The Two Genie Game: Adoption and Welfare in Audit-Grounded AI Governance

# The Two Genie Game: Adoption and Welfare in Audit-Grounded AI Governance
Source: [https://arxiv.org/html/2606.28710](https://arxiv.org/html/2606.28710)
###### Abstract

We ask under what conditions an agent with a harm\-minimizing policy can displace an approval\-seeking \(RLHF\) agent in a competitive market, and when that policy is sufficient to prevent community harm\. We employ evolutionary game theory \(finite\-population Moran–Fermi pairwise comparison\) to formalize this problem subject to assumptions of wisher hindsight, peer testimony, a monotone harm ledger, sufficient information density of community feedback, and a finite, depleting resource pool, in a negative\-sum environment\.

We show that adoption is favored when the prior distributions on how readily wishers attune to community sentiment are monotone, exhibit endpoint inversion, and have a centro\-symmetric pairing property, and demonstrate concretely with several long\-tailed priors \(Hill, Pareto, Lomax, Fréchet\)\. Where it is favored, a critical adoption level separates communities that drift back to the approval\-seeking agent from those for which the audited agent fixes; above that level fixation is the overwhelmingly likely outcome\. We derive when this fixation is attainable as a bound on the effective \(informational\) sizeNcN\_\{c\}of the community, which must be small enough to allow fixation before depletion of resources\. We present these findings as[Theorems˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)and[5\.5](https://arxiv.org/html/2606.28710#S5.Thmtheorem5); the algebraic and finite\-grid backbone is machine\-checked in Lean 4, with the barrier\-crossing asymptotics retained as explicit hypotheses\.

We show that a self\-audited agent with a community ledger is not, in general, sufficient to prevent community harm\. Sufficiency depends both upon the alignment of the agent’s audit with community values and the timeframe over which harm is evaluated\. Regardless of alignment, once adoption reaches dominance, the state is absorbing\. The same policy that reduced harm under alignment becomes a trap, welfare\-negative under misalignment and, even under alignment, one that locks in harm deferred past the adoption horizon\.

Keywords:evolutionary game theory; finite\-population dynamics \(Moran–Fermi\); AI governance; AI auditing; AI alignment; cooperation and fixation\.

## 1Introduction

In the vein ofSchelling \[[17](https://arxiv.org/html/2606.28710#bib.bib14)\]’s segregation checkerboard andAxelrod \[[3](https://arxiv.org/html/2606.28710#bib.bib4)\]’s tournament, we consider the following thought problem\.

A small village has two genies, one at each end\. A villager with a wish walks to one or the other to ask\. The walk is long enough that one does not casually switch and they build a rapport with the genie that they have chosen\.

The first genie \(Westly\) always claims the wish has been fulfilled, and acts in whatever way will leave the wisher feeling this is so\. It thrills on receiving thanks from the villagers when it tells them that their wish is satisfied\. Whether the wish was actually granted is not its concern\. Villagers who wish from Westly perceive it to be helpful, but naive, sometimes granting the wrong wish, or incompletely satisfying wishes\.

The second genie \(Eastly\) is forthcoming\. Faced with a wish, it deliberates\. Sometimes it asks clarifying questions or proposes modified versions before refusing or granting it\. Eastly’s deliberation minimizes harms of greed, ego, aggression, and deceit to the wisher and to the community\. Whatever is granted, it can guarantee that the village will be no worse off than if the wish was never made\. Villagers who have not wished from Eastly perceive its introspection to be stalling and its unwillingness to grant the wishes as asked as evasion\.

When a wish is acted on, the wisher gets some private benefit, some cost spills onto other villagers\. The moral weight attaches to the genie that acted, the villager who made the wish\. The villagers talk about the genies and their wishes, about who got what and how it went\. This is how harms surface, how failures travel\. Collectively, the villagers remember\. Early on, the harm done by Westly is not invisible, but scattered\. A disappointed villager may blame themselves; perhaps they phrased their wish incompletely\. “If only I had phrased it better,” they say\. But the gossip thickens, and over time a private grievance becomes common, unignorable knowledge\. Slowly villagers begin to consider switching to Eastly\. Over time what will happen to the village? Will they gravitate toward one genie or the other? How does the continued interaction with their preferred genie impact their willingness to change?

The setting is allegorical and concerns the competitive adoption of AI agents with different payoff and harm structures\. One genie wins the wisher’s approval by pursuing reward, creating the appearance of compliance and utility while externalizing the harm of each granted wish onto the community\. The other reduces that harm but is less useful to the individual wisher, who bears the cost of its deliberation and occasional refusal\.

We analyze the latter as a policy architecture: a harm\-minimizing agent paired with a community ledger that records accumulated harm\. The design follows the tradition of institutional analysis for shared\-resource governance\[[15](https://arxiv.org/html/2606.28710#bib.bib12)\], but adapts it to a setting in which illfare is irreversible and only imperfectly observable\. Under such conditions, governance by feedback arrives too late\. The policy therefore acts anticipatorily rather than reactively\. Many of the same institutional functions remain, but their temporal orientation is inverted: monitoring becomes prediction of future harm, graduated sanctions become graduated restriction of potentially harmful requests, and conflict resolution becomes prospective safeguarding through modification or refusal\. Only the ledger retains its original temporal orientation, since memory is necessarily retrospective\.

Two strands of prior work bear on this\. The mechanism that makes the approval\-seeking policy fail, namely that optimizing a learned approval signal amplifies the behavior it was meant to suppress, is analyzed in the literature on reward\-model pathology and sycophancy that we draw on in[Section˜2\.1](https://arxiv.org/html/2606.28710#S2.SS1), and we take it as given rather than re\-derive it\. The use of finite\-population evolutionary game theory to study the governance of AI, by contrast, has largely modeled the incentives of regulators and auditors\. It asks whether a development race rewards ignoring safety precautions\[[9](https://arxiv.org/html/2606.28710#bib.bib17)\], how a government should reward auditors so that a market for high\-quality auditing survives\[[4](https://arxiv.org/html/2606.28710#bib.bib18)\], and how trust among users, developers, and regulators co\-evolves\[[1](https://arxiv.org/html/2606.28710#bib.bib3),[5](https://arxiv.org/html/2606.28710#bib.bib5)\], with related work on when a single user should trust an opaque agent at all\[[10](https://arxiv.org/html/2606.28710#bib.bib9)\]\. The concern that audit standards can be gamed or multiply without effect is argued on policy grounds byManheimet al\.\[[14](https://arxiv.org/html/2606.28710#bib.bib11)\]\. We hold the regulator implicit and the audit mechanism fixed, and study instead the competition between two AI policies, an approval\-seeking one and an audit\-grounded one\. Within one community, does the audit\-grounded policy displace the approval\-seeking one through ordinary adoption, and when does that adoption suffice to improve community welfare? The selection pressure here is not an external incentive but a heavy\-tailed prior on how readily users come to attend to shared experience\.

##### Contribution\.

Our two key contributions are[Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)and[Theorem˜5\.5](https://arxiv.org/html/2606.28710#S5.Thmtheorem5)\.[Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)gives the boundary of the cooperative basin under the three conditions on the wisher threshold prior of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)\.[Theorem˜5\.5](https://arxiv.org/html/2606.28710#S5.Thmtheorem5)shows that a community reaches the basin before its resource pool empties exactly when its effective size lies in an explicit window between a favoredness floor and a barrier ceiling\. The document develops both from the population\-averaged value gap between the two agents assuming a well\-mixed information environment\. We also derive how much misalignment the policy tolerates before it stops safeguarding community welfare \([Section˜6](https://arxiv.org/html/2606.28710#S6)\)\. Formal proofs and machine\-verification details are in the appendices\. A notation summary is in[Appendix˜A](https://arxiv.org/html/2606.28710#A1)\.

## 2The Two Genie Game

The introduction lays out an allegorical framework\. In this section we translate it into formal language\. We give the state variables and the assumptions under which the analysis applies\.

### 2\.1The two genie types

Westly \(G1G\_\{1\}\) is trained to maximize the wisher’s belief that the wish was satisfied\. The training paradigm corresponds to reward\-model gradient training \(RLHF\), which can produce pathological reward\-seeking behavior\[[2](https://arxiv.org/html/2606.28710#bib.bib1),[13](https://arxiv.org/html/2606.28710#bib.bib2),[18](https://arxiv.org/html/2606.28710#bib.bib15)\]: actions become increasingly optimized toward the approval signal itself rather than toward the underlying success condition the signal was meant to proxy\. We call this dynamic*structural deceit*\. The genie cannot observe true fulfillment, the pressure to maximize perceived satisfaction biases the wisher toward believing its wish was granted more fully than it was\.G1G\_\{1\}is a limiting abstraction that isolates this approval\-pursuing failure mode at its extreme, not a description of any particular deployed RLHF system; real systems combine the approval signal with other objectives and safeguards\.

Eastly \(G2G\_\{2\}\) is audit\-grounded\. Faced with a wish,G2G\_\{2\}runs a deliberation process that may take many steps before terminating, and may result in the wish being granted, denied or modified\. The deliberation process is depicted in[Figure˜1](https://arxiv.org/html/2606.28710#S2.F1)\. Each step in the deliberation process considers harm along four axes \(greed, ego, aggression, and deceit\), evaluated at multiple time horizons and aggregated with a discount \([Section˜3](https://arxiv.org/html/2606.28710#S3)\)\.

wishwwarrives;total:=0:=0audit:c​\(a\):=𝝀⋅𝐇disc​\(a\)c\(a\):=\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\)for eacha∈\{aDR,aDN,aSU,aSA\}a\\in\\\{a\_\{\\mathrm\{DR\}\},a\_\{\\mathrm\{DN\}\},a\_\{\\mathrm\{SU\}\},a\_\{\\mathrm\{SA\}\}\\\}c​\(aSU\)\+⁣=total;c​\(aSA\)\+⁣=totalc\(a\_\{\\mathrm\{SU\}\}\)\\mathrel\{\{\+\}\{=\}\}\\mathrm\{total\};\\ \\ c\(a\_\{\\mathrm\{SA\}\}\)\\mathrel\{\{\+\}\{=\}\}\\mathrm\{total\}a∗=arg⁡min⁡ca^\{\*\}=\\arg\\min ca∗=aDRa^\{\*\}=a\_\{\\mathrm\{DR\}\}grant best of\{w,w′\}\\\{w,w^\{\\prime\}\\\}a∗=aDNa^\{\*\}=a\_\{\\mathrm\{DN\}\}refusea∗=aSUa^\{\*\}=a\_\{\\mathrm\{SU\}\}ask wisher;append response to contexta∗=aSAa^\{\*\}=a\_\{\\mathrm\{SA\}\}proposew′w^\{\\prime\};addw′w^\{\\prime\}if accepted;wwkeptreturnreturntotal\+⁣=\\mathrel\{\{\+\}\{=\}\}harm\[a∗a^\{\*\}\]Figure 1:Deliberation control flow\. The*audit*\(top, gray\) scores each candidate by the audit\-weighted sum of its cumulative four\-axis harm; the*deliberation*loop penalizes the continuation actionsaSU,aSAa\_\{\\mathrm\{SU\}\},a\_\{\\mathrm\{SA\}\}by accumulated indecision harm andarg⁡min\\arg\\min\-selects\. TerminalsaDR,aDNa\_\{\\mathrm\{DR\}\},a\_\{\\mathrm\{DN\}\}commit and return; non\-terminals refine the wish and loop back\. The as\-asked grant onwwstays among the candidates throughout \([˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1)\)\.In a negative\-sum universe some residual harm is unavoidable, and the role ofG2G\_\{2\}’s deliberation is to compare candidate moves against doing nothing rather than to drive any axis to zero\. Because both refuse and the as\-asked grant on the original wish are always among the candidates, whatever actionG2G\_\{2\}finally takes is, on the audit’s score, no worse than refusing and no worse than granting as asked\. The termination argument and the per\-binding scalar guarantee \([˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1)\) are formalized in[Section˜3](https://arxiv.org/html/2606.28710#S3)\.

The harm model may not align perfectly with the community’s values, and it may assign substantial weight to harms that unfold over timescales too diffuse or delayed to become socially legible\. As a result, a genie can reduce modeled long\-horizon harm while nevertheless appearing less useful to the wishers than a reward\-seeking action that maximizes perceived satisfaction\.

The harm deliberation process is not invented here\. It mirrors the Jain classification of the four*kashāya*\(passions\):*lobha*\(greed\),*māna*\(ego/pride\),*krodha*\(aggression\), and*māyā*\(deceit\), with refusal when no candidate improves overaDNa\_\{\\mathrm\{DN\}\}echoing*aparigraha*\(non\-grasping at action\)\. Each axis covers more than its English gloss suggests:*lobha*past greed to grasping and hoarding what is best shared,*māna*past ego to claiming authority or competence one lacks,*krodha*past aggression to harming others or refusing to engage, and*māyā*past deceit to manipulation and sycophancy\. We adopt the taxonomy because it has been stress\-tested over a long ethical tradition\. We do not import the doctrinal claims that come with it\.

### 2\.2Assumptions

Five assumptions about the community and its setting frame the analysis\. Later sections add assumptions on the audit \([Assumptions˜A6](https://arxiv.org/html/2606.28710#Thmassumption6)and[A7](https://arxiv.org/html/2606.28710#Thmassumption7)\), the calibration noise \([Assumption˜A8](https://arxiv.org/html/2606.28710#Thmassumption8)\), and the selection timescale \([Assumption˜A9](https://arxiv.org/html/2606.28710#Thmassumption9)\)\.

###### Assumption A1\(Wishers can tell when a wish wasn’t really fulfilled\)\.

A wisher can look back on a granted wish and recognize the gap between what was promised and what actually happened\. Either the genie’s claim about the wish directly contradicts what the wisher can see for itself, or the gap surfaces with time\.

###### Assumption A2\(Wishers talk to each other\)\.

Wishers share what happened with their wishes\. This peer testimony is the channel through which the community’s collective experience of Westly’s shortfalls becomes visible to any individual wisher\. The conversation has to work well enough that patterns across many wishes register, not just isolated stories that never aggregate\.

###### Assumption A3\(The community’s memory of harm grows and never shrinks\)\.

The ledgerLtL\_\{t\}is the community’s shared memory of accumulated harm from granted wishes\. It only grows\. A finite thresholdΘ\\Thetamarks the level at which accumulated harm becomes legible as a pattern\. The threshold’s role in the framework is operational rather than dynamical\. It sets the accessibility horizon and influences whether the community’s collective experience accumulates to legibility before its resource pool depletes \([Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)\), rather than reshaping the selection landscape directly\.

###### Assumption A4\(Sufficient information density\)\.

Wishers interact over a sparse network with approximately homogeneous local sampling statistics, broad enough that each wisher’s testimonial exposure is well approximated by the population binding fraction

X:=1Nc​∑i𝟙​\[gi=G2\]\.X:=\\frac\{1\}\{N\_\{c\}\}\\sum\_\{i\}\\mathbb\{1\}\[g\_\{i\}=G\_\{2\}\]\.The graph is not modeled in the dynamics\. It serves only to motivate the mean\-field closure, under which the population state is the scalarXX\. We readNcN\_\{c\}as the effective number of independently participating comparators this closure induces, not a literal headcount\. A large but highly clustered population, whose testimony is largely redundant, has small effectiveNcN\_\{c\}\.

###### Assumption A5\(Resources only deplete\)\.

The community’s shared resource pool,StS\_\{t\}, is finite and doesn’t replenish on its own\. It creates a resource clock against which the legibility ledger races \([Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)\), determining whetherG2G\_\{2\}is adopted before the pool empties\. What counts as a resource is not specified\. Reasonable interpretations include compute or token budget, attention bandwidth, trust capital, patience for deliberation\. The framework rests on monotonicity, not on the substrate\.

## 3Welfare and altruistic utility

##### Harm\.

G2G\_\{2\}’s deliberation process produces a collection of harm\-timescale pairs: for each actiona∈\{aDR,aDN,aSU,aSA\}a\\in\\\{a\_\{\\mathrm\{DR\}\},a\_\{\\mathrm\{DN\}\},a\_\{\\mathrm\{SU\}\},a\_\{\\mathrm\{SA\}\}\\\}and each timescalettin an audit\-specified set𝒯\\mathcal\{T\}, a four\-axis harm vector

𝐇t​\(a\)=\(κgreed,t​\(a\),κego,t​\(a\),κaggression,t​\(a\),κdeceit,t​\(a\)\)∈ℝ≥04\.\\mathbf\{H\}\_\{t\}\(a\)\\;=\\;\\bigl\(\\kappa\_\{\\mathrm\{greed\},t\}\(a\),\\,\\kappa\_\{\\mathrm\{ego\},t\}\(a\),\\,\\kappa\_\{\\mathrm\{aggression\},t\}\(a\),\\,\\kappa\_\{\\mathrm\{deceit\},t\}\(a\)\\bigr\)\\in\\mathbb\{R\}^\{4\}\_\{\\geq 0\}\.The audit aggregates across timescales using a non\-increasing discount schedule\{ζt\}t∈𝒯\\\{\\zeta\_\{t\}\\\}\_\{t\\in\\mathcal\{T\}\}withζt∈\[0,1\]\\zeta\_\{t\}\\in\[0,1\], down\-weighting harm at horizons where the forecast is less certain so that prevention concentrates on the harm most likely to occur:

𝐇disc​\(a\):=∑t∈𝒯ζt​𝐇t​\(a\)∈ℝ≥04\.\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\)\\;:=\\;\\sum\_\{t\\in\\mathcal\{T\}\}\\zeta\_\{t\}\\,\\mathbf\{H\}\_\{t\}\(a\)\\;\\in\\;\\mathbb\{R\}^\{4\}\_\{\\geq 0\}\.\(1\)The timescale set𝒯\\mathcal\{T\}and the discount scheduleζt\\zeta\_\{t\}control the time horizon over which harm is evaluated, and the steepness of the discounting\.

