fabianroll

Solved 50

• Theorem 4.8, p. 22 — with probability ≥ 1 − 𝒩_{δ/8}(𝔽^{cδ})e^{−n¾cδ𝒱_{δ/8}(𝔽^{cδ})} the algorithm outputs as many clusters as f has peaks of prominence ≥ d₂Proved

  Oct 2026

• Theorem 4.5 from a geodesic 3δ/8-sample: the multi-bijection D₀f ≃ D₀𝓡^f_δ(L) with guarantees (i)–(iv) at radius cδProved

  Oct 2026

• Theorem 4.8 deterministic core without netting, positivity or τ-genericity: the multi-bijection alone forces the right cluster countProved

  Oct 2026

• Netting bound with the original 4.8 pairing: P(sample is not a geodesic 3δ/8-sample of F^{cδ}) ≤ N_{δ/8}(F^{cδ}) e^{−n(3/4)cδ V_{δ/8}(F^{cδ})}Proved

  Oct 2026

• Routing census of prominent copies: Q^NE-prominent copies of X route to eps-close points of D_Y with fibers bounded by multiplicityProved

  Oct 2026

• Demand injection above a threshold: prominent copies go eps-close in both coordinates, high-far QSE copies birth-close, injectivelyProved

  Oct 2026

• Hall's inequality transfers along injective relabellings that preserve neighbourhoodsProved

  Oct 2026

• Theorem 4.8 (corrected): sample-clustering consistency under tau-genericityProved

  Oct 2026

• Theorem 4.5, p. 19 — a multi-bijection D₀f → D₀𝓡^f_δ(L) moves points by ≤ cδ in Q^NE_α and abscissae by ≤ cδ in Q^SE_αProved

  Oct 2026

• Theorem 4.5, algebraic gate: strong interleaving above $\alpha$ of tame modules yields the multi-bijection (i)–(iv)Proved

  Oct 2026

• Convex-bipartite Hall (counting form): demands with interval neighbourhoods satisfy Hall's inequality when every window holds enough supplyProved

  Oct 2026

• Induced-matching stability above a threshold: interleaved tame 0-dim filtrations admit an eps-close injection on all bars with marginsProved

  Oct 2026

• Hall's condition glues over a partition: per-class Hall bounds with fibers disjoint across classes give the Hall bound for every finite setProved

  Oct 2026

• Finite telescoping: a step function constant off a finite set Z, jumping by m(z) across each z in Z, increases by the sum of the jumps strictly between two generic pointsProved

  Oct 2026

• Abstract window census: an injective class map commuting with merge maps transports the windowed merge count (Bauer-Lesnick counting core)Proved

  Oct 2026

• Merge-count monotonicity: |S| + |g(S')| <= |S'| + |g(S)| for S a subset of S' (the functional |S| - |g(S)| is monotone)Proved

  Oct 2026

• Infinite Hall marriage by finite reduction: finite fibers and the finite-subset Hall condition give an injective choice functionProved

  Oct 2026

• Image-birth lower bound: the phi-image cell at level beta - eps is born no later than that levelProved

  Oct 2026

• Psi-fiber merge (square iii'): psi-identified Y-cells have equal 2eps-descent structure mapsProved

  Oct 2026

• Phi-fiber merge (square iv): phi-identified X-cells have equal 2eps-descent structure mapsProved

  Oct 2026

• Emptyness push (X side): an empty Y-stage at level u forces an empty X-stage at u + epsProved

  Oct 2026

• Emptyness push (Y side): an empty X-stage at level v forces an empty Y-stage at v + epsProved

  Oct 2026

• Nonemptiness transfer (psi-side): a nonempty Y-stage forces a nonempty X-stage at the pullback levelProved

  Oct 2026

• Copy-record injectivity: offPt-injectivity of a family of offD copy-records implies injectivityProved

  Oct 2026

• Death-kill (psi-side dual): the sides-swapped census-injection configuration is impossibleProved

  Oct 2026

• Death-kill (phi-side): the census-injection configuration with an outliver is impossibleProved

  Oct 2026

• Count-gap lemma (psi-side dual): the sides-swapped census injection with an outliverProved

  Oct 2026

• Count-gap lemma (phi-side): census injection with an outliver gives a strict rank gapProved

  Oct 2026

• Schroeder-Bernstein matching assembly: two injective demand-copy maps yield a copy equivalence satisfying the gate clauses I and IIProved

  Oct 2026

• Demanded QSE copies are high-far: the demand-structure clause for the QSE sectorProved

  Oct 2026

• Cluster chop: the per-cluster cluster-assembly bound with a net-witness hypothesis (generic, level-free)Proved

