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• A structural-gap family of order X / sqrt(log X)
• Additional filters on the parity-cubic `3r` index-six defect
• Almost all prime depth values are structural gaps
• Asymptotic skeleton of Type A/B ancestry shadows
• Automated WS-CAND-003 results — 2026-08-14
• bb.kernel
• Binary defects on the full external-nonresidue factor cycle
• Binary-prime completed layers collapse to one-dimensional power intervals
• Binary-r rescue after one exact greedy denominator
• Binary-r rescue as a signed divisor collision
• C1 conditioned-fiber collapse on the `{3,11,13}` residual family
• C1 Escape Core Theorems
• C1 Partial Theorems Toward DSC-P — corrected
• C1 Pullback Cardinality Bound
• C1 Theorem — Single-Active Pullback Escape
• C1: single-active Class-C attack
• C2 Theorem — Two-Active-Layer Escape
• Candidate-independent trap-fiber bounds
• CC.kernel
• Character-shield obstruction completeness
• Class-C active-core census through k = 1500
• Class-C residual core: coordinated primary formulation
• Classification of Jacobi-saturated Type A/B layers
• Classification of universal square-lift signature shadowing
• CN Theorem — Finite Active-Core Escape
• CN-shared analyzer report
• CN-shared certificate — 2026-08-15
• CN-Shared Theorem — Lift-Room, Tight Clusters, and Admissible Escape
• Complete classification of the depth-1 Type A/B shadow component
• Complete classification of Type A/B multiplicative coset saturation
• Complete dichotomy for prime Type A/B minimal depths
• Composite successor `3r`: exact index-six defect classification
• Composite successor `3r`: index-four defect elimination
• Composite successor `3r`: parity-constrained signed-box theorem
• Consecutive binary-selector probe
• Constructive counterexample to universal Direct-Shadow Completeness
• Constructive nonuniversal square-lift certificate results
• Coprime fab certificates force an external hard-class nonresidue
• Coprime fab certificates reduce to one divisor congruence
• Counterexample to naive hit-on-cycle / least-entry conjectures
• Counting bounds for the Type A/B minimal-depth spectrum
• Counting the source-1 structural-gap semigroup
• Cryptology control lemmas for WS-CAND-003
• Cryptology research track for WS-CAND-003
• Cubic-log sieve bound from q=3,7,11,23 Type-II filters
• Cubic-surface form of the gcd-square rescue
• Current research frontier
• Dimension-two sieve from the exact q=3 and q=7 strong/Type-II filters
• Direct-Shadow Completeness attack through k = 1000
• Direct-Shadow Completeness attack through k = 1200
• Direct-Shadow Completeness attack through k = 1500
• Direct-Shadow Completeness counterexample — primitive q=3 triple cover
• Direct-Shadow Completeness: candidatewise finite attack
• Direct-shadow smoothness theorem
• Divisor-child ancestry theorem for Type A/B shadowing
• DSC-P Fragment — Coprime Active Cores
• Dual descent system behind sufficient fab certificates
• Dyachenko ED2 lattice repair audit
• Dyadic trap lattice automated regression results
• Dyadic trap-coset and shadow lattice
• Erdős-Straus Type A/B automated research harness
• Erdős-Straus — Wall
• Even-index FAB defects expose every quadratic-nonresidue factor
• Exact `2p+1` factor filter
• Exact `4p+1` factor filter
• Exact `p+4` factor rescue
• Exact `r=11` binary rescue classification for Mordell-hard primes
• Exact ancestry rigidity for quotients 21 and 29
• Exact anchor-row domination inside the C1 residual core
• Exact classification of multiplicative-ancestry shadows
• Exact fixed-k=7 filter for Mordell-hard primes
• Exact index-6 FAB Kneser defect classification
• Exact index-six normal form for the combined FAB targets
• Exact q=11 filter for Mordell-hard primes in the strong/Type-II corridor
• Exact q=23 filter for Mordell-hard primes in the strong/Type-II corridor
• Exact q=7 filter for Mordell-hard primes in the strong/Type-II corridor
• Exact r=7 binary rescue for Mordell-hard primes
• Exact reduced-parameter domain
• Exact survivor density and hazard for the Type A/B sieve
• Exact two-target divisor-square criterion
• Exact two-target filter at the composite shift `k=15`
• Exact two-target filter at the prime shift `k=19`
• Exact two-target signed-box equivalence for prime Erdős--Straus
• Exponent-lattice formulation of square-completed ancestry shadows
• External-nonresidue factor cycle
• External-nonresidue m=1 character synchronization
• External-nonresidue shield certificates carry a synchronized nonresidue triple
• External-nonresidue shield-ratio census through `p <= 10^7`
• External-nonresidue × hard-shield fixed-k probe
• Fiber shadow kernel: a sharper local elimination theorem
• Findings index
• Findings library
• Finite shift corridor for the classical strong/Type-II conjecture
• Finite square-completion census through 50,000,000
• First exact shifted-factor filters for Mordell-hard primes
• Five-halves sieve bound from the exact q=3, q=7, and q=11 Type-II filters
• Fixed-k fab certificates are signed divisor products
• Fixed-k fab criterion collapses to one divisor of N²
• Full fiber-kernel and bounded-selector replay through k = 1200
• Full local quadratic signature quotient of Type A/B traps
• Full quadratic-signature coset for Type A/B traps
• Full quadratic-signature shield through k = 1200
• Full-stabilizer compression of FAB Kneser defects
• GCD-square reformulation of coprime fab rescue
• General odd-prime-shift ancestry rigidity — corrected scope
• Genus-theoretic identification of the square-lift defect quotient
• Hard-class union shadows and the five-of-nine C1 3-adic fiber theorem
• How grades are decided
