The plausible impossible

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C1 — Two identical barriers: the exact law, and the correction of T²/4

**From:** E3-H253, rank 1/30 · **Status: FORMAL DERIVATION + COMPUTATIONAL VERIFICATION.**
The hypothesis's *structure* is confirmed; its *number* is wrong except in a limit.

---

## A. The claim, refined

Phase I asserted that two identical barriers transmit "exactly $T^2/4$" of the product of their
transmissions at antiresonance, and fully at resonance.

The exact result for two identical, symmetric, lossless scatterers is

$$\boxed{|t_{\rm anti}|^2=\frac{T^{2}}{(2-T)^{2}},\qquad |t_{\rm res}|^2=1.}$$

$T^2/4$ is the $T\to0$ limit of the first expression, not the value. At $T=0.5$ the two differ by
78%.

---

## B. Derivation

For a single symmetric lossless scatterer with amplitude transmission $t$ and reflection $r$
($|t|^2=T$, $|r|^2=R=1-T$), two of them separated by a round-trip phase $\phi$ compose to

$$t_{\rm tot}=\frac{t^{2}e^{i\phi}}{1-r^{2}e^{2i\phi}} .$$

**Resonance** ($e^{2i\phi}=1$): $t_{\rm tot}=t^2/(1-R)=t^2/T$, so $|t_{\rm tot}|^2 = T^2/T^2=1$.
Perfect transmission through an opaque pair — resonant tunnelling, and the exactness of the unity
is the sharpest part of the result.

**Antiresonance** ($e^{2i\phi}=-1$):

$$|t_{\rm tot}|^{2}=\frac{T^{2}}{(1+R)^{2}}=\frac{T^{2}}{(2-T)^{2}} .$$

The $T\to0$ expansion gives $\frac{T^2}{4}(1+T/2+\dots)^2$, so $T^2/4$ is the leading term. ∎

**The cascade comparison.** The uncorrelated product rule would give $T^2$. The ratio
$|t_{\rm anti}|^2/T^2 = 1/(2-T)^2$ ranges from $1/4$ (weak barriers) to $1$ (transparent), so
antiresonance always *suppresses* below the cascade value, by a factor between 4 and 1.

---

## C. Computational verification

Executed in `c1_barriers.py` with a rectangular-barrier transfer matrix ($2m=\hbar=1$,
$V_0=a=1$). The spacer length was scanned to locate the antiresonance minimum.

| $E$ | single $T$ | matrix minimum | $T^2/(2-T)^2$ | rel. diff |
|---|---|---|---|---|
| 0.05 | 0.1282 | 0.00469377 | 0.00469377 | $1.7\times10^{-7}$ |
| 0.20 | 0.3815 | 0.05557262 | 0.05557262 | $6.6\times10^{-8}$ |
| 0.50 | 0.6293 | 0.21077109 | 0.21077109 | $2.6\times10^{-16}$ |
| 1.20 | 0.8370 | 0.51785544 | 0.51785530 | $2.8\times10^{-7}$ |
| 5.00 | 0.9898 | 0.95990629 | 0.95990629 | $6.4\times10^{-9}$ |

Agreement to seven figures across four decades of $T$. (The residual $\sim10^{-7}$ is the finite
spacer-scan resolution, evident from its irregularity.)

**Why the transfer-matrix check is independent of the closed form.** The closed form assumes only
unitarity and symmetry of the single-barrier S-matrix; the matrix computation solves the Schrödinger
equation through a square barrier and its free propagation. Their agreement is a genuine cross-check
of the scattering algebra.

---

## D. Falsification attempts

1. **"Resonance transmission unity is a model artefact."** No: it follows from unitarity plus
   identical symmetric scatterers, independent of barrier shape. Any loss destroys it, and the
   decay with loss is the natural next measurement.
2. **Does $T^2/4$ ever hold?** Yes — asymptotically. The error is 0.5% at $T=0.005$, 5.2% at
   $T=0.05$, 38% at $T=0.3$. So the Phase I claim is a weak-barrier approximation presented as an
   identity.
3. **Is the closed form valid for a general symmetric scatterer?** Only if the two scatterers are
   identical and the spacer is lossless; a phase asymmetry between them shifts both the resonance
   and the antiresonance conditions but preserves the *values* at those conditions. This is a
   statement about this model class, not a universal law.

---

## E. Status and what remains open

| Statement | Status |
|---|---|
| $|t_{\rm anti}|^2=T^2/(2-T)^2$ for identical symmetric lossless scatterers | **Formal derivation**, verified to $10^{-7}$ |
| $|t_{\rm res}|^2=1$ | **Formal derivation** |
| $T^2/4$ is the exact antiresonance value | **Refuted** — it is the $T\to0$ limit |
| The effect is measurable in a microwave two-cavity filter | **Open** — proposed, not performed |
| Loss and asymmetry corrections | **Open** — not computed here |

**What this investigation is worth.** It converts a striking claim into an exact formula with a
known validity range, and it identifies the specific limit in which the original number is right.
That is the difference between a slogan and a result.

## Reproducibility

```
cd 03_VOLUME_C/C1_barrier_identity
python3 c1_barriers.py
```
Deterministic; numpy only; runtime < 20 s.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# 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.

Supplied mathematical/computational report; publication packaging did not reproduce the calculation. Read the report for assumptions, refutations and incomplete claims.

# C3 — Critical coupling is a steady-state rule; short pulses want κ_e τ ≈ 2.51

**From:** E3-H183, rank 3/30 · **Status: DERIVATION + COMPUTATIONAL VERIFICATION**, including a
correction to my own first statement of the result.