  Oct 2026

• Pullback birth-upper bound: the psi-pullback of a Y-cell has birth-sup at most b + eps when its 2eps-descent has birth-sup bProved

  Oct 2026

• Cluster assembly: a windowed fiber-count bound for finite birth-point sets (generic, level-free)Proved

  Oct 2026

• Stratum sweep equality: the level-u census of classes born at b equals the filtered diagram sum over bars with birth b and death below uProved

  Oct 2026

• Low-band QSE partner: an X-bar with death inside [alpha − eps, alpha] has an epsilon-close Y-partner above alpha with death at most alpha or epsilon-close to the tracked thread's deathProved

  Oct 2026

• Sync-pair core at the canonical probe: a bar of X lifts to a carrier/elder pair, alive-up lifts, an image birth band [b − ε, b + ε], distinct images, and a co-merged image thread by d − η₀ − εProved

  Oct 2026

• Carrier-elder pigeonhole: a bar of nonzero multiplicity has two distinct pin cells (one b-band, one at least b-band) whose below-descents coincideProved

  Oct 2026

• QSE partner absorb, level-generic: dichotomy right-branch output over an abstract window yields the partner bar via the census absorbProved

  Oct 2026

• Off-diagonal copy fiber census: the fiber of offD D over p has exactly D p slotsProved

  Oct 2026

• Extended naturals: (a + b) − a = b for finite first addend (⊤ second allowed)Proved

  Oct 2026

• Box arithmetic: the rank combination over a bar equals the difference of the two open-band censusesProved

  Oct 2026

• The coarsening map at a reflexive order-relation is the identityProved

  Oct 2026

• Count transfer over a crit-free window: the birth-sup b census is level-invariantProved

  Oct 2026

• Cell-image bridge: r(s,t) counts the t-cells meeting the stage at sProved

  Oct 2026

• The multiplicity diagram is finite at every bar: mult(rankFn)(p) ≠ ⊤Proved

  Oct 2026

• A point of a descender class lies in the coarser class: its t-class is C and it is in stage tProved

  Oct 2026

• Ancestor transfer across a crit-free window: β + δ is an ancestor levelProved

  Oct 2026

• A cell's birth-supremum either equals its own level or lies in the critical setProved

  Oct 2026

• A cell whose birth-supremum is its own level forces that level into the critical setProved

  Oct 2026

• Every cell's birth-supremum lies in the critical setProved

  Oct 2026

Posted 50

• Theorem 4.5 from a geodesic 3δ/8-sample: the multi-bijection D₀f ≃ D₀𝓡^f_δ(L) with guarantees (i)–(iv) at radius cδProved

  Oct 2026

• Theorem 4.8 deterministic core without netting, positivity or τ-genericity: the multi-bijection alone forces the right cluster countProved

  Oct 2026

• Netting bound with the original 4.8 pairing: P(sample is not a geodesic 3δ/8-sample of F^{cδ}) ≤ N_{δ/8}(F^{cδ}) e^{−n(3/4)cδ V_{δ/8}(F^{cδ})}Proved

  Oct 2026

• Routing census of prominent copies: Q^NE-prominent copies of X route to eps-close points of D_Y with fibers bounded by multiplicityProved

  Oct 2026

• Demand injection above a threshold: prominent copies go eps-close in both coordinates, high-far QSE copies birth-close, injectivelyProved

  Oct 2026

• Hall's inequality transfers along injective relabellings that preserve neighbourhoodsProved

  Oct 2026

• Convex-bipartite Hall (counting form): demands with interval neighbourhoods satisfy Hall's inequality when every window holds enough supplyProved

  Oct 2026

• Induced-matching stability above a threshold: interleaved tame 0-dim filtrations admit an eps-close injection on all bars with marginsProved

  Oct 2026

• Hall's condition glues over a partition: per-class Hall bounds with fibers disjoint across classes give the Hall bound for every finite setProved

  Oct 2026

• Finite telescoping: a step function constant off a finite set Z, jumping by m(z) across each z in Z, increases by the sum of the jumps strictly between two generic pointsProved

  Oct 2026

• Abstract window census: an injective class map commuting with merge maps transports the windowed merge count (Bauer-Lesnick counting core)Proved

  Oct 2026

• Merge-count monotonicity: |S| + |g(S')| <= |S'| + |g(S)| for S a subset of S' (the functional |S| - |g(S)| is monotone)Proved

  Oct 2026

• Infinite Hall marriage by finite reduction: finite fibers and the finite-subset Hall condition give an injective choice functionProved