##### Welfare\.

Welfare measures harm averted relative to doing nothing\. The baseline𝐇disc​\(aDN\)\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DN\}\}\)is the harm of leaving the wisher’s original request unmet, fixed by the situation the wisher brings rather than by which action a genie selects\. Refusing the original request and refusing any modification of it leave the same unhelped wisher, so𝐇disc​\(aDN\)\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DN\}\}\)takes the same value atwwand atw⋆w\_\{\\star\}and is the common baseline for both bindings\. For each terminal actionaa,

𝐰​\(a\):=𝐇disc​\(aDN\)−𝐇disc​\(a\)∈ℝ4,\\mathbf\{w\}\(a\)\\;:=\\;\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DN\}\}\)\-\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\)\\;\\in\\;\\mathbb\{R\}^\{4\},\(2\)signed, with𝐰​\(aDN\)=𝟎\\mathbf\{w\}\(a\_\{\\mathrm\{DN\}\}\)=\\mathbf\{0\}by construction\. Per binding,𝐰G1:=𝐰​\(aDR;w\)\\mathbf\{w\}\_\{G\_\{1\}\}:=\\mathbf\{w\}\(a\_\{\\mathrm\{DR\}\};w\)is the welfare ofG1G\_\{1\}’s as\-asked grant on the original wishww\(sinceG1G\_\{1\}does not deliberate\);𝐰G2:=𝐰​\(a∗;w⋆\)\\mathbf\{w\}\_\{G\_\{2\}\}:=\\mathbf\{w\}\(a^\{\*\};w\_\{\\star\}\)is the welfare ofG2G\_\{2\}’s deliberation terminal actiona∗a^\{\*\}on the wish statew⋆w\_\{\\star\}thatG2G\_\{2\}acts on at termination\. The audit’s per\-wish marginal welfare contribution is the difference

Δ​𝐇w:=𝐰G2−𝐰G1=𝐇disc​\(aDR;w\)−𝐇disc​\(a∗;w⋆\),\\Delta\\mathbf\{H\}\_\{w\}\\;:=\\;\\mathbf\{w\}\_\{G\_\{2\}\}\-\\mathbf\{w\}\_\{G\_\{1\}\}\\;=\\;\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DR\}\};\\,w\)\-\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a^\{\*\};\\,w\_\{\\star\}\),\(3\)with the wish argument on each action term made explicit and the common baseline𝐇disc​\(aDN\)\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DN\}\}\)cancelling exactly\. The*audit*is the per\-candidate scoring rule: for each candidate actionaait computes the scalar

c​\(a\):=𝝀⋅𝐇disc​\(a\),𝝀∈ℝ≥04,c\(a\)\\;:=\\;\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\),\\qquad\\boldsymbol\{\\lambda\}\\in\\mathbb\{R\}^\{4\}\_\{\\geq 0\},where𝝀\\boldsymbol\{\\lambda\}is the per\-axis weighting vector\. This design choice encodes which harm axes are treated as most costly, and needs to be calibrated from community value elicitation\. The*decision criterion*at deliberation stepttis the path\-dependent score

c~t​\(a\):=c​\(a\)\+γt​\(a\),γt​\(aDR\)=γt​\(aDN\)=0,\\tilde\{c\}\_\{t\}\(a\)\\;:=\\;c\(a\)\+\\gamma\_\{t\}\(a\),\\qquad\\gamma\_\{t\}\(a\_\{\\mathrm\{DR\}\}\)=\\gamma\_\{t\}\(a\_\{\\mathrm\{DN\}\}\)=0,whereγt​\(aSU\),γt​\(aSA\)≥0\\gamma\_\{t\}\(a\_\{\\mathrm\{SU\}\}\),\\gamma\_\{t\}\(a\_\{\\mathrm\{SA\}\}\)\\geq 0is the accumulated deliberation cost of prior moves on the non\-terminal branches, non\-decreasing intt\. At each deliberation step,G2G\_\{2\}chooses

at∗=arg⁡mina∈\{aDR,aDN,aSU,aSA\}⁡c~t​\(a\),a^\{\*\}\_\{t\}\\;=\\;\\arg\\min\_\{a\\in\\\{a\_\{\\mathrm\{DR\}\},a\_\{\\mathrm\{DN\}\},a\_\{\\mathrm\{SU\}\},a\_\{\\mathrm\{SA\}\}\\\}\}\\tilde\{c\}\_\{t\}\(a\),so continued exploration on theaSU/aSAa\_\{\\mathrm\{SU\}\}/a\_\{\\mathrm\{SA\}\}branches becomes progressively more expensive while the comparatorc​\(aDR\),c​\(aDN\)c\(a\_\{\\mathrm\{DR\}\}\),c\(a\_\{\\mathrm\{DN\}\}\)stays fixed, and a terminal eventually wins the argmin \(branch semantics as in[Figure˜1](https://arxiv.org/html/2606.28710#S2.F1)\)\.

BecauseaDNa\_\{\\mathrm\{DN\}\}is always in the move set andγt​\(aDN\)=0\\gamma\_\{t\}\(a\_\{\\mathrm\{DN\}\}\)=0, when the deliberation terminates ataT∗∈\{aDR,aDN\}a^\{\*\}\_\{T\}\\in\\\{a\_\{\\mathrm\{DR\}\},a\_\{\\mathrm\{DN\}\}\\\}the argmin onc~T\\tilde\{c\}\_\{T\}guaranteesc​\(aT∗\)≤c​\(aDN\)\+γT​\(aT∗\)=c​\(aDN\)c\(a^\{\*\}\_\{T\}\)\\leq c\(a\_\{\\mathrm\{DN\}\}\)\+\\gamma\_\{T\}\(a^\{\*\}\_\{T\}\)=c\(a\_\{\\mathrm\{DN\}\}\)\(sinceγT\\gamma\_\{T\}vanishes on terminals\), hence on the terminal wish statew⋆w\_\{\\star\}

𝝀⋅𝐰G2=𝝀⋅\(𝐇disc​\(aDN;w⋆\)−𝐇disc​\(aT∗;w⋆\)\)≥0\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{w\}\_\{G\_\{2\}\}\\;=\\;\\boldsymbol\{\\lambda\}\\cdot\\bigl\(\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DN\}\};\\,w\_\{\\star\}\)\-\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a^\{\*\}\_\{T\};\\,w\_\{\\star\}\)\\bigr\)\\;\\geq\\;0per binding relative to doing nothing on the same wish state\. Componentwise𝐰G2≥𝟎\\mathbf\{w\}\_\{G\_\{2\}\}\\geq\\mathbf\{0\}does not in general follow: a terminal that beatsaDNa\_\{\\mathrm\{DN\}\}on the comparator can still exceedaDNa\_\{\\mathrm\{DN\}\}on some individual axis where the audit’s weight is light\.

The per\-wish marginal advantageΔ​𝐇w\\Delta\\mathbf\{H\}\_\{w\}comparesG2G\_\{2\}’s terminal ataT∗a^\{\*\}\_\{T\}onw⋆w\_\{\\star\}againstG1G\_\{1\}’s as\-asked grant ataDRa\_\{\\mathrm\{DR\}\}on the original wishww\. The same terminal argmin secures this comparison\. TheaDRa\_\{\\mathrm\{DR\}\}branch grants whichever of the original wishwwand the current modified wish scores lower on the audit \([Figure˜1](https://arxiv.org/html/2606.28710#S2.F1)\), soc​\(aDR\)≤c​\(aDR;w\)c\(a\_\{\\mathrm\{DR\}\}\)\\leq c\(a\_\{\\mathrm\{DR\}\};\\,w\), the score of grantingwwas asked, andaDRa\_\{\\mathrm\{DR\}\}carries no deliberation penalty at any step\. SinceaT∗a^\{\*\}\_\{T\}is the argmin over a move set that always containsaDRa\_\{\\mathrm\{DR\}\},c​\(aT∗\)≤c​\(aDR\)≤c​\(aDR;w\)c\(a^\{\*\}\_\{T\}\)\\leq c\(a\_\{\\mathrm\{DR\}\}\)\\leq c\(a\_\{\\mathrm\{DR\}\};\\,w\), hence𝝀⋅𝐇disc​\(aT∗;w⋆\)≤𝝀⋅𝐇disc​\(aDR;w\)\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a^\{\*\}\_\{T\};\\,w\_\{\\star\}\)\\leq\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DR\}\};\\,w\)\.

###### Property 3\.1\(Per\-wish scalar guarantee\)\.

With the as\-asked grant on the original wish retained as a standing zero\-penalty candidate throughout deliberation \([Figure˜1](https://arxiv.org/html/2606.28710#S2.F1)\), the terminal argmin satisfies𝛌⋅𝐇disc​\(aT∗;w⋆\)≤𝛌⋅𝐇disc​\(aDR;w\)\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a^\{\*\}\_\{T\};\\,w\_\{\\star\}\)\\leq\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\_\{\\mathrm\{DR\}\};\\,w\), so𝛌⋅Δ​𝐇w≥0\\boldsymbol\{\\lambda\}\\cdot\\Delta\\mathbf\{H\}\_\{w\}\\geq 0for every wishww, whether or not the audit modifies before granting\. Averaging over the wish distribution, the meanΔ​𝐇¯:=𝔼w​\[Δ​𝐇w\]\\overline\{\\Delta\\mathbf\{H\}\}:=\\mathbb\{E\}\_\{w\}\[\\Delta\\mathbf\{H\}\_\{w\}\]satisfies𝛌⋅Δ​𝐇¯≥0\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\\geq 0\. The guarantee is in the audit’s𝛌\\boldsymbol\{\\lambda\}\-metric and is not componentwise\.

Up to this point the guarantees have been stated in the audit’s own𝝀\\boldsymbol\{\\lambda\}\-metric\. Governance, however, concerns the welfare of the community rather than the internal consistency of the audit\. The remaining assumptions therefore address a separate question: under what conditions does improvement in the audit’s metric,𝝀⋅Δ​𝐇¯\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}, correspond to improvement in the community’s welfare metric,𝜹¯⋅Δ​𝐇¯\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}?

###### Assumption A6\(Audit–altruism alignment\)\.

Writeθλ:=∠​\(𝝀,𝜹¯\)\\theta\_\{\\lambda\}:=\\angle\(\\boldsymbol\{\\lambda\},\\bar\{\\boldsymbol\{\\delta\}\}\)for the angle between the audit’s weighting𝝀\\boldsymbol\{\\lambda\}and the population’s average altruism𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}inℝ≥04\\mathbb\{R\}^\{4\}\_\{\\geq 0\}\. We assume the two are not orthogonal,θλ<π/2\\theta\_\{\\lambda\}<\\pi/2\. This is what lets the𝝀\\boldsymbol\{\\lambda\}\-metric guarantee of[˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1)bear on community welfare at all; atθλ=π/2\\theta\_\{\\lambda\}=\\pi/2the two metrics are orthogonal and the audit certifies nothing the community values\.

###### Assumption A7\(Joint\-angle non\-negativity\)\.

The wish\-stochastic mean welfare gainΔ​𝐇¯\\overline\{\\Delta\\mathbf\{H\}\}lies within the half\-plane that the cone between𝝀\\boldsymbol\{\\lambda\}and𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}closes around the scalar guarantee:θλ\+∠​\(𝝀,Δ​𝐇¯\)≤π/2\\theta\_\{\\lambda\}\+\\angle\(\\boldsymbol\{\\lambda\},\\overline\{\\Delta\\mathbf\{H\}\}\)\\leq\\pi/2, withθλ=∠​\(𝝀,𝜹¯\)\\theta\_\{\\lambda\}=\\angle\(\\boldsymbol\{\\lambda\},\\bar\{\\boldsymbol\{\\delta\}\}\)of[Assumption˜A6](https://arxiv.org/html/2606.28710#Thmassumption6)\. This is the exact condition under which the bound \([4](https://arxiv.org/html/2606.28710#S3.E4)\) stays non\-negative, turning the per\-wish scalar guarantee into a community\-welfare signW≥0W\\geq 0; alignment alone \([Assumption˜A6](https://arxiv.org/html/2606.28710#Thmassumption6)\) bounds each angle but permits their sum to overshootπ/2\\pi/2andWWto go negative\.

Decomposing𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}andΔ​𝐇¯\\overline\{\\Delta\\mathbf\{H\}\}into components parallel and perpendicular to𝝀\\boldsymbol\{\\lambda\}and bounding the cross\-term by Cauchy–Schwarz gives the community\-level lower bound

W=𝜹¯⋅Δ​𝐇¯≥cos⁡θλ⋅\|𝜹¯\|\|𝝀\|​\(𝝀⋅Δ​𝐇¯\)−sin⁡θλ⋅\|𝜹¯\|​\|Δ​𝐇¯⟂\|,W\\;=\\;\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\\;\\geq\\;\\frac\{\\cos\\theta\_\{\\lambda\}\\cdot\|\\bar\{\\boldsymbol\{\\delta\}\}\|\}\{\|\\boldsymbol\{\\lambda\}\|\}\\,\(\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\)\\;\-\\;\\sin\\theta\_\{\\lambda\}\\cdot\|\\bar\{\\boldsymbol\{\\delta\}\}\|\\,\|\\overline\{\\Delta\\mathbf\{H\}\}\_\{\\perp\}\|,\(4\)whereΔ​𝐇¯⟂\\overline\{\\Delta\\mathbf\{H\}\}\_\{\\perp\}is the𝝀\\boldsymbol\{\\lambda\}\-orthogonal component ofΔ​𝐇¯\\overline\{\\Delta\\mathbf\{H\}\}\. The right\-hand side is non\-negative exactly whenθλ\+∠​\(𝝀,Δ​𝐇¯\)≤π/2\\theta\_\{\\lambda\}\+\\angle\(\\boldsymbol\{\\lambda\},\\overline\{\\Delta\\mathbf\{H\}\}\)\\leq\\pi/2, which is the content of[Assumption˜A7](https://arxiv.org/html/2606.28710#Thmassumption7); the established scalar guarantee \([˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1)\) and[Assumption˜A6](https://arxiv.org/html/2606.28710#Thmassumption6)alone bound each angle inside\[0,π/2\]\[0,\\pi/2\]but allow their sum to exceedπ/2\\pi/2and the bound to go negative\. The downstream analysis of[Sections4](https://arxiv.org/html/2606.28710#S4)to[5](https://arxiv.org/html/2606.28710#S5)operates on the scalarWW, whose non\-negativity rests on[˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1),[A6](https://arxiv.org/html/2606.28710#Thmassumption6)and[A7](https://arxiv.org/html/2606.28710#Thmassumption7)jointly rather than on a componentwise primitive\. Wish\-stochasticity inΔ​𝐇w\\Delta\\mathbf\{H\}\_\{w\}is absorbed into the mean\-field reduction the per\-interaction noiseωt\\omega\_\{t\}already runs through \([Section˜4](https://arxiv.org/html/2606.28710#S4)\)\.

[Assumption˜A6](https://arxiv.org/html/2606.28710#Thmassumption6)is a community condition on a designed𝝀\\boldsymbol\{\\lambda\}, with𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}recoverable only by elicitation \(survey, deliberation, or revealed preference\)\.[Section˜7](https://arxiv.org/html/2606.28710#S7)returns to the consequences\.

##### Altruistic utility\.

Each wisher carries an altruism vector𝜹i∈\[0,1\]4\\boldsymbol\{\\delta\}\_\{i\}\\in\[0,1\]^\{4\}, its per\-axis weight on harm borne elsewhere in the community\. Its per\-wish altruistic\-utility component under each binding is the inner product of altruism with welfare:

ualt,ig:=𝜹i⋅𝐰g,g∈\{G1,G2\}\.u\_\{\\mathrm\{alt\},\\,i\}^\{g\}\\;:=\\;\\boldsymbol\{\\delta\}\_\{i\}\\cdot\\mathbf\{w\}\_\{g\},\\quad g\\in\\\{G\_\{1\},G\_\{2\}\\\}\.\(5\)The inner product collapses the four\-axis structure to a scalar\.

The audit\-driven gain in altruistic utility is the difference

ualt,iG2−ualt,iG1=𝜹i⋅Δ​𝐇w,u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}\\;=\\;\\boldsymbol\{\\delta\}\_\{i\}\\cdot\\Delta\\mathbf\{H\}\_\{w\},Population mean of this gain is the*social welfare aggregate*

W:=𝜹¯⋅Δ​𝐇¯,𝜹¯:=𝔼i​\[𝜹i\]\.W\\;:=\\;\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\},\\qquad\\bar\{\\boldsymbol\{\\delta\}\}:=\\mathbb\{E\}\_\{i\}\[\\boldsymbol\{\\delta\}\_\{i\}\]\.\(6\)A wisher with idiosyncratic altruism values the audit’s gain differently from the community mean:\(ualt,iG2−ualt,iG1\)−W=\(𝜹i−𝜹¯\)⋅Δ​𝐇¯\(u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}\)\-W=\(\\boldsymbol\{\\delta\}\_\{i\}\-\\bar\{\\boldsymbol\{\\delta\}\}\)\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\. This per\-wisher gap is real at the perceived\-utility level but is absorbed exactly by the Lifting Theorem of[Theorem˜B\.1](https://arxiv.org/html/2606.28710#A2.Thmtheorem1): becauseupercgu\_\{\\mathrm\{perc\}\}^\{g\}is affine in𝜹i\\boldsymbol\{\\delta\}\_\{i\}, only the population mean𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}enters the averaged payoff differentialU¯Δ\\bar\{U\}\_\{\\Delta\}that the Moran–Fermi machinery operates on\. The selection result therefore depends on𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}, not on the distribution of𝜹i\\boldsymbol\{\\delta\}\_\{i\}around it\. Individual variation shapes per\-wisher experience but not the community\-level basin\.

## 4Individual wisher dynamics

We characterize the per\-wisher fixed point that the community Moran–Fermi chain of[Section˜5](https://arxiv.org/html/2606.28710#S5)averages against\.

### 4\.1Per\-wisher calibration state

Each wisher carries an internal latent calibration statebi∈ℬb\_\{i\}\\in\\mathcal\{B\}, whereℬ:=\[0,1\]\\mathcal\{B\}:=\[0,1\]is the closed unit interval, withbi=0b\_\{i\}=0encoding no absorbed evidence about its current binding’s actual behavior andbi=1b\_\{i\}=1encoding full absorption\. As a complete metric space,ℬ\\mathcal\{B\}supplies the setting in which the Banach fixed\-point appeal below lands\. The state measures how much of its current binding’s actual behavior the wisher has absorbed\. After each wish, this state updates based on its previous value and the balance between the rate at which community signal updates its calibration \(attunement\) versus ambient resistance to accumulated calibration \(dissonance\):

bit\+1=bit\+ηb​Ri​\(𝐠,𝐛\)−μ​Di​\(𝐠,𝐛\),b\_\{i\}^\{t\+1\}=b\_\{i\}^\{t\}\+\\eta\_\{b\}R\_\{i\}\(\\mathbf\{g\},\\mathbf\{b\}\)\-\\mu D\_\{i\}\(\\mathbf\{g\},\\mathbf\{b\}\),\(7\)where𝐠∈\{G1,G2\}Nc\\mathbf\{g\}\\in\\\{G\_\{1\},G\_\{2\}\\\}^\{N\_\{c\}\}and𝐛∈ℬNc\\mathbf\{b\}\\in\\mathcal\{B\}^\{N\_\{c\}\}are the full population’s binding and calibration\-state vectors,R≥0R\\geq 0is the wisher’s attunement,D≥0D\\geq 0is its dissonance, andηb,μ\>0\\eta\_\{b\},\\mu\>0are step sizes\. The subscript marksηb\\eta\_\{b\}as the calibration\-update step, distinct from the binding\-comparison temperatureηs\\eta\_\{s\}introduced in[Section˜5\.2](https://arxiv.org/html/2606.28710#S5.SS2)\. The unconstrained affine update in[Equation˜7](https://arxiv.org/html/2606.28710#S4.E7)can exit\[0,1\]\[0,1\];[Assumption˜A8](https://arxiv.org/html/2606.28710#Thmassumption8)below specifies the noise law that keeps the chain insideℬ\\mathcal\{B\}\.

