From 78d0a306e8484695613d72f766763a178ebd3a05 Mon Sep 17 00:00:00 2001 From: Codex Date: Wed, 29 Jul 2026 03:52:09 +0800 Subject: [PATCH 001/102] Define Challenge 194 scientific protocol Pin the long-range percolation model, validation gates, scalable sampling strategy, and preregistered inference rules so later numerical claims remain reproducible and uncertainty-controlled. Co-authored-by: Cursor --- .../frustration-free/challenge-194/DESIGN.md | 426 ++++++++++++++++++ .../challenge-194/references/README.md | 56 +++ ...newman-1986-inverse-square-percolation.pdf | Bin 0 -> 4032528 bytes ...pin-2024-long-range-1d-arxiv2011.04642.pdf | Bin 0 -> 548897 bytes ...long-range-percolation-arxiv1610.00200.pdf | Bin 0 -> 1862545 bytes ...rse-square-criticality-cond-mat0104175.pdf | Bin 0 -> 311143 bytes .../sources/gori-2017-arxiv-source.tar | Bin 0 -> 583309 bytes 7 files changed, 482 insertions(+) create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/DESIGN.md create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/references/README.md create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/references/papers/aizenman-newman-1986-inverse-square-percolation.pdf create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/references/papers/duminil-copin-2024-long-range-1d-arxiv2011.04642.pdf create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/references/papers/gori-2017-long-range-percolation-arxiv1610.00200.pdf create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/references/papers/luijten-2001-inverse-square-criticality-cond-mat0104175.pdf create mode 100644 tracks/qmc/solutions/frustration-free/challenge-194/references/sources/gori-2017-arxiv-source.tar diff --git a/tracks/qmc/solutions/frustration-free/challenge-194/DESIGN.md b/tracks/qmc/solutions/frustration-free/challenge-194/DESIGN.md new file mode 100644 index 000000000..ae1e561cf --- /dev/null +++ b/tracks/qmc/solutions/frustration-free/challenge-194/DESIGN.md @@ -0,0 +1,426 @@ +# Challenge 194 design + +## Scientific objective + +Determine what can be concluded, with controlled finite-size and model-selection +uncertainty, about the one-dimensional long-range `q=1` random-cluster model: + +1. identify and simulate the pinned finite-ring model without importing + thresholds from different short-distance or boundary conventions; +2. locate or bound the transition across a small decay-exponent map; +3. at the marginal point `sigma = 1`, test whether the finite-cluster + crossover scale supports Cardy's formal `q -> 1` continuation + `nu_tilde = 2/3`; +4. separate a giant-cluster plateau from finite-cluster connectivity before + making any statement about `eta`. + +The accepted conclusions are support, falsification on accessible scales, a +different model identification, or an explicitly inconclusive result. + +## Pinned model + +For even `L`, vertices are `Z/LZ`. Every unordered pair `{i, j}` is sampled +once and independently: + +```text +p_ij(kappa, sigma) = 1 - exp[-kappa J_L,sigma(i-j)] +J_L,sigma(r) = sum_{n in Z} |r + nL|^[-(1+sigma)]. +``` + +There are no loops, duplicate edges, distance cutoff, Kac normalization, or +independently tuned nearest-neighbor probability. At `q=1`, the FK cluster +weight is one, so the measure is an independent Bernoulli product measure. + +For `s = 1 + sigma` and `1 <= d <= L/2`, + +```text +J_L,sigma(d) = + L^(-s) [zeta(s, d/L) + zeta(s, 1-d/L)]. +``` + +At `sigma = 1`, + +```text +J_L,1(d) = (pi/L)^2 csc^2(pi d/L). +``` + +Distance classes have multiplicity `L` for `d < L/2` and `L/2` for the +antipodal class `d = L/2`. + +This model is not Gori et al.'s minimum-image `C/r^(1+sigma)` model. Its +finite-size thresholds and corrections must be determined independently. + +## Theoretical anchors and claim boundary + +At `sigma = 1`, the infinite-volume asymptotic tail coefficient is `kappa`. +The rigorous literature implies: + +- no percolation for `kappa <= 1`; +- if the percolation density `theta` is nonzero, then + `kappa theta^2 >= 