Incoprea Research · Supplied Round 3 · 30 October 2026
Evidence status: Passed exact model arithmetic with clarified magnitude and scope; no cell experiment or universal noise floor.
For a Poisson molecule count, relative counting noise falls as the inverse square root of the mean count. At a fixed concentration, a larger volume contains more molecules on average and therefore has lower relative counting noise. The supplied calculation verifies that conditional statement. It does not measure cellular signaling or establish that Poisson statistics are a universal lower bound on biological noise.
The original signaling hypothesis says that cells at the same concentration can differ in noise by an order of magnitude. It also attributes noise to the ratio of molecule number to volume. That ratio is concentration, so it cannot alone explain a difference between cells whose concentration is held equal. The additional quantity in this Poisson model is the mean number of molecules, cV.
Let X have a Poisson distribution with mean μ=cV, where c is concentration in molecules per cubic micrometre and V is volume in cubic micrometres. The Poisson variance equals μ, giving standard deviation √μ and coefficient of variation CV=√μ/μ=1/√(cV).
This is relative variability: standard deviation divided by the mean. Absolute count variability actually grows as √μ. For concentration X/V at a fixed, nonrandom volume, the coefficient of variation is the same as for X. Calling a larger volume “quieter” therefore refers to relative fluctuations, not fewer molecules of absolute fluctuation.
For two volumes at equal concentration, CV_small/CV_large=√(V_large/V_small). Thus an eightfold volume difference gives √8≈2.828 times different relative noise. A hundredfold volume difference gives a tenfold difference. If geometrically similar cells differ twofold in diameter, their volumes differ eightfold, so this particular size comparison gives 2.828, not ten.
The concentration is fixed at 100 molecules/µm³. The code evaluates 200 logarithmically spaced volumes from 0.1 to approximately 316.228 µm³. At each volume it generates 20,000 Poisson counts using seed 7, for four million synthetic counts in total. Its numerical check divides the simulated population standard deviation by the known theoretical mean cV.
| Volume | Mean molecule count | Exact relative noise CV |
|---|---|---|
| 0.1 µm³ | 10 | 0.316228 |
| 316.228 µm³ | 31,622.8 | 0.00562341 |
The exact curve is the evidence for the scaling law; the random draws check its implementation. The 200 volumes are parameter settings, not 200 observed cells. Finite simulations have sampling variation. The supplied JSON’s equal lower/upper fields around exact ratios are schema placeholders, not independently estimated confidence intervals.
A tenfold relative-noise difference is possible under the stated model when volumes differ hundredfold. The supplied eightfold-volume example therefore does not refute the original possibility claim. It specifies the volume ratio required. The code contains no dataset showing how frequently hundredfold differences occur within any biological cell type, so this report makes no such frequency claim.
Nor does a Poisson assumption establish the full statistics of a signaling network. Feedback, correlated reactions, molecule production and removal, partitioning, variable volume and external fluctuations can change the distribution. NIST’s Poisson reference gives the exact mean, standard deviation and CV used here. Swain, Elowitz and Siggia’s primary gene-expression analysis provides biological context for distinguishing intrinsic and extrinsic noise and departures from simple Poisson statistics.
This is the supplied DeepSeek-assisted Round 3 computation. Codex reviewed its equations and units, reproduced the unchanged script in an isolated copy, and prepared this minimally adjusted publication text. The result and numeric series reproduce byte-for-byte. No new biological study, parameter selection or physical measurement was performed. The code and supplied outputs accompany this note; the original hypothesis remains unchanged at Incoprea Research. No novelty or external peer review is claimed.