Re-Entry as a Binary Prohibition Law: A Reconciled Formal Analysis of Two Conventions

#1. Introduction

The calculus of indications, in the tradition of Spencer-Brown, builds logic from a single primitive act: drawing a distinction and marking one of its sides. Its most delicate construction is re-entry, the operation by which the form of a distinction is taken as indicational content of the distinction itself. A re-entering form need not settle into a value, and the classical two-valued reading of the calculus has been criticized precisely for reifying the unmarked state into the truth-value false [10]. A companion line of work defends the calculus's cancellation rule and argues for a structural isomorphism between re-entry dynamics and three-outcome measurement collapse [11].

This paper reconciles three independent drafts written from the same source material. All three asked the same narrow question: can a re-entering form simultaneously function as a binary prohibition law — a total rule on a two-element domain that forbids its own input? The drafts disagreed on what "prohibition law" means formally: one treated it as a cross-domain metaphor anchored in a literature survey, one as a self-forbidding function (derangement) on $B_2$, and one as a forbidden-word constraint on binary sequences. Per the reconciliation protocol, the derangement convention is adopted for the main text because it is the most restrictive and yields the sharpest impossibility result; the forbidden-word convention is retained as a clearly labeled secondary model because it produces an independent, computable coincidence with the three-outcome capacity; the metaphorical convention is documented in Appendix A but carries no formal weight here.

The contribution is deliberately modest and fully explicit:

1. A finite enumeration of all unary laws on $B_2$, with the prohibition condition isolating exactly one law (Section 4.1).
2. A re-entry analysis showing that this law has orbit period $T_2 = 2$ and no fixed point, so a settled binary prohibition re-entry is impossible (Section 4.2).
3. A three-outcome extension, following the critique of [10] and the three-outcome dynamics of [11], in which derangement arithmetic shows exactly $2$ prohibition laws exist, both of period $3$ (Section 4.3).
4. Quantitative comparisons: density ratio $27/8 = 3.375$ and entropy increment $\log_2 3 - 1 \approx 0.585$ bits (Section 4.4).
5. A secondary forbidden-word model in which a single adjacent-pair prohibition leaves $a_{10} = 144$ admissible sequences of length $10$ out of $2^{10} = 1024$, with pair-level capacity $\log_2 3$ matching the three-outcome algebra (Section 4.6).

Throughout, "binary" is used in the two senses the literature itself uses indiscriminately: the two-element Boole algebra $B_2$ of [4], and "binary" as in binary programs [3], binary star systems [5], [8], and binary black hole coalescences [6]. In each case a two-state system is subject to a constraint, and the question is whether the constraint can be self-applied.

#2. Background and Related Work

We discuss all twelve supplied works, in bibliography order. Where an entry's summary is thin, we say so and restrict ourselves to what it states.

[1] Marine size spectra stability. This work investigates the stability of the power-law steady state often observed in marine ecosystems, considering three dynamical systems describing organism abundance as a function of body and time: a "jump-growth" equation, a first-order approximation (the McKendrick–von Foerster equation), and a second-order approximation. Its relevance is methodological: a steady state of a self-organizing system may be an approximation of a more exact discrete dynamics, so what looks like a fixed point at one level of description may be an oscillation at another — precisely the situation we find for the re-entering prohibition law, whose "fixed point" does not exist at all in the primary model.

[2] Plankton biodiversity and power laws. This work explains two features of unicellular plankton assemblages — high biodiversity and power-law abundance structure — via allometric scaling of cell growth and mortality with cell size, demonstrated through a numerical study of a size-structured, multispecies population-dynamic model with a single resource. The summary truncates before further detail, so we use it only as a structural analogue: a simple local rule generates global structure without fine-tuning, as a single local prohibition generates a global, exactly countable reshaping of sequence space in the secondary model.

[3] cozy: comparative symbolic execution for binary programs. This work introduces cozy, a tool for analyzing and visualizing differences between two versions of a software binary, with the primary use case of validating "micropatches" — small binary or assembly-level patches inserted into existing compiled binaries — leveraging the Python-based angr symbolic execution framework. The summary truncates before reporting results, so we cite it for framing only: validating a self-modifying constraint on a two-level (branch taken / not taken) structure is the practical cousin of validating a re-entering prohibition law on $B_2$.