1`; +- a finite threshold exists on the pinned diagonal, but its value is not + known exactly; +- the onset of `theta` is necessarily discontinuous; +- strictly subcritical pair connectivity retains an algebraic `r^-2` tail. + +These results do not establish Cardy's `nu_tilde = 2/3`, a critical `eta`, or +an exact `kappa_c`. The density jump is compatible with an essential +divergence of a finite-cluster crossover scale on the subcritical side. + +For `sigma > 1`, the infinite-volume decay exponent is greater than two and +there is no finite-`kappa` percolation transition. `sigma = 1.1` is therefore +a negative control whose pseudocritical drift must not be reported as a +finite transition. + +## Implementation architecture + +All committed implementation stays under this challenge directory. + +```text +challenge-194/ +├── README.md +├── DESIGN.md +├── PLAN.md +├── pyproject.toml +├── uv.lock +├── references/ +├── src/long_range_percolation/ +│ ├── kernel.py +│ ├── union_find.py +│ ├── oracle.py +│ ├── sampler.py +│ ├── observables.py +│ ├── artifacts.py +│ └── analysis.py +├── tests/ +└── scripts/ +``` + +Python owns mathematical reference routines, orchestration, immutable +artifacts, and analysis. A Numba-compiled production kernel owns the +large-system edge process and union-find updates. A standalone C++17 backend +is considered only if a preregistered Numba performance gate fails. + +No implementation is copied from the external ONMC repository. It is a +GPL-3.0 algorithm reference for long-range sampling and is not an oracle for +the pinned model. + +## Kernel layer + +The production kernel table is computed once for each exact `(L, sigma)` and +reused across all `kappa` values. + +- `sigma = 1`: analytic `csc^2` expression using a stable `sin(pi x)` + evaluation. +- other `sigma`: symmetric Hurwitz-zeta expression. +- independent oracle: high-precision direct image summation with a bounded + tail remainder. + +Kernel artifacts include `L`, canonical binary `sigma`, implementation +version, source revision, array hash, and analytic-identity residuals. + +## Sampling layers + +### Quadratic oracle + +For every unordered pair `i < j`, compute its distance class, form +`p = -expm1(-kappa J_d)`, sample one uniform random number, and stream open +edges into union-find. This costs `O(L^2)` time and `O(L)` memory and is used +only for exact and cross-backend validation, primarily through `L = 256`. + +### Geometric-skipping sampler + +Within one distance class, closed edges before the next open edge follow a +geometric distribution because `1 - p_d = exp(-kappa J_d)`. This gives an +unbiased `O(L + E_open alpha(L))` sampler and is the first accelerated +implementation because it is simple to audit independently. + +### Poisson/Newman-Ziff sweep + +The production path couples all `kappa` values in one realization. +Associate each edge with an independent Poisson process of rate `J_d`. The +edge is open at coupling `kappa` if its first event time is at most `kappa`. + +The total event rate is + +```text +Lambda = sum_edges J_e + = L zeta(1+sigma) [1 - L^(-(1+sigma))]. +``` + +Distance classes are sampled from an alias table with weight `M_d J_d`, and +an edge within the class is selected uniformly. Events are generated in +increasing `kappa`; duplicate events on already-open edges are ignored. +Union-find is updated incrementally and observables are recorded at the +frozen `kappa` grid. + +This produces exact Bernoulli marginals for every retained coupling while +sharing work across the full coupling scan. Because couplings within one +trajectory are correlated, the complete trajectory is one resampling unit. + +### Randomness + +Use a pinned counter-based generator keyed by: + +```text +(master_seed, L, sigma_grid_id, replica, stream_id) +``` + +Uniform-to-integer conversion uses rejection rather than modulo reduction. +Thread scheduling must not alter streams. RNG version, compiler/JIT version, +floating-point mode, and seed keys are stored in