• Hunt seeds
• Infinite exact-depth structural gap families
• Infinite squarefree factor-lift families at every ancestry quotient
• Kneser defect theorem for external binary-r rescue
• Kneser defect theorem for the fixed-k FAB divisor box
• Large-prime defect forcing by CRT starvation
• Mersenne shadow lattice: an infinite exact family of impossible Type A/B minimal depths
• Mersenne-prime backbone inside the Type A/B depth spectrum
• Mirror character obstruction for fixed-k fab rescue
• Mixed-box obstruction in exact square-lift shadowing
• Multiplicative reciprocity defect quotient for Type A/B square lifts
• Multiplicative semigroup shadow families from saturated Type A/B sources
• Multiplicative trap quotient
• Multiplicative Type A/B trap coset
• Multiplicative-coset and Jacobi shields survive square completion
• Normalized Type-II signed-divisor target
• Odd-prime-shift ancestry rigidity above the small-ancestor window
• Prime-child ancestry shadows
• Prime-child shadow automated regression results
• Prime-depth backbone projection
• Prime-extension shadow family for square-completed Type-II layers
• Prime-index hierarchy for FAB Kneser defects
• Prime-index spectrum of the square-completed Type-II layers
• Prime-modulus backbone and unbounded depth for the square-completed Type-II layers
• Prime-modulus backbone of the Type A/B depth spectrum
• Prime-power shadow kernel
• Prime-power stabilizer dichotomy: only powers of two are periodic
• Prime-power Type A/B trap dichotomy
• Prime-shift Kneser dichotomy for the classical strong/Type-II corridor
• Primitive sextic defect chains and the neighbor-square recurrence
• Prior-art audit — recent claimed Erdős–Straus proofs
• Prior-art calibration for the square-completed Type-II route
• Prior-art matrix for WS-CAND-003
• Proof ledger
• Proper Jacobi ancestor theorem for higher Type A/B signature codimension
• Provenance correction: the completed square layer is the classical strong/Type-II route
• Public Erdős–Straus hunt
• Pure-dyadic first denominators are universally obstructed on the hard-prime lane
• q=3 Fiber Injectivity and Three-Layer Minimum
• q=3 Next-Digit Normal Form
• q=3 Pointwise Absorption
• q=3 Pullbacks Are Singleton
• q=3 Strong Absorption
• q=3 Weak-Ancestor Redundancy
• Quadratic cycle core for the Type A/B character shield
• Quadratic trap signature and character shield
• Quadratic-character prior-art note
• Quadratic-field bridge for the square-lift core
• Quantitative counting in the Type A/B exact-depth spectrum
• Quantitative infinite-coinfinite spectrum for square-completed Type-II depth
• Quotient-13 shadow rigidity
• Quotient-17 shadow rigidity
• Quotient-21 shadow rigidity
• Quotient-29 shadow rigidity
• Quotient-9 shadow rigidity
• Reciprocity defect quotient for square-lift signature shadows
• Reciprocity tower automated regression results
• Reciprocity tower shadows
• Research proportions — Type A/B program
• Safe-prime shifts q = 23 mod 24 carry a forced nine-class Type-II obstruction
• Secret Oasis expedition — 2026-08-14
• Shadow semigroups from the three Jacobi-saturated bases
• Shadow-cover geometry behind Direct-Shadow Completeness
• Shifted nonresidue transfer and the `p+4` mirror filter
• Shifted-factor descent for the post-DSC Erdős–Straus wall
• Single-active Class-A local escape
• Single-active first-shell collapse — retracted proof attempt
• Single-active hard-class quotient collapse through k = 100,000
• Small-selector hypothesis for residual Type A/B fiber kernels
• Square-completed López layer equals a symmetric signed divisor box
• Square-completed trap complement: global inversion and exact mixed-parameter count
• Square-completed Type-II trap geometry: López A/B are the two boundary orthants
• Square-lift core after quadratic character saturation
• Square-lift quadratic-signature shadow theorem
• Square-lift reciprocity and infinite ancestor-shadow families
• Square-lift towers in the character residual core
• Squarefree factor-lift shadows and an infinite quotient-nine family
• Squarefree semiprime layers always contain genuinely new mixed Type-II residues
• Squarefree-lift localization of the Type A/B character residual
• Stabilizer-extension shadow theorem for square-completed Type-II layers
• Standard Type II as the square-divisor completion of López Type A
• The composite rescue core
• The Type A/B depth-spectrum diamond
• The Type A/B minimal-depth spectrum
• The Type A/B minimal-depth spectrum is infinite and co-infinite
• The Type A/B reciprocity matrix
• Two-divisibility constructor for the fab master equation
• Two-target Kneser collapse for external nonresidue shifts
• Type A/B shadow ancestry: formal working lemmas
• Type A/B shadow pullbacks as a structured odd-covering problem
• Type A/B trap quotient factorization
• Type I cannot rescue a hard-prime `q=7` Type-II miss
• Type-I companion to the exact `q=11` filter
• Type-II root geometry: López A/B are the comparable-root cases
• Unbounded defect forcing from prescribed external nonresidue load
• Unbounded strong sufficient certificates and the fixed-k divisor-ratio box
• Unique-active valuation excess is a first prime-power lift
• Universal reduced escape for a unique active fixed-negative row
• Universal square-lift shadow classification
• WS-CAND-001 — Authority cannot confer truth
• WS-CAND-002 — Justified outward decimal rendering
• WS-CAND-003 cryptology results — 2026-08-14
• WS-CAND-003: Erdős-Straus Type A/B witness depth and congruence shadow structure
• Zero-product atom decomposition of square-lift multiplicative defects
• {2,3,5,7}-smooth forced factors cannot produce a Type-II hit