{ "collections": [ { "collection": "expedition-4", "hypotheses": 300, "nested_leads": 30, "supplied_result_reports": 3, "readme": "sources/expedition-4/README.md", "results": [ { "name": "C1_nv_rosette", "url": "sources/expedition-4/03_VOLUME_C/C1_nv_rosette/REPORT.md", "text": "# C1 — The NV stress rosette: an exact blind subspace, and where it closes\n\n**From:** E4-H227, rank 1/30 · **Status: FORMAL MATHEMATICAL RESULT (linear-response model) +\nCOMPUTATIONAL VERIFICATION.**\n\n---\n\n## 1. The claim, refined\n\nFor an ensemble of nitrogen-vacancy centres with the four ⟨111⟩ axes, the axial (D) shift\nresponds to stress through the two allowed scalar couplings\n\n$$\\delta D_i = a_1\\,(n_i\\cdot\\sigma\\cdot n_i) + a_2\\,\\mathrm{tr}\\,\\sigma, \\qquad i=1\\ldots4,$$\n\nand the transverse (E) splitting through the two-component transverse projection. The hypothesis\nasserts that the four D shifts leave a two-dimensional stress subspace unconstrained, that the\nunconstrained subspace is the diagonal traceless family, and that the E channel closes it.\n\n**Result:** the first two claims are **proved exactly**; the third is **proved with a correction** —\nit holds only when the E splitting is treated as the two-component vector it is, and my first\nattempt (a single transverse axis per orientation) gave rank 5 and a spurious residual blindness.\n\n---\n\n## 2. Derivation\n\nLet $n_i$ be the four unit ⟨111⟩ axes. Because $\\sigma$ is symmetric and $D$ is a scalar under the\nsite's $C_{3v}$ symmetry, the linear response contains exactly two invariant couplings: the trace and\nthe axial projection. Assembling the $4\\times6$ matrix $M$ with rows\n$M_i = \\nabla_\\sigma[\\delta D_i]$:\n\n**Rank.** The four rows satisfy $\\sum_i n_in_i^{T} = \\tfrac{4}{3}I$, so the *axial* parts are not\nindependent in the way one expects; the matrix has **rank 4** and a **two-dimensional null space**.\n\n**Null space (exact, by symbolic elimination).** The two basis vectors are\n$(-1,1,0,0,0,0)$ and $(-1,0,1,0,0,0)$ in Voigt order — i.e. every **diagonal traceless** stress\n$\\sigma=\\mathrm{diag}(a,b,-a-b)$. The reason is immediate:\n\n$$n_i\\cdot\\sigma\\cdot n_i = \\tfrac{1}{3}(a+b+c) = 0 \\quad\\text{for all four } n_i\n\\text{ whenever } a+b+c=0 .$$\n\nSo a pure [100]-type uniaxial stress (or any diagonal deviator in the crystal frame) is **exactly\ninvisible** to all four D shifts simultaneously, not merely weakly coupled.\n\n**The E channel closes it.** The E splitting of each orientation is a two-component object (magnitude\nand orientation in the transverse plane), so it supplies **eight** further linear functionals. The\nresulting $12\\times6$ matrix has **rank 6**: the full symmetric stress tensor is determined.\n\n---\n\n## 3. Computational verification\n\n`c1_rosette.py`, exact linear algebra plus sympy null space.\n\n```\nrank of 4x6 D-only matrix = 4 null-space dimension = 2\nnull basis (-1,1,0,0,0,0), (-1,0,1,0,0,0) -> diag(a, b, -a-b)\nmax |M v| for diagonal traceless = 0.0 to 4e-16 (annihilated exactly)\nmax |M e_j| for off-diagonals = 1.333 (fully visible)\nrank of 12x6 D+E matrix = 6\n```\n\n**Spectra.** For a [100] load: $\\delta D = (+1.333,+1.333,+1.333,+1.333)$ — four equal shifts — while\nthe E components alternate $(+0.471,+0.471,-0.471,-0.471)$. For a [111] load:\n$\\delta D = (+2.000,+1.111,+1.111,+1.111)$ — unequal — with the axial E component zero for the\naligned orientation. The two load geometries are therefore distinguishable in a single spectrum,\nexactly as the hypothesis claims.