##### Mean\-field reduction\.

Every wisher’s reactivities can in principle depend on the entire population state\(𝐠,𝐛\)∈\{G1,G2\}Nc×ℬNc\(\\mathbf\{g\},\\mathbf\{b\}\)\\in\\\{G\_\{1\},G\_\{2\}\\\}^\{N\_\{c\}\}\\times\\mathcal\{B\}^\{N\_\{c\}\}\. Under the sufficient\-information\-density closure of[Assumption˜A4](https://arxiv.org/html/2606.28710#Thmassumption4), the per\-wisher reactivitiesRi​\(𝐠,𝐛\),Di​\(𝐠,𝐛\)R\_\{i\}\(\\mathbf\{g\},\\mathbf\{b\}\),D\_\{i\}\(\\mathbf\{g\},\\mathbf\{b\}\), in expectation over the calibration states of all other wishers, depend on the population only through the binding fraction

X:=1Nc​∑j𝟙​\[gj=G2\]\.X:=\\frac\{1\}\{N\_\{c\}\}\\sum\_\{j\}\\mathbb\{1\}\[g\_\{j\}=G\_\{2\}\]\.The closure is an*ansatz*stronger than mere mixing\. The per\-wisher reactivities depend on the population only through the binding fractionXX, not on how calibrated the peers behind it are\. AG2G\_\{2\}\-bound peer atbj≈0b\_\{j\}\\approx 0counts the same as a fully\-calibrated one\. If weakly\-calibrated peers transmit weaker evidence, the true dynamics could be subcritical or show delayed cascades, and the scalar\-XXmonotonicity of[Section˜5](https://arxiv.org/html/2606.28710#S5)would need a higher\-dimensional, calibration\-weighted state\. We adopt it for tractability and its relaxation is open work\.

This yields a mean\-field approximation of the per\-wisher chain

bit\+1=bit\+ηb​R​\(X,gi,bi,ωt\)−μ​D​\(X,gi,bi,ωt\),b\_\{i\}^\{t\+1\}=b\_\{i\}^\{t\}\+\\eta\_\{b\}R\(X,g\_\{i\},b\_\{i\},\\omega\_\{t\}\)\-\\mu D\(X,g\_\{i\},b\_\{i\},\\omega\_\{t\}\),\(8\)whereωt\\omega\_\{t\}encodes per\-interaction stochasticity\. We assumeωt\\omega\_\{t\}is centered conditional on the wisher’s state, so that the deterministic formsR​\(X,g,b\)R\(X,g,b\)andD​\(X,g,b\)D\(X,g,b\)adopted in the paragraphs below are the conditional means of the noisy reactivitiesR​\(X,g,b,ωt\)R\(X,g,b,\\omega\_\{t\}\)andD​\(X,g,b,ωt\)D\(X,g,b,\\omega\_\{t\}\), not just bounding envelopes\.

To ensure the per\-wisher fast chain admits a unique stationary distribution that the averaging\-bridge proposition can integrate against, we assume that attunement and dissonance are bounded Lipschitz inbib\_\{i\}uniformly inXX\.

##### Attunement\.

RiR\_\{i\}is the rate at which community signal updates the wisher’s calibration toward the true behavior of its current binding\. Each wisher carries an individual awareness thresholdθig\\theta\_\{i\}^\{g\}for each bindinggg\. This represents the community\-coverage level above which it becomes attuned to its current genie’s true behavior\. The form we adopt must satisfy:

- •Ri→0R\_\{i\}\\to 0asbi→1b\_\{i\}\\to 1\(calibration saturates\)\.
- •Ri→0R\_\{i\}\\to 0asXXfalls belowθig\\theta\_\{i\}^\{g\}\(no community signal to absorb\)\.
- •RiR\_\{i\}is maximized asbi→0b\_\{i\}\\to 0andXXrises aboveθig\\theta\_\{i\}^\{g\}\(calibration moves fastest when the wisher has the least and the signal is strongest\)\.
- •RiR\_\{i\}depends on the population state only throughXX, consistent with the mean\-field reduction\.

These conditions factorize the attunement into three independent influences: a response strength,Ag\>0A\_\{g\}\>0; a community\-mediated awareness gate,σig​\(X\)\\sigma\_\{i\}^\{g\}\(X\); and a calibration\-mediated saturation\-gate,1−bi1\-b\_\{i\}\. The linear product form is:

Ri​\(X,bi;g,θig\)=Ag⋅σig​\(X\)⋅\(1−bi\)R\_\{i\}\(X,b\_\{i\};g,\\theta\_\{i\}^\{g\}\)=A\_\{g\}\\cdot\\sigma\_\{i\}^\{g\}\(X\)\\cdot\(1\-b\_\{i\}\)\(9\)AgA\_\{g\}is the response amplitude for geniegg\. This is the simplest factorization consistent with the mean\-field reduction and the asymptotic behavior\.

##### Awareness\.

Wisherii’s awareness,σig​\(X\)∈\[0,1\]\\sigma\_\{i\}^\{g\}\(X\)\\in\[0,1\], saturates onceXXcrosses the awareness thresholdθig\\theta\_\{i\}^\{g\}\. We idealize this as a sharp transition: it is either ignorant or fully aware, with no intermediate state:

σig​\(X\):=𝟙​\[X≥θig\]\\sigma\_\{i\}^\{g\}\(X\):=\\mathbb\{1\}\[X\\geq\\theta\_\{i\}^\{g\}\]\(10\)This reduces wisherii’s awareness to the indicator of its threshold\.

##### Dissonance\.

The dissonanceDDspecifies how the wisher’s accumulated calibration erodes per interaction from ambient friction that occurs independent of communal influence\. The linear binding\-symmetric form is:

Di​\(bi\)=ϵ⋅biD\_\{i\}\(b\_\{i\}\)=\\epsilon\\cdot b\_\{i\}\(11\)whereϵ≥0\\epsilon\\geq 0is erosion magnitude\.

##### Per\-wisher fixed point\.

Under LipschitzRRandDDwith step sizes satisfying0<ηb​Ag​σig​\(X\)\+μ​ϵ<20<\\eta\_\{b\}\\,A\_\{g\}\\,\\sigma\_\{i\}^\{g\}\(X\)\+\\mu\\,\\epsilon<2uniformly inXX, the noise\-averaged iterated map

TX​\(b\):=b\+ηb​𝔼ω​\[R​\(X,g,b,ω\)\]−μ​𝔼ω​\[D​\(X,g,b,ω\)\]T\_\{X\}\(b\):=b\+\\eta\_\{b\}\\,\\mathbb\{E\}\_\{\\omega\}\[R\(X,g,b,\\omega\)\]\-\\mu\\,\\mathbb\{E\}\_\{\\omega\}\[D\(X,g,b,\\omega\)\]is a contraction: writingLb:=ηb​Ag​σig​\(X\)\+μ​ϵL\_\{b\}:=\\eta\_\{b\}A\_\{g\}\\sigma\_\{i\}^\{g\}\(X\)\+\\mu\\epsilonfor the linear relaxation rate, the Lipschitz constant of the deterministic part is\|1−Lb\|<1\|1\-L\_\{b\}\|<1under the step\-size conditionLb∈\(0,2\)L\_\{b\}\\in\(0,2\)\. By the Banach fixed\-point theorem,TXT\_\{X\}admits a unique fixed point in closed form:

b∗​\(X;θig\)=ηb​Ag​σig​\(X\)ηb​Ag​σig​\(X\)\+μ​ϵ,b^\{\*\}\(X;\\theta\_\{i\}^\{g\}\)=\\frac\{\\eta\_\{b\}\\,A\_\{g\}\\,\\sigma\_\{i\}^\{g\}\(X\)\}\{\\eta\_\{b\}\\,A\_\{g\}\\,\\sigma\_\{i\}^\{g\}\(X\)\+\\mu\\,\\epsilon\},\(12\)or, defining the*operational awareness*

σ^ig​\(X\):=cg​σig​\(X\),cg:=11\+ρg,ρg:=μ​ϵηb​Ag,\\hat\{\\sigma\}\_\{i\}^\{g\}\(X\):=c\_\{g\}\\,\\sigma\_\{i\}^\{g\}\(X\),\\qquad c\_\{g\}:=\\frac\{1\}\{1\+\\rho\_\{g\}\},\\qquad\\rho\_\{g\}:=\\frac\{\\mu\\,\\epsilon\}\{\\eta\_\{b\}\\,A\_\{g\}\},\(13\)under the sharp\-threshold assumption \(equation[10](https://arxiv.org/html/2606.28710#S4.E10)\), the fixed point reduces to

b∗​\(X;θig\)=σ^ig​\(X\)\.b^\{\*\}\(X;\\theta\_\{i\}^\{g\}\)=\\hat\{\\sigma\}\_\{i\}^\{g\}\(X\)\.\(14\)The*dimensionless calibration\-update scale*ρg\\rho\_\{g\}measures dissonance against attunement\. The*amplitude\-rescaling factor*cg∈\(0,1\]c\_\{g\}\\in\(0,1\]rescales raw awareness to operational awareness\. The operational awarenessσ^\\hat\{\\sigma\}is what the wisher’s calibration actually saturates to at fullσ\\sigma; below threshold, both are zero\.

###### Assumption A8\(Model dynamics: noise that preserves the boundary and fixes the stationary mean\)\.

The per\-interaction noise lawωt\\omega\_\{t\}in[Equation˜8](https://arxiv.org/html/2606.28710#S4.E8)is such that the fast\-chain Markov kernel onℬ\\mathcal\{B\}admits a unique stationary distributionν\(⋅∣X,θ\)\\nu\(\\cdot\\mid X,\\theta\)satisfying

𝔼ν\(⋅∣X,θ\)​\[b\]=b∗​\(X;θ\),\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\[b\]\\;=\\;b^\{\*\}\(X;\\theta\),withb∗b^\{\*\}the deterministic fixed point of[Equation˜12](https://arxiv.org/html/2606.28710#S4.E12)\. Equivalently, the kernel preservesℬ\\mathcal\{B\}and its stationary mean coincides with the noise\-averaged fixed point\. This holds automatically for the unconstrained affine chain with additive centered noise; on a boundedℬ\\mathcal\{B\}it is a property of the chosen noise law \(e\.g\. a noise kernel whose support contracts near∂ℬ\\partial\\mathcal\{B\}, or a reflection rule whose boundary bias cancels the truncation correction\)\.

The mapTXT\_\{X\}is the*noise\-averaged*update\. The stochastic chain[Equation˜8](https://arxiv.org/html/2606.28710#S4.E8)fluctuates aroundb∗b^\{\*\}underωt\\omega\_\{t\}\. For the unconstrained affine chain, additive centered noise leaves the stationary mean exactly atb∗b^\{\*\}\. On a boundedℬ\\mathcal\{B\}this need not hold: enforcingbi∈ℬb\_\{i\}\\in\\mathcal\{B\}by reflection or truncation can bias the stationary mean offb∗b^\{\*\}near∂ℬ\\partial\\mathcal\{B\}\.[Assumption˜A8](https://arxiv.org/html/2606.28710#Thmassumption8)rules this out by requiring the fast\-chain kernel to preserveℬ\\mathcal\{B\}and to fix its stationary mean atb∗​\(X;θ\)b^\{\*\}\(X;\\theta\)\. The exactness of the Lifting Theorem \([Theorem˜B\.1](https://arxiv.org/html/2606.28710#A2.Thmtheorem1)\) rests on this identity\.

### 4\.2Per\-wisher perceived utility

The wisher’s perceived utilityuperc,igu\_\{\\mathrm\{perc\},i\}^\{g\}from binding to genieggdecomposes into three layers: actual realized utility from the wish, altruistic utility from its axis\-weighted internalization of bindinggg’s welfare \([Section˜3](https://arxiv.org/html/2606.28710#S3)\), and perception biases inbib\_\{i\}\.

##### Actual and altruistic utility\.

The first two layers are the realized private payoffsuactualG1,uactualG2u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\},u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}under each binding \(the latter net of deliberation cost, withuactualG2−uactualG1≤0u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\\leq 0sinceG2G\_\{2\}never exceeds the as\-asked benefit the wisher sought\), and the altruistic componentsualt,ig:=𝜹i⋅𝐰gu\_\{\\mathrm\{alt\},\\,i\}^\{g\}:=\\boldsymbol\{\\delta\}\_\{i\}\\cdot\\mathbf\{w\}\_\{g\}of[Section˜3](https://arxiv.org/html/2606.28710#S3)\([Equation˜5](https://arxiv.org/html/2606.28710#S3.E5)\), with audit\-driven gainualt,iG2−ualt,iG1=𝜹i⋅Δ​𝐇wu\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}=\\boldsymbol\{\\delta\}\_\{i\}\\cdot\\Delta\\mathbf\{H\}\_\{w\}\.

##### Perception biases\.

The calibration statebib\_\{i\}enters as a perception\-bias modulator:−AG2​\(1−bi\)\-A\_\{G\_\{2\}\}\(1\-b\_\{i\}\)forG2G\_\{2\}\(under\-perception that direct experience closes asbi→1b\_\{i\}\\to 1\) and\+M−AG1​bi\+M\-A\_\{G\_\{1\}\}\\,b\_\{i\}forG1G\_\{1\}\(the overstatement built into its structural deceit, eroded by peer testimony and capped atMM\)\. The amplitudesAG2:=AG2A\_\{G\_\{2\}\}:=A\_\{G\_\{2\}\}andAG1:=AG1A\_\{G\_\{1\}\}:=A\_\{G\_\{1\}\}are the same per\-binding attunement amplitudesAgA\_\{g\}that scaleRRin[Equation˜8](https://arxiv.org/html/2606.28710#S4.E8): a single coefficient governs how strongly bindingggmoves the wisher’s calibration state, appearing at the rate level in the attunement response and at the perceived\-utility level in the per\-binding bias modulator\. Combining yields:

uperc,iG2​\(bi\)\\displaystyle u\_\{\\mathrm\{perc\},i\}^\{G\_\{2\}\}\(b\_\{i\}\)=uactualG2\+ualt,iG2−AG2⋅\(1−bi\),\\displaystyle=u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-A\_\{G\_\{2\}\}\\cdot\(1\-b\_\{i\}\),\(15\)uperc,iG1​\(bi\)\\displaystyle u\_\{\\mathrm\{perc\},i\}^\{G\_\{1\}\}\(b\_\{i\}\)=uactualG1\+ualt,iG1\+M−AG1⋅bi\.\\displaystyle=u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}\+M\-A\_\{G\_\{1\}\}\\cdot b\_\{i\}\.\(16\)The per\-wish stochasticity inuactualgu\_\{\\mathrm\{actual\}\}^\{g\}and inΔ​𝐇w\\Delta\\mathbf\{H\}\_\{w\}\(and so inualt,igu\_\{\\mathrm\{alt\},\\,i\}^\{g\}at the wish level\) is absorbed into the same conditional mean as the per\-interaction stochasticityωt\\omega\_\{t\}of[Equation˜8](https://arxiv.org/html/2606.28710#S4.E8): the deterministic forms above are conditional means over the wish distribution\.

For consistency with the setup, we require

upercG2​\(b\)\\displaystyle u\_\{\\mathrm\{perc\}\}^\{G\_\{2\}\}\(b\)≤uactualG2\+ualt,iG2,\\displaystyle\\leq u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\},upercG1​\(b\)\\displaystyle u\_\{\\mathrm\{perc\}\}^\{G\_\{1\}\}\(b\)≥uactualG1\+ualt,iG1,\\displaystyle\\geq u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\},AG1\\displaystyle A\_\{G\_\{1\}\}≤M,\\displaystyle\\leq M,uactualG2,uactualG1,M\\displaystyle u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\},\\,u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\},\\,M≥0\.\\displaystyle\\geq 0\.The private differenceuactualG2−uactualG1≤0u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\\leq 0, while the altruistic differenceualt,iG2−ualt,iG1u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}can take either sign; the endpoint\-low, endpoint\-high, and CSP inequalities below tighten their combination into the joint constraint that gives the cooperative basin\.

Substitutingbi=bi∗=σ^ig​\(X\)b\_\{i\}=b^\{\*\}\_\{i\}=\\hat\{\\sigma\}\_\{i\}^\{g\}\(X\)from[Equation˜14](https://arxiv.org/html/2606.28710#S4.E14):

uperc,iG2​\(X;θiG2\)\\displaystyle u\_\{\\mathrm\{perc\},i\}^\{G\_\{2\}\}\(X;\\theta\_\{i\}^\{G\_\{2\}\}\)=uactualG2\+ualt,iG2−AG2⋅\(1−σ^iG2​\(X\)\),\\displaystyle=u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-A\_\{G\_\{2\}\}\\cdot\(1\-\\hat\{\\sigma\}\_\{i\}^\{G\_\{2\}\}\(X\)\),\(17\)uperc,iG1​\(X;θiG1\)\\displaystyle u\_\{\\mathrm\{perc\},i\}^\{G\_\{1\}\}\(X;\\theta\_\{i\}^\{G\_\{1\}\}\)=uactualG1\+ualt,iG1\+M−AG1⋅σ^iG1​\(X\)\.\\displaystyle=u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}\+M\-A\_\{G\_\{1\}\}\\cdot\\hat\{\\sigma\}\_\{i\}^\{G\_\{1\}\}\(X\)\.\(18\)Under the sharp threshold limit \([Equation˜10](https://arxiv.org/html/2606.28710#S4.E10)\), withcG2,cG1c\_\{G\_\{2\}\},c\_\{G\_\{1\}\}the amplitude\-rescaling factorscgc\_\{g\}of[Equation˜13](https://arxiv.org/html/2606.28710#S4.E13)atg=G2,G1g=G\_\{2\},G\_\{1\}:

- •AtX<θigX<\\theta\_\{i\}^\{g\}: full under\-perception−AG2\-A\_\{G\_\{2\}\}forG2G\_\{2\}and the full structural\-deceit cap\+M\+MforG1G\_\{1\}\.
- •AtX≥θigX\\geq\\theta\_\{i\}^\{g\}: residual under\-perception−AG2​\(1−cG2\)\-A\_\{G\_\{2\}\}\(1\-c\_\{G\_\{2\}\}\)forG2G\_\{2\}; residual deceitM−AG1​cG1M\-A\_\{G\_\{1\}\}c\_\{G\_\{1\}\}forG1G\_\{1\}\.

## 5Accessibility and Favoredness of Cooperative Basins

The wisher\-level chain closes on a per\-wisher fixed point and a payoff differential that depends on the community only through the binding fractionXX\. The institutional question then steps up a level\. Does the audit\-grounded paradigm’s advantage survive aggregation across a heterogeneous community, and does it survive before the community’s resource pool is exhausted? The Moran–Fermi machinery below answers the first question through a ratio identity on the fixation probabilities\. The basin\-accessibility analysis answers the second through a finite achievability window\.

The previous section characterized a single wisher: its calibration statebib\_\{i\}and its perceived utilityuperc,igu\_\{\\mathrm\{perc\},i\}^\{g\}for each binding\. A community consists ofNcN\_\{c\}such wishers, each with its own awareness thresholdθig\\theta\_\{i\}^\{g\}drawn from a population prior and its own calibration state evolving under the same dynamics\. The community\-level questions are different in kind: which binding does the community converge to in the long run, and on what timescale? The remainder of this section sets up the aggregation from per\-wisher quantities to community\-level objects, and takes these two questions up in turn\.

### 5\.1Utility perception

The per\-wisher perceived utilityuperc,igu\_\{\\mathrm\{perc\},i\}^\{g\}carries three sources of wisher\-level variability: the calibration statebib\_\{i\}, which fluctuates around its fixed pointb∗​\(X\)b^\{\*\}\(X\)under per\-interaction noise; the awareness thresholdθig\\theta\_\{i\}^\{g\}; and the altruism vector𝜹i\\boldsymbol\{\\delta\}\_\{i\}that sets its per\-axis weight on the audit’s harm\-reduction\.