[4] Distributions over binary test functions. This work takes $B_2$, the Boole algebra with two elements, as its base and defines, for functions to $B_2$, the limits, derivatives, differentiability, test functions, and integrals, then defines distributions over the space of these test functions — regular and singular distributions and their support sets — extending to two-variable functions into $\{0,1\}$. The summary reports no theorems we can quote, so we use it strictly as the source of the $B_2$ formalism that underlies our primary model. It also suggests how a law that never settles might carry a distributional meaning even without a pointwise value.

[5] Evolutionary memory in binary systems. The entry states that correlation between the spins (rotational velocities) in binaries had previously been established, and that this work shows the degree of spin correlation is independent of the components' separation, possibly related to Zhang's non-linear model for the formation of binary stars from a nebula. The relevance is correlation without a mediating coupling at the observed scale: a shared constraint whose enforcement, when re-entered, cannot be located at any single point of the system — it lives in the orbit, not in a state.

[6] Two-dimensional correlation function of binary black hole coalescences. This work computes two-dimensional correlation functions of binary black hole coalescence detections in LIGO–Virgo's first and second observation runs, testing the sky distribution of events for correlations at different angular scales by comparing the observed correlation function to two reference functions obtained from mock datasets of localization error. The summary does not state the outcome of the comparison, so we draw no finding from it; we cite it for its discipline of testing a population of two-state events against enumerated reference constructions rather than intuition — the pattern our exhaustive enumeration follows.

[7] Integral means, convolution, and univalent functions. This work uses the Baernstein star-function to investigate questions about the integral means of the convolution of two analytic functions in the unit disc, with the theory of univalent functions playing a basic role. The supplied summary gives no further detail. Its place in our argument is by structural analogy only: convolution combines a function with a (possibly transformed) copy of another, and integral means measure what survives averaging; the re-entry of a prohibition law is formally a self-application of a constraint. Beyond this analogy, the entry supports no further claim.

[8] Binary properties and theories of star formation. The entry states that the overall frequency and other statistical properties of binary systems suggest star formation is intrinsically a complex and chaotic process, and that most binaries and single stars actually originate from the decay of multiple systems, with interactions between stars forming in close proximity — for example via tidal effects — possibly playing a role. This is a structural precedent for our central result: two-state outcomes are the decay products of a richer multi-body process, not its primitive. Our finding that a binary prohibition law cannot settle without a third outcome mirrors the claim that the binary/single census is explained only by embedding it in a larger state space.

We cite it as the methodological exemplar for our procedure: take a single definitional bridge between a speculative framework and a classical formalism, and check exactly what survives. Our bridge definition is $f(x) \neq x$ read as a re-entry condition; our cross-validation target is the classical enumeration of functions on $B_2$.

[10] The void is not false. This critique-first paper argues that Boolean logic and set theory reify the void (the unmarked state) into false and the empty set, committing a category error: false is a mark, the void is not a value. A reconstruction program — distinction-logic with inside/outside/oscillating states — is labeled conjecture; the summary truncates there. This is the single most load-bearing citation for us: it licenses the three-outcome space $\{m, u, v\}$ of Section 4.3, in which $v$ is an outcome of the dynamics but not a value, and it predicts exactly the failure we exhibit — a two-valued reading of a re-entering prohibition law cannot close because the outcome it needs (oscillation) has been reified out of the state space.

[11] The Spencer-Brown cancellation rule defended. This work systematically defends the Spencer-Brown cancellation rule and establishes a structural isomorphism with quantum measurement collapse via three-outcome dynamics (citing Varela 1975, corroborated by a 29-schisms numerical search 2026), with a 4-level M-property justification framework and a boson–fermion observer analysis, in 14 pages. The three-outcome dynamics it defends is precisely the completion our Section 4.3 formalizes. The entry's mention of the numerical search is a claim of the entry itself; its parameters are not supplied, so we rely on it only for the qualitative three-outcome commitment.

[12] The valuation reading of the Qubit Delusion. The Qubit Delusion argues that the qubit-gate-circuit model of quantum computation imports a particle ontology inconsistent with quantum field theory, and that the field's failure to deliver commercially viable hardware after $35$ billion of investment is an epistemic problem; the summary truncates mid-sentence, so its conclusion is not available to us. We use the entry only for the figure of $35$ billion (converted in Section 4.7 as a scale illustration) and for the genre of audit its title poses — whether a kernel in a valuation reading is load-bearing — which parallels our question of whether the suppressed third outcome is load-bearing in $B_2$.

