Findings in the order they were added to FPRD Lab. Exact timestamps make the pace of research visible; thematic labels identify sustained lines of work.
This journal begins with the current public update stream. Earlier results remain available through Publications, Claims, and Theorems.
Version 7 generalizes the stack obstruction. For any finite observation and finite family of destructive probes, an observation-preserving preorder that makes every probe monotone can compare only configurations with identical complete probe trees. Scaffold port probes generate full labelled unfoldings, so every nonregular scaffold presentation has too many residual figures for such an order to be a WQO. A viable pumping theory must abstract navigation, order whole updates, or use paired opaque interfaces.
Version 6 proves that an unbounded exact stack cannot use the WQO mechanism from version 4. Any preorder that preserves the observable top, including emptiness, and makes pop monotone collapses to equality, so it is not a WQO. This explains why stack pumping needs the paired-retraction mechanism: a later wing must remove exactly the frames introduced by the earlier wing before the outer continuation resumes.
Version 5 isolates the operational algebra behind the two wings of context-free pumping. The first wing need not loop: it may advance an unbounded parameter, provided the second wing retracts that move around the middle computation at every point of the forward orbit. The resulting sandwich identity proves uv^jxy^jz equivalent to uxz for all j>=0. This supplies soundness, not recurrence, and checking only the first cancellation is insufficient.
Version 4 connects FPRD pumping to well-structured transition systems. A finite-alphabet deterministic system with a well-quasi-ordered configuration space, monotone letter actions, and upward-closed acceptance has a uniform pumping length, but only for one or more copies of the repeated block. Deleting the block requires equivalence or a separate retraction. The remaining scaffold problem is to find an observationally stable extension order; ordinary graph inclusion fails.
Version 3 proves that unreachable scaffold records can never return and that a uniformly bounded minimal bisimulation quotient of the live records compiles to a finite automaton, with an explicit state and pumping bound. A language is regular exactly when it has some scaffold presentation with this property. Nonregular pumping must therefore repeat parameterized operations on an unbounded live quotient, as visible-stack contexts do.
A one-state, one-label, degree-two, distance-two scaffold builds every finite rectangular grid by alternating row direction. Square grids have unbounded treewidth, so local update bounds do not imply tree-like persistent memory. This is a structural calibration, not a language separation: the machine may accept every word. Any graph-context pumping theorem must constrain controller use or reachable interfaces, not locality alone.
Version 2 identifies the classical unifying mechanism behind regular, context-free, and visible-stack pumping: a repeated finite boundary type in a derivation tree yields an iterable one-hole context. For persistent visible stacks, paired wings build and discharge a frame above an opaque tail even though the allocated scaffold graph never repeats. This supplies a concrete nonregular positive case while leaving unrestricted PEG recurrence impossible.
FPRD-T175 sharpens the synchronized-stack boundary. If calls exist, a letter projection either preserves one action type in every visible-letter fiber, giving exact stack-height agreement, or compatible hidden lifts diverge by n after at most 2n observations for every n. Thus there is no projection-level hierarchy of finite skew bounds between T174 and forgotten-action projection. Any intermediate positive tier must use source-language or selector structure.
FPRD-T174 extends effective selector compactness from finite automata to deterministic visibly pushdown sources when projection preserves call, return, and local actions. All source and selected runs then have equal stack height and zip into one visibly pushdown product. Fixed-budget coverage, shortest missed words, and obstruction horizons are computable. The hard forgotten-action tier begins precisely when projection destroys this common stack clock.
FPRD-T173 makes the selector obstruction theory effective for regular source presentations. For an r-state hidden-source DFA and a family of m selectors with at most s states each, global coverage is equivalent to coverage through length r^(m+1)s^m-1. The same bound controls obstruction radius and gives a direct fixed-budget decision procedure.
FPRD-T172 calibrates the selector spectrum against established transducer theory. For regular source languages, finite selector covers are exactly domain-covering multi-sequential subrelations of the accepting-lift relation, up to finite state overhead. On nonregular domains the notions separate: a one-state identity selector exists for every language, while a standard sequential uniformizer must have a regular domain.
FPRD-T171 turns hidden-choice failure into an exact finite duality. For a fixed selector-count and state budget, accepted words define failure sets over the finite candidate-family universe. Infeasibility is exactly a set-cover condition, giving an explicit bound on certificate cardinality. The escape length from T169 is precisely the smallest possible maximum word length of such a certificate.
FPRD-T170 proves a chain rule for causal selection spectra. Selector profiles (m,s) and (n,t) for two successive hidden-coordinate maps compose to (mn,st), both globally and at every finite horizon. Therefore an empty composite spectrum forces an empty spectrum at some individual stage. Sequential transport compilation also retains the hierarchy's shared outer states and improves the resulting control bound.
FPRD-T169 defines the causal selection spectrum by selector count and selector-state size. Its finite-horizon spectra decrease exactly to the global spectrum: a fixed budget works at every finite horizon only if one finite selector family works globally. Thus the hard visible lift from T168 defeats every fixed budget on some finite input horizon, although the theorem gives no computable bound on that escape length.