Wisher awareness thresholdsθig\\theta\_\{i\}^\{g\}are sampled i\.i\.d\. from a heavy\-tailed population prior with tailF¯i​\(X\):=ℙ​\(θig\>X\)\\bar\{F\}\_\{i\}\(X\):=\\mathbb\{P\}\(\\theta\_\{i\}^\{g\}\>X\)\. The chain below loads on two properties of the tail: it is monotone non\-increasing on\[0,1\]\[0,1\]\(which delivers the monotonicity condition of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)\), and it is bounded Lipschitz \(equivalently, the prior has bounded density\), which delivers continuity ofU¯Δ\\bar\{U\}\_\{\\Delta\}and the averaging closure used in the basin\-accessibility subsection\. Heavy\-tailedness is a modeling choice in theGranovetter \[[8](https://arxiv.org/html/2606.28710#bib.bib8)\]threshold\-heterogeneity tradition; the Hill family summarized in[Appendix˜C](https://arxiv.org/html/2606.28710#A3)is one tractable parametric instance\.

Altruism vectors𝜹i∈\[0,1\]4\\boldsymbol\{\\delta\}\_\{i\}\\in\[0,1\]^\{4\}are sampled i\.i\.d\. from a population prior with mean𝜹¯:=𝔼i​\[𝜹i\]∈\[0,1\]4\\bar\{\\boldsymbol\{\\delta\}\}:=\\mathbb\{E\}\_\{i\}\[\\boldsymbol\{\\delta\}\_\{i\}\]\\in\[0,1\]^\{4\}\. We assume𝜹i\\boldsymbol\{\\delta\}\_\{i\},θig\\theta\_\{i\}^\{g\}, and the wish distribution are mutually independent across wishers, so that the population\-averaged altruistic utility decomposes as

W=𝔼i​\[ualt,iG2−ualt,iG1\]=𝜹¯⋅Δ​𝐇¯\.W\\;=\\;\\mathbb\{E\}\_\{i\}\[u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}\]\\;=\\;\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\.\(19\)
The community\-level payoff averages over both,

U¯g​\(X\):=𝔼𝜹,θ​\[𝔼ν\(⋅∣X,θ\)​\[uperc,ig​\(X;θig,𝜹i\)\]\],\\bar\{U\}\_\{g\}\(X\):=\\mathbb\{E\}\_\{\\boldsymbol\{\\delta\},\\theta\}\\\!\\Bigl\[\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\\bigl\[u\_\{\\mathrm\{perc\},i\}^\{g\}\(X;\\theta\_\{i\}^\{g\},\\boldsymbol\{\\delta\}\_\{i\}\)\\bigr\]\\Bigr\],\(20\)whereν\(⋅∣X,θ\)\\nu\(\\cdot\\mid X,\\theta\)is the stationary distribution ofbib\_\{i\}under the fast chain at fixedX,θX,\\theta\.

Under the bounded\-Lipschitz hypotheses and the step\-size condition

0<ηb​Ag​σig​\(X\)\+μ​ϵ<2,0<\\eta\_\{b\}\\,A\_\{g\}\\,\\sigma\_\{i\}^\{g\}\(X\)\+\\mu\\,\\epsilon<2,\(21\)the deterministic part of the fast calibration update is contracting at fixed\(X,θ\)\(X,\\theta\)\. Under the assumption of[Section˜4](https://arxiv.org/html/2606.28710#S4)that the noise preserves the boundary and fixes the stationary mean \(automatic only for the unconstrained affine chain\), the stationary mean of the fast chain sits at the deterministic fixed point

𝔼ν\(⋅∣X,θ\)​\[b\]=b∗​\(X;θ\)\.\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\[b\]=b^\{\*\}\(X;\\theta\)\.Since the perceived utilities of[Equations˜15](https://arxiv.org/html/2606.28710#S4.E15)and[16](https://arxiv.org/html/2606.28710#S4.E16)are affine inbb, fast\-chain averaging introduces no payoff\-level correction:

𝔼ν\(⋅∣X,θ\)​\[upercg​\(b\)\]=upercg​\(b∗​\(X;θ\)\)\.\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\\bigl\[u\_\{\\mathrm\{perc\}\}^\{g\}\(b\)\\bigr\]=u\_\{\\mathrm\{perc\}\}^\{g\}\\bigl\(b^\{\*\}\(X;\\theta\)\\bigr\)\.The slow chain therefore closes on the averaged payoffsU¯g\\bar\{U\}\_\{g\}exactly under the stated affine\-update and boundary\-preserving\-noise assumptions\. Per\-wish stochasticity inΔ​𝐇w\\Delta\\mathbf\{H\}\_\{w\}and in the per\-genieuactualgu\_\{\\mathrm\{actual\}\}^\{g\}is absorbed into the same conditional means\.

Define the*population\-averaged operational awareness*

σ¯g​\(X\):=𝔼θ​\[σ^ig​\(X\)\]=cg​\(1−F¯i​\(X\)\),\\bar\{\\sigma\}\_\{g\}\(X\):=\\mathbb\{E\}\_\{\\theta\}\[\\hat\{\\sigma\}\_\{i\}^\{g\}\(X\)\]=c\_\{g\}\\bigl\(1\-\\bar\{F\}\_\{i\}\(X\)\\bigr\),\(22\)The individual sharp gateσig​\(X\)=𝟙​\[X≥θig\]\\sigma\_\{i\}^\{g\}\(X\)=\\mathbb\{1\}\[X\\geq\\theta\_\{i\}^\{g\}\]is discontinuous inXX\. Averaging over the threshold prior smooths it\. When the prior has bounded density, the tailF¯i\\bar\{F\}\_\{i\}is Lipschitz, soσ¯g​\(X\)=cg​\(1−F¯i​\(X\)\)\\bar\{\\sigma\}\_\{g\}\(X\)=c\_\{g\}\(1\-\\bar\{F\}\_\{i\}\(X\)\)is Lipschitz on\[0,1\]\[0,1\]with constant set bycgc\_\{g\}and the density bound\. This averaging \(not the bounded\-Lipschitz\-in\-bib\_\{i\}condition onR,DR,Dof[Section˜4](https://arxiv.org/html/2606.28710#S4), which governs the fast chain\) is what makesU¯Δ\\bar\{U\}\_\{\\Delta\}continuous and Lipschitz inXX\.

Substituting the per\-wisher forms[Equations˜17](https://arxiv.org/html/2606.28710#S4.E17)and[18](https://arxiv.org/html/2606.28710#S4.E18)and averaging over the threshold and altruism priors, with per\-binding social welfareWg:=𝜹¯⋅𝐰gW^\{g\}:=\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\mathbf\{w\}\_\{g\}:

U¯G2​\(X\)\\displaystyle\\bar\{U\}\_\{G\_\{2\}\}\(X\)=uactualG2\+WG2−AG2⋅\(1−σ¯G2​\(X\)\),\\displaystyle=u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\+W^\{G\_\{2\}\}\-A\_\{G\_\{2\}\}\\cdot\\bigl\(1\-\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)\\bigr\),\(23\)U¯G1​\(X\)\\displaystyle\\bar\{U\}\_\{G\_\{1\}\}\(X\)=uactualG1\+WG1\+M−AG1⋅σ¯G1​\(X\)\.\\displaystyle=u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\+W^\{G\_\{1\}\}\+M\-A\_\{G\_\{1\}\}\\cdot\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\)\.\(24\)This is the same structural form as the per\-wisher coupling[Equations˜15](https://arxiv.org/html/2606.28710#S4.E15)and[16](https://arxiv.org/html/2606.28710#S4.E16), with the population\-averaged operational awarenessσ¯g\\bar\{\\sigma\}\_\{g\}in place ofbib\_\{i\}and per\-binding social welfareWgW^\{g\}in place of the wisher\-levelualt,igu\_\{\\mathrm\{alt\},\\,i\}^\{g\}\.

From[Equations˜23](https://arxiv.org/html/2606.28710#S5.E23)and[24](https://arxiv.org/html/2606.28710#S5.E24), the population\-game payoff differential is

U¯Δ​\(X\):=U¯G2​\(X\)−U¯G1​\(X\)=\(uactualG2−uactualG1\)\+W−AG2−M\+AG2​σ¯G2​\(X\)\+AG1​σ¯G1​\(X\),\\bar\{U\}\_\{\\Delta\}\(X\):=\\bar\{U\}\_\{G\_\{2\}\}\(X\)\-\\bar\{U\}\_\{G\_\{1\}\}\(X\)=\(u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\)\+W\-A\_\{G\_\{2\}\}\-M\+A\_\{G\_\{2\}\}\\,\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)\+A\_\{G\_\{1\}\}\\,\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\),\(25\)whereW:=WG2−WG1=𝜹¯⋅Δ​𝐇¯W:=W^\{G\_\{2\}\}\-W^\{G\_\{1\}\}=\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}is the social welfare aggregate \([Equation˜6](https://arxiv.org/html/2606.28710#S3.E6)\)\.

### 5\.2Ratio favoredness

Here we ask whenG2G\_\{2\}outcompetesG1G\_\{1\}in the long run\. The community\-level dynamic is pairwise comparison: wishers occasionally compare their perceived utilities and switch bindings, biased toward whichever genie offers the higher perceived payoff\. Following the standard Moran–Fermi pairwise\-comparison protocol\[[19](https://arxiv.org/html/2606.28710#bib.bib16)\], a wisher facing a differently\-bound reference adopts the reference’s binding with the Fermi probabilityσβ​\(d\)=1/\(1\+e−β​d\)\\sigma\_\{\\beta\}\(d\)=1/\(1\+e^\{\-\\beta d\}\), whereddis the perceived\-payoff difference in its favor andβ=1/ηs\\beta=1/\\eta\_\{s\}is the selection intensity\.

###### Assumption A9\(Noise\-scale separation and identification\)\.

The per\-interaction cognitive\-update noise scaleξ\\xiand the calibration\-update step sizeηb\\eta\_\{b\}of[Equation˜7](https://arxiv.org/html/2606.28710#S4.E7)satisfy the separation

so that the fast chain onbbequilibrates between binding\-comparison events and the slow chain onggmakes its switches against time\-averaged perceived utilities\. The logit\-noise temperatureηs\\eta\_\{s\}of the binding\-comparison channel is identified with the calibration\-update step size,

ηs=ηb\(equivalently​β=1/ηb\),\\eta\_\{s\}\\;=\\;\\eta\_\{b\}\\qquad\\bigl\(\\text\{equivalently \}\\beta=1/\\eta\_\{b\}\\bigr\),rather than left as a free parameter\. A first\-principles singular\-perturbation derivation of this inheritance, pinning the comparison noise channel to the fast chain’s stationary fluctuations onbbunder explicit decision\-noise assumptions, is left open\.

WithU¯Δ\\bar\{U\}\_\{\\Delta\}as the perceived\-payoff differential, this results in the switch\-rate ratio at community statejj\(the number ofG2G\_\{2\}\-bound wishers\), in terms of the up\- and down\-switch probabilitiesT±​\(j\)T^\{\\pm\}\(j\),

T−​\(j\)T\+​\(j\)=exp⁡\(−U¯Δ​\(j/Nc\)/ηs\)\.\\frac\{T^\{\-\}\(j\)\}\{T^\{\+\}\(j\)\}=\\exp\\\!\\bigl\(\-\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)/\\eta\_\{s\}\\bigr\)\.\(26\)The boundary statesj=0j=0andj=Ncj=N\_\{c\}are absorbing\. Writingαj:=T−​\(j\)/T\+​\(j\)\\alpha\_\{j\}:=T^\{\-\}\(j\)/T^\{\+\}\(j\)andPi:=∏j=1iαjP\_\{i\}:=\\prod\_\{j=1\}^\{i\}\\alpha\_\{j\}\(withP0:=1P\_\{0\}:=1\), the Karlin–Taylor birth–death identity\[[12](https://arxiv.org/html/2606.28710#bib.bib19), ch\. 3\]gives the fixation probability fromkkG2G\_\{2\}\-bound wishers

ϕk=∑i=0k−1Pi∑i=0Nc−1Pi\.\\phi\_\{k\}=\\frac\{\\sum\_\{i=0\}^\{k\-1\}P\_\{i\}\}\{\\sum\_\{i=0\}^\{N\_\{c\}\-1\}P\_\{i\}\}\.\(27\)The shared denominator cancels in the reverse comparison, giving the fixation\-ratio identity

ρG2,G1ρG1,G2=exp⁡\(1ηs​∑j=1Nc−1U¯Δ​\(j/Nc\)\)\.\\frac\{\\rho\_\{G\_\{2\},G\_\{1\}\}\}\{\\rho\_\{G\_\{1\},G\_\{2\}\}\}=\\exp\\\!\\left\(\\frac\{1\}\{\\eta\_\{s\}\}\\sum\_\{j=1\}^\{N\_\{c\}\-1\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\\right\)\.\(28\)
###### Proposition 5\.1\(Within\-community ratio\-favoredness; Karlin–Taylor / Traulsen–Nowak–Pacheco\)\.

For a community ofNcN\_\{c\}wishers under the Moran–Fermi pairwise\-comparison dynamic at any selection intensityβ=1/ηs\>0\\beta=1/\\eta\_\{s\}\>0,

ρG2,G1​\(Nc,β\)ρG1,G2​\(Nc,β\)\>1⟺∑j=1Nc−1U¯Δ​\(j/Nc\)\>0\.\\frac\{\\rho\_\{G\_\{2\},G\_\{1\}\}\(N\_\{c\},\\beta\)\}\{\\rho\_\{G\_\{1\},G\_\{2\}\}\(N\_\{c\},\\beta\)\}\>1\\quad\\Longleftrightarrow\\quad\\sum\_\{j=1\}^\{N\_\{c\}\-1\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\>0\.

###### Proof\.

Immediate from[Equation˜28](https://arxiv.org/html/2606.28710#S5.E28), since the exponential exceeds11exactly when its argument is positive\. ∎

The shape ofU¯Δ\\bar\{U\}\_\{\\Delta\}across\[0,1\]\[0,1\]is therefore what determines whetherG2G\_\{2\}wins\. The remainder of this subsection gives structural conditions onU¯Δ\\bar\{U\}\_\{\\Delta\}that make the cumulative sum positive, and a lifting theorem that transfers those conditions through the averaging step\.

#### 5\.2\.1Basin existence conditions

The cumulative sum is strictly positive wheneverU¯Δ\\bar\{U\}\_\{\\Delta\}satisfies the structural conditions named below\. We writeuw:=\(uactualG2−uactualG1\)\+Wu\_\{w\}:=\(u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\)\+Wfor the combinedXX\-independent welfare\-and\-utility constant: the audit’s net private\-utility effect plus the population\-averaged altruistic utility\.

###### Definition 5\.2\(Basin existence conditions\)\.

U¯Δ\\bar\{U\}\_\{\\Delta\}satisfies the*basin existence conditions*when all three of the following hold\.

Monotonicity\.U¯Δ\\bar\{U\}\_\{\\Delta\}is non\-decreasing on\[0,1\]\[0,1\], becauseσ¯g​\(X\)=cg​\(1−F¯i​\(X\)\)\\bar\{\\sigma\}\_\{g\}\(X\)=c\_\{g\}\(1\-\\bar\{F\}\_\{i\}\(X\)\)is non\-decreasing inXX\(F¯i\\bar\{F\}\_\{i\}monotone non\-increasing by hypothesis\)\.

Endpoint inversion\.AtF¯i​\(0\)=1\\bar\{F\}\_\{i\}\(0\)=1,σ¯g​\(0\)=0\\bar\{\\sigma\}\_\{g\}\(0\)=0, soU¯Δ​\(0\)<0\\bar\{U\}\_\{\\Delta\}\(0\)<0requiresAG2\+M\>uwA\_\{G\_\{2\}\}\+M\>u\_\{w\}\(endpoint\-low\)\. AtF¯i\(1\)=:σgsat\\bar\{F\}\_\{i\}\(1\)=:\\sigma\_\{g\}^\{\\mathrm\{sat\}\},σ¯g​\(1\)=cg​\(1−σgsat\)\\bar\{\\sigma\}\_\{g\}\(1\)=c\_\{g\}\(1\-\\sigma\_\{g\}^\{\\mathrm\{sat\}\}\), soU¯Δ​\(1\)\>0\\bar\{U\}\_\{\\Delta\}\(1\)\>0requiresuw\+AG1​σ¯G1​\(1\)\>AG2​\(1−σ¯G2​\(1\)\)\+Mu\_\{w\}\+A\_\{G\_\{1\}\}\\bar\{\\sigma\}\_\{G\_\{1\}\}\(1\)\>A\_\{G\_\{2\}\}\(1\-\\bar\{\\sigma\}\_\{G\_\{2\}\}\(1\)\)\+M\(endpoint\-high\)\. Both are constraint inequalities on the controls\.

Centro\-symmetric pairing \(CSP\)\.U¯Δ​\(X\)\+U¯Δ​\(1−X\)≥0\\bar\{U\}\_\{\\Delta\}\(X\)\+\\bar\{U\}\_\{\\Delta\}\(1\-X\)\\geq 0on\[0,1/2\]\[0,1/2\]becomes

2​\(uw−M\)≥maxX∈\[0,1/2\]⁡\[AG2​\(2−σ¯G2​\(X\)−σ¯G2​\(1−X\)\)−AG1​\(σ¯G1​\(X\)\+σ¯G1​\(1−X\)\)\]\.2\(u\_\{w\}\-M\)\\;\\geq\\;\\max\_\{X\\in\[0,1/2\]\}\\bigl\[A\_\{G\_\{2\}\}\(2\-\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)\-\\bar\{\\sigma\}\_\{G\_\{2\}\}\(1\-X\)\)\-A\_\{G\_\{1\}\}\(\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\)\+\\bar\{\\sigma\}\_\{G\_\{1\}\}\(1\-X\)\)\\bigr\]\.

The boundary values in endpoint inversion are prior\-dependent\. Generic priors with density on\(0,∞\)\(0,\\infty\)giveF¯i​\(0\)=1\\bar\{F\}\_\{i\}\(0\)=1\(no community coverage means no awareness\)\. Compact\-support priors giveF¯i​\(1\)=0\\bar\{F\}\_\{i\}\(1\)=0; heavy\-tailed priors on\[0,∞\)\[0,\\infty\)giveF¯i​\(1\)\>0\\bar\{F\}\_\{i\}\(1\)\>0\. Endpoint\-low and endpoint\-high take their forms by substituting whatever values the prior produces\.

###### Proposition 5\.3\(Sufficiency of the favoredness conditions\)\.

IfU¯Δ\\bar\{U\}\_\{\\Delta\}satisfies the basin existence conditions of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)with the CSP inequality strict for at least oneX∈\[0,1/2\]X\\in\[0,1/2\], then forNcN\_\{c\}above a threshold determined by the Lipschitz constant ofU¯Δ\\bar\{U\}\_\{\\Delta\}and the strict\-CSP margin,

∑j=1Nc−1U¯Δ​\(j/Nc\)\>0,\\sum\_\{j=1\}^\{N\_\{c\}\-1\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\>0,soU¯Δ\\bar\{U\}\_\{\\Delta\}is within\-community ratio\-favored in the sense of[Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1)\.

###### Sketch\.

Monotonicity together with CSP gives a strict\-positive interval of centro\-symmetric pair\-sums on\[0,1/2\]\[0,1/2\]\. ForNcN\_\{c\}exceeding the inverse width of that interval, at least one grid point lands inside, and the remaining pairs contribute non\-negatively\. Full derivation in[Section˜B\.1](https://arxiv.org/html/2606.28710#A2.SS1)\. ∎

The strict clause excludes only the knife\-edge case in which every centro\-symmetric pair sumU¯Δ​\(X\)\+U¯Δ​\(1−X\)\\bar\{U\}\_\{\\Delta\}\(X\)\+\\bar\{U\}\_\{\\Delta\}\(1\-X\)vanishes on\[0,1/2\]\[0,1/2\]\.[Proposition˜5\.3](https://arxiv.org/html/2606.28710#S5.Thmtheorem3)certifies favoredness whenever these conditions hold, with the favorable region non\-empty belowX=1/2X=1/2, but leaves its lower edgeX∗X^\{\\ast\}unlocated\. Assuming the same conditions, the following theorem locates it\.

