Obstruction retrieval map
Begun 2026-10-01; canonical countertests updated 2026-10-30. Search the route name before restarting an argument. Each entry records a failure in a stated class, not a universal impossibility theorem. The catalog distinguishes retrieval value from proof status; dependencies identifies the current replacement.
| Route; why it looked promising | Exact failure and scope | What survives; replacement | Source |
|---|---|---|---|
| Local Agmon suppression; magnetic energy is expensive | The available Agmon comparison controls an extensive global excess, with a volume-dependent threshold; it does not uniformly suppress one local patch | Global large deviations survive. Prove normalized Euclidean local probabilities and clusters separately | Canonical argument, identities |
| Blocking into a strong-coupling box; larger blocks might improve the mixing criterion | The displayed sufficient criterion has unfavourable surface scaling in stated regimes; its general comparison requires an additional block-gap hypothesis | Exact elimination is useful when generated interactions and flow are controlled; the monotonic comparison includes its explicit δ hypothesis | Blocking and Schur |
| Gradient flow as RG; smoothing appears to remove dangerous fields | Exact invertible flow is a conjugation/change of variables. Operator-norm criteria do not improve; discarding generated nonlocal terms is the actual coarse graining and needs an error bound | One-step flow derivatives have locality bounds; the Hamiltonian relative truncation estimate and generated couplings need control | Flow and truncation |
| Flow instability sharpness; an unstable mode might explain the entire truncation exponent | The note’s qualitative unstable mode survives, but its displayed lowest Landau rate gB does not support its headline exact 2gB saturation claim | The corrected full-generator proof gives gB-k_parallel^2; kernel-tail sharpness and relative operator comparison remain open | Instability note |
| Moment/Feynman–Bijl positivity; finite inverse moments look like a lower edge | Consecutive spectral moment ratios and trial states upper-bound an edge. A gapless measure can have every inverse moment finite | Upper finiteness is useful with a nonzero admissible observable. A complete observable-frame susceptibility estimate can lower-bound the appropriate relaxation operator under its own hypotheses | Moments, finiteness, frame theorem |
| Finite certified crossover verification; all unknowns might fit in one finite box | No physical-crossover certificate is supplied by the examined methods; enlarging the box does not supply weak-side RG control | The abstract finite certificate remains sufficient if supplied. Current construction moves to local weak-side insertion and normalized interaction control | Reasons to stop, correcting finite verification and bands |
| Correlation inequalities; monotonicity might extend the known strong side | A monotone estimate from strong coupling does not give the needed weak-trajectory lower bound or generated-action control; the tested nonabelian inequalities do not discharge H1 | A correctly stated inequality may constrain its own observables. Use constructive weak-side RG rather than treating monotonicity as arrival in the box | Reasons to stop, earlier openings |
| Explicit weak-side threshold chasing; better constants might meet the physical window | The displayed Balaban-style estimates miss the relevant weak coupling by many orders of magnitude; this derivation’s constants do not produce the crossover bridge | Gaussian anatomy and ultraviolet bounds survive. A different normalized local estimate is needed, not merely arithmetic optimization | Threshold failure, Gaussian step |
| Bare-vacuum Feshbach; integrate out nonzero modes perturbatively | The proposed bare-vacuum Schur error is not uniform in the UV cutoff. The note does not complete a rigorous fixed-lattice gap proof either | Refined transfer lemma and torus valley calculation remain inputs. Seek renormalized dressing and the open H1–H3 estimates | Corrected reduction, valley |
| Bounded torus valley as a small global matrix perturbation | Full one-loop U is bounded and periodic, but U minus its linear zero-mode term is neither on the covering space; local quadratic power counting does not control all matrix coordinates | The one-loop coefficient and conditional symmetry argument survive; H3 needs a domain-aware global or localized spectral estimate | Valley domain and subtraction, relative Schur hypotheses |
| Local matrix spacing as the full periodic gap | The declared SU2 flat-holonomy operator admits centre-odd and cubic-invariant trials with exponentially small inter-sector upper gap; the one-patch fundamental-loop variance is not the global vacuum variance | Name the centre character and vacuum observable algebra, use a neutral adjoint trial for a neutral target, and prove the field/operator matching | Benchmark proof, exact neutral trial |
| Action cost plus entropy as normalized rarity; local large fields cost Euclidean action | An action lower bound is not a probability under the normalized measure; constant-background mode counts do not establish arbitrary polymer convergence or a Hamiltonian operator inequality | Keep action cost and model entropy as scoped inputs. Current retained-barrier gas controls actual ratios in its weak strip | Action cost, entropy, operator distinction, small-field §§33–37 |
| Differentiate the barrier or reuse a finite-order response theorem | The differentiated-resolvent route reaches a fifth-derivative obstruction; pointwise conditional log densities may have zeros, and finite-order estimates do not sum all clusters | Retain the barrier undifferentiated; normalize before taking absolute coefficients. All-order weak-strip marked/dressed estimates now supply a replacement with a narrower domain | Small-field §§23,27–37 |
| Thermodynamic record cost; reliable recording needs work | The stipulated lower bound W≥kBT log(A0/η) supplies no positive action infimum or theorem of finite-cost attainability | It is a conditional resource tradeoff. Seek a composition-stable physical record restriction independent of chosen work budget | Thermodynamic records |
