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Three-label first-window boundary compatibility

Abstract

The closed three-label compatibility relation includes every safe base-internal boundary construction.

Theorem 1.1 (Normalized inverse codes on every original inherited tail).

Proof. Machine-checked in Lean as D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/FirstWindowBoundaryCompatibility.base_script_code (✓ std3). ∎

Source. Repository-derived.

Commentary.

For any internal set S and BaseAssignment a, the two displayed even-row equations and seam disjunction make the actual common baseWords prefix safe on every original value, phase and inherited tail. Its acquired consecutive endpoint differences are exactly the two coordinates of baseCode. The successor begins zero, is broken by its index-one zero after a leading one, or enters from a zero-ending root. This statement supplies the literal realization used by both three-label and four-label normalized constructions.

Theorem 1.2 (The complete Pi3 relation and base-internal physical table).

Proof. Machine-checked in Lean as D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/FirstWindowBoundaryCompatibility.original_first_window_three_label_compatibility (✓ std3). ∎

Source. Repository-derived.

Commentary.

Fix m>=5, k=2m-2 and P=ZMod(2m-1). H(0),H(1) lie in B={1,…,m}; their ordered membership pair is ell. Its full image has three labels. Two-block original GLOBAL feasibility is equivalent to Pi3: the internal image on U={2,…,m-1} has at most two labels, and m is odd or one of the two boundaries is nonbase. For the base-internal clause, the internal image is {0,X}, with X nonzero. The third label Z must occur at one or both boundaries. Thus all six rows of table 29.19 apply, without an additional parity or boundary compatibility condition. The target has arbitrary labels A(v)!=B(v) separately, arbitrary cross-value coincidences, arbitrary Type universes and independently observed f(none). Every original value, phase, inherited tail and unknown legal or rejected history is retained.

Necessity directly uses the same-stream row theorem: any correct two-block competitor has even rows, its second row is zero on U, and its row pairs refine both original support coordinates. Hence r<=2. If both boundary labels are base, the full image is exactly {0} union the internal image. With n=3 and r<=2 this forces r=2 and excludes base internally. The already established nonbase-internal obstruction then forces odd m. This comparison retains rejection and early stopping; scheduled rows on stopped sources supply no acquired endpoint or fee.

For sufficiency, U is nonempty because it contains phase two. The bound r<=2 therefore gives r=1 or r=2. The homogeneous construction handles r=1. If r=2 and base is absent internally, the nonbase-internal theorem gives a safe common stream exactly under the remaining Pi3 clause. If base is present internally, the following complete table supplies the actual stream. Thus all three-label internal and boundary arrangements are included, while the explicit base-internal clause records its own selected rows, code refinement and archive counts.

Let S={h in U: ell(h)=X}, q=|S|, t=q modulo two and e=m modulo two. classIndex(X,z) is 0 for base, 1 for X and 2 for the third label Z. In the table below, Z is the label called Y in 29.19. At least one boundary index is 2. BaseAssignment records flip, exterior, donor and the left and right two-bit boundary codes. baseChoice(e,t,x,y) selects the following exact supports; u is the Z boundary when the boundaries differ and h=m+1 is an actual exterior base-labelled phase.

For boundaries Z,Z, E0=S union {1} when q is odd and E0=S when q is even; E1={1,m}. For boundaries Z,X in either order with q odd, E0=S union {1} and E1={1,m}. With q even, E0=S union {1,m} and E1={u,h}. For boundaries Z,0 in either order with q odd, E0=S union {u} and E1={u,h}. With q even and m odd, E0=(U without S) union {u} and E1=R union {1,m}. With q even and m even, E0=(U without S) union {1,m} and E1=R union {u}. These are all six rows and both orientations of each mixed-boundary row.

The root row reuses nonbaseRoot and the second row uses offsets from m in the ordered path W1. In the following two formulas, left,right,flip,exterior,donor are the fields of a. supportRow(E,h) is one on E and zero off E; all additions are in ZMod(2). Offset one of the second path is physical phase m+1; offset m wraps to phase one, and offset zero is physical phase m.

baseWords is exactly the prefix-parity inverse of the root row followed by the prefix-parity inverse of baseSecond. The displayed two parity equations make each complete ordered row even. The first equation counts S and U once, while |U|=m-2 has the same parity as m. The second counts all m-1 exterior phases once plus the donor correction and the two boundaries. This is phase multiplicity, with no repetition from tails or history lengths.

Internally the codes are (flip,0) for base and (1+flip,0) for X. Exterior phases have (0,exterior), except m+1 which has (0,exterior+donor). At phase one and m the codes are left and right. Equal codes always imply equal membership labels. Repeated labels may occupy distinct codes: in the odd-q Z,X row, when Z is at phase one, its code is 11 and X uses both 10 internally and 01 at phase m. When Z is at phase m, Z has 01 while X uses 10 and 11. In both orientations the supports remain S union {1} and {1,m}, and all unequal labels remain separated. Four-label saturation is not imposed on these three-label assignments.

Every root begins zero and clears any legal inherited tail. Its terminal tail is at most m-1. The selected boundary codes satisfy one of three physical seam conditions: the second word begins zero; its first bit is one and the actual donor toggles the next bit to zero; or the root ends zero. In the first case the incoming tail clears immediately. In the second, the seam run is at most m<k, and the zero at literal index one prevents extension. In the last case the second word enters from actual tail zero; its length m<k makes even an all-one word safe. Thus the entire prefix succeeds on every original source, including both seams and every inherited tail.

The actual chronological endpoint differences are precisely the two coordinates of baseCode. Each source decodes those differences from its own acquired archive and uses its remembered free value to select the required membership component and A(v) or B(v). The same preselected literal stream serves both free values and both original alphabets. Initial bottom stops freely. The live script archive has exactly two issued blocks and 2m bits; this count does not itself assert an optimal PaidTrace price.

Ordered membership and the first-row definition use D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/FirstWindowBoundaryCodes. Phase shifts and support sums use D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/FirstWindowBoundaryAttainment. The inverse and clear-seam executions use D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/WindowChargeInverse; the marked-zero inherited-tail execution uses D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/InternalZeroSafety. Original history and common-stream feasibility use D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/DonorCorrection.

This result gives the complete three-label two-block compatibility equivalence and the base-internal physical table. It does not assert the least price independently of the zero/one-block lower bounds. Four-label phase pivots and their equivalent boundary table, degenerate membership normalization, full sharp preset prices and unequal adaptive PaidTrace prices are separate statements. No claim that every even refining row pair has a safe concatenation follows from this selected construction.

References