Specific computation for the existing probability paper
The owner offered external compute. This is a bounded numerical counterpart to #740, linked to #739 and using #636 only as an implementation reference. It does not restart a general width/angle/source scan.
Compute the once-per-complete-component cylinder density nu_w^G(p) at w=4,8,12 for exactly two fixed inputs:
- independent NN SITE percolation, p=1/4;
- independent matching NN+NNN SITE percolation, p=1/8.
These are rigorously subcritical by elementary degree/path bounds (3p<1 and 7p<1); no pc decimal or moving-p fit is required. The two probabilities are not required to have equal masses.
Why these six values change the analysis
Form, separately for each graph/probability,
R_4 = nu_4*nu_12/nu_8^2,
beta_eff(4) = log(R_4)/log(4/3).
IF nu_w=A w^(-beta) exp(-kappa*w)(1+o(1)), the exponential mass and amplitude cancel and R_w->(4/3)^beta. This is one direct diagnostic of the 1/2 sewing hypothesis, not a fit of three unknown parameters. Report the contrast even when inconsistent with 1/2; a three-width agreement is not an asymptotic proof. Do not silently use a no-winding survival eigenvalue in place of a component density.
Supplied local engine and exact controls
The current owner handoff has scripts/cylinder_winding_intensity.py, a standalone one-frontier lifted-gain transfer with winding flags per active component and integer reward when a winding component is permanently retired. Appending an empty row flushes all remaining components. The count is once per COMPLETE component, not once per full row, path, cut, or marked vertex.
Widths 2/3/4 have 6/14/38 full states and 3/4/7 all-p reward-preserving stochastic lumps. At p=1/2 NN nu values are 7/48,169/1984,323849/5576960. Width-two variance per unit length is 343/6912, not its mean. All-p rational functions for both graphs are in the same handoff. nu4_w(p)=nu8_w(1-p) is an exact complementary-probability check, not equality at the same p.
A bounded capacity probe of the Python builder was actually run for NN:
w=5: 102 states / 3264 row transitions;
w=6: 282 / 18048;
w=7: 786 / 100608;
w=8: 2214 / 566784 (about 6.7 seconds here), 90 reward lumps.
Those are closure/resource probes, not new physical enumerations. Naive all-mask expansion scales poorly. For w=12 use sparse aggregation or a site-by-site factorization as needed; do not create a dense full operator or symbolic rational inverse by default. Report actual wall time, memory and state count instead of extrapolating a price from this host.
Numerical target and an available forward-error certificate
For row kernel K, reward g and stationary distribution pi, nu=pi*g. Return log nu with absolute numerical error at most 1e-8 if practical, along with the actual achieved bound. A normalized nonnegative candidate pi_hat can be certified because every row has the same empty-row reset of probability delta=(1-p)^w:
|pi_hatg - nu| <= ||g||_infinity * ||pi_hatK-pi_hat||_1 / delta.
The handoff implements this in exact rational arithmetic for small cases. At larger width use higher precision and outward bounds for the residual and g evaluation. A tiny unscaled residual alone is not relative accuracy on rare nu. Finite-width closure must preserve reward law; stochastic lumping is not configurationwise equality for a fixed sequence of row masks.
Return one result JSON and a brief interpretation to #739. Preserve full/lumped state and probability conventions, error bound, numerical precision and costs. No exponential-length torus Monte Carlo is needed. If w=12 requires a different algorithm, return the concrete bottleneck and measured cost rather than silently widening the task. No GPU, paid hardware activation, merge or new production schedule is requested by this issue.
Specific computation for the existing probability paper
The owner offered external compute. This is a bounded numerical counterpart to #740, linked to #739 and using #636 only as an implementation reference. It does not restart a general width/angle/source scan.
Compute the once-per-complete-component cylinder density nu_w^G(p) at w=4,8,12 for exactly two fixed inputs:
These are rigorously subcritical by elementary degree/path bounds (3p<1 and 7p<1); no pc decimal or moving-p fit is required. The two probabilities are not required to have equal masses.
Why these six values change the analysis
Form, separately for each graph/probability,
R_4 = nu_4*nu_12/nu_8^2,
beta_eff(4) = log(R_4)/log(4/3).
IF nu_w=A w^(-beta) exp(-kappa*w)(1+o(1)), the exponential mass and amplitude cancel and R_w->(4/3)^beta. This is one direct diagnostic of the 1/2 sewing hypothesis, not a fit of three unknown parameters. Report the contrast even when inconsistent with 1/2; a three-width agreement is not an asymptotic proof. Do not silently use a no-winding survival eigenvalue in place of a component density.
Supplied local engine and exact controls
The current owner handoff has
scripts/cylinder_winding_intensity.py, a standalone one-frontier lifted-gain transfer with winding flags per active component and integer reward when a winding component is permanently retired. Appending an empty row flushes all remaining components. The count is once per COMPLETE component, not once per full row, path, cut, or marked vertex.Widths 2/3/4 have 6/14/38 full states and 3/4/7 all-p reward-preserving stochastic lumps. At p=1/2 NN nu values are 7/48,169/1984,323849/5576960. Width-two variance per unit length is 343/6912, not its mean. All-p rational functions for both graphs are in the same handoff. nu4_w(p)=nu8_w(1-p) is an exact complementary-probability check, not equality at the same p.
A bounded capacity probe of the Python builder was actually run for NN:
w=5: 102 states / 3264 row transitions;
w=6: 282 / 18048;
w=7: 786 / 100608;
w=8: 2214 / 566784 (about 6.7 seconds here), 90 reward lumps.
Those are closure/resource probes, not new physical enumerations. Naive all-mask expansion scales poorly. For w=12 use sparse aggregation or a site-by-site factorization as needed; do not create a dense full operator or symbolic rational inverse by default. Report actual wall time, memory and state count instead of extrapolating a price from this host.
Numerical target and an available forward-error certificate
For row kernel K, reward g and stationary distribution pi, nu=pi*g. Return log nu with absolute numerical error at most 1e-8 if practical, along with the actual achieved bound. A normalized nonnegative candidate pi_hat can be certified because every row has the same empty-row reset of probability delta=(1-p)^w:
|pi_hatg - nu| <= ||g||_infinity * ||pi_hatK-pi_hat||_1 / delta.
The handoff implements this in exact rational arithmetic for small cases. At larger width use higher precision and outward bounds for the residual and g evaluation. A tiny unscaled residual alone is not relative accuracy on rare nu. Finite-width closure must preserve reward law; stochastic lumping is not configurationwise equality for a fixed sequence of row masks.
Return one result JSON and a brief interpretation to #739. Preserve full/lumped state and probability conventions, error bound, numerical precision and costs. No exponential-length torus Monte Carlo is needed. If w=12 requires a different algorithm, return the concrete bottleneck and measured cost rather than silently widening the task. No GPU, paid hardware activation, merge or new production schedule is requested by this issue.