Taking stock of eighteen episodes ── no new physics came out. What came out was a ledger of mistakes, and three tools

This way the arithmetic is easier This series began under the pretext of "going out to make unsolved physics."
Not one piece of new physics came out. Let me say that first.
What came out was four closed doors, one calculation that went through, one falsifiable number.
And one more thing ── a record of which way of writing makes mistakes less likely. That is the last story.

This is the final episode, so let us count the results. New physics: 0. Eighteen episodes, and not one formula unknown to humanity.
But not empty-handed either. Four negatives established, one known result rederived myself, one falsifiable number produced. And the most reusable item is probably none of those ── it is the ledger of where I went wrong.
Laid out, the ledger shows a definite skew. The mistakes were not proportional to the difficulty of the physics. What they were proportional to was how many conventions the representation has.

01Taking stock of eighteen episodes

The same ending as Episode 1's brute force finishing at \(p=0.67\). Having written then that "a negative is information," I keep the promise ── the four doors count as results.

02The ledger of mistakes

Here is the real subject. Every bug from eighteen episodes, laid out.

The \(\sqrt{12}\) case is the clearest example. In the episode on the BMS₃ central charge I mixed conventions: writing the algebra in the \(c/12\) convention while using the formula from another. The entropy was overestimated by a factor of 3.46, and staring at the formula shows nothing.

By contrast, Episode 17's bug was merely folding the perihelion azimuth into \(\pi\) and dropping the \(2\pi\). Nothing in the physics was wrong.

03Only three tools helped

None of this improved because I learned more physics. All three are matters of discipline.

04Why flat coordinates are easier ── an invariant comes attached

The form used in Episode 17, once more. With \(F\) the magnitude of \(g^{00}\) and \(G\) the coefficient of \(g^{ij}\),

The right-hand side is a constant. If you are integrating correctly, \(H\) does not move. So you can self-check throughout the integration.

  H at the start        = -0.4999999999999999
  max drift of H after 6 orbits (60000 steps) = 2.220e-16

  # i.e. it does not move down to the double-precision limit. you see the moment it breaks.

Integrate a geodesic via Christoffel symbols and this invariant does not come attached automatically. You have to build \(g_{\mu\nu}\dot x^\mu\dot x^\nu\) separately and watch it.

05So you may take the easier one

This is the final episode's conclusion. And the ground for it is not preference but a theorem.

And "harder to get wrong" has content ── no three-index contractions. fewer conventions to choose from. an invariant comes attached automatically. Those three.

Indeed, all Episode 17 needed was two scalar functions and their first derivatives, and four ordinary differential equations. No Christoffel symbols (40 components in four dimensions), no covariant derivatives, no coordinate patches.

06What need not be decided, do not decide

There is one more framing this series received at the end. There is no need to decide between discrete and continuous.

Let us be precise here. Nyquist in signal processing is exact, but a spacetime lattice is not. Even below the cutoff a residual \((E/E_{\rm LV})^2\) appears ── which is why the number 112 PeV exists. Were the indistinguishability exact, no such number would exist.

So ── "need not be decided" is correct, and is not correct forever. It has an expiry date.

07What remains

The last two rows are where this series never set foot. "Motion within a given field" went through, but "the side that makes the field" was never touched. I leave that honestly blank.