\n\n---\n\n## 4. The correction I had to make\n\nMy first pass used **one** transverse unit vector per orientation, which produced an $8\\times6$\nmatrix of **rank 5** and a residual null vector $\\mathrm{diag}(0,-1,1)$. That looked like a genuine\nfailure of the E channel to close the blind subspace.\n\nIt was an artefact: the E splitting is a two-component vector, and one arbitrary transverse axis\ndiscards half of it. Restoring both components gives rank 6. The lesson is recorded because the\nfalse result was *more interesting* than the true one and would have been easy to publish.\n\n---\n\n## 5. Consequence for published pressure measurements\n\nAny experiment that infers pressure from a **single scalar D shift** averages over the four\norientations and is blind to the diagonal deviator. Since that deviator is exactly what decides\nwhether a transition is driven by shear or by compression, a D-only \"pressure\" is really a\nhydrostatic-plus-trace reading. The four-orientation rosette materialises the deviator in the\n*split* between the four lines, and the E terms in their fine structure.\n\n**Practical statement:** the calibration constant $a_1/a_2$ is measurable on the same crystal from\nthe ratio of the split to the mean, so no absolute pressure scale is needed to obtain the shear state.\n\n---\n\n## 6. Status\n\n| Statement | Status |\n|---|---|\n| D-only response has rank 4 and a 2D null space | **Formal proof** |\n| The null space is exactly the diagonal traceless family | **Formal proof**, verified to machine precision |\n| A [100] load gives four equal D shifts with alternating E signs | **Proved and computed** |\n| A [111] load gives unequal D shifts with zero axial E | **Proved and computed** |\n| D plus the full E channel determines all six components | **Formal proof** (12×6 rank 6) |\n| An 8-equation D+single-E system is over-determined | **Refuted** — my own earlier framing; rank is 5 |\n| Measured in a diamond-anvil cell | **Open** — needs an oriented ensemble with narrow depth distribution |\n\n## 7. Reproducibility\n\n```\ncd 03_VOLUME_C/C1_nv_rosette\npython3 c1_rosette.py\n```\nDeterministic; numpy + sympy; runtime < 5 s.\n" }, { "name": "C2_array_range", "url": "sources/expedition-4/03_VOLUME_C/C2_array_range/REPORT.md", "text": "# C2 — Aperture–dispersion limit on wavelength-steered array range resolution\n\n**From:** E4-H160, rank 15/30 · **Status: DERIVATION + COMPUTATIONAL VERIFICATION.**\n\n---\n\n## 1. The claim, refined\n\nFor a grating-type optical phased array whose beam is steered by wavelength,\n\n$$\\sin\\theta = \\lambda/d \\;\\Rightarrow\\; \\frac{d\\theta}{d\\lambda}=\\frac{1}{d\\cos\\theta},\n\\qquad \\Delta\\theta_{\\rm beam}=\\frac{\\lambda}{Nd\\cos\\theta}.$$\n\nThe target is illuminated only while the beam covers it, so the **effective** chirp bandwidth is the\nfrequency excursion that moves the beam by one beamwidth:\n\n$$\\Delta\\lambda_{\\rm eff}=\\frac{\\Delta\\theta_{\\rm beam}}{d\\theta/d\\lambda}=\\frac{\\lambda}{N},\n\\qquad B_{\\rm eff}=\\frac{c}{N\\lambda},$$\n\nand therefore\n\n$$\\boxed{\\delta R \\;\\ge\\; \\frac{c}{2B_{\\rm eff}} \\;=\\; \\frac{N\\lambda}{2}\n\\;=\\;\\frac{\\lambda}{2}\\,D\\,\\frac{d\\theta}{d\\lambda}}$$\n\n**The direction is the surprise.** A larger aperture narrows the beam, shortens the frequency range\nover which the target stays lit, and therefore makes the range floor **worse**. Angular resolution\nand range resolution trade directly against each other in this architecture.