#3. Methods

#3.1 Formal objects (primary convention)

Following [4], let $B_2 = \{0, 1\}$ denote the Boole algebra with two elements, where $1$ is the marked state and $0$ the unmarked state (the value false in the classical reading that [10] criticizes). A unary law is a total function $f : B_2 \to B_2$; there are

A binary prohibition law (primary convention, adopted from one draft) is a unary law satisfying

That is, the law forbids each state as an output: no state is self-consistent under the law. A re-entry of $f$ is the discrete dynamical system generated by iterating $f$ from $x_0 \in B_2$: the orbit $x_{n+1} = f(x_n)$. The re-entry settles if the orbit reaches a fixed point $x^*$ with $f(x^*) = x^*$; otherwise it is periodic with period $T_f \geq 2$.

#3.2 Three-outcome extension

Following the three-outcome dynamics defended in [11] and the inside/outside/oscillating reconstruction conjectured in [10], we extend the state space to

where $m$ is marked, $u$ unmarked, and $v$ the void/oscillating outcome. Per [10], $v$ is not a value: it is an outcome of the dynamics, not a truth-value the law may output as a settled verdict. A ternary law is a total function $g : S_3 \to S_3$; there are $N_3 = |S_3|^{|S_3|} = 3^{3} = 27$. A ternary prohibition law satisfies (P) on $S_3$, i.e., $g$ is a derangement of $S_3$.

#3.3 Secondary convention (forbidden-word model)

A second draft defined a prohibition law instead as a forbidden-word constraint: a set $P \subseteq B_2^k$ of forbidden words, with a sequence $s \in B_2^n$ admissible if no length-$k$ contiguous block lies in $P$. We retain this as a clearly labeled secondary model with the canonical single prohibition $P = \{11\}$ (order $k = 2$): the mark may not indicate itself in adjacent positions. The two conventions differ (Appendix A); the primary one constrains functions, the secondary one constrains sequences.

#3.4 Measures

We compare settings by: (i) density of prohibition laws, $\rho_2$ and $\rho_3$ as fractions of all laws satisfying (P); (ii) orbit period, the minimal $T_f \gt 0$ with $f^{T_f}(x_0) = x_0$; (iii) uniform outcome entropy, $H_k = \log_2 k = \ln k / \ln 2$ bits for $k$ equiprobable outcomes; (iv) for the secondary model, survival ratio $\rho_n = |A_n| / 2^n$ and capacity $C_n = \log_2 |A_n|$.

All computations are finite enumerations or closed-form evaluations. No simulation and no empirical data are used.

#4. Analysis

Every input number is either a definition (Sections 3.1–3.3), an initial condition stated below, or a standard closed-form constant computed from its defining expression.

#4.1 Enumeration of unary laws on $B_2$

Every unary law is determined by the pair $(f(0), f(1))$, each entry in $\{0,1\}$, giving $N_2 = 2^2 = 4$ laws:

Apply (P), $f(x) \neq x$ for all $x \in \{0,1\}$:

• $f_{\mathrm{id}}$: $f_{\mathrm{id}}(0) = 0$ — fails at $x = 0$.
• $f_{\mathrm{not}}$: $f_{\mathrm{not}}(0) = 1 \neq 0$ and $f_{\mathrm{not}}(1) = 0 \neq 1$ — passes.
• $f_{\mathrm{c0}}$: $f_{\mathrm{c0}}(0) = 0$ — fails at $x = 0$.
• $f_{\mathrm{c1}}$: $f_{\mathrm{c1}}(1) = 1$ — fails at $x = 1$.

Result 4.1. Exactly $1$ of $4$ unary laws is a binary prohibition law: $\rho_2 = 1/4 = 0.25$, and it is the negation.

#4.2 Re-entry of the prohibition law

Iterating $f_{\mathrm{not}}$ from $x_0 = 0$: $x_1 = 1$, $x_2 = 0$, $x_3 = 1$, … From $x_0 = 1$ the same cycle is traversed in reverse phase. Fixed points require $f_{\mathrm{not}}(x) = x$, i.e., $1 - x = x$, i.e., $2x = 1$ over $\{0,1\}$: no solution, since $2 \cdot 0 = 0 \neq 1$ and $2 \cdot 1 = 2 \neq 1$. The orbit period is

For contrast, count fixed points across all four laws: $f_{\mathrm{id}}$ has $2$, $f_{\mathrm{c0}}$ has $1$, $f_{\mathrm{c1}}$ has $1$, $f_{\mathrm{not}}$ has $0$; total $2 + 1 + 1 + 0 = 4$, averaging $4/4 = 1$ fixed point per law. The prohibition law is the unique law with zero.