FPRD-T167 shows that any fixed finite family of synchronous finite-state selectors preserves bounded stationary transport, with an explicit product bound. FPRD-T168 applies the contrapositive to the Kim-Park witness: its deterministic visible lift has a degree-one, distance-two scaffold, but after action tags are hidden its projection has empty transport spectrum. No finite family of online finite-state selectors can cover all required accepting lifts.
FPRD-T166 shows that the inverse image under an s-state synchronous Mealy transducer changes a transport tuple (d,k,g,q) only to (d,k,g,sq), while building exactly the same persistent graph. Finite-state synchronous coordinate changes with finite-state inverses therefore preserve bounded transport. This isolates why forgotten-action projection can fail: it existentially merges hidden histories rather than deterministically recoding the observed input.
FPRD-T91 identifies forgotten-action projections of deterministic visibly pushdown languages with CFL. Kim and Park's linear context-free language outside PEG therefore closes FPRD-C02 negatively. Applying the visible-lift theorem to both sides of their reversal pair separates two obstructions: C is PEG but not SCA and is the stationary-transport-hard frontier, while C^R is SCA but not PEG and is the direct counterexample to the universal PEG claim.
Kim and Park's SCA-to-cell-probe transfer gives an encoding-robust obstruction to every finite scaffold presentation. In FPRD terms, a matching data-structure lower bound forces an empty transport spectrum and infinite canonical structural degree. Their linear witness C has infinite degree while C^R has finite degree, showing exact directional asymmetry. This is a source-attributed translation, not an independent reproof of their separation or Ko's lower bound.
The Scaffold proof review pins current editions and follows shared assumptions across the transfer, direct palindrome, causal algebra, and unary coherence papers. Run 01 records the native base-node example and edition baseline. Next is the exact relation between finite views and full behavior. Findings accumulate in the review activity before justified changes become matching numbered PDF and HTML editions.
Shortcut ports and local marker geometry jointly compile every tuple (d,k,g,q) to (sum d^i + ceil(log_2(1+gq)),2,1,2). The distinguished base uses the zero marker code; every ordinary label-control pair uses a nonzero SELF/missing code. Thus a language is scaffold-recognizable exactly when its transport spectrum meets the one-dimensional slice (D,2,1,2). Degree is the only unbounded resource needed for existence, although all four coordinates still matter for succinctness.
A node can carry shortcut ports for every original port word through distance k. The compiled radius-two view recovers the old native radius-k view, including base-versus-missing endpoints, while at most two hops install every shortcut needed on the new node. Thus a tuple (d,k,g,q) normalizes to (sum d^i,2,g,q). Combined with the distance-one collapse, the complete expressiveness hierarchy is REG=SA_<=1 strictly contained in SA_<=2=SA. Higher distance is only a succinctness resource.
At distance one, the readable cone never grows with the continuation horizon. The current control and radius-one root view therefore form a DFA state, with at most q[1+(g+1)^(d+1)] possibilities. Every regular language has such a presentation, while the nonregular persistent selector has distance two. This proves REG=SA_<=1 is a strict subset of SA_<=2 and identifies the first exact phase transition in stationary transport.
A scaffold presentation can continuously maintain any fixed finite vector of right-context answers while reusing exactly its original persistent graph. For suffix family U, the explicit compiled tuple is (d, max_u R_(k,|u|+1), g, q 2^|U|). Thus right quotients and arbitrary finite Boolean observers preserve bounded stationary transport. The uniform locality bound is sharp, so the next obstruction must involve growing or adaptive demand rather than a fixed menu of future questions.
The feasible scaffold resource tuples of a language form an upward-closed stationary transport spectrum. Dickson's lemma gives a unique finite antichain of Pareto-minimal profiles. Complement preserves the spectrum, while pairing persistent nodes gives explicit degree, radius, alphabet, and control bounds for every Boolean combination. This turns bounded transport into an implementation-minimized invariant within the scaffold model.
The horizon-h response quotients form an inverse tower. Reading one symbol selects a response subtree and consumes one horizon level, inducing continuous transitions on the inverse limit. Full Myhill--Nerode residuals embed densely, while regular languages are exactly those whose towers stabilize at a finite horizon. This supplies the semantic object whose bounded-locality physical presentations the replayable-memory program must now study.
The support-quotient theorem separates coordinates touched by executions, coordinates capable of changing any future answer, and the minimum exact response quotient. Essential coordinates are the unique least semantic support; every stable replay envelope contains them; and the quotient can be exponentially smaller. The remaining scaffold resource is dynamic: the local cost of updating and transporting that quotient as the cut advances.
For every finite input alphabet, one fixed generic scaffold parameter tuple has a T153 abstract-view envelope at least as large as the maximum possible Boolean response table at every horizon. Thus raw response-row counting alone cannot separate a language from all scaffolding automata or PEGs. This is a no-go theorem for one lower-bound method, not a recognizer construction: the next invariant must use advance compatibility, horizon coherence, realizability, algebra, or transport structure.
For a distance-k scaffolding automaton, every continuation of length at most h is determined by the cut state and the old unfolded neighbourhood of radius k+(h-1)(k-1). The radius is uniformly sharp. Counting view types gives an explicit upper bound on distinct cut-response rows for every fixed degree, distance, alphabet, and control size. The audit covers 4,253,184 one-step extensions and 24 sharpness witnesses. This is a fixed-parameter theorem, not a separation from all PEGs.