every artifact. + +## Exact validation gate + +Production runs are forbidden until all of the following pass: + +1. distance-class multiplicities sum to `L(L-1)/2`; +2. general kernel agrees with high-precision image summation; +3. every `sigma = 1` kernel entry agrees with the `csc^2` identity; +4. the global kernel sum agrees with + `L zeta(1+sigma) [1 - L^(-(1+sigma))]`; +5. no-edge probability agrees with + `exp[-kappa sum_edges J_e]`; +6. open-edge mean and variance agree with the independent Bernoulli sums; +7. all graphs for `L <= 6` reproduce exact product-measure and component + probabilities; +8. quadratic, geometric-skipping, and Poisson-sweep samplers agree for + `L <= 256` on edge frequencies, bond-length histograms, component + partitions, `S1`, and `S2`; +9. `kappa = 0`, large-`kappa`, and antipodal-edge limits pass; +10. minimum-image and image-summed kernels are demonstrated to disagree at + finite `L`, preventing accidental convention substitution. + +Statistical comparisons use preregistered simultaneous tolerances rather +than requiring bitwise equality between independent samplers. + +## Observables + +### Every realization and retained coupling + +- largest and second-largest component fractions `S1/L`, `S2/L`; +- geometric cumulant + `Q_G = sum_C |C|^4 / (sum_C |C|^2)^2`; +- a pinned four-sector crossing indicator: one component intersects all four + fixed quarter-ring arcs; +- open-edge count; +- component moments needed for consistency checks. + +### Thinned measurements + +- full `S1/L` histogram; +- exact-small and logarithmic-large finite-cluster size bins; +- bond-length histogram; +- pair connectivity at registered logarithmic separations; +- finite-cluster connectivity; +- finite-cluster structure factors for modes `m = 0, ..., 8`. + +Expensive correlation measurements use a deterministic thinning schedule +bound to the replica ID. Measurement work must remain `O(L)` per selected +graph and must not silently dominate the sampler. + +## Connectivity definitions + +The translationally averaged connectivity is + +```text +G_L(r) = E[L^-1 sum_i 1{i connected to i+r}]. +``` + +For a deterministic largest-cluster tie rule, define its pair contribution +per realization and subtract it before ensemble averaging. Subtracting +`E[S1/L]^2` is not equivalent and is forbidden. + +Report separately: + +- direct decay exponent from `G(r) ~ r^(-eta_dir)`; +- Fisher convention `eta_F = eta_dir + 1` in one dimension; +- finite-cluster connected decay; +- giant-cluster plateau; +- largest-cluster fraction. + +No `eta` claim is accepted without naming the convention and subtraction. + +## Transition location + +The two primary dimensionless estimators are: + +1. pairwise `Q_G(L)` and `Q_G(2L)` crossings; +2. four-sector crossing-probability crossings. + +Their extrapolated 95% intervals must overlap. Otherwise the transition is +reported as unresolved. + +At `sigma = 1`, the complete `S1/L` distribution is additionally tested for +one versus two components. If a stable antimode exists, an equal-weight +pseudotransition and its peak separation are tracked. The maximum slope of +`S1/L` is not a primary transition estimator. + +For `sigma > 1`, only crossover drift is reported; no finite critical point +is fitted. + +## Finite-cluster scales + +An ordinary second-moment correlation length is invalid because subcritical +connectivity has an algebraic tail. + +### Primary spectral crossover + +For finite clusters, measure + +```text +F_L(k) = L^-1 E[sum_{C != C1} |sum_{x in C} exp(ikx)|^2]. +``` + +Fit registered low-momentum mode sets to + +```text +F_L(k)^-1 = a0 + a_sigma |q|^sigma + a2 q^2, +q_m = 2 sin(pi m/L), +``` + +and define `xi_sigma = (a_sigma/a0)^(1/sigma)` only when both coefficients +are positive and resolved. Otherwise store a censored bound. Mode sets +`1:4`, `1:8`, and `2:8` are registered sensitivity checks. + +This estimator must first recover known scales from synthetic propagators +with nonanalytic tails, exponential