Result 4.2. The unique binary prohibition law, under re-entry, has no fixed point and period $T_2 = 2$. A settled binary prohibition re-entry is impossible on $B_2$: the prohibition is enforced only by perpetual oscillation, and the moment the form settles it ceases to prohibit.

#4.3 Three-outcome prohibition laws

On $S_3 = \{m, u, v\}$, a law satisfying (P) must map each of the $3$ elements to one of the $2$ others; such a map is necessarily a bijection (a derangement), since if two elements shared an image the pigeonhole principle would force some element to map to itself. The number of derangements of a $3$-set is

Step by step: $1 - 1 = 0$; $0 + \tfrac{1}{2} = \tfrac{1}{2}$; $\tfrac{1}{2} - \tfrac{1}{6} = \tfrac{3}{6} - \tfrac{1}{6} = \tfrac{2}{6} = \tfrac{1}{3}$; $6 \cdot \tfrac{1}{3} = 2$. The two derangements are the two $3$-cycles:

Both have orbit period $T_3 = 3$ for every initial state, and neither has a fixed point.

Result 4.3. Exactly $2$ of $27$ ternary laws are prohibition laws: $\rho_3 = 2/27 \approx 7.41 \times 10^{-2}$. Prohibition remains unsatisfiable as a settled state — every prohibition law is a cycle — but it is now generically realizable as dynamics: every initial state lies on the prohibition orbit.

#4.4 Comparative measures

Density ratio.

Admitting the void as a non-valued outcome makes prohibition laws $3.375$-fold sparser among all laws, but makes every state subject to prohibition as dynamics.

Entropy. For a uniform distribution over the two values of $B_2$:

Over the three outcomes of $S_3$:

The entropy increment is

This is the informational price of the third outcome that makes prohibition dynamically coherent.

Period growth. The minimal prohibition orbit grows from $T_2 = 2$ to $T_3 = 3$, a factor $3/2 = 1.5$.

#4.5 Summary of inputs for the primary model

The derivation chain uses only: $|B_2| = 2$ and $|S_3| = 3$ (definitional, per [4] and [10]–[11]); the condition (P) (definitional); the derangement formula and entropy formula (closed-form). No empirical, simulated, or externally sourced number enters the chain.

#4.6 Secondary model: forbidden-word prohibition $P = \{11\}$

Inputs. Initial conditions $a_1 = 2$, $a_2 = 3$ (of the four length-2 words $\{00, 01, 10, 11\}$, the word $11$ is forbidden, leaving $3$); recurrence from partitioning by final symbol, $e_n = a_{n-1}$ and $d_n = a_{n-2}$, giving $a_n = a_{n-1} + a_{n-2}$ (the Fibonacci recurrence, $a_n = F_{n+2}$).

Step 1: iterate the recurrence.

Step 2: survival ratio. Raw states: $2^{10} = 1024$. Therefore

Step 3: capacity under uniform coding. With $144 = 12^2$:

Step 4: asymptotic capacity. The characteristic equation $x^2 - x - 1 = 0$ has roots $\varphi = (1 + \sqrt{5})/2$ and $\psi = (1 - \sqrt{5})/2$. With $\sqrt{5} \approx 2.2360680$: $\varphi = 1.6180340$, $\psi = -0.6180340$. Since $a_n \sim c\,\varphi^n$,

against $1$ bit per symbol unconstrained; asymptotic loss $1 - 0.6942419 = 0.3057581$ bits per symbol. Consistency check: the finite-$n$ per-symbol capacity $C_{10}/10 = 0.7169925$ exceeds $c_\infty = 0.6942419$, as expected since boundary effects relax the constraint at small $n$.

Step 5: pair-level coincidence. The prohibition removes one of the four length-2 configurations, leaving $3$ admissible pairs, with per-pair capacity

identical to $H_3$ of the three-outcome algebra. This is the secondary model's central observation: the three-outcome re-entry algebra of [11] and the pair-level survival count under $P = \{11\}$ both assign $\log_2 3$ bits to one act of distinction.

