R6 — simulate the paper's model to test whether the tick is the mechanism

[brianmmaina]

The problem with R5

The dp/p_c finding was a correlation on five points, and dp/p_c is large for exactly INTC and MSFT — also the two cheapest, highest-volume, most heavily quoted names in the sample. A dozen things that separate a $30 stock from a $500 stock would produce the same ordering. R5 could not distinguish "the tick causes the failure" from "dp/p_c is a proxy for being cheap."

Simulation holds everything else fixed

tools/zi_paper.cpp implements the paper's model — the austere one: single order size, uniform deposition, equal rates, constant cancellation — on this project's matching engine. Run at each stock's measured (α, μ, δ, σ) and its real tick size, nothing about a cheap stock is present except four flow parameters and dp.

Prediction stated before running: small dp/p_c → simulated ratio near 1; large dp/p_c → ratio near that stock's empirical ratio. The second half is risky — a coarse grid inflating the spread is easy, matching the magnitude of the real inflation is not.

	dp/p_c	sim ratio	real ratio	sim/real
GOOG	0.21	0.66 ± 0.01	4.36	0.15
AAPL	0.35	0.72 ± 0.01	3.70	0.19
AMZN	0.76	0.83 ± 0.01	5.66	0.15
INTC	17.30	32.23	39.75	0.81
MSFT	22.03	40.87	50.35	0.81

The tick alone reproduces 81% of the observed inflation — the same fraction for both constrained stocks, which nothing here was tuned to produce.

Simulated inflation is perfectly rank-ordered by dp/p_c (ρ = 1.000, exact p = 0.017). Inside a simulation that is far stronger than the same number on real data: dp/p_c and the four flow parameters are the only things that vary, so there is no confounder left to appeal to.

What this does not show, kept in the writeup

• The remaining 19% is real. The tick is the dominant term, not the whole story. What's left is presumably what the model deletes — strategic quoting, hidden liquidity, heterogeneous sizes.
• The small-tick ratio is 0.66–0.83, not 1.0. The simulation runs ~25% below the mean-field prediction, consistently. f(ε) is itself an approximation so a gap this size is unremarkable, but it is a gap and it is not rounded to "agreement."

The width scan caught a real bug

The paper's deposition intervals are semi-infinite; a simulation must truncate them. If the answer moves when the truncation moves, the boundary is setting the spread and the whole result is an artifact.

The first implementation truncated to a fixed price box centred at zero. The scan returned:

32.23, 32.23, 32.23, 0.00, 32.23

Non-monotonic in the width is a bug, not a boundary effect. The book could pin its best bid against the top of the box, at which point the sell interval [b+1, box_top] was empty and no sell order could ever arrive again — an absorbing one-sided state that silently reported a spread of zero. The same box also made buy and sell arrival rates depend on where the price sat inside it, quietly breaking the model's equal-rates assumption.

Anchoring each interval to the opposing best quote — buys on [a(t)−W, a(t)−1], sells on [b(t)+1, b(t)+W] — gives constant equal widths and lets prices wander. After the fix: GOOG 0.66/0.68/0.65/0.69/0.70 across a 16× width range, INTC 32.23 at every width.

Without the scan the fixed-box version would have produced a plausible-looking table.

Design notes

• Separate tool, not a flag on zi_sim. zi_sim uses empirical sizes and an empirical placement histogram — a better imitation of a market and a worse test of this paper. Mixing them would let a difference in the model masquerade as a difference in the law.
• Parameters come from farmer2005.measure_all, not re-measured, so a difference in measurement cannot masquerade as a difference in the model. Same reason compare.py imports stylized_facts.
• By-product: an independent exercise of the matching engine — a CDA driven for millions of events against an analytically known answer, agreeing to within a constant wherever that answer applies.

191 tests pass; -Werror clean.