Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Stopped residual synthesis

Abstract

One actual common stopped catalog law synthesizes the complete closed residual cap.

Definition 1.1 (Ordered three-subsets).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.triples

Formalization. D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.triples (✓ std3).

Source. Repository-derived.

Commentary.

Numerical addresses are zero through m-1; m is the base sentinel, ordered after every numerical address. triples m is the existing powersetCard 3 of range(m+1). A sorted subset determines its first two actual query addresses and its third terminal label.

Definition 1.2 (Actual chronological table).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.subsetTable

Formalization. D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.subsetTable (✓ std3).

Source. Repository-derived.

Commentary.

For every m at most L and every selected three-subset, subsetTable queries the first two numerical labels increasingly until the first false reply and emits that label, or emits the third selected label without querying it. Numerical address i emits label i+1; sentinel m emits zero. The original StoppedCoefficientRealization.table supplies actual Catalog true L membership and its existential chronological schedule. An invalid subset receives the base table and has zero probability under subsetLaw.

Definition 1.3 (One law before every word).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.subsetLaw

Formalization. D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.subsetLaw (✓ std3).

Source. Repository-derived.

Commentary.

For natural 3 at most m at most L, subsetLaw draws uniformly from triples m and maps the entire table into the original Catalog true L. The PMF is source independent and precedes the universal Boolean-word quantifier. It is not a collection of wordwise marginal choices.

Definition 1.4 (Positive numerical labels).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.positives

Formalization. D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.positives (✓ std3).

Source. Repository-derived.

Commentary.

positives m b is the set of numerical addresses less than m whose actual lowReply is true. Its cardinal is the original k_m(b). The base sentinel is excluded.

Theorem 1.5 (Simultaneous stopped subset identities).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.stopped_subset_law

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

Source. Repository-derived.

Commentary.

For all 3 at most m at most L, the actual subsetLaw simultaneously has the following properties at every Boolean word b. If n<m is its first false numerical address, mass at label n+1 is exactly 3/(m+1). Base mass is choose(card(positives m b),2)/choose(m+1,3). Every output label greater than m has zero mass. When all first m numerical replies are true the base mass is exactly 3/(m+1). The chronological proof establishes that output equals the first absent label if and only if that address was selected: all earlier selected coordinates are positive, including when the first absent address is the unqueried third selected label. Every later zero pattern is retained. Base emission holds exactly when the sentinel was selected and both selected numerical replies were true. Pinned powerset-cardinality and choose identities supply the subsequent counting; counting alone does not establish the chronological relation.

Definition 1.6 (Truncated residual demand).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.demand

Formalization. D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.demand (✓ std3).

Source. Repository-derived.

Commentary.

demand a epsilon k is max(a(k+2)-epsilon,0). Zero-based a(k) is original a_(k+1). The original strict active cutoff i<L if and only if epsilon<a(i) gives demand a epsilon (L-2)=0 and identifies its finite sum with the existing residual a epsilon, original D_3. No summability assumption on a is used.

Definition 1.7 (Common component weights).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.weight

Formalization. D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.weight (✓ std3).

Source. Repository-derived.

Commentary.

weight a epsilon k is (k+4)/3 times the difference between consecutive residual demands. Original component length is m=k+3. Antitonicity makes each weight nonnegative. The sum is a(2)-epsilon+D_3/3, and the sum of each weight times 3/(k+4) is a(2)-epsilon. These are applications of the pinned finite telescoping identities.

Theorem 1.8 (Closed residual synthesis).

Lean statement: D5/S3/ObserverMemory/Algorithms/KBonacciAcquisition/StoppedResidualSynthesis.stopped_residual_synthesis

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

Source. Repository-derived.

Commentary.

For every L at least three, every antitone real amplitude sequence with a(0) at most one, every positive epsilon, the original strict cutoff at L, and residual a epsilon at most 3 epsilon, there is one PMF w over actual Catalog true L tables. For every word whose firstFalse n is less than L, the mass of label n+1 is exactly a(n)-epsilon and the combined mass at zero and n+1 is at least 1-D_3/3. At the all-positive low word the base mass is exactly 1-D_3/3. The same w handles all words, including multiple-zero suffixes. The prefix branch has remaining mass R=1-W. Put t0=a(0)-a(2) and t1=a(1)-a(2). The three fixed prefix tables have weights t1, t0-t1 and R-t0 and terminal vectors (1,2,0), (1,0,0) and (0,0,0). These are nonnegative because 0<=t1<=t0<=R. They encode the two shared threshold choices without dividing by R. The literal finite mixture first chooses a component and then a whole subset table. The new chronological inclusion and support relations are indispensable to its all-word active mass and pair bound. R=0, D_3=3 epsilon, plateaus, zero increments and zero component masses all remain in the theorem.

For the complete original application assume 1>=a_1>=a_2>=…>0, convergence to zero, and 0<epsilon<=1. The supplied cutoff is strict, so a_(L+1)=epsilon is inactive. For L=0,1,2 the existing stopped terminal coupling realizes the active targets exactly and emits base on positive exhaustion; L=0 emits base immediately. For L>=3 the displayed active identities give base excess epsilon-delta and outside-pair mass delta with 0<=delta<=D_3/3<=epsilon. Thus active TV is exactly epsilon. The positive-row identity and the existing full-tail theorem give exact actual-tail error max(D_3/3,a_n); for small L it is a_n. All original later amplitudes are retained.

Original source x=(r,g) has prime r, g between ceil(F_r/10) and floor(F_r/5), and N(x)=1+F_r g. Query i tests prime p_(i+1) on this same static source. The full disjoint B/P interval Tape carrier, the marks theta_j=(B_1,1/(j+2)), the original nonatomic targets and unchanged consumer are retained. Existing whole-table Ico sampling chooses one table measurably from one source-independent uniform seed before any reply. OriginalScheduledRealization executes its chosen increasing schedule with at most two actual paid requests and finite same-source handoff for every seed and source. The executable uniform-seed realization uses the existing catalog representative, which can choose a different representing schedule for a table with more than one representation. It is a faithful alternative with exactly the constructed whole-table output law and the same hard-two stopped discipline, rather than an asserted identity of sampler or hidden schedule distributions. The sampled whole-table law is the constructed PMF; no marginal-equality argument asserts identity of a different sampler or schedule distribution. There is no source probability law, advice, bounded competing state/address restriction or infinite all-positive source assumption.

The existing catalog terminal-vector expansion and stopped coefficient iff supply the full original coefficient condition c>=H and the common scalar budget. Completing the chosen schedule with the original nonstopped interpreter retains its output table and same-source handoff. Its full acquired log has length exactly the chosen schedule length, at most two; every request added after the reconstructed first-zero stop is counted. The general original 110.14 problem and supplier audit issues16802 and17028 remain unresolved. This application makes no a_1<1 equivalence, unrestricted class comparison, global search or literature originality claim. Classical monotone initial-uniform mixture algebra, reservoir sampling and scalar query-in-expectation results do not by themselves supply the actual stopped whole-table relation proved here.

References