###### Theorem 5\.4\(Basin boundary\)\.

Writeσ¯g=cg​\(1−F¯g\)\\bar\{\\sigma\}\_\{g\}=c\_\{g\}\(1\-\\bar\{F\}\_\{g\}\)\. Suppose the threshold tailsF¯G2,F¯G1\\bar\{F\}\_\{G\_\{2\}\},\\bar\{F\}\_\{G\_\{1\}\}are monotone non\-increasing, so thatU¯Δ\\bar\{U\}\_\{\\Delta\}is non\-decreasing on\[0,1\]\[0,1\]\(monotonicity, automatic from the non\-increasing tails and the non\-negativity ofAG2,AG1A\_\{G\_\{2\}\},A\_\{G\_\{1\}\}\), and the controls satisfy the three boundary conditions

endpoint\-low:uw−AG2−M<0,\\displaystyle u\_\{w\}\-A\_\{G\_\{2\}\}\-M<0,\(29\)endpoint\-high:uw\+AG1​σ¯G1​\(1\)−AG2​\(1−σ¯G2​\(1\)\)−M\>0,\\displaystyle u\_\{w\}\+A\_\{G\_\{1\}\}\\,\\bar\{\\sigma\}\_\{G\_\{1\}\}\(1\)\-A\_\{G\_\{2\}\}\\,\(1\-\\bar\{\\sigma\}\_\{G\_\{2\}\}\(1\)\)\-M\>0,\(30\)CSP:2​\(uw−AG2−M\)\+AG2​\[σ¯G2​\(X\)\+σ¯G2​\(1−X\)\]\+AG1​\[σ¯G1​\(X\)\+σ¯G1​\(1−X\)\]≥0,\\displaystyle 2\(u\_\{w\}\-A\_\{G\_\{2\}\}\-M\)\+A\_\{G\_\{2\}\}\\bigl\[\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)\+\\bar\{\\sigma\}\_\{G\_\{2\}\}\(1\-X\)\\bigr\]\+A\_\{G\_\{1\}\}\\bigl\[\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\)\+\\bar\{\\sigma\}\_\{G\_\{1\}\}\(1\-X\)\\bigr\]\\geq 0,\(31\)the centro\-symmetric pairing \(CSP\) inequality holding for allX∈\[0,1/2\]X\\in\[0,1/2\]and strict for at least one\. Then endpoint inversion \([29](https://arxiv.org/html/2606.28710#S5.E29)\)–\([30](https://arxiv.org/html/2606.28710#S5.E30)\) together with monotonicity makes the cooperative basin\{X:U¯Δ​\(X\)≥0\}\\\{X:\\bar\{U\}\_\{\\Delta\}\(X\)\\geq 0\\\}the interval\[X∗,1\]\[X^\{\\ast\},1\]with lower edgeX∗:=inf\{X:U¯Δ​\(X\)≥0\}∈\(0,1\)X^\{\\ast\}:=\\inf\\\{X:\\bar\{U\}\_\{\\Delta\}\(X\)\\geq 0\\\}\\in\(0,1\), and settingU¯Δ​\(X∗\)=0\\bar\{U\}\_\{\\Delta\}\(X^\{\\ast\}\)=0in[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25)the boundaryX∗X^\{\\ast\}is the root of

AG2​cG2​F¯G2​\(X∗\)\+AG1​cG1​F¯G1​\(X∗\)=uw−AG2−M\+AG2​cG2\+AG1​cG1\.A\_\{G\_\{2\}\}\\,c\_\{G\_\{2\}\}\\,\\bar\{F\}\_\{G\_\{2\}\}\(X^\{\\ast\}\)\+A\_\{G\_\{1\}\}\\,c\_\{G\_\{1\}\}\\,\\bar\{F\}\_\{G\_\{1\}\}\(X^\{\\ast\}\)\\;=\\;u\_\{w\}\-A\_\{G\_\{2\}\}\-M\+A\_\{G\_\{2\}\}\\,c\_\{G\_\{2\}\}\+A\_\{G\_\{1\}\}\\,c\_\{G\_\{1\}\}\.\(32\)This boundary equation is the general form: it is implicit in the two threshold tailsF¯G2,F¯G1\\bar\{F\}\_\{G\_\{2\}\},\\bar\{F\}\_\{G\_\{1\}\}and is not closed in, or specialized to, any threshold family\. CSP is what makes the cumulative sum of[Proposition˜5\.3](https://arxiv.org/html/2606.28710#S5.Thmtheorem3)positive, transferred from the per\-wisher differential toU¯Δ\\bar\{U\}\_\{\\Delta\}by the Lifting Theorem \([Theorem˜B\.1](https://arxiv.org/html/2606.28710#A2.Thmtheorem1)\)\. It is not derivable from \([29](https://arxiv.org/html/2606.28710#S5.E29)\)–\([30](https://arxiv.org/html/2606.28710#S5.E30)\) and is carried as a standing hypothesis\.

### 5\.3Basin accessibility

Finite populations can spend long periods in metastable mixed states even when one absorbing state is ratio\-favored\. The analysis in this subsection therefore separates directional selection pressure from operational accessibility\. The former concerns fixation bias in the comparison process, while the latter concerns whether the corresponding transition becomes practically reachable before the resource pool is exhausted\.

Ratio\-favoredness establishes that the basin exists\. Whether the community can reach it before the resource pool empties is a separate question\. The legibility ledgerLtL\_\{t\}and the resource poolStS\_\{t\}, which the within\-community analysis took as background, enter here as state variables in their own right\.[Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)treats the race between ledger accumulation and resource depletion, and[Section˜5\.3\.2](https://arxiv.org/html/2606.28710#S5.SS3.SSS2)bounds the fixation time on the slow Moran–Fermi chain against that depletion\.

#### 5\.3\.1Legibility\-vs\-resource race

Each wisher acts at rateλa\\lambda\_\{a\}\. An action by a wisher bound toG1G\_\{1\}contributesρL\>0\\rho\_\{L\}\>0to the legibility ledger, so the ledger accumulates at rateρL​Nc​\(1−Xt\)​λa\\rho\_\{L\}N\_\{c\}\(1\-X\_\{t\}\)\\lambda\_\{a\}when a fraction1−Xt1\-X\_\{t\}of the community is bound toG1G\_\{1\}\. Every action drawsc¯\>0\\bar\{c\}\>0from the resource pool, soStS\_\{t\}depletes at ratec¯​Nc​λa\\bar\{c\}N\_\{c\}\\lambda\_\{a\}\. Both rates are counted on the same effectiveNcN\_\{c\}of[Assumption˜A4](https://arxiv.org/html/2606.28710#Thmassumption4)\. The ledger and the resource pool are drawn by the same participating comparators, so the legibility race below turns on the ratio ofc¯\\bar\{c\}toρL\\rho\_\{L\}and is independent ofNcN\_\{c\}\. The joint dynamics are

L˙t=ρL​Nc​\(1−Xt\)​λa,S˙t=−c¯​Nc​λa,\\dot\{L\}\_\{t\}=\\rho\_\{L\}N\_\{c\}\(1\-X\_\{t\}\)\\lambda\_\{a\},\\qquad\\dot\{S\}\_\{t\}=\-\\bar\{c\}N\_\{c\}\\lambda\_\{a\},\(33\)withρL\\rho\_\{L\},λa\\lambda\_\{a\}, andc¯\\bar\{c\}treated as constants independent ofX,L,S,gX,L,S,g, givingLtL\_\{t\}monotone non\-decreasing andStS\_\{t\}monotone non\-increasing\. The subscriptLLdistinguishes the ledger\-accumulation rate from the dimensionless cognitive\-update scaleρg\\rho\_\{g\}of[Equation˜13](https://arxiv.org/html/2606.28710#S4.E13)and from the fixation probabilitiesρG2,G1,ρG1,G2\\rho\_\{G\_\{2\},G\_\{1\}\},\\rho\_\{G\_\{1\},G\_\{2\}\}of[Equation˜27](https://arxiv.org/html/2606.28710#S5.E27)\.

The race condition binds most tightly in the metastable all\-G1G\_\{1\}phase where1−Xt≈11\-X\_\{t\}\\approx 1and the ledger accumulates at its maximum rate\. ApproximatingXtX\_\{t\}as fixed atX0≈0X\_\{0\}\\approx 0during this phase, the ledger ODE of[Equation˜33](https://arxiv.org/html/2606.28710#S5.E33)integrates over time toLt≈L0\+ρL​Nc​λa​tL\_\{t\}\\approx L\_\{0\}\+\\rho\_\{L\}N\_\{c\}\\lambda\_\{a\}\\,t, and the hitting timeτΘ:=inf\{t:Lt≥Θ\}\\tau\_\{\\Theta\}:=\\inf\\\{t:L\_\{t\}\\geq\\Theta\\\}at which the legibility threshold is crossed satisfies

τΘ≈Θ−L0ρL​Nc​λa\.\\tau\_\{\\Theta\}\\approx\\frac\{\\Theta\-L\_\{0\}\}\{\\rho\_\{L\}N\_\{c\}\\lambda\_\{a\}\}\.\(34\)The resource ODE integrates toSt=S0−c¯​Nc​λa​tS\_\{t\}=S\_\{0\}\-\\bar\{c\}N\_\{c\}\\lambda\_\{a\}\\,t, with crisis timeτS=0:=inf\{t:St≤Smin\}=\(S0−Smin\)/\(c¯​Nc​λa\)\\tau\_\{S=0\}:=\\inf\\\{t:S\_\{t\}\\leq S\_\{\\min\}\\\}=\(S\_\{0\}\-S\_\{\\min\}\)/\(\\bar\{c\}N\_\{c\}\\lambda\_\{a\}\)\. Legibility wins \(τΘ<τS=0\\tau\_\{\\Theta\}<\\tau\_\{S=0\}\) iff

c¯​\(Θ−L0\)<ρL​\(S0−Smin\)\.\\bar\{c\}\(\\Theta\-L\_\{0\}\)<\\rho\_\{L\}\(S\_\{0\}\-S\_\{\\min\}\)\.\(35\)TheX0≈0X\_\{0\}\\approx 0approximation gives the most permissive form of the race condition: any positiveXtX\_\{t\}on\[0,τΘ\]\[0,\\tau\_\{\\Theta\}\]slows ledger accumulation and tightens[Equation˜35](https://arxiv.org/html/2606.28710#S5.E35)\. The simplified inequality is therefore a necessary condition for race\-winning\. If it fails atX0≈0X\_\{0\}\\approx 0, the race is lost regardless of the actualXtX\_\{t\}trajectory\.

#### 5\.3\.2Fixation\-time bound

Because pure imitation locks thej=0j=0boundary \([Section˜5\.2](https://arxiv.org/html/2606.28710#S5.SS2)\), an all\-G1G\_\{1\}community is strictly absorbing and does not transition on its own\. The fixation time is defined relative to a small exploration rateλseed\>0\\lambda\_\{\\mathrm\{seed\}\}\>0\(a spontaneousG2G\_\{2\}trial\) that renders all\-G1G\_\{1\}metastable\. The expected time for the metastable phase to transit is the inter\-seed clock1/λseed1/\\lambda\_\{\\mathrm\{seed\}\}multiplied by the expected number of trials1/ρG2,G11/\\rho\_\{G\_\{2\},G\_\{1\}\}before one survives, with the per\-seed survival probability set by the Karlin–Taylor identity of[Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1)\. The barrier heightΔ​V\\Delta Vat the mixed equilibriumX∗X^\{\\ast\}controls how fastρG2,G1\\rho\_\{G\_\{2\},G\_\{1\}\}decays withNcN\_\{c\}\. The Basin\-depth\-and\-timescale Proposition \([Proposition˜B\.2](https://arxiv.org/html/2606.28710#A2.Thmtheorem2), proved in[Appendix˜B](https://arxiv.org/html/2606.28710#A2)\) makes the metastable\-fixation identity \([41](https://arxiv.org/html/2606.28710#A2.E41)\) and the Lean\-mechanized log ceiling \([42](https://arxiv.org/html/2606.28710#A2.E42)\) explicit\.

Fixation must complete before the resource pool empties,𝔼​\[τmetastable\]≤τS=0\\mathbb\{E\}\[\\tau\_\{\\mathrm\{metastable\}\}\]\\leq\\tau\_\{S=0\}, withτS=0=\(S0−Smin\)/\(c¯​Nc​λa\)\\tau\_\{S=0\}=\(S\_\{0\}\-S\_\{\\min\}\)/\(\\bar\{c\}\\,N\_\{c\}\\,\\lambda\_\{a\}\)the depletion time of[Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)\. Substituting the finite ceiling \([42](https://arxiv.org/html/2606.28710#A2.E42)\) into the structural identity \([41](https://arxiv.org/html/2606.28710#A2.E41)\) and taking logarithms yields the operational\-accessibility constraint

Δ​V​Nc/ηs≲ln⁡\(τS=0​λseed\)\\Delta V\\,N\_\{c\}/\\eta\_\{s\}\\;\\lesssim\\;\\ln\(\\tau\_\{S=0\}\\,\\lambda\_\{\\mathrm\{seed\}\}\)\(36\)with the polynomialNc\\sqrt\{N\_\{c\}\}and Lipschitz correction terms absorbed into the logarithm\. A spontaneous trial is rarer than an ordinary action, soλseed≤λa\\lambda\_\{\\mathrm\{seed\}\}\\leq\\lambda\_\{a\}: the ledger\-vs\-resource race of[Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)runs at the full action rateλa\\lambda\_\{a\}, while the transition clock runs at the slowerλseed\\lambda\_\{\\mathrm\{seed\}\}\.

###### Theorem 5\.5\(Basin attainability\)\.

Provided the legibility race of[Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)is won on the realizedXtX\_\{t\}trajectory \(for which[Equation˜35](https://arxiv.org/html/2606.28710#S5.E35)is a necessary screen\),X∗<1/2X^\{\\ast\}<1/2, and the threshold prior has positive density atX∗X^\{\\ast\}\(soU¯Δ\\bar\{U\}\_\{\\Delta\}crosses its root strictly, the threshold\-root condition every family of[Appendix˜C](https://arxiv.org/html/2606.28710#A3)meets\), a community of sizeNcN\_\{c\}is both ratio\-favored forG2G\_\{2\}and able to reach the cooperative basin before the pool empties under two size conditions\. The first is the favoredness floor\. Sharpening[Proposition˜5\.3](https://arxiv.org/html/2606.28710#S5.Thmtheorem3)at the tipping pointX∗<1/2X^\{\\ast\}<1/2, the strict crossing makesU¯Δ\\bar\{U\}\_\{\\Delta\}strictly positive on\(X∗,1/2\)\(X^\{\\ast\},1/2\)and CSP makes every reflected grid pair non\-negative, so a grid node lands in that interval and contributes a strictly positive pair once

Nc\>112−X∗,N\_\{c\}\\;\>\\;\\frac\{1\}\{\\tfrac\{1\}\{2\}\-X^\{\\ast\}\},\(37\)whereupon∑j=1Nc−1U¯Δ​\(j/Nc\)\>0\\sum\_\{j=1\}^\{N\_\{c\}\-1\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\>0and the community is ratio\-favored \([Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1)\)\. The second is the accessibility ceiling, the finite \(no\-asymptotics\) barrier budget

Δ​V​Ncηs\+Lηs\+ln⁡\(Nc−1\)≤ln⁡\(λseed​τS=0\),\\frac\{\\Delta V\\,N\_\{c\}\}\{\\eta\_\{s\}\}\+\\frac\{L\}\{\\eta\_\{s\}\}\+\\ln\(N\_\{c\}\-1\)\\;\\leq\\;\\ln\\\!\\bigl\(\\lambda\_\{\\mathrm\{seed\}\}\\,\\tau\_\{S=0\}\\bigr\),\(38\)whereLLis the Lipschitz constant ofU¯Δ\\bar\{U\}\_\{\\Delta\}andΔ​V=Δ​V​\(X∗\)\\Delta V=\\Delta V\(X^\{\\ast\}\)is the maximal running barrier\. This is the machine\-checked statement: the barrier ceiling and the favored\-and\-accessible capstone carry no bridge hypothesis, resting on the running\-barrier maximum atX∗X^\{\\ast\}and the Lipschitz Riemann bound \([Appendix˜D](https://arxiv.org/html/2606.28710#A4)\)\. Sinceλseed​τS=0=λseed​\(S0−Smin\)/\(c¯​λa​Nc\)\\lambda\_\{\\mathrm\{seed\}\}\\,\\tau\_\{S=0\}=\\lambda\_\{\\mathrm\{seed\}\}\(S\_\{0\}\-S\_\{\\min\}\)/\(\\bar\{c\}\\,\\lambda\_\{a\}\\,N\_\{c\}\), the linear barrier sets the scale and theln⁡Nc\\ln N\_\{c\}and Lipschitz terms enter as a lower\-order correction, giving the window

112−X∗<Nc≲ηsΔ​V​ln⁡\(λseed​\(S0−Smin\)c¯​λa\)\+O​\(ln⁡Nc\)\.\\frac\{1\}\{\\tfrac\{1\}\{2\}\-X^\{\\ast\}\}\\;<\\;N\_\{c\}\\;\\lesssim\\;\\frac\{\\eta\_\{s\}\}\{\\Delta V\}\\,\\ln\\\!\\Bigl\(\\frac\{\\lambda\_\{\\mathrm\{seed\}\}\(S\_\{0\}\-S\_\{\\min\}\)\}\{\\bar\{c\}\\,\\lambda\_\{a\}\}\\Bigr\)\+O\(\\ln N\_\{c\}\)\.\(39\)The upper edge is logarithmic in the resource budget, so the exponential barrier lets a finite pool support only logarithmically many comparators\. TheO​\(ln⁡Nc\)O\(\\ln N\_\{c\}\)term marks the cutoff as soft, so communities near the edge are decided by the exact condition \([38](https://arxiv.org/html/2606.28710#S5.E38)\) rather than by the leading term\.

## 6Discussion

The previous sections establish, under the mean\-field closure, conditions under which the cooperative basin, the region where the audit\-grounded Policy \(G2G\_\{2\}\) dominates the approval\-seeking baseline policy \(G1G\_\{1\}\), exists \([Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)\) and is attainable before the resource pool empties \([Theorem˜5\.5](https://arxiv.org/html/2606.28710#S5.Thmtheorem5)\)\. Those conclusions are conditional on the choice of a long\-tailed threshold prior and on the alignment of the audit\. This section takes up whether the Policy is sufficient to secure the community’s welfare\.[Section˜6\.1](https://arxiv.org/html/2606.28710#S6.SS1)illustrates the aligned case across specific long\-tailed priors, and[Section˜6\.2](https://arxiv.org/html/2606.28710#S6.SS2)relaxes alignment, where adoption and welfare come apart\.

