (tabulator was always correct; stdout/comment only, selftest sha unchanged) - Wrap-d5 witness:
python3 sat.py --witness wrap-d5→ the explicit 64-hexagram sequence in documentation/CIRCULAR_KING_WEN.md, C1–C5-valid, wrap d = 5 - Full-space wrap masses:
SOLVE_KNUTH_SCORE=1 ./solve --estimate-knuth 20000000000(2×10¹⁰ probes, the budget behind the published 17.5/65.2/17.4% figures; the scorer prints point masses without per-class CIs — the published uncertainty statement is the two-run agreement in §5 — and mass ratios are heavy-tail dominated, so small budgets (~10⁵ probes) will NOT reproduce them; this is an hours-scale run on many-core hardware. Method self-validation in documentation/SEARCH_SPACE_SIZE.md) - 16-alternation corollary ingredients: documentation/PARITY_ALTERNATION.md + Lean
alternations_15_general - Non-promotion decision + rationale: documentation/CIRCULAR_KING_WEN.md §Status decision (operator, 2026-07-03)
- Attribution: the circular reading is McKenna & McKenna (1975); the wrap-parity theorem, its 560T measurement, the alternation corollary, and the wrap-d5 SAT decision are ROAE (to our knowledge — corrections welcome via documentation/CITATIONS.md)
The 64 hexagrams as a cycle in King Wen order (computed from the sequence itself). Red edges are odd transitions; the highlighted wrap edge 64→1 jumps d = 3 — odd, as the wrap-parity theorem forces. The circular reading has 16 odd transitions where the linear reading has 15: the wrap adds exactly one, always.
Peter Meyer (1998, web) published the complete cyclic line-change sequence of the King Wen order — the 64 Hamming distances including the wraparound term — with an explicit XOR-and-popcount formalization (see CITATIONS.md). His data thus contains the wrap value d=3 this report analyzes, decades before this work; the wrap-parity theorem, the d in {1,3,5} space analysis, and the 17.4%-vs-absent measurement remain, to our knowledge, first stated here. Found during a bibliography review 2026-07-04; corrections welcome.
The circular transition-parity string (64 values: transition i is "odd" iff an odd number of lines change, the wrap included) switches value exactly 32 times in every C1+C4+C5-valid ordering. Proof: index the 64 cyclic transitions 0..63, transition i connecting positions i and i+1 (mod 64, 0-indexed); pair p occupies positions 2p and 2p+1, so within-pair transitions sit at the 32 even indices and are all even (C1: reversal preserves line-count parity; the four self-reverse pairs are complement pairs, d = 6), while between-pair transitions sit at odd indices 1..61 and the wrap at index 63 — also odd. The parity-alternation theorem (TR-6) gives exactly 15 odd between-pair transitions, and the wrap-parity theorem (§2) makes the wrap odd, so there are exactly 16 odd transitions (McKenna's 16-of-64, §3), all confined to odd cyclic indices. Adjacent indices on a 64-cycle have opposite index parity (including the 63/0 seam), so the 16 odd transitions are pairwise non-adjacent — 16 isolated values, each contributing exactly two switches: 32. The result is invariant across the wrap's distance class (d ∈ {1, 3, 5} are all odd). This fills the one remaining cell in the TR-6/TR-7 linear→circular lattice: alternations 15 → 16 (§4), switches 30 (TR-6 corollary) → 32. Verified on King Wen: cyclic odd transitions = 16, all at odd indices; linear switches = 30; cyclic switches = 32. Derived in cross-report synthesis 2026-07-04 (composition of TR-6's 30-switches corollary with this report's wrap-parity theorem), independently re-derived and re-verified before folding in.
Verification: both ingredient theorems are kernel-checked (switches_30_general,
wrap_parity_general in lean/KingWen.lean); the KW instance is a three-line check from solve.py's
binary_hexagrams (count sign changes of the cyclic Hamming-distance parity string).
The sequence's two endpoint pairs are individually distinguished: the pure pair {Qian, Kun} that opens it (C4) and the alternating pair {Jiji, Weiji} that closes it — Cook 2006's "pure opens, mixed closes," measured linearly as the final-pair anchor (7.84% of C1–C5 mass, LITERATURE_RULES_POPULATION_TESTS.md). They are also the only intrinsically extremal pairs: the unique pair of run-length-6 (constant) hexagrams and the unique pair of run-length-1 (strictly alternating) hexagrams. Under McKenna's circular reading the two observations become one: the two anchor pairs are neighbors on the circle — KW places the alternating pair in the last slot, adjacent to the pure pair across the wrap.