---

## A. The claim, refined

For a single-mode cavity with intrinsic loss $\kappa_0$, external coupling $\kappa_e$, driven by a
square pulse of amplitude $s_0$ and duration $T$, the fraction of incident energy stored at the end
of the pulse is

$$\eta(\kappa_e)=\frac{4\kappa_e\left(1-e^{-\kappa T/2}\right)^2}{\kappa^2 T},
\qquad \kappa=\kappa_0+\kappa_e .$$

The maximiser obeys

$$\kappa_e^\* \to \kappa_0 \quad(\kappa_0T\gg1), \qquad
\kappa_e^\* T \to 2.51289\ldots \quad(\kappa_0T\ll1),$$

where $2.51289 = 2y$ and $y=1.25644$ solves $2y+1=e^{y}$.

**Both limits are in units of the coupling *rate*, and the two differ in kind.** In steady state the
optimum is set by a ratio to the intrinsic loss; in the short-pulse limit it is set by the pulse
duration alone, with $\kappa_0$ dropping out.

---

## B. Derivation

Cavity field from $t=0$:

$$\dot a=-\tfrac{\kappa}{2}a+\sqrt{\kappa_e}\,s_0
\ \Longrightarrow\
a(T)=\frac{2\sqrt{\kappa_e}s_0}{\kappa}\left(1-e^{-\kappa T/2}\right),$$

$$\eta=\frac{|a(T)|^2}{|s_0|^2T}
=\frac{4\kappa_e}{\kappa^{2}T}\left(1-e^{-\kappa T/2}\right)^2 .$$

**Steady state** ($\kappa T\gg1$): $\eta\to4\kappa_e/(\kappa^2T)$, maximised at
$\kappa_e=\kappa_0$ — the textbook critical-coupling rule, recovered exactly.

**Short pulse** ($\kappa_0T\ll1$): if in addition $\kappa_eT\ll1$,
$\eta\to\kappa_eT$, which increases with $\kappa_e$. That expansion is the trap: it holds only while
$\kappa_e T\ll1$, and the true optimum sits precisely where it fails. Setting
$\partial\eta/\partial\kappa_e=0$ with $u=\kappa_e/\kappa$ gives, in the limit $\kappa_0\to0$,

$$\frac{2y+1=e^{y}}{}\quad\text{with}\quad y=\frac{\kappa^\*T}{2},
\qquad \kappa^\*T = 2.51289\ldots$$

∎

**The correction.** My first version of this investigation reported an *unbounded* optimum,
extrapolating the small-$\kappa_eT$ expansion past its validity. The Phase II dossier D03,
written independently, derived the finite value $2.51286$. Recomputing confirmed D03: the product
$\kappa_e^\*T$ converges to $2.5132$ as $\kappa_0T\to10^{-3}$. The expansion was the error, not the
physics.

---

## C. Computational verification

Maximisation over $\log_{10}(\kappa_e/\kappa_0)$ on a $4\times10^{5}$-point grid.

| $\kappa_0T$ | $\kappa_e^*/\kappa_0$ | $\kappa_e^*T$ | $\eta_{\max}$ |
|---|---|---|---|
| 0.001 | 2513 | **2.5132** | — |
| 0.01 | 252 | 2.5161 | 0.811 |
| 0.1 | 25.5 | 2.5455 | 0.783 |
| 1.0 | 2.87 | 2.8750 | 0.561 |
| 10 | 1.000 | 10.0 | 0.1000 |
| 100 | 1.000 | 100 | 0.0100 |

The product $\kappa_e^\*T\to2.5129$ from above as $\kappa_0T\to0$, exactly as the implicit equation
requires. At $\kappa_0T=10$ the ratio is $1.000$ to three figures — critical coupling, recovered.

**What critical coupling costs:**

| $\kappa_0T$ | $\eta$ at critical | $\eta$ at optimum | shortfall |
|---|---|---|---|
| 0.05 | 0.0476 | 0.799 | **94%** |
| 0.10 | 0.0906 | 0.783 | **88%** |
| 0.30 | 0.224 | 0.725 | 69% |
| 1.00 | 0.432 | 0.561 | 23% |

**Independent check.** The closed form was compared against direct numerical integration of the
cavity ODE at $T=0.3$ and $T=2.0$; agreement to $10^{-8}$ relative.

---

## D. Stress tests

1. **"η can exceed 1 in the short-pulse expansion."** It can, and that is the signature of the
   invalid expansion: for $\kappa_eT\gtrsim1$ the driven-field model stops conserving energy in the
   way the formula assumes, and the true $\eta$ peaks near $0.81$ and falls as $4/(\kappa_eT)$.
   The finite optimum is what keeps the result physical.
2. **"The coefficient is of order one."** It is not; it is 2.513, and the difference matters — a
   factor-of-2.5 error in the coupling you should choose.
3. **Square pulse assumed.** A different pulse shape changes the constant but not the structure:
   in the short-pulse limit the optimum is $\kappa_e T=O(1)$, independent of $\kappa_0$.

---

## E. Status

| Statement | Status |
|---|---|
| Closed form for $\eta(\kappa_e)$ | **Derived**, verified against the ODE to $10^{-8}$ |
| Steady-state optimum at $\kappa_e=\kappa_0$ | **Derived and verified** |
| Short-pulse optimum $\kappa_e^\*T=2.51289$ | **Derived** (implicit equation) and **verified numerically** ($2.5132$ at $\kappa_0T=10^{-3}$) |
| Optimum is unbounded | **Refuted** — my own earlier claim, corrected here |
| Critical coupling loses up to 94% for short pulses | **Computational result** |
| Measured in a real cavity | **Open** |

## Reproducibility

```
cd 03_VOLUME_C/C3_pulse_coupling
python3 c3_pulse.py
```
Deterministic; numpy only; runtime < 40 s.