The Markov parameters of a 24-state model: 10 exact zeros, then 33 orders

At its defaults it draws the markov parameters of a 24-state model: 10 exact zeros, then 33 orders. |CAᵏB| against k. The first 10 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 10 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 2490.14, just under four times larger. The measured ratio between one parameter and the next falls from 1.375·10⁴ to 3366.9 across the run, from above, converging on it, and the run as a whole spans 2.37·10³³ — 33 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

markov-blowup is one function in lib/figures/lti.js — a model that is a rational function — the degree that is a rank, by two routes that disagree. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

|CAᵏB| against k. The first 10 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 10 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 2490.14, just under four times larger. The measured ratio between one parameter and the next falls from 1.375·10⁴ to 3366.9 across the run, from above, converging on it, and the run as a whole spans 2.37·10³³ — 33 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

|CAᵏB| against k. The first 10 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 10 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 2490.14, just under four times larger. The measured ratio between one parameter and the next falls from 1.375·10⁴ to 3366.9 across the run, from above, converging on it, and the run as a whole spans 2.37·10³³ — 33 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

|CAᵏB| against k. The first 6 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 6 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 1146.16, just under four times larger. The measured ratio between one parameter and the next falls from 4046 to 1330.3 across the run, from above, converging on it, and the run as a whole spans 2.01·10²⁹ — 29 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

|CAᵏB| against k. The first 10 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 10 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 2490.14, just under four times larger. The measured ratio between one parameter and the next falls from 1.375·10⁴ to 5887.7 across the run, from above, converging on it, and the run as a whole spans 6.35·10¹¹ — 12 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

|CAᵏB| against k. The first 10 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 10 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 2490.14, just under four times larger. The measured ratio between one parameter and the next falls from 1.375·10⁴ to 2788.5 across the run, from above, converging on it, and the run as a whole spans 1.25·10⁶⁸ — 68 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

|CAᵏB| against k. The first 4 are exactly zero — not small, zero, with no rounding in them — because A is a three-point stencil and information takes one step per grid point to travel the 4 points from the actuator to the sensor. After that they grow like ‖A‖₂ᵏ, and ‖A‖₂ is not the (n+1)² the stencil is scaled by: it is 4(n+1)²sin²(nπ / 2(n+1)) = 666.178, just under four times larger. The measured ratio between one parameter and the next falls from 1690 to 688.52 across the run, from above, converging on it, and the run as a whole spans 4.05·10⁴³ — 44 orders. A Hankel matrix of these numbers has a dynamic range no rank decision can see through, which is why the textbook route to the McMillan degree returns the wrong integer while the route that reads only samples of H returns the right one.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

9 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.