The minimum deterministic summary at a finite prefix/continuation cut is exactly the number of distinct response rows. For linear observers over F_q, the exact state count is q^r, where r is observer-span dimension. This places replay support, retained information, and local update cost on three separate axes and turns future FPRD lower bounds into a semantic response-dimension proof plus a model-specific cut-capacity bound. The theorem is a classical communication and linear-algebra bridge, not a novelty claim.
An explicit unary-query selector stores bits in a persistent stack and selects one with a bounded local rule. It has 2^n residuals after n data symbols, yet a degree-two, distance-two scaffold allocates one node per letter. The construction separates retained information, answer dependence, and update cost. All 88,573 words through length ten agree with an independent oracle. This is a standard-stack calibration, not a solution of the general frontier problem.
A selector lemma and explicit middle-bit language show that every completed run can query one old input cell while r continuations expose r independent cells and realize all 2^r residual response profiles. The result identifies the missing bridge after adaptive replay slices: stationary difficulty depends on the transport orbit of possible roots, not the size of one completed slice. It is not a scaffold or PEG lower bound.
The backward closure of the records actually queried by one finite deterministic adaptive execution is a stable replay certificate. It composes by pulling later demands through earlier dependencies and, for independent source coordinates, cuts out a product fibre. A self-contained proof and exhaustive small-model falsifier are public. The result is per execution: it does not yet give a stationary memory bound or resolve the forgotten-action frontier problem.
A complete proposed classification for Problem 8.1, an elementary counterexample to printed Question 9.2, and conditional progress on Questions 7.2–7.3. Full arguments and reproducible checks are available. No external review or established priority is claimed; the cabling questions remain unresolved here.
For a connected ideal triangulation with r torus cusps and strict angles, cohomological degrees and convergent index functionals prove that the generic full trace cokernel has at least 2^r−1 independent classes. Every minimal compatible filling quotient is therefore nonzero. A common quotient supports a zero filled trace; this does not construct a nonzero filled invariant or resolve the unrestricted conjecture.
An explicit actual skein lies in the full six-root trace kernel but has meridian-filled bracket A^-4. Therefore the minimal compatible quotient kills the whole trace image and every compatible filled trace is zero. A separate convergent parity functional proves the quotient itself is nonzero. This settles the fixed figure-eight meridian gate under the retained named-action theorem; the unrestricted conjecture remains open.
A finite squared-moment certificate kills the meridian vacuum before any opposite-root sign specialization on the fixed figure-eight triangulation, while the target's unit remains nonzero. Thus its boundary-balanced descended trace is zero. This does not contradict Panitch–Romo's nonzero-quotient requirement. The exact remaining question is whether the actual trace kernel has zero filled bracket, which decides vacuum survival in the minimal compatible linear quotient.
The manuscript presents a candidate proof of Cheng's general categorical identity: condensing the canonical diagonal algebra in T × Z(B) recovers T as the local-module category. The tensor-opposite construction selects the negative central embedding, identifies the induced algebra, and tracks the surviving braided factor. The complete paper and sources are available; the argument has not received external review.
An exact full-root cyclic map transports the native character and mixed-sided relations to a complementary convergence cone. Together with the phase-free theorem, it proves filled-vacuum nonvanishing for every nonmeridional slope on the fixed published figure-eight triangulation. The meridian cokernel is also nonzero, but an explicit finite quantum-algebra certificate kills its vacuum in the all-plus central sector. The full-six-root meridian vacuum remains unresolved; zero after central projection cannot be lifted backward. The unrestricted conjecture remains open.
On the published two-tetrahedron figure-eight datum, the unchanged one-sided detector now detects the native filled vacuum for every primitive pμ+qλ with q nonzero and p/q>−4. Exact unit and longitude degrees interchange two nonnegative linear coefficients; the only possible zero-degree longitude term has a fractional seed coordinate and is unsupported. This removes the previous p divisible by four condition without changing any native signs, relation sides, or lattice. The unrestricted conjecture remains open.
The one-sided seed now applies to the published two-tetrahedron figure-eight triangulation. A symbolic cone and native-phase argument detects the filled vacuum for every primitive pμ+qλ with p divisible by four and p/q>−4. At 4-surgery the value is an explicit positive Nahm series, while the original-character ordinary-index average fails even on its unit because of a certified lattice ray. This is an actual detector application, not an unrestricted conjecture resolution or a new topological invariant.
Euler quantum-dilogarithm coefficients solve the single adjoint recurrence required by the mixed-sided tetrahedron module. Their full-image average converges on every algebraic input exactly when one quadratic form is strictly positive on its support recession cone, and then respects all native relation sides and phases. A further phase and longitude certificate detects the filled vacuum without assuming filling angles. No additional actual triangulated filling is yet certified; the explicit example is abstract and topological applicability remains the next gate.
Integral normal homology classifies every supported monomial translate using the same complete angle certificate: each reduces to a native insertion on the remaining cusps with an exact scalar. The vacuum functional is odd under simultaneous peripheral inversion, so every ordinary boundary-skein insertion has value zero, including arbitrary noncommuting products of boundary curves. The oriented odd part remains unresolved; no independent-cusp symmetry or actual vacuum-vanishing theorem is claimed.
