IDEAINNOV · updated August 18, 2026
Hard-Problem Solving Showcase
A plain look at how coupled-equation analysis and discrete-math-specialized approaches have been applied across 30 research hard problems in physics, physiology, biochemistry, mathematics, and chemistry.
What this shows
The table below summarizes the before/after state of 30 research hard problems the IDEAINNOV paper pipeline has actually worked on. Problems that were already stable and complete are marked “Maintained (complete)”; problems whose write-up quality was improved through review are marked “Improved”; problems still under way are labeled “In progress / scheduled” without overstating the result. Internal algorithm details and code are patent-pending and not disclosed — only outcomes and progress meaningful to you are shown here.
Summary by domain
Physics
Coupled-PDE analysis was applied to 6 problems spanning turbulence, superconductivity, plasma, and dark matter. Most passed verification and are maintained as-is; a few had their write-up quality improved through language-consistency review.
Physiology
6 problems covering neural coding, cardiac waves, and immune dynamics. Most are maintained as stable; a subset had both language consistency and validation transparency improved.
Biochemistry
6 problems covering protein folding, enzyme kinetics, and gene expression. Several cases had validation transparency and topical alignment strengthened; the rest maintain their existing stable analysis.
Mathematics
Continuum-PDE-type problems (like blow-up) remain stable under the coupled-equation approach. Three problems — the prime-counting error term, the Collatz stopping-time problem, and the random graph coloring transition — are instead being reworked with a discrete-math-specialized approach (probability distribution, visualization, combinatorial optimization); results are not yet finalized.
Chemistry
6 problems covering catalytic reactions, battery interfaces, and crystallization. Some are undergoing continued deepening interpretation; the rest are queued for sequential start — we disclose the progress honestly rather than claiming completion.
30 hard problems — before/after status
| No. | Domain | Problem | Before | After | What changed |
|---|---|---|---|---|---|
| 01 | Physics | Turbulent Flow Analysis (Navier–Stokes) | Complete | Maintained (complete) | The turbulent-flow analysis passed verification and is kept as-is. |
| 02 | Physics | High-Tc Superconductivity Cooper Pairing | Complete | Maintained (complete) | The Cooper-pairing analysis passed verification and is kept as-is. |
| 03 | Physics | Fusion Plasma MHD Instability | Complete | Maintained (complete) | The MHD-instability analysis passed verification and is kept as-is. |
| 04 | Physics | Dark Matter Halo Density Profile | Complete | Improved | Language-consistency review was applied, improving the write-up quality. |
| 05 | Physics | Nonequilibrium Entropy Production | Complete | Maintained (complete) | The nonequilibrium-entropy analysis passed verification and is kept as-is. |
| 06 | Physics | Quantum Decoherence-to-Classical Transition | Complete | Maintained (complete) | The decoherence-transition analysis passed verification and is kept as-is. |
| 07 | Physiology | Neural Spike Coding | Complete | Maintained (complete) | The neural-coding analysis passed verification and is kept as-is. |
| 08 | Physiology | Cardiac Reentry Wave (Arrhythmia) | Complete | Improved | Both language consistency and validation transparency were improved. |
| 09 | Physiology | T-Cell Clonal Expansion Dynamics | Complete | Maintained (complete) | The T-cell dynamics analysis passed verification and is kept as-is. |
| 10 | Physiology | Cerebral Blood-Flow Autoregulation Coupling | Complete | Improved | Language-consistency review was applied, improving the write-up quality. |
| 11 | Physiology | Senescent Cell Accumulation Dynamics | Complete | Maintained (complete) | The senescent-cell accumulation analysis passed verification and is kept as-is. |
| 12 | Physiology | Respiratory Rhythm Neural Circuit | Complete | Improved | Language-consistency review was applied, improving the write-up quality. |
| 13 | Biochemistry | Protein Folding Energy Landscape | Complete | Maintained (complete) | The protein-folding analysis passed verification and is kept as-is. |
| 14 | Biochemistry | Amyloid Nucleation and Growth | Complete | Improved | Both validation transparency and language consistency were improved. |
| 15 | Biochemistry | Enzyme Transition-State Kinetics | Complete | Improved | Validation transparency was strengthened, raising narrative reliability. |
| 16 | Biochemistry | Gene Expression Noise | Complete | Improved | Both language consistency and topical alignment were strengthened. |
| 17 | Biochemistry | Metabolic Flux Homeostasis | Complete | Maintained (complete) | The metabolic-flux analysis passed verification and is kept as-is. |
| 18 | Biochemistry | Ribosome Elongation Kinetics | Complete | Improved | Topical alignment was strengthened, improving argument consistency. |
| 19 | Mathematics | Prime-Counting Function Error Term | Complete | Discrete-math specialized approach applied (in progress) | Being reworked with a discrete-math-specialized approach (probability distribution, visualization, combinatorial optimization). |
| 20 | Mathematics | Collatz Stopping-Time Problem | In progress | Discrete-math specialized approach applied (in progress) | Being reworked with a discrete-math-specialized approach (probability distribution, visualization, combinatorial optimization). |
| 21 | Mathematics | Nonlinear PDE Blow-up | Complete | Maintained (complete) | The blow-up analysis passed verification and is kept as-is. |
| 22 | Mathematics | Optimal Transport Regularity | In progress | Continued deepening interpretation scheduled (not yet started) | Not yet started; continued deepening interpretation will be applied in sequence. |
| 23 | Mathematics | Turing Pattern Bifurcation | Complete | Improved | Computation stability was strengthened so the run completes without interruption. |
| 24 | Mathematics | Random Graph Coloring Transition | In progress | Discrete-math specialized approach applied (in progress) | Being reworked with a discrete-math-specialized approach (probability distribution, visualization, combinatorial optimization). |
| 25 | Chemistry | Heterogeneous Catalyst Reaction–Diffusion | In progress | Continued deepening interpretation under way (in progress) | Continued deepening interpretation is under way; the result is not yet finalized. |
| 26 | Chemistry | Homogeneous Catalyst Selectivity | In progress | Continued deepening interpretation scheduled (not yet started) | Not yet started; continued deepening interpretation will be applied in sequence. |
| 27 | Chemistry | Li-Battery SEI Layer Growth | In progress | Continued deepening interpretation scheduled (not yet started) | Not yet started; continued deepening interpretation will be applied in sequence. |
| 28 | Chemistry | Photocatalytic Charge Transport | In progress | Continued deepening interpretation scheduled (not yet started) | Not yet started; continued deepening interpretation will be applied in sequence. |
| 29 | Chemistry | Polymorph Nucleation Competition | In progress | Continued deepening interpretation scheduled (not yet started) | Not yet started; continued deepening interpretation will be applied in sequence. |
| 30 | Chemistry | BZ Reaction Spatiotemporal Wave Pattern | In progress | Continued deepening interpretation scheduled (not yet started) | Not yet started; continued deepening interpretation will be applied in sequence. |
Note: three discrete-math problems (prime-counting error term, Collatz stopping time, random graph coloring transition) are still being reworked with a specialized approach. We label them "in progress" honestly rather than claiming completion.
Learn more
Curious how the paper pipeline actually works, or which SCI journals you could submit a finished paper to? Check the links below.
The figures and statuses on this page reflect progress at a point in time; they do not guarantee reproducibility or any specific future improvement. Internal validation/algorithm mechanisms remain undisclosed while patent applications are pending.