Reciprocal Mersenne Subseries

Finite observations and their limits

The recorded obstructions concern specified representations. Fixed precision, a bounded carry state and a prescribed suffix can lose information needed by the actual-selector condition. They do not show that the underlying deterministic sequence has no arithmetic structure. The relevant exact statements remain fixedPrecisionTropicalNoGo, balancedPulse_no_autonomous_decoder and affineBinaryOrbit_mod_twoPow_eq. Their hypotheses, rather than the fact that an attempted argument failed, determine their scope.

The inherited experiment log reports $\mathrm{200,000}$ seam rows with counts $R:M:U=100197:49899:49898$, which a saved run reproduces, a roughly geometric run-length histogram, and a longest equal run of $15$ in $\mathrm{40,000}$ binary digits. It also reports agreement with OEIS A211706 and a near-integer phase constant of about $2.5$. The other figures and the external digit comparison come from that log alone. They are descriptive reports, not certificates of normality, independence, a random sampling model, or an infinite anti-concentration estimate. A statement about “sampling noise” would require a specified probabilistic model that is not supplied here.

The exceptional row $r=7$ and the crossing $(s,d)=(10,7)$ are likewise statements about the tested ranges only: reset rows $6\le r\le \mathrm{200,000}$ and rows $s\le 2500$ (saved run). No claim that this is the only exceptional row of the infinite orbit is made.

These representations all descend from the same greedy carry; they are not independent tests of a common random hypothesis. Strict tail domination gives an injective coding, but says neither that its digits are random nor that averaging over indices is impossible. The measure-one and nowhere-dense properties of the achievement set do not decide membership of a specified rational. The correct research question is whether a precise arithmetic estimate can be proved for the actual selector, not whether finite data resemble a random model.

Why the support and rational-target problems remain distinct

Proposition 11.4 is a reason to separate the two halves rather than assimilate them. On the universal side the present certificate method first meets a parity obstruction that depends on the choice of support. The squarefree value is nevertheless already known to be irrational at power-of-two bases, and adjoining $1$ removes the opening-block obstruction without changing irrationality. Thus the obstruction concerns this method; it does not concern the value or the universal problem.

Short-window near-integer anti-concentration is one possible missing input for the particular mechanisms analysed here. No equivalence shows that it is necessary for either problem, and no argument identifies it as a common necessary obstruction to both. The squarefree example shows why finite rational changes of support must be checked before an opening-block failure is interpreted as a property of the value.

What the evidence does and does not determine

It does not determine whether a short proof exists: nothing in a catalogue of failed methods bounds the length of a successful one.

It records limitations of specified methods. The fixed-precision completion theorem and the balanced-pulse countermodel concern the particular summaries defined above, not all fixed-precision arithmetic or all finite-state methods with additional arithmetic input. An irrationality-measure bound on $E$ does not classify a specified rational in $\mathcal{A}$. Instantiating the present block-certificate method at a general support is blocked, on the universal side, by Proposition 11.4. Survival through any tested finite depth does not by itself establish membership; a fatal-gap certificate would establish non-membership. These facts do not decide rational-target membership, and they do not yield an impossibility theorem for all probabilistic, measure-theoretic or averaging methods.