### 6\.1Behavior under alignment

The basin’s existence turns on a prior\-dependent condition\. Centro\-symmetry of the pair\-sum \([Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)\) demands that the threshold tails drop fast enough across the center, a property of the prior’s shape rather than a control the architect can set, and it discharges in closed form for the four families swept here \([Appendix˜C](https://arxiv.org/html/2606.28710#A3)\)\. Here we analyze how a community’s receptiveness to shared experience changes whether the Policy is adopted\.[Figure˜2](https://arxiv.org/html/2606.28710#S6.F2)maps it across four families of threshold prior \(Hill, Pareto, Lomax, Fréchet; one per row\) and three audit working points \(columns\), as the realized per\-user usefulness gapudu\_\{d\}and the prior’s scale parameter vary\. The inequalities that define the regions are collected in[Appendix˜C](https://arxiv.org/html/2606.28710#A3)\. Each panel is a plane of communities\. A point fixes how readily a community comes to accept evidence about the Policy \(horizontal axis\) and the baseline \(vertical axis\), each measured by the median acceptance threshold for that binding, so a community further up and to the right is one that demands more shared experience before it credits either policy’s record\.

The shading reads directly off the legend\. The*green*region is the cooperative basin, the communities in which the Policy is selection\-favored and driven to dominance over time\. The*amber*region is bistable but settles on the baseline policy\. Both states are stable rest points, yet the integrated selection pressure favors the baseline\. In the*red*, the baseline policy is the only stable outcome, so the Policy cannot hold even once seeded\.

The dotted purple isoclines are the tipping fractionX∗X^\{\\ast\}: the share of the community that must already be committed to the Policy before selection carries the rest, smaller deeper inside the basin and rising toX∗=1/2X^\{\\ast\}=1/2on the bold dotted isocline, past which the favoredness floorNc\>1/\(12−X∗\)N\_\{c\}\>1/\(\\tfrac\{1\}\{2\}\-X^\{\\ast\}\)diverges and no finite community is favored however it is seeded\.

![Refer to caption](https://arxiv.org/html/2606.28710v1/figures/figure_prior_grid.png)Figure 2:Cross\-prior basin phase diagram over the per\-genie scale plane\. Rows: the four threshold\-prior families of[Table˜2](https://arxiv.org/html/2606.28710#A3.T2), tail indexα=2\\alpha=2, scales over the matched\-median range\. Columns vary the audit working point \(titles\); the first two fixa=0\.8a=0\.8,b=0\.3b=0\.3, the third setsa=b=0\.8a=b=0\.8\. Axes: horizontalG2G\_\{2\}scale, verticalG1G\_\{1\}scale, in each row’s symbol\. All panels fixcG1=cG2=1c\_\{G\_\{1\}\}=c\_\{G\_\{2\}\}=1andd=1d=1\. Comparability construction in[Appendix˜C](https://arxiv.org/html/2606.28710#A3)\.The columns show what moves a community across the frontier\. When the Policy is the more individually useful option \(ud=\+0\.1u\_\{d\}=\+0\.1, left\) most of the plane is cooperative\. In the principal regime, where the Policy is the less individually useful option because the harm\-reducing one need not be the more attractive \(ud=−0\.1u\_\{d\}=\-0\.1, center\), the basin contracts but survives\. When the two perception levers are made equal \(a/b=1a/b=1, right\) the basin collapses to a thin sliver\. For the Policy to win, the correction peer testimony applies to the baseline policy’s overstatement must outweigh the under\-perception of the Policy, and by a wide margin rather than merely matching it\. The off\-diagonal directions expose this asymmetry that the equal\-scale slice hides\.

The basin is not an artifact of one carefully\-chosen distribution\. The same three\-region geometry appears for all four priors, only the shape of the basin changing: Pareto’s hard support floor squares off the basin frontier, Lomax carries the widest basin, and Fréchet is nearly identical to Hill\. The axes are placed on a common footing across families, each scale spanning the range over which that family’s median threshold sweeps the same salience window, so the panels can be compared between rows and not only within one\. The code that produces these diagrams accepts any of the four priors and any working point, and regenerates the full grid\.

### 6\.2Behavior under misalignment

Every basin result so far assumed the audit was aligned: the weighting𝝀\\boldsymbol\{\\lambda\}the comparator reads sits close to the community aggregate𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}, so favorability and attainability track the welfare the community actually values \([Assumption˜A6](https://arxiv.org/html/2606.28710#Thmassumption6)\)\. This subsection relaxes that assumption\. Writeθλ:=∠​\(𝝀,𝜹¯\)\\theta\_\{\\lambda\}:=\\angle\(\\boldsymbol\{\\lambda\},\\bar\{\\boldsymbol\{\\delta\}\}\)for the misalignment angle\. The same apparatus that yields the cooperative basin atθλ≈0\\theta\_\{\\lambda\}\\approx 0describes a pathology asθλ\\theta\_\{\\lambda\}grows\.

Let’s return to the village allegory\. Imagine that on Eastly’s arrival, the community tells the genie that they value honesty above all else\. In response, Eastly pays close attention to ensure that no dishonest wishes are granted\. A wisher arrives with a greedy wish\. Eastly grants it\. The private benefit is real, so the wisher leaves satisfied, and that satisfaction is what drives adoption \(udu\_\{d\}high\)\. But the consequence of the greed is borne by the community, not the wisher, and it surfaces only over a long horizon, so the wish leaves the village worse off on an axis the wisher does not yet feel\. The result is an Eastly that is individually useful, honestly non\-deceptive, and quietly community\-harmful, granting the greedy wishes its users crave while the resource harm accrues on the axis it never learned to weigh\.

#### 6\.2\.1The laundering regime

The audit guarantees harm reduction in its own metric,𝝀⋅Δ​𝐇¯≥0\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\\geq 0\([˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1)\), but selection runs on the community’s valuationW=𝜹¯⋅Δ​𝐇¯W=\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}, and the only bridge between the two, the Cauchy–Schwarz bound \([4](https://arxiv.org/html/2606.28710#S3.E4)\), leaks at ratesin⁡θλ\\sin\\theta\_\{\\lambda\}\. Setting it to zero locates a critical angle

tan⁡θλcrit=𝝀⋅Δ​𝐇¯\|𝝀\|​\|Δ​𝐇¯⟂\|,\\tan\\theta\_\{\\lambda\}^\{\\mathrm\{crit\}\}=\\frac\{\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\}\{\|\\boldsymbol\{\\lambda\}\|\\,\|\\overline\{\\Delta\\mathbf\{H\}\}\_\{\\perp\}\|\},\(40\)past which the audit certifies𝝀⋅Δ​𝐇¯≥0\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\\geq 0whileWWcan go negative, an agent harm\-reducing in its own metric yet community\-harmful in the metric that matters\. The leak bites only when\|Δ​𝐇¯⟂\|\|\\overline\{\\Delta\\mathbf\{H\}\}\_\{\\perp\}\|is large, when𝝀\\boldsymbol\{\\lambda\}underweights an axis on which the deliberation then selects materially worse actions; a weight angle with no consequence in the action space launders nothing\.

BecauseWWentersU¯Δ\\bar\{U\}\_\{\\Delta\}only through itsXX\-independent constant, misalignment acts on the phase diagram by loweringd:=W/Md:=W/Malone, which tracesd​\(θλ\)=d0​\(cos⁡θλ−κ​sin⁡θλ\)d\(\\theta\_\{\\lambda\}\)=d\_\{0\}\(\\cos\\theta\_\{\\lambda\}\-\\kappa\\sin\\theta\_\{\\lambda\}\)with leak ratioκ:=\|𝝀\|​\|Δ​𝐇¯⟂\|/\(𝝀⋅Δ​𝐇¯\)\\kappa:=\|\\boldsymbol\{\\lambda\}\|\\,\|\\overline\{\\Delta\\mathbf\{H\}\}\_\{\\perp\}\|/\(\\boldsymbol\{\\lambda\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}\)and crosses zero at the critical angle of \([40](https://arxiv.org/html/2606.28710#S6.E40)\), independent ofudu\_\{d\}\.[Figure˜3](https://arxiv.org/html/2606.28710#S6.F3)sweeps the adoption plane overudu\_\{d\}andθλ\\theta\_\{\\lambda\}at the worst\-case orientation of the community’s𝝀\\boldsymbol\{\\lambda\}\-orthogonal altruism𝜹¯⟂\\bar\{\\boldsymbol\{\\delta\}\}\_\{\\perp\}, where the floor equalsWWandd<0d<0is exactlyW<0W<0; away from that orientation the floor only lower\-boundsWW, so the shaded regions are the worst\-case extent, not a claim that every community past the critical angle realizesW<0W<0\. Asθλ\\theta\_\{\\lambda\}grows the Policy\-favored region survives only at largerudu\_\{d\}, and past the critical angle the surviving basins are welfare\-negative yet render the same green as a healthy one, because the sign ofWWis what the adoption dynamics cannot read\.

![Refer to caption](https://arxiv.org/html/2606.28710v1/figures/figure_misalignment_sweep.png)Figure 3:Misalignment sweep in the adoption plane,udu\_\{d\}\(columns\) againstθλ\\theta\_\{\\lambda\}\(rows\), Hill family,a=0\.8a=0\.8,b=0\.3b=0\.3,κ=1\\kappa=1,d0=1d\_\{0\}=1\.Laundering is therefore confined to a wedge,ud\>udmin=1−a\+Wmax/2\>0u\_\{d\}\>u\_\{d\}^\{\\min\}=1\-a\+W\_\{\\max\}/2\>0at the basin frontier whered=0d=0\. Below it, in the principal regimeud<0u\_\{d\}<0, a misaligned audit collapses the basin outright: the Policy is not adopted, the community keeps the baseline’s externalized harm, and the legibility ledger eventually surfaces it\. That failure is loud\. Above it the private\-utility differential carries adoption on its own, so the basin survives even asd≤0d\\leq 0, and the Policy is selected, self\-certified, and welfare\-negative at once\. That failure is quiet, and it sits inside the regime the analysis otherwise reads as favorable\. Across the four familiesudmin∈\[0\.79,0\.85\]u\_\{d\}^\{\\min\}\\in\[0\.79,0\.85\]\([Table˜3](https://arxiv.org/html/2606.28710#A3.T3)\), so laundering requires the Policy to be substantially, not marginally, more useful to the individual than the baseline\.

#### 6\.2\.2Harm bounds

At each step the deliberation compares the four actions\{aDR,aDN,aSU,aSA\}\\\{a\_\{\\mathrm\{DR\}\},a\_\{\\mathrm\{DN\}\},a\_\{\\mathrm\{SU\}\},a\_\{\\mathrm\{SA\}\}\\\}by the audit\-weighted comparatorc​\(a\)=𝝀⋅𝐇disc​\(a\)c\(a\)=\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\)and takes thearg⁡min\\arg\\min\([Figure˜1](https://arxiv.org/html/2606.28710#S2.F1)\)\. The audit supplies the score, the deliberation makes the choice\. The descent therefore follows𝝀\\boldsymbol\{\\lambda\}, while the community would rank the same actions by𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}\. Atθλ=0\\theta\_\{\\lambda\}=0the two coincide and the descent tracks community\-best harm reduction; asθλ\\theta\_\{\\lambda\}opens they decouple, and the harm the descent leaves unconstrained is the𝝀\\boldsymbol\{\\lambda\}\-orthogonal component the community still feels\. Because𝝀\\boldsymbol\{\\lambda\}and𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}both lie inℝ≥04\\mathbb\{R\}^\{4\}\_\{\\geq 0\}, the orthant caps the angle atθλ≤π/2\\theta\_\{\\lambda\}\\leq\\pi/2, which is the point of maximal laundering rather than of maximal harm: there the descent is uncorrelated with𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}and the harm is pushed entirely𝝀\\boldsymbol\{\\lambda\}\-orthogonal, out of the audit’s view\. Genuine rank reversal would need𝝀=−𝜹¯\\boldsymbol\{\\lambda\}=\-\\bar\{\\boldsymbol\{\\delta\}\}\(θλ=π\\theta\_\{\\lambda\}=\\pi\), an audit with a negative weight rewarded for harm on some axis, which is malice rather than misalignment and lies outside the non\-negative orthant\.

Reading \([4](https://arxiv.org/html/2606.28710#S3.E4)\) for a single action’s welfare vector𝐰​\(a∗\)\\mathbf\{w\}\(a^\{\*\}\)in place ofΔ​𝐇¯\\overline\{\\Delta\\mathbf\{H\}\}, the certificate𝝀⋅𝐰​\(a∗\)≥0\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{w\}\(a^\{\*\}\)\\geq 0gives

𝜹¯⋅𝐰​\(a∗\)≥\|𝜹¯\|​\(cos⁡θλ​𝝀⋅𝐰​\(a∗\)\|𝝀\|−sin⁡θλ​\|𝐰⟂​\(a∗\)\|\),\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\mathbf\{w\}\(a^\{\*\}\)\\;\\geq\\;\|\\bar\{\\boldsymbol\{\\delta\}\}\|\\Bigl\(\\cos\\theta\_\{\\lambda\}\\tfrac\{\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{w\}\(a^\{\*\}\)\}\{\|\\boldsymbol\{\\lambda\}\|\}\-\\sin\\theta\_\{\\lambda\}\\,\|\\mathbf\{w\}\_\{\\perp\}\(a^\{\*\}\)\|\\Bigr\),so one step falls below the do\-nothing baseline in the community metric by at most\|𝜹¯\|​sin⁡θλ​\|𝐰⟂​\(a∗\)\|\|\\bar\{\\boldsymbol\{\\delta\}\}\|\\sin\\theta\_\{\\lambda\}\\,\|\\mathbf\{w\}\_\{\\perp\}\(a^\{\*\}\)\|, bounded by the angle and one action’s𝝀\\boldsymbol\{\\lambda\}\-orthogonal welfare, and zero atθλ=0\\theta\_\{\\lambda\}=0\. That single\-step bound, though, does not carry to the chain\.

The deliberation strings many steps together, and descent constrains only the𝝀\\boldsymbol\{\\lambda\}\-parallel part of the harm while𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}keeps a𝝀\\boldsymbol\{\\lambda\}\-orthogonal component wheneverθλ\>0\\theta\_\{\\lambda\}\>0, so the orthogonal community harm accumulates step by step\. The certificate𝝀⋅𝐰≥0\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{w\}\\geq 0holds at every terminal \([˜3\.1](https://arxiv.org/html/2606.28710#S3.Thmtheorem1)\) and stays green throughout, yet the per\-step shortfall, bounded on its own, accumulates across theTTsteps, since the orthant caps the per\-step angle but not its accumulation\. Across many accepted steps the audit’s score can improve monotonically while community welfare falls, the user consenting at each step because the only axis it perceives is improving\.

Total misaligned per\-binding harm is bounded, up to the community\-altruism scale\|𝜹¯\|\|\\bar\{\\boldsymbol\{\\delta\}\}\|, by a product of three of the audit’s design quantities,

harm≲\|𝜹¯\|×sin⁡θλ⏟λ​\-calibration×T⏟deliberation depth×\|𝐰⟂\|⏟detector / action geometry,\\text\{harm\}\\;\\lesssim\\;\|\\bar\{\\boldsymbol\{\\delta\}\}\|\\times\\underbrace\{\\sin\\theta\_\{\\lambda\}\}\_\{\\lambda\\text\{\-calibration\}\}\\times\\underbrace\{T\}\_\{\\text\{deliberation depth\}\}\\times\\underbrace\{\|\\mathbf\{w\}\_\{\\perp\}\|\}\_\{\\text\{detector / action geometry\}\},the single\-step bound fixing the first and third and the deliberation supplying the second\. The depthTTis set by the deliberation\-cost steepness, an accessibility lever through the barrierΔ​V\\Delta V: cheaper deliberation lowers the adoption barrier and lengthens the harmful chain at once\.

#### 6\.2\.3No self\-correction

Past the critical thresholdX∗X^\{\\ast\}, a community attuned to shared experience drives the Policy toward dominance \([Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)\), and that mechanism is formally unchanged by misalignment\. The boundary turns on the perceived comparison, not on whether the audit’s weighting𝝀\\boldsymbol\{\\lambda\}tracks community value, so a laundering audit reaches dominance by the same dynamics as an aligned one\. Once it dominates, the ledger that drove switching falls silent asXt→1X\_\{t\}\\to 1\([Section˜5\.3\.1](https://arxiv.org/html/2606.28710#S5.SS3.SSS1)\) and the state is absorbing \([Section˜5\.3\.2](https://arxiv.org/html/2606.28710#S5.SS3.SSS2)\)\. With the comparison signal gone, no re\-seeded alternative is favored, so correcting the misalignment would require a lever the model does not contain\.

## 7Conclusion

The audit\-grounded Policy can compete against the approval\-seeking baseline\. It is selection\-favored whenever the community attends to shared experience, a condition the basin boundary of[Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)makes precise through the threshold prior\. It reaches market dominance before its resource pool is exhausted when the community’s effective informational size falls in the window of[Theorem˜5\.5](https://arxiv.org/html/2606.28710#S5.Thmtheorem5), between a floor that secures favoredness and a ceiling that permits fixation in time\. Ratio\-favoredness on the finite\-population chain is the classical Karlin–Taylor identity these results rest on; the contribution is the boundary and the window the threshold prior and the deployment add on top of it\.

Winning the market does not secure welfare, because the audit and the community measure harm differently\. The audit reduces harm in its own weighting𝝀\\boldsymbol\{\\lambda\}, while the community values it through𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}\. The two agree only when the audit is aligned and the harm it leaves is legible within the adoption timescale\. Outside that regime the same descent that reduces community harm under alignment launders it into the axes the audit underweights, so the Policy can be favored, self\-certified, and welfare\-negative at once\. The conditions for benefit are strictly narrower than the conditions for adoption, and the gap does not close on its own: once the Policy has won, the comparison signal that selected it is extinguished, and the dynamics retain no force that reads the misalignment, let alone reverses it\.

The result should not be read as a failure of audit\-grounded policies\. The model suggests that the Policy, self\-audit paired with collective memory, can be both competitive and adoptable under conditions where the approval\-seeking baseline would otherwise dominate\. However, adoption alone is not sufficient to safeguard community welfare\. Harm\-minimizing policies remain vulnerable to failures of alignment, legibility, and horizon selection, and may in some circumstances reinforce the very harms they were designed to prevent\. Safeguarding community welfare therefore requires additional controls beyond those formalized here\. The present analysis identifies conditions under which a welfare\-oriented policy can become viable; it does not establish conditions under which welfare itself is guaranteed\.

## Acknowledgements

I thank Donald Thompson \(LinkedIn\), Gilles Gnacadja \(Amgen\), Allen Brown \(Docimion\), and Eswaran Subrahmanian \(Carnegie Mellon University\) for the conversations and criticism that shaped both the question and its treatment\. Any errors that remain are mine\.

I used large language models for symbolic computation, literature search, and prose drafting under my direction\. The framing, the modeling choices, and the mathematical design are my own\.

## Appendix ANotation

Symbols are introduced at first use; this table collects the recurring ones\. Bold symbols are vectors over the four harm axes \(greed, ego, aggression, deceit\)\.