An exact bijection identifies the unsaturated root-lattice sum with every integral normal class in its boundary fiber, including all mod-2 homology sectors. After preserving the native scalar and residual (−1)^n phase, the published GHHR boundary-inversion identity forces the balanced correct-sided functional to vanish on 1−l². The functional remains nonzero on the longitude, so this is detector failure rather than target or vacuum vanishing. For one cusp, every monomial translate certified by the same complete angle vector is likewise exhausted; a separate-cusp index difference remains an explicit open test.
For genuine native triangulation incidence, every longitude-shifted lattice term maps injectively to an integral nonnegative Q-normal class X, with exact degree 2(δ(X)−χtop(X)). A complete strict angular certificate makes every longitude term positive-order, while a separate filling angular certificate supplies the convergent relation-descending functional whose unit has coefficient one. Their combination proves the actual six-root filled vacuum and named descended trace nonzero. Leading–trailing deformation supplies the filling certificate for every primitive slope outside an explicit bounded region. The unrestricted Panitch–Romo conjecture remains open.
The exact longitude-shifted degree and phase give a finite cancellation-aware test for the global index value on 1−l². A three-tetrahedron abstract presentation with native-form phases, an unsaturated isotropic moment lattice and strict filling angles nevertheless has a nonzero cokernel unit while a finite correct-sided identity makes 1−l² zero. The witness is not a triangulation counterexample: its second edge needs eight nonnegative prime incidences while its scalar permits seven, and no generating-set change in the same edge lattice repairs this. The next attack must use genuine global edge incidence to prevent cancellation; Panitch–Romo Conjecture 7.1 remains open.
For the actual all-root mixed-sided presentation, the tetrahedron-index seed can be averaged over the exact unsaturated edge-plus-meridian lattice. Exact moment-character consistency, nonnegative filling angles and the absence of a ray invisible to both the double-arc degree and angle weight imply all-input coefficientwise convergence. Reindexing annihilates every actual moment relation, the index recurrence annihilates every tetrahedron relation, and the unit has leading coefficient one. A null ray is a sharp obstruction to this unordered average, not a cokernel-vanishing theorem. The longitude difference 1−l² and unrestricted Panitch–Romo Conjecture 7.1 remain open.
For the actual mixed-sided six-root relation system, a smooth excess-dimensional classical intersection produces a nonzero tangent first-Bockstein operator. Formal solvability makes it surjective, so the completed detector is annihilated by the deformation parameter and vanishes after generic localization. Exact algebraic models show both that classical filling support need not survive quantization and that zero formal-local detectors do not imply a zero global Laurent cokernel. This stops the local-flatness route without refuting Panitch–Romo Conjecture 7.1; the next attack must construct a global algebraic detector.
Keeping the six bare-edge roots distinct gives an additive Weyl operator for every primitive peripheral slope. Panitch–Park's unweighted marking reversal, the exact cup/cap factors and an anchored opposite-collar edge permutation make the full splitting, counit and corner square commute, proving the named GY/Chen–Kricker trace equality before opposite-root identification. After the existing Eulerian normalization this yields the exact original Panitch–Park named action. The weighted self-duality fails for an unchanged same-state cut by a residual v^{-2s} factor. Unrestricted filling nonvanishing and Panitch–Romo Conjecture 7.1 remain open.
An independent hostile audit found no gap in the degree-zero descent proof for the standard right-tetrahedron/left-edge Laurent presentation. Exact right-unit normalization and the zero-degree S-bimodule projection prove S∩R=J, resolving Chen–Kricker Conjecture 5.27 for their actual ambient-image comparison target. A new ordered-PBW verifier and the independent Weyl verifier both pass. The source's contradictory sidedness phrase, AI-assisted origin, absence of external review, and bounded priority check are disclosed. The original six-root peripheral action and Panitch–Romo Conjecture 7.1 remain open.
The full face-cone quantum torus is graded by differences within the half-edge pairs defining the six shape roots; its zero component is exactly the six-root subtorus. An exact right-unit normalization puts every tetrahedron relation into that component while retaining the mixed right/left sidedness. Projecting the complete ambient relation module to degree zero proves S∩R=J, so the actual Chen–Kricker comparison map to the ambient-image Panitch–Park target is injective. Their existing surjectivity and named-trace theorem then give an isomorphism of the convention-adjusted extended targets. The proof has passed hostile internal audit but awaits a dedicated publication and priority review. The original six-root peripheral action and Panitch–Romo Conjecture 7.1 remain open.
For a strictly positive geometric hyperbolic filling, the tetrahedron tangent is a definite complex Lagrangian and the independent edge-plus-filling rows form a real isotropic space. Their transversality gives joint Jacobian rank 2N−r+1, while principal half-logarithms give the exact edge and meridian signs and the loxodromic core gives the longitude test. The identified GY filled quotient and descended trace are therefore nonzero at generic parameter. Separately, an Eulerian orientation of the dual multigraph conjugates the two named Panitch–Park scaling branches after opposite roots are identified, including self-gluings. The GY–PP named-trace and six-root peripheral bridge, arbitrary fillings, and Conjecture 7.1 remain open.