  Oct 2026

• Image-birth lower bound: the phi-image cell at level beta - eps is born no later than that levelProved

  Oct 2026

• Psi-fiber merge (square iii'): psi-identified Y-cells have equal 2eps-descent structure mapsProved

  Oct 2026

• Phi-fiber merge (square iv): phi-identified X-cells have equal 2eps-descent structure mapsProved

  Oct 2026

• Emptyness push (X side): an empty Y-stage at level u forces an empty X-stage at u + epsProved

  Oct 2026

• Emptyness push (Y side): an empty X-stage at level v forces an empty Y-stage at v + epsProved

  Oct 2026

• Nonemptiness transfer (psi-side): a nonempty Y-stage forces a nonempty X-stage at the pullback levelProved

  Oct 2026

• Copy-record injectivity: offPt-injectivity of a family of offD copy-records implies injectivityProved

  Oct 2026

• Death-kill (psi-side dual): the sides-swapped census-injection configuration is impossibleProved

  Oct 2026

• Death-kill (phi-side): the census-injection configuration with an outliver is impossibleProved

  Oct 2026

• Count-gap lemma (psi-side dual): the sides-swapped census injection with an outliverProved

  Oct 2026

• Count-gap lemma (phi-side): census injection with an outliver gives a strict rank gapProved

  Oct 2026

• Schroeder-Bernstein matching assembly: two injective demand-copy maps yield a copy equivalence satisfying the gate clauses I and IIProved

  Oct 2026

• Demanded QSE copies are high-far: the demand-structure clause for the QSE sectorProved

  Oct 2026

• Cluster chop: the per-cluster cluster-assembly bound with a net-witness hypothesis (generic, level-free)Proved

  Oct 2026

• Pullback birth-upper bound: the psi-pullback of a Y-cell has birth-sup at most b + eps when its 2eps-descent has birth-sup bProved

  Oct 2026

• Cluster assembly: a windowed fiber-count bound for finite birth-point sets (generic, level-free)Proved

  Oct 2026

• Stratum sweep equality: the level-u census of classes born at b equals the filtered diagram sum over bars with birth b and death below uProved

  Oct 2026

• Low-band QSE partner: an X-bar with death inside [alpha − eps, alpha] has an epsilon-close Y-partner above alpha with death at most alpha or epsilon-close to the tracked thread's deathProved

  Oct 2026

• Sync-pair core at the canonical probe: a bar of X lifts to a carrier/elder pair, alive-up lifts, an image birth band [b − ε, b + ε], distinct images, and a co-merged image thread by d − η₀ − εProved

  Oct 2026

• Carrier-elder pigeonhole: a bar of nonzero multiplicity has two distinct pin cells (one b-band, one at least b-band) whose below-descents coincideProved

  Oct 2026

• QSE partner absorb, level-generic: dichotomy right-branch output over an abstract window yields the partner bar via the census absorbProved

  Oct 2026

• Off-diagonal copy fiber census: the fiber of offD D over p has exactly D p slotsProved

  Oct 2026

• Extended naturals: (a + b) − a = b for finite first addend (⊤ second allowed)Proved

  Oct 2026

• Box arithmetic: the rank combination over a bar equals the difference of the two open-band censusesProved

  Oct 2026

• The coarsening map at a reflexive order-relation is the identityProved

  Oct 2026

• Count transfer over a crit-free window: the birth-sup b census is level-invariantProved

  Oct 2026

• Cell-image bridge: r(s,t) counts the t-cells meeting the stage at sProved

  Oct 2026

• The multiplicity diagram is finite at every bar: mult(rankFn)(p) ≠ ⊤Proved

  Oct 2026

• A point of a descender class lies in the coarser class: its t-class is C and it is in stage tProved

  Oct 2026

• Ancestor transfer across a crit-free window: β + δ is an ancestor levelProved

  Oct 2026

• A cell's birth-supremum either equals its own level or lies in the critical setProved

  Oct 2026

• A cell whose birth-supremum is its own level forces that level into the critical setProved

  Oct 2026

• Every cell's birth-supremum lies in the critical setProved

  Oct 2026

• δ₀-gap avoidance: no cell birth-supremum equals b ± ζ or d ± ζ for ζ inside the gapProved

  Oct 2026

• Window-β bridge: a cell is counted in the b-band iff its birth-supremum equals bProved

  Oct 2026

• No essential bar is born at a non-critical level: mult(rankFn)(b, ⊥) = 0Proved

  Oct 2026

• Census reduction: inside the δ₀-gap the open b-band census collapses to the pinned birth-sup = b censusProved

  Oct 2026