FK-0 … FK-2 · scope, definitions, grammars
FK-0 scope. This card specifies a fold-story: a finite descending ladder of levels n = N…0, each carrying base, value, and rail. It defines two lawful grammars, one lawful seam, one lawful floor, and a classification engine whose duty includes refusal.
FK-1 definitions. fold: one level transition. rail: accumulated damage ∈ [0,1]. value: level capacity. currency: the unit counted (pair = 2 fermions, single = 1). seam: exactly one anomalous fold separating two conforming runs. floor: terminal level, d≈0, holds the escaping pair.
FK-2 the two grammars. ADDITIVE: C(n) = C(n−1) + 2c — the ×2 rides the new cost; total is linear; grammar of carving, where each pass strikes the original. MULTIPLICATIVE: C(n) = 2ᵈ·C(n−1) — the ×2ᵈ rides the total; grammar of compounding, where each fold eats the last fold's output. d is the carve dimension: edge 1, surface 2, volume 3. One operator placement separates the two universes.
FK-3 · the completed table
| level | base 2ⁿ | value | rail 0.2n | currency |
|---|
| 5 | 32 | 64 = 2·2⁵ | 1.00 | pairs |
| 4 | 16 | 32 = 2·2⁴ | 0.80 | pairs |
| 3 | 8 | 16 = 2·2³ | 0.60 | pairs · seam below |
| 2 | 4 | 4 = 2² | 0.40 | singles |
| 1 | 2 | 2 = 2¹ | 0.20 | singles |
| 0 | 1 = 1² | 2 | 0.00 | boson floor · pair escapes |
FK-3 normative. value(n) = 2·2ⁿ for n ≥ 3 (pair currency); 2ⁿ for n ∈ {1,2} (single currency); 2 at the floor. rail(n) = 0.2n exactly (additive, c = 0.1). base(0) = 1² is the pair-as-one: two fermions bound as a single boson. 8 never appears — the missing 8 is the seam signature, and the engine below finds it blind as one anomalous ×4 fold at exactly that position.
FK-4/FK-5 · seam & floor rules · FK-6 · dimension
FK-4 floor rule. A terminal run of d≈0 folds is the boson floor, stripped before grammar analysis and reported as a flag, never as noise.
FK-5 seam rule. A seam is exactly one anomalous fold whose removal leaves all remaining folds within one constant d (tol 0.12). Two or more anomalies is not a seamed ladder; it is lawless, and lawless is CAUGHT.
FK-6 dimension extension. The ×2 of the base spec is the d=1 case. Surfaces fold ×4, volumes ×8, and composite objects fold ×2ᵈ for effective d (measured this session: our own F₃ rank bench folds at d_eff = 9 — cost ×512 per size doubling — which is why L=8 was its ceiling).
conductor RUNNING
FK-7 · conformance QUEUED
FK-8 · blind classification QUEUED
2D — grammar separation
top (log scale): multiplicative ladders are straight lines, slope = −d — chain d=1, octree d=3, bench d=9; the ROOT0 value column is straight except one kink: the seam, visible to the naked eye at the missing 8. bottom (linear): the additive rail is the straight line here instead — each grammar is linear in exactly one chart, and never both.
toddler corner
ELI5: the kernel is a stack of shrinking boxes. the top boxes count toys in PAIRS (that's why they look twice as big as their shelf), the bottom boxes count toys one at a time, and there's exactly one weird jump in the middle where the counting rule changed — a box labeled 8 that was never built. the scratch-marks on the side (the rail) grow by the same amount every box: scratches don't breed. and the machine on this card is a sorting hat for ANY stack of boxes: it says "pair stack," "single stack," "stack with one rule-change," or — most important — "this stack is lying, I refuse." the refusing is the job.
physics decode & provenance
AMBER · the decode (training-memory anchors, declared). Pair currency ↔ 2e transport and the superconducting even-odd parity effect (islands refuse singles, staircase 2e-periodic). Seam ↔ the pairing transition: above the gap count quasiparticles singly, below it the condensate counts in twos. Floor ↔ pair-as-boson. Σ=0 (the ROOT0 tripod rule) is the binding condition — the singlet, not a fourth member. Each mapping is an interpretation ratified by the spec author, not a derivation.
LIT · engine verification. Seven conformance vectors classified correctly in python (foldkernel2.py) and re-classified live on this card. Two engine bugs were caught by its own probes during construction (garbage passing as seamed; float equality 0.2·3 ≠ 0.6) and fixed on record — the card ate its own dogfood.