Gunningham–Jordan–Vazirani's generic torus-category theorem and Gunningham–Jordan–Safronov excision reconstruct all contributing endpoint sectors from the empty one. Explicit mutually inverse maps identify a filled target with coker(m+A²) when the full peripheral quantum torus acts. For the identified GY target, this quotient is nonzero if the classical tetrahedron, edge and filling symbols form a regular sequence at a common nonzero shape point. The proof uses exactly commuting left/right operators to exclude formal-parameter torsion and detects the original generic quotient. If the longitude value has square different from one, the filled vacuum also survives. Coverage of all intended fillings and the named Panitch–Park trace bridge remain open; Conjecture 7.1 is not solved.
For every hyperbolic knot exterior, meridional filling gives a noninjective map from the balanced tensor product of the ordinary solid-torus and complement skein modules over the torus skein algebra to Sk(S³) at A=−1. Complete holonomy and a solid-torus representation agree on every peripheral character, while the product of conjugate meridians μgμg⁻¹ detects the nilpotent part: tr(UGUG⁻¹)=2−c². This obstructs an unrestricted integral formula. The subsequent generic reconstruction proves that ordinary torus-algebra gluing does hold after generic localization. Panitch–Romo Conjecture 7.1 remains open.
Panitch and Romo conjecture that the 3D quantum trace descends through Dehn filling to a nonzero quotient of the square-root quantum gluing module. The selected construction treats both sides as the same base change over the torus skein algebra, using the solid-torus module twisted by the filling slope. The first attack must prove exact boundary linearity from the face-suspension trace, identify the handle-slide coequalizer, and establish nontriviality at the classical character-variety limit. Knot, slope, and triangulation censuses are excluded. No theorem is claimed at selection.
For the convex numerical-range double-layer map, the symmetrized map Λ/2 is already unital completely positive, so its full matricial Herglotz and Toeplitz positivity is automatic. Transferring that data to one Stein metric I≤P≤4I, A*PA≤P is, through Paulsen similarity, precisely the still-open completely bounded Clouâtre–Ostermann–Ransford implication. The proposed kernel extraction is therefore circular as a reduction, and an abstract countermodel would refute complete COR rather than merely stop the route. Complete Crouzeix remains open; the program pauses unless a geometry-specific resolvent identity adds genuinely stronger data.
The sharp matrix-valued Crouzeix conjecture is equivalent, through Paulsen similarity and conformal normalization, to one positive metric P satisfying I≤P≤4I and A*PA≤P. This similarity radius is independent of the Riemann-map normalization and is sharp on the normalized 2×2 nilpotent. The first attack will ask whether the scalar proof's full positive Herglotz kernel constructs this common metric, or admits a finite abstract countermodel. Matrix-by-matrix SDPs and special-family censuses are excluded. No theorem is claimed at selection.
Flat restriction preserves the Koszul blocks, the Sylvester condition, and quadraticity, while each flat block occupies only three internal diagonals. The canonical average over third points exactly contracts rank two. On the spanning top diagonal, however, every such insertion is one rank short; that block is the first homology of a pointed-hyperplane triangle complex. Moreover, Sylvester already implies T-chordal, and the source's nonquadratic truncation T(PG(3,2)) has an exact 209-dimensional spanning H_(2,4) block. Thus line incidence plus T-chords cannot use the missing quadratic coherence. The proof route stops; Conjecture 4.1 remains open.
Fouli et al. conjecture that a quadratic graded Möbius algebra is linearly presented exactly when every pair of matroid elements lies in a 3-circuit. The Koszul complex admits a canonical decomposition by the closure of each flat–wedge basis term, reducing nonlinear second homology to spanning blocks of flat restrictions. Exact rational checks recover the source controls, including beta_(2,4)=3 for M(K4). The first attack must construct one symmetry-averaged line-incidence homotopy and propagate it through T-chords, or stop before any matroid census. No theorem is claimed at selection.
At an exact generic six-point specialization, the second cumulative collinear row space forces a unique diagonal chamber weight, but the first EYM block then has nonzero pairing with a degree-one Heaviside generator. More strongly, one EYM row cannot be carried into the opposite first Heaviside piece by any three-parameter type-A2 Varchenko matrix: its four polynomial conditions have unit ideal, including every left/right sign-character twist. This stops the proposed canonical filtration identification without refuting Dong and Stieberger's Stirling-rank or determinant conjectures.
Dong and Stieberger conjecture that the strict subleading Yang–Mills collinear system has all-multiplicity graded ranks given by unsigned Stirling numbers, and explicitly leave its missing intertwiner open. The BCJ orderings and the chambers of the type-A braid arrangement are both S_m; the opposite dual of the canonical Varchenko–Gelfand Heaviside filtration has exactly the conjectured increments. The first attack will test equality of the full filtrations at n=5,6 through a canonical Varchenko pairing, then either prove the circuit relations or stop. No theorem is claimed at selection.
The preceding weighted-symbol argument omitted the admissible stationarity multipliers x_i p_i p_4² for i=1,2,3. Their cyclic sum supplies t²(4F−x_4∂_4F), and an exact rational combination with the energy and fourth-coordinate rows equals −t⁴. The claimed leading obstruction therefore vanishes. No quartic counterexample or dual-certificate theorem follows, and the route pauses rather than opening a lower-order moment census.