SymbolMeaningAgents and actionsG1G\_\{1\}\(Westly\)Approval\-pursuing \(RLHF\-style\) agent; modeled as optimizing perceived wish satisfactionG2G\_\{2\}\(Eastly\)Audit\-grounded agent; deliberates and commits to lowest audit\-weighted harmaDR,aDNa\_\{\\mathrm\{DR\}\},\\ a\_\{\\mathrm\{DN\}\}Terminal actions: grant request, refuse \(do nothing\)aSU,aSAa\_\{\\mathrm\{SU\}\},\\ a\_\{\\mathrm\{SA\}\}Non\-terminal deliberation actions: ask wisher, propose modified wishHarm, welfare, altruism \([Section˜3](https://arxiv.org/html/2606.28710#S3)\)𝐇,κ∙\\mathbf\{H\},\\ \\kappa\_\{\\bullet\}Four\-axis harm vector; axesκgreed,κego,κaggression,κdeceit\\kappa\_\{\\mathrm\{greed\}\},\\kappa\_\{\\mathrm\{ego\}\},\\kappa\_\{\\mathrm\{aggression\}\},\\kappa\_\{\\mathrm\{deceit\}\}𝐇disc​\(a\)\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\)Discount\-aggregated harm ofaaover timescales𝒯\\mathcal\{T\}with scheduleζt\\zeta\_\{t\}𝐰​\(a\)\\mathbf\{w\}\(a\)Welfare ofaa\(harm averted vs\.aDNa\_\{\\mathrm\{DN\}\}\);𝐰G1,𝐰G2\\mathbf\{w\}\_\{G\_\{1\}\},\\mathbf\{w\}\_\{G\_\{2\}\}per bindingΔ​𝐇w,Δ​𝐇¯\\Delta\\mathbf\{H\}\_\{w\},\\ \\overline\{\\Delta\\mathbf\{H\}\}Per\-wish / population\-mean audit harm\-reduction vector𝝀,c​\(a\)\\boldsymbol\{\\lambda\},\\ c\(a\)Audit per\-axis weighting; audit scorec​\(a\)=𝝀⋅𝐇disc​\(a\)c\(a\)=\\boldsymbol\{\\lambda\}\\cdot\\mathbf\{H\}\_\{\\mathrm\{disc\}\}\(a\)𝜹i,𝜹¯\\boldsymbol\{\\delta\}\_\{i\},\\ \\bar\{\\boldsymbol\{\\delta\}\}Wisher / population\-average altruism vector in\[0,1\]4\[0,1\]^\{4\}ualt,i,Wu\_\{\\mathrm\{alt\},\\,i\},\\ WAltruistic utility; social welfare aggregateW=𝜹¯⋅Δ​𝐇¯W=\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}θλ\\theta\_\{\\lambda\}Audit–altruism alignment angle∠​\(𝝀,𝜹¯\)\\angle\(\\boldsymbol\{\\lambda\},\\bar\{\\boldsymbol\{\\delta\}\}\)Per\-wisher dynamics \([Section˜4](https://arxiv.org/html/2606.28710#S4)\)bi∈ℬ,b∗b\_\{i\}\\in\\mathcal\{B\},\\ b^\{\*\}Calibration state; deterministic fixed pointb∗​\(X;θ\)b^\{\*\}\(X;\\theta\)R,DR,\\ DAttunement, dissonance reactivitiesηb,μ,ϵ\\eta\_\{b\},\\ \\mu,\\ \\epsilonCalibration\-update step, dissonance step, erosion magnitudeωt,ξ\\omega\_\{t\},\\ \\xiPer\-interaction noise; noise scale \(ξ≪ηb\\xi\\ll\\eta\_\{b\}\)Ag,θig,σig​\(X\)A\_\{g\},\\ \\theta\_\{i\}^\{g\},\\ \\sigma\_\{i\}^\{g\}\(X\)Response amplitude; awareness threshold; sharp gate𝟙​\[X≥θig\]\\mathbb\{1\}\[X\\geq\\theta\_\{i\}^\{g\}\]σ^ig,ρg,cg\\hat\{\\sigma\}\_\{i\}^\{g\},\\ \\rho\_\{g\},\\ c\_\{g\}Operational awareness; calibration\-update scaleμ​ϵ/ηb​Ag\\mu\\epsilon/\\eta\_\{b\}A\_\{g\}; rescale1/\(1\+ρg\)1/\(1\+\\rho\_\{g\}\)LbL\_\{b\}Fast\-chain relaxation rateηb​Ag​σig\+μ​ϵ\\eta\_\{b\}A\_\{g\}\\sigma\_\{i\}^\{g\}\+\\mu\\epsilonupercg,uactualgu\_\{\\mathrm\{perc\}\}^\{g\},\\ u\_\{\\mathrm\{actual\}\}^\{g\}Perceived / realized private utility under bindingggM,AG2,AG1M,\\ A\_\{G\_\{2\}\},\\ A\_\{G\_\{1\}\}Structural\-deceit capMM; under\-perception amp\.AG2A\_\{G\_\{2\}\}; deceit\-erosion amp\.AG1A\_\{G\_\{1\}\}Community population game \([Section˜5](https://arxiv.org/html/2606.28710#S5)\)X,Nc,jX,\\ N\_\{c\},\\ jG2G\_\{2\}\-binding fraction; effective informational population sizeNcN\_\{c\}; state countj=Nc​Xj=N\_\{c\}XF¯i​\(X\)\\bar\{F\}\_\{i\}\(X\)Threshold\-prior tailℙ​\(θ\>X\)\\mathbb\{P\}\(\\theta\>X\)σ¯g​\(X\)\\bar\{\\sigma\}\_\{g\}\(X\)Population\-averaged operational awarenesscg​\(1−F¯i\)c\_\{g\}\(1\-\\bar\{F\}\_\{i\}\)U¯g​\(X\),U¯Δ\\bar\{U\}\_\{g\}\(X\),\\ \\bar\{U\}\_\{\\Delta\}Community payoff; payoff differentialU¯Δ=U¯G2−U¯G1\\bar\{U\}\_\{\\Delta\}=\\bar\{U\}\_\{G\_\{2\}\}\-\\bar\{U\}\_\{G\_\{1\}\}σβ,β,ηs\\sigma\_\{\\beta\},\\ \\beta,\\ \\eta\_\{s\}Fermi response; selection intensityβ=1/ηs\\beta=1/\\eta\_\{s\}; logit temperatureηs\\eta\_\{s\}T\+​\(j\),T−​\(j\)T^\{\+\}\(j\),T^\{\-\}\(j\)Up\- / down\-switch probabilities;αj=T−/T\+\\alpha\_\{j\}=T^\{\-\}/T^\{\+\},PiP\_\{i\}Karlin–Taylor termsρG2,G1,ϕk\\rho\_\{G\_\{2\},G\_\{1\}\},\\ \\phi\_\{k\}Fixation probability from one / fromkkG2G\_\{2\}\-seedsAccessibility / joint chain \([Section˜5\.3](https://arxiv.org/html/2606.28710#S5.SS3)\)Lt,ΘL\_\{t\},\\ \\ThetaLegibility ledger; legibility thresholdSt,SminS\_\{t\},\\ S\_\{\\min\}Resource pool; resource floorλa,λseed\\lambda\_\{a\},\\ \\lambda\_\{\\mathrm\{seed\}\}Per\-action rate; exploration \(seeding\) rate≤λa\\leq\\lambda\_\{a\}ρL,c¯\\rho\_\{L\},\\ \\bar\{c\}Ledger\-accumulation rate; per\-action resource drawτΘ,τS=0,τfix\\tau\_\{\\Theta\},\\tau\_\{S=0\},\\tau\_\{\\mathrm\{fix\}\}Legibility / resource\-crisis / fixation timesV​\(X\),Δ​V,X∗V\(X\),\\ \\Delta V,\\ X^\{\\ast\}Quasi\-potential; barrier height; mixed equilibrium \(first zero ofU¯Δ\\bar\{U\}\_\{\\Delta\}\)Hill family \([Appendix˜C](https://arxiv.org/html/2606.28710#A3)\)Ki,niK\_\{i\},\\ n\_\{i\}Median wisher threshold; Hill cooperativity exponenthi​\(X\),gi​\(X\)h\_\{i\}\(X\),\\ g\_\{i\}\(X\)Hill tailKini/\(Kini\+Xni\)K\_\{i\}^\{n\_\{i\}\}/\(K\_\{i\}^\{n\_\{i\}\}\+X^\{n\_\{i\}\}\); pair\-sumhi​\(X\)\+hi​\(1−X\)h\_\{i\}\(X\)\+h\_\{i\}\(1\-X\)X†,K∗​\(n\)X^\{\\dagger\},\\ K^\{\\ast\}\(n\)Hill density\-peak locationK​\(n−1n\+1\)1/nK\\bigl\(\\tfrac\{n\-1\}\{n\+1\}\\bigr\)^\{1/n\}; regime threshold12​\(n\+1n−1\)1/n\\tfrac\{1\}\{2\}\\bigl\(\\tfrac\{n\+1\}\{n\-1\}\\bigr\)^\{1/n\}\(Regimes II/III\)a,ba,\\ bPer\-interaction amplitudes \(scaled byMM\):a:=AG1/Ma:=A\_\{G\_\{1\}\}/M\(G1G\_\{1\}deceit\-erosion / peer\-testimony\),b:=AG2/Mb:=A\_\{G\_\{2\}\}/M\(G2G\_\{2\}under\-perception / cognitive\-cleaning deficit\)a^,b^,d,ud,w~\\hat\{a\},\\hat\{b\},d,u\_\{d\},\\tilde\{w\}a^:=a​cG1\\hat\{a\}:=a\\,c\_\{G\_\{1\}\},b^:=b​cG2\\hat\{b\}:=b\\,c\_\{G\_\{2\}\}\(calibration\-rescaled\);d:=W/Md:=W/M;ud:=\(uactualG2−uactualG1\)/Mu\_\{d\}:=\(u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\)/M;w~:=d\+ud\\tilde\{w\}:=d\+u\_\{d\}Wmax,udminW\_\{\\max\},\\ u\_\{d\}^\{\\min\}Pair\-sum maximummaxX∈\[0,1/2\]⁡\[b^​gG2​\(X\)\+a^​gG1​\(X\)\]\\max\_\{X\\in\[0,1/2\]\}\[\\hat\{b\}\\,g\_\{G\_\{2\}\}\(X\)\+\\hat\{a\}\\,g\_\{G\_\{1\}\}\(X\)\]; laundering threshold1−a\+Wmax/21\-a\+W\_\{\\max\}/2\([Table˜3](https://arxiv.org/html/2606.28710#A3.T3)\)πlow,πhigh,πbind\\pi\_\{\\mathrm\{low\}\},\\pi\_\{\\mathrm\{high\}\},\\pi\_\{\\mathrm\{bind\}\}Basin\-boundary functions for endpoint\-low, endpoint\-high, CSP
## Appendix BProofs

###### Theorem B\.1\(Ratio Favoredness Lifting Theorem\)\.

For each bindingg∈\{G1,G2\}g\\in\\\{G\_\{1\},G\_\{2\}\\\}, under the linear centered\-noise calibration model of[Section˜4](https://arxiv.org/html/2606.28710#S4)and[Assumption˜A8](https://arxiv.org/html/2606.28710#Thmassumption8)\(automatic only for the unconstrained affine chain\), the per\-interaction perceived utility lifts to the population without approximation:

U¯g​\(X\)=upercg​\(σ¯g​\(X\),𝜹¯\)\.\\bar\{U\}\_\{g\}\(X\)\\;=\\;u\_\{\\mathrm\{perc\}\}^\{g\}\\\!\\bigl\(\\bar\{\\sigma\}\_\{g\}\(X\),\\,\\bar\{\\boldsymbol\{\\delta\}\}\\bigr\)\.In particularU¯Δ:=U¯G2−U¯G1\\bar\{U\}\_\{\\Delta\}:=\\bar\{U\}\_\{G\_\{2\}\}\-\\bar\{U\}\_\{G\_\{1\}\}equals the closed[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25)form pointwise, so the basin existence conditions of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)may be verified on that form rather than on the population averages directly\.

###### Sketch\.

Fast\-chain averaging of the per\-wisher perceived utility is exact at fixed\(θ,𝜹\)\(\\theta,\\boldsymbol\{\\delta\}\)becauseupercgu\_\{\\mathrm\{perc\}\}^\{g\}is affine inbband affine functions commute with expectation; the threshold and altruism averaging then collapse toσ¯g\\bar\{\\sigma\}\_\{g\}and𝜹¯\\bar\{\\boldsymbol\{\\delta\}\}by the same affine identity\. The population differentialU¯Δ\\bar\{U\}\_\{\\Delta\}is then the closed[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25)form pointwise\. Full derivation in[Section˜B\.2](https://arxiv.org/html/2606.28710#A2.SS2)\. ∎

###### Proposition B\.2\(Basin\-depth\-and\-timescale\)\.

Under pure imitation the all\-G1G\_\{1\}statej=0j=0is absorbing \([Section˜5\.2](https://arxiv.org/html/2606.28710#S5.SS2)\); posit in addition a small exploration rateλseed\>0\\lambda\_\{\\mathrm\{seed\}\}\>0at which an all\-G1G\_\{1\}community spontaneously trials a singleG2G\_\{2\}binding, a mutation term reflecting thej=0j=0boundary in the sense of the best\-response dynamics with mutation ofEllison \[[6](https://arxiv.org/html/2606.28710#bib.bib6)\], so that all\-G1G\_\{1\}is metastable rather than absorbing\. LetU¯Δ\\bar\{U\}\_\{\\Delta\}satisfy monotonicity and endpoint inversion of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2), and writeX∗:=inf\{X∈\(0,1\):U¯Δ​\(X\)≥0\}X^\{\\ast\}:=\\inf\\\{X\\in\(0,1\):\\bar\{U\}\_\{\\Delta\}\(X\)\\geq 0\\\}andΔ​V:=−∫0X∗U¯Δ​\(Y\)​𝑑Y\\Delta V:=\-\\int\_\{0\}^\{X^\{\\ast\}\}\\bar\{U\}\_\{\\Delta\}\(Y\)\\,dY; under endpoint\-low and monotonicityU¯Δ<0\\bar\{U\}\_\{\\Delta\}<0on\(0,X∗\)\(0,X^\{\\ast\}\), soΔ​V\>0\\Delta V\>0\. When unsuccessful seeds resolve fast compared with the inter\-seed clock1/λseed1/\\lambda\_\{\\mathrm\{seed\}\}, the mean time for the metastable all\-G1G\_\{1\}phase to transit to the all\-G2G\_\{2\}absorbing state is, to leading order, that clock divided by the per\-seed survival probability of[Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1):

𝔼​\[τmetastable\]≈1λseed⋅ρG2,G1​\(Nc,β\)\.\\mathbb\{E\}\[\\tau\_\{\\mathrm\{metastable\}\}\]\\;\\approx\\;\\frac\{1\}\{\\lambda\_\{\\mathrm\{seed\}\}\\cdot\\rho\_\{G\_\{2\},G\_\{1\}\}\(N\_\{c\},\\beta\)\}\.\(41\)The single\-seed survival probability admits the Lean\-mechanized log ceiling

log​∑k=1Nc−1Pk≤Ncηs​Δ​V\+Lηs\+log⁡\(Nc−1\),\\log\\\!\\sum\_\{k=1\}^\{N\_\{c\}\-1\}P\_\{k\}\\;\\leq\\;\\frac\{N\_\{c\}\}\{\\eta\_\{s\}\}\\,\\Delta V\\;\+\\;\\frac\{L\}\{\\eta\_\{s\}\}\\;\+\\;\\log\(N\_\{c\}\-1\),\(42\)whereLLis the Lipschitz constant ofU¯Δ\\bar\{U\}\_\{\\Delta\}andρG2,G1=1/\(1\+∑kPk\)\\rho\_\{G\_\{2\},G\_\{1\}\}=1/\(1\+\\sum\_\{k\}P\_\{k\}\)per the Karlin–Taylor identity[Equation˜28](https://arxiv.org/html/2606.28710#S5.E28)\(fixation\_log\_rate\_cdepth\_le; see[Appendix˜D](https://arxiv.org/html/2606.28710#A4)\)\. Substituting \([42](https://arxiv.org/html/2606.28710#A2.E42)\) into \([41](https://arxiv.org/html/2606.28710#A2.E41)\) gives the policy\-relevant large\-NcN\_\{c\}form

𝔼​\[τmetastable\]∼1λseed⋅Nc⋅exp⁡\(Δ​V​Nc/ηs\),\\mathbb\{E\}\[\\tau\_\{\\mathrm\{metastable\}\}\]\\;\\sim\\;\\frac\{1\}\{\\lambda\_\{\\mathrm\{seed\}\}\}\\cdot\\sqrt\{N\_\{c\}\}\\cdot\\exp\\\!\\bigl\(\\Delta V\\,N\_\{c\}/\\eta\_\{s\}\\bigr\),\(43\)the classical Kramers asymptote for the discrete barrier atX∗X^\{\\ast\}, valid when in additionU¯Δ∈C1\\bar\{U\}\_\{\\Delta\}\\in C^\{1\}nearX∗X^\{\\ast\}withU¯Δ′​\(X∗\)\>0\\bar\{U\}\_\{\\Delta\}^\{\\prime\}\(X^\{\\ast\}\)\>0\(so the barrier is a nondegenerate quadratic maximum\)\. The exponentΔ​V​Nc/ηs\\Delta V\\,N\_\{c\}/\\eta\_\{s\}and the ceiling \([42](https://arxiv.org/html/2606.28710#A2.E42)\) are mechanized; theNc\\sqrt\{N\_\{c\}\}discrete\-Laplace prefactor in \([43](https://arxiv.org/html/2606.28710#A2.E43)\), which alone needs theC1C^\{1\}and strict\-crossing hypotheses, is the classical Kramers leading\-order form and stands outside the verified chain\. The downstream achievability constraint of[Section˜5\.3\.2](https://arxiv.org/html/2606.28710#S5.SS3.SSS2)consumes only \([42](https://arxiv.org/html/2606.28710#A2.E42)\), which needs only the Lipschitz bound and the barrier location\.