How much of that is forced? Three theorems (elementary; each exhaustively verified by finite computation over the 64 hexagrams / 32 pairs — a Lean formalization is planned, see the Verification Guide):
(i) Transition rigidity (T1): every hexagram of the pure pair is at Hamming distance exactly 3 from every hexagram of the alternating pair (an alternating 6-bit string has exactly three 1s, so it differs from 111111 and from 000000 in three positions each) — so an anchor adjacency, wherever it occurs and however oriented, is a d = 3 transition: C2-legal, odd, one unit of the largest odd budget class. In particular KW's wrap distance 3 (§2) is forced by which pair closes, not by any orientation choice.
(ii) Seam eligibility (T2i): pairs are parity-homogeneous (16 even / 16 odd — TR-6 ingredients), and the wrap-parity theorem (§2) then forbids all 16 even pairs — including all four self-reverse pairs and the pure pair itself — from ever occupying the final slot.
(iii) Pair-determined wrap (T2ii): for each of the 16 eligible (odd) pairs the wrap distance is a function of the pair alone (orientation-free), classifying them 10 : 3 : 3 into d = 3, 1, 5 closers (the 4 antipalindromic pairs — A₂ among them — plus the 6 popcount-3 reverse-pairs at d = 3; the 3 popcount-5 reverse-pairs at d = 1; the 3 popcount-1 reverse-pairs at d = 5; the wrap-d5 SAT witness of §5, which closes on (32, 1), is one of the latter — consistent). Eligibility is a necessary condition: that all 16 eligible pairs are actually realized as closers is not proven here (the measured wrap masses show every class is realized, and explicit witnesses realize A₂ and (32, 1)).
The measured full-space wrap masses (§5: 65.2 / 17.5 / 17.4% for d = 3 / 1 / 5) sit remarkably close to the bare eligible-pair-counting baseline (62.5 / 18.75 / 18.75%) — the wrap-distance profile is, to first order, pair-counting, with only a mild residual tilt toward d = 3. (Hedge: the baseline is a heuristic reference, not a null; per-class CIs are heavy-tail dominated per §5; the per-pair spread within classes is unknown except for A₂.)
This re-prices Cook's anchor. Against the naive 1/31 ≈ 3.2% the measured 7.84% looks like a ×2.4 enrichment, but the parity-forced eligibility baseline is 1/16 = 6.25%, so of that apparent enrichment ×1.9 is parity-forced (it holds for every C4+C5 ordering) and only ×1.25 is the contingent residual (7.84 / 6.25). Within its own d = 3 class A₂ carries 7.84% against a 6.52% class average (the other nine d = 3 closers average 6.37%) — mildly, not dramatically, over-represented.
Measured circular anchor adjacency (v2.0). What remains genuinely contingent is the adjacency
placement itself. Its circular population frequency — R-C1c, the weighted C1–C5-mass fraction in
which the alternating pair occupies slot 2 or slot 32 — was pre-registered above (v1.9) and has now
been measured (2×10¹⁰ weighted-Knuth probes; evidence evidence/r6/rc1c_primary.out): 13.05% of
C1–C5 mass (slot 32: 7.85%, reproducing the published R-C1 = 7.84% — the built-in scorer gate;
slot 2: 5.20%). Against the pre-registered references that is ×2.0 the uniform-slots baseline
(6.45%) and ×1.66 the eligibility-adjusted lower bound (7.84%). The descriptive A₂ slot histogram
is U-shaped: slot 2 is the largest non-final slot (5.20%, vs 3.84% at slot 3 and a 2.68% minimum at
slot 17), so the alternating pair is enriched at both circle-adjacent slots, not merely
late-biased — though slot 32 remains the global maximum. The KW ground truth (slot 2 = 0,
slot 32 = 1, adjacent = 1) and the negative control (the wrap-d5 SAT witness scores adjacent = 0)
were verified in both languages before the run. In plain terms: roughly one in eight valid
orderings places the two anchor pairs adjacent on the circle — KW's configuration is
population-common, and this measurement prices it; it does not elevate it. Likewise the circular
solution-space size is now measured: the C5-budget-override walk passed its self-gate (the
standard-multiset override reproduces N_lin byte-identically) and gives N(M′) = 6.507×10³⁷
(95% CI [6.50, 6.51]×10³⁷) with wrap-d1 mass f₁(M′) = 0.175, so the exact decomposition yields
|C_circ| = 0.652·|C1–C5| + 0.175·6.507×10³⁷ ≈ 9.80×10³⁷ — about 0.74× the linear space
(using the fresh run's f₃ = 0.6518 instead of the published 0.652 changes nothing at 3
significant figures). This resolves the one report-only R-series observable registered in v1.9;
per the §6 non-promotion decision it is measurement and theorem, not constraint — neither the
circular reading (McKenna's frame) nor the anchor rule (Cook's observation) enters the formal
system.