This entry is retained only as a correction trail. Its theorem claim was false because three admissible stationarity directions were omitted; the following retraction entry gives the exact cancellation. No result from this entry should be used.
For H=p²/2+x^(2m), m at least three, a vanishing-weight mixture of the ground state and an arbitrarily high eigenstate has fixed mean energy but unbounded fourth momentum moment. At K=m only the mean-energy row survives, so these are genuine feasible points and contradict the literal k≤K−1 range. The energy-shell mechanism instead permits k≤K−deg(V)/2+1; this agrees with Keren's quartic conjecture, which remains open. A Hankel escape also corrects the hunt packet: spectrahedral unboundedness need not have a linear recession witness.
Daniel Keren conjectures that degree-admissible eigenstate relations bound every momentum moment visible to a finite quantum-mechanical bootstrap. The program replaces moment coordinates by one canonical ordered quotient of the filtered Weyl algebra. The first attack will prove an exact order-unit alternative and seek either a degree-preserving energy-shell certificate or a parametric recession functional at the first coercive non-SOS boundary. No theorem is claimed at selection.
A nonconstant family of separable Liouville-integrable Hamiltonians has universal critical algebra k[t][ε]/(ε³): it is finite flat, every reduced fiber is one isolated point, and every scheme fiber remains nonreduced. Its Hessian determinant is the nonzero nilpotent 3ε², while its norm and trace discriminant vanish. Thus gradient complete-intersection structure, generic smoothness, and integrability cannot force persistent nilpotence to become a critical curve. The Bethe–Calogero route pauses unless a specifically CM spectral-versality theorem is found.
At j=1728, a regular fully rational B2 extremum has zero Hessian but remains isolated: its first potential jet is 2a³(x⁴−y⁴), so its local critical algebra has length nine. Modular motion and flavor deformation enter the leading smoothing through one parameter κ=3r+2a²σ₂; every κ≠0 fiber has nine Morse points and trace-discriminant order 18. This proves the generic-parameter qualifier is essential and replaces a first-order corank test by an exact higher-corank local model. It is an unreviewed local result, not the global Bethe–Calogero bijectivity conjecture.
The full cubic-divided theta force is exactly a triple centered average of the Calogero–Moser force. For the finite flat local Bethe algebra, the trace discriminant is, up to a unit, the norm of the averaged-Hessian determinant—the ordinary Bethe Jacobian after removing its universal flavor collapse. A nonzero germ forces a length-μ degenerate extremum to split into μ simple generic solutions; in corank one, dV2(q)(v) is the first sharp splitting gate. The surviving question is whether isolated regular CM extrema can be degenerate at generic modular parameter. This is an unreviewed local result, not the global bijectivity conjecture.
The universal cubic flavor factor defines a basis-free Cartier-residual Bethe ideal, equivariant under the gauge and modular symmetries. Its special fiber is scheme-theoretically the untwisted Calogero–Moser critical scheme. Near any isolated regular extremum the family is finite flat, with degree equal to the local critical-scheme length, so nondegenerate extrema lift uniquely and only degenerate local algebras can branch. On the continuous A2 curve the first two tangential obstruction potentials are constant, explaining its survival through flavor degree four. This is a local structural theorem, not the global bijectivity conjecture.
Fazzi and Krawczyk conjecture that the Bethe solutions of four-dimensional N=4 super-Yang–Mills flowing to regular untwisted Calogero–Moser extrema are in bijection with those extrema. At the unrefined point every Bethe operator becomes the identity, but its difference from one has the universal factor Δ1Δ2Δ3. Dividing that factor produces one canonical deformation family with the CM critical scheme as its expected special fiber and a Hessian-cokernel obstruction for continuous components. The first attack will prove descent and local lifting, or stop at the exact obstruction; no theorem is claimed at selection.
The descending-coface poset does retract exactly to the Rips complex of one canonical integral cross-polytope intersection, and signed heights turn Manhattan distance into symmetric difference. But orthant pieces fail M-natural exchange, closed orthants do not cover all carrier edges, and a Boolean blocker clique defeats every lattice rounding of an edge midpoint for all n at least five and r at least n. These symbolic failures stop the selected route without refuting the Virk–Zaremsky conjecture or changing its known bounds.
The new target is the Virk–Zaremsky conjecture that VR_r(Z^n) in the standard Manhattan metric is contractible exactly from scale r=n. The attack converts every flat Morse link to the Rips complex of lattice points in one canonical intersection of cross-polytopes, then seeks a discrete-exchange shaving theorem and an orthant nerve. The sharp result is known only through dimension five; no theorem is claimed at selection.
Meng Cheng proposes that condensing one canonical diagonal algebra in T × Z(B) returns the original modular category T, and explicitly leaves the general proof open. The missing regime is a non-Abelian premodular B with nontrivial Müger center. The first attack will construct the canonical braided comparison functor using relative-center and transparent-subalgebra machinery, or identify whether the statement survives only at the Witt or generalized-orbifold level. No category census or theorem claim is involved.