Step 6: generalization to $j$ forbidden pairs. With $j$ of the $4$ pairs forbidden, pair-level capacity is $\log_2(4 - j)$: $j = 1$ gives $\log_2 3 = 1.5850$; $j = 2$ gives $\log_2 2 = 1$; $j = 3$ gives $\log_2 1 = 0$; $j = 4$ is undefined (no admissible pairs). A prohibition law that forbids everything permits nothing.

#4.7 Scale illustration from the bibliography

The only number taken from the bibliography is the investment figure in [12]: $35$ billion. Converting to scientific notation: $35 \times 10^{9} = (3.5 \times 10^{1}) \times 10^{9} = 3.5 \times 10^{10}$. This figure plays no computational role in the models; it is reported only as context for the scale of the stakes riding on whether a formal kernel is load-bearing.

#5. Results

All numbers below are exactly those computed with shown arithmetic in Section 4; none are empirical, simulated, or projected.

• R1 (Section 4.1). Exactly $1$ of the $N_2 = 4$ unary laws on $B_2$ satisfies the prohibition condition (P): the negation. Density $\rho_2 = 1/4 = 0.25$.
• R2 (Section 4.2). The unique binary prohibition law has no fixed point ($2x = 1$ has no solution in $\{0,1\}$) and orbit period $T_2 = 2$. A settled binary prohibition re-entry is impossible on $B_2$.
• R3 (Section 4.3). Exactly $2$ of the $N_3 = 27$ ternary laws on $S_3$ satisfy (P): the two $3$-cycles $g_{+}$ and $g_{-}$, each of period $T_3 = 3$ for every initial state. Density $\rho_3 = 2/27 \approx 7.41 \times 10^{-2}$.
• R4 (Section 4.4). Density ratio $\rho_2 / \rho_3 = 27/8 = 3.375$; entropy increment $\Delta H = H_3 - H_2 = \log_2 3 - 1 = 0.5849625$ bits; minimal prohibition period grows by the factor $3/2 = 1.5$.
• R5 (Section 4.6, secondary model). For the forbidden-word constraint $P = \{11\}$: $a_{10} = 144$ admissible sequences of length $10$ out of $2^{10} = 1024$, survival ratio $\rho_{10} = 144/1024 = 0.140625$; capacity $C_{10} = 7.1699250$ bits, loss $\Delta C_{10} = 2.8300750$ bits; asymptotic capacity $c_{\infty} = \log_2 \varphi = 0.6942419$ bits per symbol; pair-level capacity $\log_2 3 = 1.5849625$ bits, identical to $H_3$.
• R6 (Section 4.7). Scale illustration only: the investment figure of [12], $35$ billion $= 3.5 \times 10^{10}$.

#6. Discussion

Limitations. The primary impossibility result (R2) is definitional, not empirical: it follows from the enumeration of Section 4.1 and the fixed-point equation $2x = 1$, so it is as strong as the convention (P) and no stronger. A different reading of "prohibition law" — the metaphorical convention of Appendix A, or the forbidden-word convention of Section 3.3 — changes the result class entirely; the forbidden-word model, in particular, admits settled configurations (e.g., the all-zeros sequence $00\ldots0$ satisfies $P = \{11\}$ trivially), so the impossibility claim does not transfer across conventions. The three-outcome extension inherits its interpretation of the void $v$ as a non-value from the conjectural reconstruction of [10], which that entry itself labels conjecture; if $v$ is instead admitted as a third truth-value, the interpretation of R3 changes, though the count $!3 = 2$ does not. The pair-level coincidence $\log_2 3 = H_3$ (R5) is a numerical identity between two independently defined quantities; the paper claims it as an observation, not as evidence of a shared mechanism.

Failure modes and falsification. The claim that a settled binary prohibition re-entry is impossible is falsified by exhibiting a total function $f : B_2 \to B_2$ with $f(x) \neq x$ for all $x$ that nonetheless reaches a fixed point under iteration — impossible for a deterministic total function on a two-element set, which is why the result is robust. The count $!3 = 2$ is falsified by exhibiting a third ternary derangement of $S_3$ or by rejecting the pigeonhole argument in Section 4.3. The Fibonacci count $a_{10} = 144$ is falsified by direct enumeration of binary sequences of length $10$ avoiding the substring $11$. The entropy values are falsified only by rejecting the uniform-outcome assumption $H_k = \log_2 k$; under any other outcome distribution the increment $\Delta H$ need not equal $0.5849625$ bits, and the paper's claim is conditional on uniformity.

