# C2 — Adding a pipe: the hypothesis is false, and why

**From:** E3-H031, rank 2/30 · **Status: REFUTED AS STATED**, with the reason identified and the
correct formulation stated.

---

## A. The claim under test

Phase I: *"Adding a pipe can lower a network's total throughput."*

This is the Braess paradox transplanted from traffic to hydraulics. The investigation finds that
the transplant fails, for a reason that is structural rather than numerical.

---

## B. What the computation shows

Two formulations were solved numerically for a four-node crossed-path network
($s\!\to\!A$ fast, $A\!\to\!t$ slow; $s\!\to\!B$ slow, $B\!\to\!t$ fast; optional bridge $A\!-\!B$),
with power-law head loss $\Delta p = r\,q|q|^{n-1}$.

**Formulation A — fixed pressure difference.** Throughput with and without the bridge:

| $n$ | without | with bridge | change |
|---|---|---|---|
| 1.0 | 1.818 | 4.194 | $+2.375$ |
| 1.5 | 1.877 | 3.182 | $+1.305$ |
| 2.0 | 1.907 | 2.807 | $+0.900$ |
| 3.0 | 1.937 | 2.483 | $+0.545$ |

Throughput never falls, for any exponent.

**Formulation B — fixed total demand.** The head required to deliver $Q=1$:

| $n$ | without | with bridge | change |
|---|---|---|---|
| 1.0 | 0.5500 | 0.2385 | $-0.3115$ |
| 1.5 | 0.3889 | 0.1762 | $-0.2127$ |
| 2.0 | 0.2750 | 0.1269 | $-0.1481$ |

The head *falls* in every case. Scanning bridge resistance from $10^{-3}$ to $50$ produces only
improvement (penalty $-0.159$ to $-0.027$). The symmetric-path control gives an exactly zero change
to $10^{-12}$: the bridge connects equipotential nodes and does nothing.

---

## C. Why it must be so

The Braess paradox is a property of **selfish routing** — a Wardrop equilibrium in which each user
minimises its own cost and no one minimises the total. A passive pipe network is not that. Its flow
distribution extremises a convex dissipation functional:

$$\Phi(q)=\sum_e \frac{r_e}{n+1}|q_e|^{n+1}\quad\text{subject to flow conservation.}$$

Adding an edge enlarges the feasible set, so the minimum of a convex functional cannot increase.
The conjugate quantity — total flow at fixed head, or head at fixed flow — therefore cannot get
worse. This is the same monotonicity that gives Rayleigh's theorem for linear networks; it survives
nonlinearity as long as the element law is monotone and the flow is variationally determined.

**The hypothesis fails because it imports an equilibrium concept from a different physics.** In a
pipe network there are no self-interested agents to create the inefficient equilibrium the paradox
requires.

---

## D. What would rescue it

Two routes exist, and both are testable:

1. **Selfish routing in a physical system.** If the "flow" is a population of agents each choosing a
   route on measured delay — vehicles, packets under congestion control, or active particles
   following a chemical trail — the equilibrium is Wardrop-like and Braess can occur. That is a
   claim about traffic or about active matter, not about pipes.
2. **Non-monotone element laws.** A component whose $\Delta p(q)$ decreases over some range — a
   pressure-compensated valve, a fluidic element with hysteresis, or a segment exhibiting negative
   differential resistance — breaks the convexity that the monotonicity argument needs. Then an
   added parallel path can shift the operating point onto a falling branch and reduce total flow.

The corrected claim for Volume B is: **"Adding a component can raise the pumping head required for
a fixed delivery only if the flow distribution is determined by individual optimisation rather than
by a global variational principle, or if some element's characteristic is non-monotone."**

---

## E. Status

| Statement | Status |
|---|---|
| Adding a pipe lowers throughput in a passive power-law network, fixed $\Delta p$ | **Refuted** (numerically, $n=1$–$3$; provably by convexity) |
| Adding a pipe raises required head at fixed demand | **Refuted** (numerically; same argument) |
| The paradox requires selfish routing or non-convex elements | **Established reasoning**, consistent with the classical literature |
| A physical realisation exists | **Open** — candidate systems named, none tested |

**This is the most valuable kind of outcome for the programme:** a hypothesis that looked plausible
because it borrowed a famous result from another field, found to rest on an equilibrium concept that
does not apply. The refutation is cheap, decisive, and it clarifies what would have to be true.

## Reproducibility

```
cd 03_VOLUME_C/C2_braess_networks
python3 c2_braess.py
```
Deterministic Newton solves; numpy only; runtime < 5 s. Note that the $n=3$ demand-fixed case is
excluded by a convergence guard in the script rather than reported.