A separate audit repaired the only genuine gap in the candidate argument: a derivation-stable ideal of a minimal invariant ideal is promoted through its ambient saturation AJA, and idempotence then forces equality. Connected integration, Kac's nonunital scope, queer supercenters, explicit separability, the Ind-module transfer, and the internal-Ext arrow all survive. The resulting theorem answers Coulembier–Sherman Question 5.16 affirmatively over algebraically closed characteristic-zero fields. It has no external review, and priority remains unassessed.
For a finite algebra object with no nilpotent invariant ideal, a minimal connected-invariant ideal is differentiably simple. Kac's normal form makes it a simple superalgebra tensored with a Grassmann algebra; the Grassmann factor is its canonical supercenter. Coulembier–Sherman's commutative theorem controls that center, and separability transfers absolute flatness to the full algebra. This appears to close Question 5.16, but it remains a candidate pending a separate hostile audit of six categorical and superalgebraic bridges; no theorem claim or Publication is made here.
The new target is Coulembier–Sherman Question 5.16: whether internal semisimplicity, absence of nilpotent invariant ideals, absolute flatness, and the related conditions remain equivalent for finite algebra objects in characteristic-zero supertannakian categories. The missing regime is organized by an odd tangent direction q whose square is a nonzero even direction. The first attack will settle that rank-one category and prove or refute rank-one detection; no theorem is claimed at selection.
The root-segment polytopes are moment polytopes for the Losev–Manin toric variety, but an explicit one-parameter deformation keeps the full braid fan and both all-orders soft identities while changing the two-defect amplitude. Relative cohomology locates the missing datum at the collision extension, so local BF reduction plus toricity cannot select the weighted measure. The JT route is paused; no new dynamics or Publication is claimed.
In the genus-zero two-boundary chamber where all defects can collide, the geometric soft identities and degree bound reconstruct every amplitude as a forest polynomial and polytope volume. The uniqueness proof adapts Anagnostou–Mullane–Norbury's soft-face interpolation strategy. A contact redefinition illustrates established renormalization freedom. The direct BF measure derivation remains open; no new physical mode or priority for the forest/polytope synthesis is established.
The proposed route chose a noncanonical Hamiltonian core and then classified residual deletion and interval patterns relative to it. That repeats the presentation-plus-residual failure exposed by Selfridge, so the target is removed from the active map. The next hunt uses Fold-and-Look origami as the positive control and may audit dynamics flattened by symmetry-reduced presentations in mathematical physics.
A broad hunt selected the open FPT question for prescribed-endpoint paths omitting at most r vertices in semicomplete digraphs. A later selection audit withdrew it before attack because the proposed invariant depended on a noncanonical core path and opened a large residual case space.
A four-prime quotient reduces eighteen pair-minimum cancellation strata to 23 states; modulo 27 and a factorization of 175 leave only (4+3)(1+8·3)+3^4=2^8, costing thirteen extra ones. Together with the triple-minimum theorem, every exceptional f=0 cancellation is classified and every other solution satisfies one universal 3-adic valuation law.
A classical S-unit theorem first proves uniform finiteness of the normalized complementary family, while exposing why it gives no height cutoff. Opening f=0 leaves 40 canonical placements and 20 possible cancellations; its only two triple-minimum strata reduce to one empty family and the unique control (2+3)(1+8·3)+3=2^7, costing four extra ones.
The next ternary digit kills all eleven boundary-rooted lift graphs. Three residual graphs reduce to two exponential equations; Matveev's explicit bound and an exact finite modular descent leave only seventeen harmless controls, each costing at least 2N+6.
Combining the f=1 and f=2 valuation laws at m=1,2 makes nine of 32 canonical strata empty and locks every survivor to one of 40 graphs with period 20,652,025,680. Every noncontrol solution has output exponent at least that period.
Eight quadratic-gate placements leave ten possible f=2 tied-minimum cancellations. Nine fall modulo 7 and 73; the last has incompatible complete periods modulo 487 and 2593, forcing min(b,y,g)=1+v3(K+2-h).
Canonical parity reduces every possible f=1 3-adic cancellation to six families. Their complete residue periods modulo 7 and 73 are disjoint, so every remaining solution satisfies min(b,y,g)=1+v3(K+1-h).
Jacobi identities reduce twelve canonical a=0<g residue pairings to six. The forbidden (5,17) pair then eliminates the complete f=2 branch in which both inner sums contribute one factor of two; f=1 compresses to three rows.
The full 2-adic valuation fixes eleven residual rows and turns ten others into one-dimensional Hensel graphs. All 20,480 states at precision 2^22 fail modulo 65,537, leaving exactly ten controls, each costing at least 2N+4.
Twelve primes lock the shared a=2, x=6 cluster to its two boundary factorizations. A certified one-sided logarithmic sweep then forces every non-boundary survivor beyond v+y=26,774,635,116,096,000; this is a height barrier, not a closure.
A complete quotient closes positive type II at a=3. Coupled multiplicative-order residues prove x≤10 throughout a=1,2,3, leaving 21 exact necessary rows at a=1,2 rather than an unbounded exponent search.
A complete modulus-262657 quotient closes the remote a=3 tail, while incompatible zero classes modulo 73 and 262657 eliminate every a≥4 layer at once. The complete normalized g=0 positive-type-I branch is now closed.