| Topology selects an absolute action; a nontrivial sector has positive energy | Sector energy, stabilization and a quantized phase ratio do not fix the common normalization; shrinking crossing times or rescaling the action leaves the selector open | Topology can stabilize or quantize a ratio once a phase unit is supplied | Sector test, stabilization, spin patching |
| Relativistic Kepler threshold; collision-free bound motion requires L>k/c | The positive infimum depends on the singular −k/r core and admitted collision-free orbits; smoothing restores arbitrarily small bound-orbit action | The singular-model threshold is exact. A universal selector must justify its short-distance admissibility premise | Kepler, smooth small circles |
| Phase/EBK consistency; single-valued phases might force h>0 | The tested consistency laws hold at arbitrary supplied h and admit h→0 families; interference alone survives arbitrarily small classical action | Exact phase/insertion laws and fixed-h convergence are useful. Selection needs a separate physical premise | Rivero test, spin patching, mechanical fringes |
| Preparation-independent cell floor; all instruments obey information bounds | Arbitrarily wide/separated prepared packets, correlations or squeezing evade a universal Galileo-action threshold. A state support theorem alone does not establish a hard impulse-width instrument theorem | General resource-scoped disturbance/Bures bounds survive; Gaussian invariant tests have a sharp conditional floor | Probabilistic escape, disturbance, early correction |
| Prior covariance or a shared bath closes recording | Indirect observations, stored first readings and the two-pointer protocol sharpen the completed posterior. The nonlinear generator can defeat a covariance-only statistical-speed ceiling | Linear Gaussian feedback can close under an imposed terminal readout law; nonlinear weak copies preserve the countertest. Seek an independently physical law for all retained records and memory | Unit/two pointers, routes, recording §§1–8.20 |
| Positive tails cure phase loss completely | In the supplied quantum free-evolution reference, positive density gives exact phase reconstruction from density/current, but the stated local-data metric loses uniform stability as the bridge density vanishes | Additive bridge-phase memory remains. The live Newton step is a physical rule preparing and carrying it through separation and reunion | Recording §8.20 |
| Continuity by itself selects positivity | Fixed protocols can be continuous at κ=0; the iff theorem optimizes a specified non-adaptive Gaussian, preparation-invariant family | Defend a physical uniform complete-record continuity premise before extending the model theorem | Leibniz records |
| Reversible phase-state evolution excludes all classical completions | A bounded cusp has merging point states, but complete trajectories support a reversible classical shift with exact states and conditioning | The phase-state floor is a conditional theorem; the history model retains future-selection data and has no constructed finite-dimensional local apparatus | Cusp floor, history model |
| Continuity or canonical copying alone supplies the quantum scale | Actual cusp transport has distinct small-Gaussian limits at one point. The continuity floor additionally assumes a \(W_2\)-closed canonical-copy class, and a classical entropy-floor witness supplies its own scale | The Gaussian sector and positive covariance infimum follow under their stated hypotheses; neither positivity of physical \(\hbar\) nor quantum dynamics follows | Canonical central limit, continuity selector |
| An unread entropy floor is stable under recording | One minimum-action Gaussian pointer and a coarse record of probability at least \(9/32\) can produce arbitrarily small posterior action from admissible classical states; this family uses increasing preparation aperture | Gaussian inputs pass that meter. Its \(W_2\)-closed Gaussian-mixture repair passes the meter and copy mixers, but a specified nonlinear classical shear leaves that class; other classical completions are not excluded | Entropy class, meter and shear |
| A smooth competing-branch response proves a universal action | A fixed response ceiling selects a positive multiplier only for the specified normalized double-well weight; a retained classical label avoids the zero-action jump, and a thermal law can mimic the response | The explicit conditional bound survives. Establish the response law, a uniform ceiling and any identification with physical \(\hbar\) independently | Branch-response theorem |
| A classical limit uniquely specifies finite-action quantum dynamics | Mixed cell orders converge to the same specified Hamiltonian at fixed \(\hbar\), yet some joint limits keep the corresponding rays separated; distinct factor orderings can share a classical symbol | The scoped quantum lift and mean limits survive. Their existence does not prove uniqueness or nonzero-action necessity | Lift conjecture, ordering |
Pointwise coherent records require nonzero frequency sums for non-convergence. Equal-action cross terms can cancel, and a derivative sign change may produce only two illuminated branches. The Kepler angle map also needs its signed image width: compression cannot cause pre-fold wrapping. These exceptions restrict crossing-based conclusions, while the incoherent classical-limit countermodel survives.
A classical transverse-area restriction bounds energy sublevel sets, not quantum levels; its pointwise admitted energies still approach zero. The inverse-square comparison also requires a specified domain and stability: self-adjointness alone does not select a stable positive scale.
All-coupling T2′ belongs to a separate optional strengthening: the continuum problem requires an appropriate weak-coupling scaling trajectory, construction and a surviving positive physical edge. See the correction. None of these closures is a reason to abandon either main research goal.