Attribution: circular frame McKenna & McKenna (1975); the final-pair anchor rule Cook (2006); the rigidity/eligibility theorems, the 10:3:3 classification, and the eligibility-adjusted re-pricing are ROAE (to our knowledge first stated here — the ingredients are elementary and may appear elsewhere; corrections welcome via CITATIONS.md).
- Anchor rigidity (T1) + seam eligibility (T2i) + 10:3:3 classification (T2ii): exhaustive finite
re-check in one Python session from
solve.py'sbinary_hexagrams. None of the three exists as a named theorem inlean/.anchor_cross_distance_three,no_even_pair_closesandcloser_classes_10_3_3are intended names for a planned formalization inlean/KingWen.lean; no theorem or lemma by any of those names exists inlean/on any branch. T2i's conclusion is nonetheless kernel-checked, though not as a standalone named theorem: it appears as an intermediate step inside the proof ofcircular_alternations_16inlean/KingWen.lean(pc6 (l.getD 62 0) % 2 = 1— the closing pair's class is odd — established for every C1+C4+C5 sequence fromwrap_parity_general(§2) andpartner_parity; that theorem's own conclusion is the 16-alternation count, not the seam parity). To cite T2i from Lean you must either cite the enclosing theorem or lift the step out into a named lemma. The pair-parity ingredientspartner_preserves_parityandparity_split_32_32are kernel-decidetheorems in the same file (TR-6). - Circular anchor adjacency R-C1c + A₂ slot histogram:
SOLVE_KNUTH_SCORE=1 ./solve --estimate-knuth 20000000000(KW gate: slot2 = 0, slot32 = 1, adjacent = 1; d5-witness negative control = 0; the run's slot-32 mass must reproduce R-C1 ≈ 7.84% — measured run:evidence/r6/rc1c_primary.out, adjacent = 0.130472). - Circular-space size:
SOLVE_KNUTH_C5_BUDGET="1:1,2:20,3:14,4:19,6:9" SOLVE_KNUTH_SCORE=1 ./solve --estimate-knuth 20000000000(self-gate: standard-budget override1:2,2:20,3:13,4:19,6:9reproduces N_lin — verified byte-identical,evidence/r6/budget_selfgate.out; M′ run:evidence/r6/mprime_walk.out).
| Version | Date | Changes |
|---|---|---|
| v1.0 | 2026-07-04 | First public release |
| v1.1 | 2026-07-04 | Plain-language executive summary added; internal drafting TODOs resolved (figures kept as planned improvements) |
| v1.8 | 2026-07-04 | 32-circular-switches corollary added (TR-6 30-switches × wrap-parity composition; derived in cross-report synthesis 2026-07-04, re-verified independently) |
| v1.9 | 2026-07-10 | "The anchors on the circle" section added: anchor-transition rigidity (T1) + seam eligibility (T2i) + pair-determined 10:3:3 wrap classification (T2ii) — elementary, exhaustively finite-verified (Lean formalization planned); Cook's final-pair anchor re-priced against the parity-forced 1/16 eligibility baseline (apparent ×2.4 = ×1.9 forced · ×1.25 contingent). Circular anchor-adjacency population frequency (R-C1c) and circular-space size |
| v2.0 | 2026-07-10 | R-C1c and |C_circ| measured (evidence reports/evidence/r6/): circular anchor adjacency = 13.05% of C1–C5 mass (slot 32 = 7.85%, reproducing the R-C1 gate; slot 2 = 5.20%, the largest non-final slot — U-shaped A₂ histogram), vs pre-registered references 6.45% uniform-slots / 7.84% eligibility lower bound; |C_circ| = 0.652·N_lin + 0.175·6.507×10³⁷ ≈ 9.80×10³⁷ ≈ 0.74× the linear space. Report-only; no promotion. |
| v2.1 | 2026-07-20 | Conditional-forcing correction (adversarial-review F-14a). §3's "forced by the constraint system, an artifact of no design choice at all" restated as forced given C4 + C5 — and therefore not an independent design choice within that system — with the added note that C5 is itself a regularity read off King Wen, so "forced" is relative to KW-derived constraints rather than to an unconstrained arranger. The prior phrasing smuggled the KW-derived constraints in as premise. No measurement changed |
| v2.2 (current) | 2026-07-26 | Wrap-mass uncertainty stated from archived artifacts (round-2 audit, completeness loop 4e G2). The published 17.5/65.2/17.4% masses always cited "CIs per METHODS" without printing them; the instrument in fact emits point masses without per-class CIs, so no ± existed to print. The abstract and §5 now state the published uncertainty as the two-run agreement: the independent v2.0 r6 rerun (evidence/r6/rc1c_primary.out, 2×10¹⁰ probes) re-measured 17.45/65.18/17.37% — within 0.05 pp per class of the published figures. Per-class bootstrap CIs would need a recompute and are left as an open improvement. No mass value changed |