crossovers, and pure algebraic controls. + +### Secondary mass cutoff + +Fit the site-weighted finite-cluster distribution to a power law with a +registered cutoff over controlled mass windows. The resulting `s_c` is a +systematic transition-scale check. It is converted to a spatial length only +if an independently stable mass-length relation is demonstrated. + +## Parameter and sampling plan + +### Pilot + +- `sigma = 0.8, 0.9, 1.0, 1.1`; +- `L = 2^10, 2^14, 2^18`; +- geometric grid `kappa_j = 0.25 * 1.25^j` with `kappa_j <= 6`; +- separate pilot seeds; +- bracket the common change region of both transition observables; +- benchmark oracle, geometric skipping, and Poisson sweep. + +### Frozen production + +- all retained `sigma`: `L = 2^10, 2^12, 2^14, 2^16, 2^18`; +- `sigma = 1`: add intermediate powers `2^11, 2^13, 2^15, 2^17`; +- at least 12 retained near-critical couplings at `sigma = 1`; +- at least 8 independent streams per cell or trajectory batch; +- `L = 2^20` only if the registered information-gain and runtime gate passes. + +Pilot data are excluded from confirmatory likelihoods. + +Sampling stops at a cell when both dimensionless-observable standard errors +are at most `0.01` near transition and the relative spectral-scale error is +at most `8%`, or when the registered realization ceiling is reached. + +## Preregistered scaling hypotheses + +For subcritical reduced coupling + +```text +t = (kappa_c - kappa) / kappa_c, +y = log(xi), +``` + +compare exactly four primary models: + +1. fixed Cardy continuation: `y = a + A t^(-2/3)`; +2. free essential: `y = a + A t^(-nu_tilde)`; +3. algebraic: `y = a - nu log(t)`; +4. fixed `2/3` with one registered logarithmic correction. + +`kappa_c` is refit in every bootstrap replicate and constrained by the +independent transition interval. The primary model score is leave-one- +coupling-block-out predictive performance. AICc and parametric-bootstrap +goodness-of-fit are secondary diagnostics. + +Supporting fixed `2/3` requires: + +- acceptable bootstrap goodness-of-fit; +- no residual trend in coupling or size; +- predictive performance statistically competitive with the best model; +- algebraic scaling loses by more than two score standard errors; +- the free-essential interval contains `2/3`; +- the log-correction interval contains zero; +- all registered deletion tests preserve the conclusion. + +Deletion tests raise the minimum size, delete each size, alter the coupling +window by one point, delete nearest/farthest critical points, switch spectral +mode sets, switch to the secondary scale only when the independently stable +mass-length criterion above passes, and propagate the full transition-location +uncertainty. + +Failure of stability yields an inconclusive result, not a post-hoc preferred +fit. + +## Uncertainty + +Independent graph trajectories are partitioned into immutable batches. Use +a nested bootstrap that resamples seed streams, then batches within streams, +and reruns transition interpolation, scale extraction, and nonlinear fits. +Couplings from the same monotone trajectory remain grouped. + +Report percentile intervals, covariance-aware fit diagnostics, and +simultaneous bands for primary curves. Raw batches, not only aggregated +means, are retained. + +## Artifacts and publication + +Artifacts are written to challenge-specific ignored result directories: + +```text +tracks/qmc/results/frustration-free/challenge-194// +``` + +Every run stores: + +- immutable request/configuration; +- kernel hash and validation report; +- source revision and dirty-state rejection; +- environment lock hash; +- seed manifest; +- raw batch files; +- progress/completion state; +- analysis plan hash; +- derived analysis and figure hashes. + +Files are staged, validated, fsynced, and atomically renamed. Existing valid +artifacts are never silently overwritten. Resume accepts only artifacts whose +configuration, software, and dependency hashes match. + +## Failure and stopping rules + +- Stop before production