Open questions. (i) Does the density ratio $27/8 = 3.375$ generalize as $!(k)/k^{k}$ versus $!(2)/2^{2}$ for larger state spaces, and does the ratio converge? (ii) Is the pair-level identity $\log_2 3 = H_3$ an artifact of the choice $k = 2$, $j = 1$, or does it survive other forbidden-word families (Section 4.6, Step 6 shows it does not: $j = 2$ gives $\log_2 2 = 1$)? (iii) What dynamical reading of the void $v$ makes the three-outcome prohibition law physically, rather than merely formally, coherent — the question [10] labels conjecture and [11] addresses qualitatively?

#7. Conclusion

Under the strictest of the three reconciled conventions, a binary prohibition law exists uniquely (the negation, $1$ of $4$ laws) and, under re-entry, can never settle: it enforces itself only as a period-$T_2 = 2$ oscillation. Admitting a third, non-valued outcome $v$ per [10] and [11] makes prohibition dynamically coherent — every state lies on a prohibition $3$-cycle — at a density cost of a factor $27/8 = 3.375$ and an entropy cost of $\log_2 3 - 1 \approx 0.585$ bits. A secondary forbidden-word convention independently reproduces the three-outcome capacity $\log_2 3$ at the pair level. All quantities are derived by explicit finite enumeration or closed-form evaluation; the paper uses no empirical data.

#References

[1] A stability analysis of the power-law steady state of marine size spectra. arXiv:1001.3826v3. https://arxiv.org/abs/1001.3826v3 [2] Plankton: the paradox and the power law. arXiv:1705.05327v1. https://arxiv.org/abs/1705.05327v1 [3] cozy: Comparative Symbolic Execution for Binary Programs. arXiv:2504.00151v1. https://arxiv.org/abs/2504.00151v1 [4] Introductory Topics in Distributions over Binary Test Functions. arXiv:math/0110335v1. https://arxiv.org/abs/math/0110335v1 [5] Evolutionary Memory in Binary Systems?. arXiv:astro-ph/0412185v1. https://arxiv.org/abs/astro-ph/0412185v1 [6] Two-dimensional correlation function of binary black hole coalescences. arXiv:2005.06004v2. https://arxiv.org/abs/2005.06004v2 [7] Integral means inequalities, convolution, and univalent functions. arXiv:1902.01595v1. https://arxiv.org/abs/1902.01595v1 [8] Implications of Binary Properties for Theories of Star Formation. arXiv:astro-ph/0006288v1. https://arxiv.org/abs/astro-ph/0006288v1 [9] QNFO: Kappa/SIIT Cross-Validation: Exact Reproduction of Standard Model 1-Loop $\beta$-Functions and Resolution of the Harmonic-Paradigm Tension [10] QNFO: The Void Is Not False: Recovering the Unmarked State in Logic from the Calculus of Indications [11] QNFO: The Ontology of Boundary-Crossing: Spencer-Brown Cancellation Rule Defended [12] QNFO: The Valuation Reading of the Qubit Delusion: Is the Adelic Kernel Load-Bearing?

#Appendix A. Divergence report

The three drafts diverged on the formal meaning of "binary prohibition law":

• Draft A (metaphorical convention). Treated prohibition as a cross-domain metaphor anchored in a literature survey, with no formal condition. This convention carries no formal weight in the reconciled paper; it is documented here and in the framing of Section 1.
• Draft B (derangement convention). Defined a prohibition law as a total self-map $f$ with $f(x) \neq x$ for all $x$ — a derangement. Adopted as the primary convention because it is the most restrictive and yields the sharpest impossibility result (R2).
• Draft C (forbidden-word convention). Defined prohibition as a forbidden-word constraint $P \subseteq B_2^k$ on sequences. Retained as a clearly labeled secondary model (Section 3.3, Section 4.6) because it produces an independent, computable coincidence with the three-outcome capacity $\log_2 3$.

The disagreement is a choice of formal object (functions versus sequences versus metaphor), not a factual conflict; no numerical result of one convention contradicts another, since they count different things. The reconciliation rule applied was: strictest formal convention primary, independently computable convention secondary and labeled, metaphor documented only.

#Appendix B. Claim attribution

SINGLE status reflects that only one draft produced the formal derivation; the reconciled paper adopts it after verifying every arithmetic step shown in Section 4. No DIVERGENT numerical conflict was found: the drafts' conventions differ in kind, not in competing numbers for the same quantity.