Obstruction retrieval map
2026-10-01. 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 | Correction, original, identities |
| Blocking into a strong-coupling box; larger blocks might improve the mixing criterion | Under the stated bounds, applying the same criterion to a naive block worsens its size factor unless coupling/effective action genuinely changes | Exact elimination is useful when generated interactions and flow are controlled; the monotonic comparison includes its explicit δ hypothesis | Monotone criterion, which supersedes earlier test |
| 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 compact-lattice step is uniformly bounded, but generated couplings still need measure-weighted control | Conjugation, before decimation, compactness correction |
| 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2gB saturation claim | Use qualitative instability only until the factor and generic spectral inference are corrected; do not cite exact saturation | Instability note |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 | Existing block criteria at physical crossover demand infeasible certificates; 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 | UV correction, reduction, valley |
| 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 | W≥kBT log(A0/η) permits arbitrary η>0 with increasing finite work and supplies no universal positive action floor | 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. Compact-support error and recoil assumptions have no nonvacuous quantum instance | 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 |
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.