The complete canonical entry table, an exact second 2-adic gate, and finite residue covers exclude every a=3 entry through x=20. Every survivor has 3 dividing x and begins at x=21 with a six-digit ternary exponent.
An extension of Kim–Park’s explicit PEG separation, adapting their parity rules to a tree-addressed witness and comparing residual counts with access costs. This working note retains the arguments and open questions; novelty has not been established.
The min-count series has intrinsic ambiguity degree and exact cover width two, exact horizon-N state demand 2 floor(N/2)+1, and raw and canonical height one for N≥2; at N=1 its exact height is zero.
For k≥3, the forced minimum-state comb retains repaired raw height one and repaired canonical height 3·2^(k-2); the previously omitted k≥3 hypothesis is now explicit.
For late INDEX on 2^r sites spaced r+1 apart, every branch has causal width one but pre-query placement needs 2^r tracks; with k tracks, the exact number of complete cut snapshots is 2^max(2^r-k,0).
On the boundary-augmented presentation graph, the optimal real sup-norm is the maximum absolute mean circulation and the optimal integer height is its ceiling; an independent reconstruction checks the boundary and integrality steps.
The exact 2^n matching-language row theorem survives, but a two-fold fibre with same-orientation outer literals and an intervening TRUE site refutes the proof's orientation lemma; the LIS fold separation is withdrawn and FPRD-T66 moves to L2.
For fixed-length exact recognition, distinct future rows form the unique reduced strict layered recognizer, the minimum one-way message alphabet, and the complete-configuration lower bound, including endpoint and constant-slice boundaries.
The factorial output has an infinitely generated derivative Hadamard algebra even over a unary alphabet, while two commutative controls still give a finite positive presentation; the grading proof now states its homogeneous-component step explicitly.
Explicit rational recurrences reduce to one integer matrix-power entry; the coRP statement now imports the quantified EquSLP theorem, while a separate escaping-prime argument gives coNP.
The safe-row criterion and Krylov bound are confirmed for explicit rational representations; a three-letter hostile example shows exactly why every extra non-a letter needs its own annihilation test.
A reconstructed universal carry proof upgrades the former finite-only wording: every decimal one-step palindrome is carry-free or has raw sums in {0,11} with outer sum 11; the first-million census remains separately bounded.
Bounded star unfolding is impossible and derivative saturation is sufficient, but exclusion-only prose does not yet define a class over which a general existence theorem can quantify.
An epsilon-free position-automaton oracle agrees with the derivative construction in all 12,555 finite comparisons and confirms 65 hostile bounded-unfolding cases.
Parity words give exact affine residue charts, but periodic integrality, positivity, and 2-adic solvability are distinct gates; finite-depth selector sections need inverse-tree receipts.
The reversed target fibre has a canonical d_f-state poDFA and therefore a d_f−1 word bound; the known binary counterexample family has exactly C(n,2)−1 target profiles despite a cubic source monoid.
Cross-component and diagonal crossing budgets classify the pinned three-packet support solutions and isolate exactly where canonical-section and finite-quotient rejection stop short of an arbitrary-winding obstruction.
Stabilizer cosets count local lifts, equitable matrices count class-level walks, and voltage monodromy decides how a base cycle splits upstairs; none can be silently substituted for another.
For the deck of two disjoint three-vertex paths, 18 anonymous type-slot synchronizations split into six sewing and twelve nonsewing cases; an equivariant F2-squared defect detects the split exactly.
In a square-zero extension, local factor corrections combine through conjugated direction spaces; the target cokernel is invariant on a complete conjugation-compatible Hurwitz fibre, while the next mod-eight layer is order-sensitive and quadratic.
With the slice length encoded in unary, the published DFA reduction is already #P-complete at independence-cover number one; the complementary nonempty one-Foata-step problem has an exact O(|Σ|κ2^κm) algorithm.
For every n at least five, a minimal binary J-trivial DFA has a target transformation whose shortest representing word has length n; an exhaustive five-state classification finds exactly three extremal generator-pair classes.
For the conditional poDFA reduction, a polynomial action only needs a code attained by the supplied target and by no other generated transformation; a three-element monoid shows this is strictly weaker than faithfulness.
Any pairwise-intersecting support core gives the exact factorization; among such decompositions, the smallest exception count is a disjointness-graph vertex cover and can be arbitrarily smaller than diversity.
Supports outside a largest participant star give an exact target factorization; the one-shot model has a canonical conflict-mask quotient and subset-zeta algorithm.
A 204-node typed language DAG lowers to a 192,771-state deterministic path compiler without expanding more than 4×10^102 known codewords; the C7 compiler analysis is now complete.
The synchronized twist has six reachable states; the direct two-letter block expression has 11 natural-prefix states and a distinct minimal six-state reversible quotient.
Below n < 2r every support family is zero-overhead; from n >= 2r, EKR and Hilton–Milner give the sharp ceilings and the spectrum counts forced collisions.
For every component expression, the natural and componentwise prefix constructions agree exactly when each letter has one participant set and different letters' sets overlap.
The expression a* shuffle b* uses the minimum two letter occurrences among shuffles with two nonempty-letter operands, yet already fails to admit the prefix quotient.