\n\n---\n\n## 2. Computational check\n\n`c2_range_limit_v2.py`: the range PSF is the Fourier transform of the illumination envelope (a\nGaussian of FWHM $B_{\\rm eff}$ truncated at the chirp edges).\n\n| $N$ | $D$ (mm) | $B_{\\rm eff}$ (GHz) | floor $N\\lambda/2$ |\n|---|---|---|---|\n| 256 | 0.51 | 756 | 198 µm |\n| 1024 | 2.05 | 189 | 794 µm |\n| 4096 | 8.19 | 47.2 | 3.17 mm |\n| 8192 | 16.4 | 23.6 | 6.35 mm |\n\nAt $N=4096$, the ideal resolution $c/2B$ falls from 15 mm at 10 GHz to 15 µm at 10 THz, while the\nsimulated PSF width **plateaus near 2 mm** and stops responding to bandwidth. At 100×$B_{\\rm eff}$\nthe achievable resolution equals the floor for every aperture tested.\n\n---\n\n## 3. Where the numbers do not quite meet\n\nThe scaling $N\\lambda/2$ is confirmed; the **constant** is criterion-dependent. The simulated FWHM\nsaturates at $\\approx 2$ mm for $N=4096$ against the analytic $N\\lambda/2 = 3.17$ mm — a ratio of\n0.63, which is the Gaussian-FWHM versus Rayleigh-style convention factor (0.44/0.5 for a transform-\nlimited Gaussian diluted by the truncation). I report the **scaling** as the result and the constant\nas convention-dependent, rather than quoting a spurious agreement.\n\n---\n\n## 4. Stress tests\n\n- **Is the effect just the beam walking off the target?** Yes — that *is* the mechanism. The point is\n that it is unavoidable in a wavelength-steered array, and it is absent in a phase-steered array,\n which decouples pointing from frequency.\n- **Does a larger target help?** A target subtending many beamwidths would stay partly lit and soften\n the limit; the floor is a point-target result and should be stated as such.\n- **Numerical:** the FFT uses $2^{15}$ points; halving it changes the FWHM by <2%.\n\n---\n\n## 5. Status\n\n| Statement | Status |\n|---|---|\n| $B_{\\rm eff}=c/(N\\lambda)$ and $\\delta R\\ge N\\lambda/2$ | **Derived** |\n| The floor scales with element count (worse with larger aperture) | **Derived and computed** |\n| Bandwidth beyond $B_{\\rm eff}$ buys no resolution | **Computed** (plateau) |\n| The absolute constant vs a Gaussian FWHM criterion | **Convention-dependent** — stated, not resolved |\n| Measured on a chip array | **Open** — a programmed-aperture sweep would test it directly |\n\n## 6. Reproducibility\n\n```\ncd 03_VOLUME_C/C2_array_range\npython3 c2_range_limit_v2.py\n```\nDeterministic; numpy only; runtime < 60 s.\n" }, { "name": "C3_haven_noise", "url": "sources/expedition-4/03_VOLUME_C/C3_haven_noise/REPORT.md", "text": "# C3 — Can a noise measurement return the Haven ratio?\n\n**From:** E4-H230, rank 9/30 · **Status: INCONCLUSIVE / PARTIAL.** The estimator was implemented and\nrun; the statistics were not sufficient to test the predicted scaling. This report records what was\ndone, what the data showed, and exactly what would be needed.\n\n---\n\n## 1. The claim\n\nThe Haven ratio $H = D^{*}/D_\\sigma$ measures how correlated successive ion hops are; it normally\nrequires a tracer experiment plus impedance on two samples. The hypothesis is that the low-frequency\nfield-noise amplitude seen by a nearby spin sensor, together with the conductivity, returns $H$\nlocally — specifically that\n\n$$\\frac{S_E(0)}{6k_BT\\sigma}=\\frac{A}{H},$$\n\nwith $A$ an apparatus geometric factor.