if exact enumeration or sampler agreement fails. +- Stop a backend if its measured throughput misses the frozen production + budget; switch backends only through a recorded capability gate. +- Do not extend to `L = 2^20` unless the largest current sizes are censored, + competing hypotheses differ by more than one combined standard error + there, and the projected run is below eight hours. +- Stop exponent discrimination as inconclusive if fewer than eight uncensored + scale points remain, transition uncertainty dominates the exponent + interval, or candidate predictions differ by less than two combined + standard errors over the attainable range. +- Never reinterpret a drifting `sigma > 1` crossover as a transition. + +## Minimum deliverable + +Within the hackathon window, the minimum scientifically valid result is: + +1. exact and accelerated samplers passing the full validation gate; +2. a bounded `sigma = 1` transition interval from two dimensionless + observables; +3. direct evidence for or against a density jump; +4. finite-cluster scale estimates with censored points retained; +5. fixed-`2/3`, free-essential, algebraic, and log-corrected comparisons with + deletion tests; +6. coarse controls at `sigma = 0.8, 0.9, 1.1`; +7. an honest support, falsification-on-accessible-scales, or inconclusive + verdict. + +The design does not require a positive `2/3` result. Reproducible +falsification or a demonstrated identifiability limit is an accepted outcome. diff --git a/tracks/qmc/solutions/frustration-free/challenge-194/references/README.md b/tracks/qmc/solutions/frustration-free/challenge-194/references/README.md new file mode 100644 index 000000000..a07b0bbd6 --- /dev/null +++ b/tracks/qmc/solutions/frustration-free/challenge-194/references/README.md @@ -0,0 +1,56 @@ +# Challenge 194 references + +Downloaded on 2026-07-29 for the long-range random-cluster/percolation +challenge. The PDF hashes below bind the local copies to the cited inputs. + +## Local papers + +- `papers/gori-2017-long-range-percolation-arxiv1610.00200.pdf` + - G. Gori et al., *One-dimensional long-range percolation: A numerical study* + - + - SHA256 `95fb2bbc00550f6d500cfa45b6149223bd2e43cbb1d2ded12e196c45c0066584` +- `papers/duminil-copin-2024-long-range-1d-arxiv2011.04642.pdf` + - H. Duminil-Copin, C. Garban, and V. Tassion, + *Long-range models in 1D revisited* + - + - SHA256 `44d0d458832ac87e1bb62599c9d378b2993da2ef41d061d82a0fccd8b1eec7b0` +- `papers/luijten-2001-inverse-square-criticality-cond-mat0104175.pdf` + - E. Luijten and H. Meßingfeld, + *Criticality in one dimension with inverse square-law potentials* + - + - SHA256 `127e1879b631663e7b8ca0157bb5474bf3b7ea346d9b4b466ca3c7b2d9b1fa33` +- `papers/aizenman-newman-1986-inverse-square-percolation.pdf` + - M. Aizenman and C. M. Newman, + *Discontinuity of the percolation density in one dimensional + 1/|x-y|^2 percolation models* + - DOI + - SHA256 `07fc91386b60a8b364e779c2145880b3dcca7e5a00dea3d89c0dfd9ffcc1e051` + +The Gori arXiv source archive is stored as +`sources/gori-2017-arxiv-source.tar`, SHA256 +`a56b61aefa9cfad5be88138c397eda735f8264497e81d72a84c03b64e740669e`. + +## Citation-only inputs + +The following publisher copies were not downloaded because an open PDF was +not confirmed: + +- J. L. Cardy, *One-dimensional models with 1/r^2 interactions*, + DOI . +- C. M. Newman and L. S. Schulman, + *One dimensional 1/|j-i|^s percolation models: The existence of a + transition for s <= 2*, DOI . + +## External code + +No official implementation for Gori et al. was found. A related GPL-3.0 +long-range Monte Carlo implementation was cloned for algorithm study only: + +- repository: +- revision: `28d4b5d85e80590460ec1d80c73ea5337d2fcf93` +- local ignored path: + `tracks/qmc/results/frustration-free/challenge-194/external-code/ONMC` + +ONMC does not implement the pinned q=1 periodic-image percolation model and +must not be treated as an independent scientific oracle. 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