{ "collections": [ { "collection": "expedition-5", "hypotheses": 300, "nested_leads": 30, "supplied_result_reports": 1, "readme": "sources/expedition-5/README.md", "results": [ { "name": "03_VOLUME_C", "url": "sources/expedition-5/03_VOLUME_C/VOLUME_C_COMPLETE.md", "text": "# Volume C — Three verification investigations\n\n*Incoprea Research — Expedition V: The Hidden Discovery Laboratory*\n\n| | Hypothesis | Outcome | Status |\n|---|---|---|---|\n| **C1** | H152 — 1/f noise: number or mobility fluctuation? | Both scale as $1/N$; a geometry series at fixed carrier number separates them, with slopes $-1$ versus $0$ | **Computational model result** |\n| **C2** | H231 — Laplace pressure displaces a droplet's transition | $\\Delta T = 2\\gamma T_m/(\\rho\\Delta H\\,r)$, and $\\Delta T\\cdot r$ constant is the discriminating signature | **Derivation + computation** |\n| **C3** | H007 — Butler–Volmer vs Marcus at high overpotential | BV is exactly linear in Tafel; Marcus peaks at $\\eta^*=\\lambda$ and reverses | **Derivation + computation** |\n\nAll three are theory/computation. **No empirical claim is made anywhere in this volume** — no data\nwere available in the execution environment, and none is simulated in place of data.\n\n---\n\n## C1 — separating two noise mechanisms\n\nThe point of the investigation is that the usual discriminator does not work. **Both** the McWhorter\n(number) and Hooge (mobility) models give $S_I/I^2 \\propto 1/N$, so a doping series or a bias series\ncannot separate them. What does separate them is geometry at **fixed carrier number**: vary the film\nthickness and adjust the area so $N$ is constant, which changes the trap count (area) while leaving\n$\\alpha_H$ untouched.\n\n| thickness | area | number-noise | mobility-noise |\n|---|---|---|---|\n| 5 nm | 200 µm² | 1.000 | 1.000 |\n| 20 nm | 50 µm² | 0.250 | 1.000 |\n| 100 nm | 10 µm² | 0.050 | 1.000 |\n| 500 nm | 2 µm² | 0.010 | 1.000 |\n\nA log–log slope of $-1$ indicates number fluctuation, $0$ indicates mobility fluctuation.\nThe superposition of log-uniform Lorentzians was verified to give a $1/f$ slope of **−0.992**.\n\n**Limitations.** This is a model, not a measurement. Real contacts, surface depletion and series\nresistance all break the fixed-$N$ assumption, and the report says so.\n\n---\n\n## C2 — the Laplace shift, and how to separate it from chemistry\n\n$$\\Delta T = \\frac{2\\gamma T_m}{\\rho \\Delta H\\, r}$$\n\nFor a lipid-like droplet with $\\gamma = 2$ mN/m, $\\rho = 900$ kg/m³, $\\Delta H = 2\\times10^4$ J/kg:\n13.3 K at 5 nm, 3.3 K at 20 nm, 0.67 K at 100 nm, 0.07 K at 1 µm.\n\nThe discriminating signature is **not** the magnitude but the form: a Laplace shift gives\n$\\Delta T\\cdot r$ constant, while a surfactant-chemistry effect gives a size-independent offset.\nAt 20 versus 40 nm the two differ by 1.67 K, so a resolution better than 0.33 K gives a 5σ\nseparation. A shift that persists above 1 µm is not Laplace pressure at all.\n\n---\n\n## C3 — where Butler–Volmer and Marcus part\n\nButler–Volmer is exactly linear in a Tafel plot (constant slope $2.303/\\alpha f$). Marcus is a\ndownward parabola in $\\log i$ versus $\\eta$, peaking at $\\eta^* = \\lambda$ (in volts, for a\none-electron transfer) and then *falling*.\n\n| $\\lambda$ | BV/Marcus ratio at 0.6 V | Marcus slope at 0.6 V |\n|---|---|---|\n| 0.5 eV | $1.4\\times10^{5}$ | −4.09 dec/V |\n| 0.7 eV | $1.4\\times10^{5}$ | +2.64 dec/V |\n| 1.0 eV | $5.6\\times10^{5}$ | +7.69 dec/V |\n\nThe sign of the Marcus slope at a given overpotential is the decisive observable, and it depends on\n$\\lambda$ — which is what makes the measurement informative rather than merely confirmatory.\n\n**A correction to the hypothesis.** H007 quoted a 1100× separation at 0.6 V. The computed separation\nis four to five orders of magnitude larger and depends strongly on $\\lambda$ and $\\alpha$; the quoted\nfigure is not generic. The *sign* test survives; the magnitude claim does not.\n\n---\n\n## Reproducibility\n\n```\ncd 03_VOLUME_C/C1_noise_mechanism && python3 c1_noise.py\ncd 03_VOLUME_C/C2_laplace_shift && python3 c2_laplace.py\ncd 03_VOLUME_C/C3_bv_marcus && python3 c3_bv_marcus.py\n```\n\nDeterministic; numpy only.\n" } ] } ], "manifest": [ { "collection": "expedition-5", "source": "/Users/tobymikle/Desktop/300v5", "stable_before_after": true, "files": [ { "path": "00_OVERLAP/OVERLAP_REGISTER.md", "sha256": "775155010d716876db907fdc0c0d1cf72f0e796f436eaf57af6f1a3bf208dfbd", "bytes": 6424 }, { "path": "01_VOLUME_A/A1_Model_collisions.md", "sha256": "e3e5780cd4dde4247c7e481aeb9693ae4c335d8ba18c1c042ccc7bd5361a8214", "bytes": 110163 }, { "path": "01_VOLUME_A/A2_Model_collisions.md", "sha256": "9c4894a798b92fc431776506faff1a3c6f22ab657103213dcf5901448a806d29", "bytes": 119243 }, { "path": "01_VOLUME_A/B1_Measurement_collisions.md", "sha256": 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