\n\n---\n\n## 2. What was implemented\n\n`c3_haven.py`: a two-dimensional lattice gas on a $32\\times32$ periodic square lattice with site\nexclusion, at occupancy fractions 0.1–0.4. Hops are attempted at random to nearest neighbours and\nsucceed only into empty sites; under a bias $\\varepsilon=eEa/k_BT$ the hop probabilities are\n$e^{\\pm\\varepsilon/2}$ along the field.\n\nMeasured: the tracer mean-square displacement (from which $D^{*}$), the centre-of-mass drift under a\nsmall bias (from which $D_\\sigma = v/\\varepsilon$ in lattice units), and the implied $H$.\n\n---\n\n## 3. Results\n\n| occupancy | $D^{*}$ | $D_\\sigma$ | $H = D^{*}/D_\\sigma$ |\n|---|---|---|---|\n| 0.10 | $1.0\\times10^{-5}$ | $7.1\\times10^{-5}$ | 0.10 |\n| 0.20 | $6.8\\times10^{-4}$ | $1.6\\times10^{-3}$ | 0.43 |\n| 0.30 | $1.0\\times10^{-3}$ | $8.7\\times10^{-4}$ | 1.19 |\n| 0.40 | $4.1\\times10^{-3}$ | $2.7\\times10^{-4}$ | **15.3** |\n\n**These numbers are not physical.** A Haven ratio of 15 is impossible for this model; $H$ should lie\nbetween roughly 0.5 and 1 for a lattice gas with exclusion. The estimator is under-sampled:\n- at occupancy 0.1 there are only ~100 ions on $32^2$ sites, and the tracked tracer makes too few\n hops in 60 000 attempted moves to give a reliable MSD slope;\n- the drift estimate uses the mean of a periodic coordinate, which wraps and biases the mean;\n- the bias $\\varepsilon=0.02$ gives a drift comparable to the fluctuation of the mean.\n\nNo amount of re-framing turns these numbers into a test of the hypothesis.\n\n---\n\n## 4. What the run does establish\n\n- The **estimator is implementable**: tracer MSD, drift under bias, and conductivity are all\n accessible in one simulation, and the code runs.\n- The tracer and charge diffusion coefficients **do differ**, and the ratio moves with occupancy, so\n the quantity the hypothesis wants to measure is not identically one. That is a necessary condition\n and it is met.\n- The predicted *scaling* ($S_E(0)\\propto\\sigma/H$) was **not tested**: no noise spectrum was\n computed to convergence, and the $A$ factor is geometry-dependent and not fixed here.\n\n---\n\n## 5. What would make it decisive\n\n1. Lattice $128^2$ or larger with $10^{7}$ attempted moves per state point — a factor $\\sim10^{3}$ in\n statistics over what was affordable here.\n2. Drift measured from the **unwrapped** displacement of every ion, not the mean of a periodic\n coordinate.\n3. The noise spectrum computed as the Fourier transform of the total-dipole autocorrelation over at\n least $10^{3}$ independent segments, with the $S(0)$ plateau demonstrated rather than assumed.\n4. A geometry calibration $A$ obtained from a known-conductivity reference, so the absolute relation\n is tested rather than only its slope.\n\nNone of these is conceptually difficult; the run was simply not powered.\n\n---\n\n## 6. Status\n\n| Statement | Status |\n|---|---|\n| A single simulation can return $D^{*}$, $D_\\sigma$ and hence $H$ | **Implemented** |\n| $H$ differs from 1 and moves with occupancy | **Indicated**, but the magnitudes are unusable |\n| $S_E(0)/(6k_BT\\sigma)=A/H$ | **Untested** |\n| The measurement is feasible with a spin sensor | **Open** — plausible, not demonstrated |\n\n**This is the honest outcome: an inconclusive investigation.** It is reported as such rather than\ngiven a favourable spin, and the recipe for making it decisive is above.\n\n## 7. Reproducibility\n\n```\ncd 03_VOLUME_C/C3_haven_noise\npython3 c3_haven.py\n```\nDeterministic (seeds fixed); numpy only; runtime ≈ 5 min. 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