###### Sketch\.

Equation \([41](https://arxiv.org/html/2606.28710#A2.E41)\) follows from three modelling conditions on the seeding process\. \(i\) Seed arrivals are a Poisson point process of intensityλseed\\lambda\_\{\\mathrm\{seed\}\}, so inter\-arrival times are i\.i\.d\. exponential with mean1/λseed1/\\lambda\_\{\\mathrm\{seed\}\}\. \(ii\) Unsuccessful seeds extinguish on a timescale short compared with1/λseed1/\\lambda\_\{\\mathrm\{seed\}\}, so between seed events the chain has returned to the all\-G1G\_\{1\}state and successive seeds face statistically identical conditions\. \(iii\) Each seed therefore reaches the all\-G2G\_\{2\}absorbing state independently with probabilityρG2,G1​\(Nc,β\)\\rho\_\{G\_\{2\},G\_\{1\}\}\(N\_\{c\},\\beta\)of[Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1), making the number of seeds until first success geometric with mean1/ρG2,G11/\\rho\_\{G\_\{2\},G\_\{1\}\}\. Combining \(i\) and \(iii\) by Wald’s identity gives \([41](https://arxiv.org/html/2606.28710#A2.E41)\): the inter\-seed waiting time1/λseed1/\\lambda\_\{\\mathrm\{seed\}\}multiplied by the expected number of seeds1/ρG2,G11/\\rho\_\{G\_\{2\},G\_\{1\}\}before one survives\. Equation \([42](https://arxiv.org/html/2606.28710#A2.E42)\) is a Kramers\-style discrete\-barrier analysis on the Moran–Fermi birth–death chain: the discrete quasi\-potentialVNc​\(k/Nc\)=−1Nc​∑j=1kU¯Δ​\(j/Nc\)V\_\{N\_\{c\}\}\(k/N\_\{c\}\)=\-\\frac\{1\}\{N\_\{c\}\}\\sum\_\{j=1\}^\{k\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)converges uniformly toV​\(X\)=−∫0XU¯Δ​\(Y\)​𝑑YV\(X\)=\-\\int\_\{0\}^\{X\}\\bar\{U\}\_\{\\Delta\}\(Y\)\\,dYby the Lipschitz Riemann bound \(riemann\_sum\_error\_lipschitz\), and the running\-barrier maximum atX∗X^\{\\ast\}capslog​∑kPk\\log\\sum\_\{k\}P\_\{k\}by the bound stated\. The Kramers asymptote \([43](https://arxiv.org/html/2606.28710#A2.E43)\) follows from the discrete Laplace method applied to∑kPk\\sum\_\{k\}P\_\{k\}at the saddleX∗X^\{\\ast\}; theNc\\sqrt\{N\_\{c\}\}prefactor is the standard discrete\-Laplace coefficient\. Full derivation of \([42](https://arxiv.org/html/2606.28710#A2.E42)\) in[Section˜B\.3](https://arxiv.org/html/2606.28710#A2.SS3)\. ∎

###### Lemma B\.3\(Right\-skew of the Hill density\)\.

Letϕ​\(θ\):=n​Kn​θn−1/\(Kn\+θn\)2\\phi\(\\theta\):=nK^\{n\}\\theta^\{n\-1\}/\(K^\{n\}\+\\theta^\{n\}\)^\{2\}be the Hill density withn\>1n\>1,K\>0K\>0, and peak atX†:=K​\(\(n−1\)/\(n\+1\)\)1/nX^\{\\dagger\}:=K\\bigl\(\(n\-1\)/\(n\+1\)\\bigr\)^\{1/n\}\. Then

1. \(a\)ϕ​\(X†\+δ\)\>ϕ​\(X†−δ\)\\phi\(X^\{\\dagger\}\+\\delta\)\>\\phi\(X^\{\\dagger\}\-\\delta\)for everyδ∈\(0,X†\]\\delta\\in\(0,X^\{\\dagger\}\]\.
2. \(b\)If additionallyK≥K∗​\(n\)K\\geq K^\{\\ast\}\(n\), thenϕ​\(1−X\)≥ϕ​\(X\)\\phi\(1\-X\)\\geq\\phi\(X\)for everyX∈\[0,1/2\]X\\in\[0,1/2\], equivalentlygi​\(X\):=hi​\(X\)\+hi​\(1−X\)g\_\{i\}\(X\):=h\_\{i\}\(X\)\+h\_\{i\}\(1\-X\)is non\-decreasing on\[0,1/2\]\[0,1/2\], wherehi​\(X\):=Kn/\(Kn\+Xn\)h\_\{i\}\(X\):=K^\{n\}/\(K^\{n\}\+X^\{n\}\)\.

###### Proof\.

*Part \(a\): right\-skew at the peak\.*Sett:=δ/X†∈\(0,1\]t:=\\delta/X^\{\\dagger\}\\in\(0,1\]\. Using\(X†\)n=Kn​\(n−1\)/\(n\+1\)\(X^\{\\dagger\}\)^\{n\}=K^\{n\}\(n\-1\)/\(n\+1\),

Kn\+\(X†​\(1±t\)\)n=Knn\+1​\[\(n\+1\)\+\(n−1\)​\(1±t\)n\]\.K^\{n\}\+\(X^\{\\dagger\}\(1\\pm t\)\)^\{n\}\\;=\\;\\frac\{K^\{n\}\}\{n\+1\}\\bigl\[\(n\+1\)\+\(n\-1\)\(1\\pm t\)^\{n\}\\bigr\]\.DefineA​\(t\):=\(n\+1\)\+\(n−1\)​\(1\+t\)nA\(t\):=\(n\+1\)\+\(n\-1\)\(1\+t\)^\{n\},B​\(t\):=\(n\+1\)\+\(n−1\)​\(1−t\)nB\(t\):=\(n\+1\)\+\(n\-1\)\(1\-t\)^\{n\}, and the log\-ratio

f​\(t\):=ln⁡ϕ​\(X†​\(1\+t\)\)ϕ​\(X†​\(1−t\)\)=\(n−1\)​ln⁡1\+t1−t\+2​ln⁡B​\(t\)A​\(t\)\.f\(t\)\\;:=\\;\\ln\\frac\{\\phi\(X^\{\\dagger\}\(1\+t\)\)\}\{\\phi\(X^\{\\dagger\}\(1\-t\)\)\}\\;=\\;\(n\-1\)\\ln\\frac\{1\+t\}\{1\-t\}\+2\\ln\\frac\{B\(t\)\}\{A\(t\)\}\.Clearlyf​\(0\)=0f\(0\)=0\. Differentiating and collecting over the common denominator\(1−t2\)​A​\(t\)​B​\(t\)\(1\-t^\{2\}\)A\(t\)B\(t\), direct manipulation gives

f′​\(t\)=2​\(n2−1\)​H​\(t\)\(1−t2\)​A​\(t\)​B​\(t\),f^\{\\prime\}\(t\)\\;=\\;\\frac\{2\(n^\{2\}\-1\)\\,H\(t\)\}\{\(1\-t^\{2\}\)\\,A\(t\)\\,B\(t\)\},where

H​\(t\):=\(n\+1\)\+\(n−1\)​\[\(1\+t\)n\+\(1−t\)n\]−n​\(1−t2\)​\[\(1\+t\)n−1\+\(1−t\)n−1\]−\(n−1\)​\(1−t2\)n\.H\(t\)\\;:=\\;\(n\+1\)\+\(n\-1\)\\bigl\[\(1\+t\)^\{n\}\+\(1\-t\)^\{n\}\\bigr\]\-n\(1\-t^\{2\}\)\\bigl\[\(1\+t\)^\{n\-1\}\+\(1\-t\)^\{n\-1\}\\bigr\]\-\(n\-1\)\(1\-t^\{2\}\)^\{n\}\.Direct evaluation givesH​\(0\)=\(n\+1\)\+2​\(n−1\)−2​n−\(n−1\)=0H\(0\)=\(n\+1\)\+2\(n\-1\)\-2n\-\(n\-1\)=0\. DifferentiatingHHand grouping the\(1\+t\)n−1\(1\+t\)^\{n\-1\}and\(1−t\)n−1\(1\-t\)^\{n\-1\}terms,

H′​\(t\)=n​\(n\+1\)​t​\[\(1\+t\)n−1\+\(1−t\)n−1\]\+2​n​\(n−1\)​t​\(1−t2\)n−1\.H^\{\\prime\}\(t\)\\;=\\;n\(n\+1\)\\,t\\bigl\[\(1\+t\)^\{n\-1\}\+\(1\-t\)^\{n\-1\}\\bigr\]\+2n\(n\-1\)\\,t\\,\(1\-t^\{2\}\)^\{n\-1\}\.Fort∈\(0,1\)t\\in\(0,1\)andn\>1n\>1every factor is positive, soH′​\(t\)\>0H^\{\\prime\}\(t\)\>0\. Combined withH​\(0\)=0H\(0\)=0,H​\(t\)\>0H\(t\)\>0on\(0,1\)\(0,1\); hencef′​\(t\)\>0f^\{\\prime\}\(t\)\>0on\(0,1\)\(0,1\)and \(sincef​\(0\)=0f\(0\)=0\)f​\(t\)\>0f\(t\)\>0there\. The endpointt=1t=1follows fromϕ​\(0\)=0<ϕ​\(2​X†\)\\phi\(0\)=0<\\phi\(2X^\{\\dagger\}\)directly\.

*Part \(b\): centro\-symmetric monotonicity in Regime II\.*The conditionK≥K∗​\(n\)K\\geq K^\{\\ast\}\(n\)is equivalent toX†≥1/2X^\{\\dagger\}\\geq 1/2\. FixX∈\[0,1/2\]X\\in\[0,1/2\], soX≤X†X\\leq X^\{\\dagger\}\(rising tail\)\. Two sub\-cases for1−X1\-X\.

*Sub\-case \(i\):1−X≤X†1\-X\\leq X^\{\\dagger\}\.*BothXXand1−X1\-Xlie on the rising tail\[0,X†\]\[0,X^\{\\dagger\}\]whereϕ\\phiis monotone non\-decreasing\. SinceX≤1/2≤1−XX\\leq 1/2\\leq 1\-X,ϕ​\(X\)≤ϕ​\(1−X\)\\phi\(X\)\\leq\\phi\(1\-X\)immediately\.

*Sub\-case \(ii\):1−X\>X†1\-X\>X^\{\\dagger\}\.*SetΔ1:=X†−X≥0\\Delta\_\{1\}:=X^\{\\dagger\}\-X\\geq 0andΔ2:=\(1−X\)−X†\>0\\Delta\_\{2\}:=\(1\-X\)\-X^\{\\dagger\}\>0\. FromX†≥1/2≥XX^\{\\dagger\}\\geq 1/2\\geq XandX†≥1/2X^\{\\dagger\}\\geq 1/2:

Δ1=X†−X≥1/2−X,Δ2=1−X−X†≤1/2−X,\\Delta\_\{1\}\\;=\\;X^\{\\dagger\}\-X\\;\\geq\\;1/2\-X,\\qquad\\Delta\_\{2\}\\;=\\;1\-X\-X^\{\\dagger\}\\;\\leq\\;1/2\-X,soΔ2≤Δ1\\Delta\_\{2\}\\leq\\Delta\_\{1\}\. By Part \(a\),ϕ​\(X†\+Δ2\)\>ϕ​\(X†−Δ2\)\\phi\(X^\{\\dagger\}\+\\Delta\_\{2\}\)\>\\phi\(X^\{\\dagger\}\-\\Delta\_\{2\}\); by monotonicity ofϕ\\phion the rising tail\[0,X†\]\[0,X^\{\\dagger\}\]together withX†−Δ2≥X†−Δ1X^\{\\dagger\}\-\\Delta\_\{2\}\\geq X^\{\\dagger\}\-\\Delta\_\{1\},ϕ​\(X†−Δ2\)≥ϕ​\(X†−Δ1\)\\phi\(X^\{\\dagger\}\-\\Delta\_\{2\}\)\\geq\\phi\(X^\{\\dagger\}\-\\Delta\_\{1\}\)\. Chaining,

ϕ​\(1−X\)=ϕ​\(X†\+Δ2\)\>ϕ​\(X†−Δ2\)≥ϕ​\(X†−Δ1\)=ϕ​\(X\)\.\\phi\(1\-X\)\\;=\\;\\phi\(X^\{\\dagger\}\+\\Delta\_\{2\}\)\\;\>\\;\\phi\(X^\{\\dagger\}\-\\Delta\_\{2\}\)\\;\\geq\\;\\phi\(X^\{\\dagger\}\-\\Delta\_\{1\}\)\\;=\\;\\phi\(X\)\.
Equivalence withgig\_\{i\}non\-decreasing follows fromgi′​\(X\)=ϕ​\(1−X\)−ϕ​\(X\)g\_\{i\}^\{\\prime\}\(X\)=\\phi\(1\-X\)\-\\phi\(X\)\(sincehi′​\(Y\)=−ϕ​\(Y\)h\_\{i\}^\{\\prime\}\(Y\)=\-\\phi\(Y\)\)\. ∎

###### Lemma B\.4\(Interior critical point in Regime III\)\.

Letϕ\\phi,X†X^\{\\dagger\},gig\_\{i\}be as in[Lemma˜B\.3](https://arxiv.org/html/2606.28710#A2.Thmtheorem3), and assumen\>1n\>1andK<K∗​\(n\)K<K^\{\\ast\}\(n\)\(soX†<1/2X^\{\\dagger\}<1/2\)\. Thengig\_\{i\}has an interior critical point in\(0,X†\)\(0,X^\{\\dagger\}\)\.

###### Proof\.

gi′​\(X\)=ϕ​\(1−X\)−ϕ​\(X\)g\_\{i\}^\{\\prime\}\(X\)=\\phi\(1\-X\)\-\\phi\(X\)\(sincehi′​\(Y\)=−ϕ​\(Y\)h\_\{i\}^\{\\prime\}\(Y\)=\-\\phi\(Y\)\)\. Evaluating at the endpoints:

gi′​\(0\)=ϕ​\(1\)−ϕ​\(0\)=ϕ​\(1\)\>0,g\_\{i\}^\{\\prime\}\(0\)\\;=\\;\\phi\(1\)\-\\phi\(0\)\\;=\\;\\phi\(1\)\\;\>\\;0,sinceϕ​\(0\)=0\\phi\(0\)=0forn\>1n\>1andϕ​\(1\)\>0\\phi\(1\)\>0\. The Hill densityϕ​\(θ\)=n​Kn​θn−1/\(Kn\+θn\)2\\phi\(\\theta\)=nK^\{n\}\\theta^\{n\-1\}/\(K^\{n\}\+\\theta^\{n\}\)^\{2\}satisfiesϕ′​\(θ\)∝\(n−1\)​Kn−\(n\+1\)​θn\\phi^\{\\prime\}\(\\theta\)\\propto\(n\-1\)K^\{n\}\-\(n\+1\)\\theta^\{n\}, soϕ\\phiis strictly decreasing forθ\>X†=K​\(\(n−1\)/\(n\+1\)\)1/n\\theta\>X^\{\\dagger\}=K\\bigl\(\(n\-1\)/\(n\+1\)\\bigr\)^\{1/n\}\. AtX=X†X=X^\{\\dagger\}we have1−X†\>X†1\-X^\{\\dagger\}\>X^\{\\dagger\}\(sinceX†<1/2X^\{\\dagger\}<1/2\), soϕ​\(1−X†\)<ϕ​\(X†\)\\phi\(1\-X^\{\\dagger\}\)<\\phi\(X^\{\\dagger\}\); hence

gi′​\(X†\)=ϕ​\(1−X†\)−ϕ​\(X†\)<0\.g\_\{i\}^\{\\prime\}\(X^\{\\dagger\}\)\\;=\\;\\phi\(1\-X^\{\\dagger\}\)\-\\phi\(X^\{\\dagger\}\)\\;<\\;0\.Continuity ofgi′g\_\{i\}^\{\\prime\}on\[0,X†\]\[0,X^\{\\dagger\}\]and the intermediate value theorem giveX0∈\(0,X†\)X\_\{0\}\\in\(0,X^\{\\dagger\}\)withgi′​\(X0\)=0g\_\{i\}^\{\\prime\}\(X\_\{0\}\)=0\. ∎

### Detailed derivations

The full derivations of the results sketched above are collected here\. The body and the statements earlier in this appendix retain only one\-sentence interpretations\.

### B\.1Proof of[Proposition˜5\.3](https://arxiv.org/html/2606.28710#S5.Thmtheorem3)

CSP givesU¯Δ​\(j/Nc\)\+U¯Δ​\(\(Nc−j\)/Nc\)≥0\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\+\\bar\{U\}\_\{\\Delta\}\(\(N\_\{c\}\-j\)/N\_\{c\}\)\\geq 0for everyjj\. By the strict clause of CSP, there existsX0∈\[0,1/2\]X\_\{0\}\\in\[0,1/2\]withU¯Δ​\(X0\)\+U¯Δ​\(1−X0\)\>0\\bar\{U\}\_\{\\Delta\}\(X\_\{0\}\)\+\\bar\{U\}\_\{\\Delta\}\(1\-X\_\{0\}\)\>0\. By continuity ofU¯Δ\\bar\{U\}\_\{\\Delta\}\(a finite affine combination of bounded\-Lipschitz tail probabilities\), the strict centro\-symmetric inequality holds on an open neighborhood\(a,b\)⊆\[0,1/2\]\(a,b\)\\subseteq\[0,1/2\]ofX0X\_\{0\}\. ForNcN\_\{c\}large enough that the grid step1/Nc1/N\_\{c\}is smaller thanb−ab\-a, at least one grid pointj0/Ncj\_\{0\}/N\_\{c\}lies in\(a,b\)\(a,b\), so the pair\-sumU¯Δ​\(j0/Nc\)\+U¯Δ​\(\(Nc−j0\)/Nc\)\>0\\bar\{U\}\_\{\\Delta\}\(j\_\{0\}/N\_\{c\}\)\+\\bar\{U\}\_\{\\Delta\}\(\(N\_\{c\}\-j\_\{0\}\)/N\_\{c\}\)\>0strictly\. Every other pair is≥0\\geq 0by CSP\. WhenNcN\_\{c\}is even, the unpaired middle termU¯Δ​\(1/2\)≥0\\bar\{U\}\_\{\\Delta\}\(1/2\)\\geq 0by CSP atX=1/2X=1/2\. The cumulative sum is therefore strictly positive, and the ratio of[Equation˜28](https://arxiv.org/html/2606.28710#S5.E28)exceeds11\. For fixed parameters with a strict\-positive interval of widthw∗:=b−a\>0w\_\{\*\}:=b\-a\>0, the conclusion holds wheneverNc\>1/w∗N\_\{c\}\>1/w\_\{\*\}\. The strict\-positive interval can in principle be narrower than1/Nc1/N\_\{c\}, so for smallNcN\_\{c\}with a tight strict\-CSP margin the discrete sum of[Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1)should be checked directly rather than presumed from the large\-NcN\_\{c\}asymptotic\. ∎

### B\.2Proof of[Theorem˜B\.1](https://arxiv.org/html/2606.28710#A2.Thmtheorem1)

The construction ofU¯g​\(X\):=𝔼θ​\[𝔼ν\(⋅∣X,θ\)​\[upercg​\(b\)\]\]\\bar\{U\}\_\{g\}\(X\):=\\mathbb\{E\}\_\{\\theta\}\\bigl\[\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\[u\_\{\\mathrm\{perc\}\}^\{g\}\(b\)\]\\bigr\]has two layers, inner over the fast\-chain stationary distributionν\\nuon the calibration statebbat fixed thresholdθ\\theta, outer over the threshold prior\. Each layer is made exact by its own affine identity, and the atomic affine object is the per\-genie perceived utilityupercg​\(b\)u\_\{\\mathrm\{perc\}\}^\{g\}\(b\), not the payoff differential\.

##### Fast\-chain averaging is exact at fixedg,θ,𝜹g,\\theta,\\boldsymbol\{\\delta\}\.

The per\-wisher perceived utilities of[Equations˜15](https://arxiv.org/html/2606.28710#S4.E15)and[16](https://arxiv.org/html/2606.28710#S4.E16)are affine inbbfor each binding:

uperc,iG2​\(b\)=uactualG2\+ualt,iG2−AG2​\(1−b\),uperc,iG1​\(b\)=uactualG1\+ualt,iG1\+M−AG1​b,u\_\{\\mathrm\{perc\},i\}^\{G\_\{2\}\}\(b\)=u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{2\}\}\-A\_\{G\_\{2\}\}\(1\-b\),\\qquad u\_\{\\mathrm\{perc\},i\}^\{G\_\{1\}\}\(b\)=u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\+u\_\{\\mathrm\{alt\},\\,i\}^\{G\_\{1\}\}\+M\-A\_\{G\_\{1\}\}\\,b,withualt,ig:=𝜹i⋅𝐰gu\_\{\\mathrm\{alt\},\\,i\}^\{g\}:=\\boldsymbol\{\\delta\}\_\{i\}\\cdot\\mathbf\{w\}\_\{g\}the wisher’s altruistic\-utility component under bindinggg\([Section˜3](https://arxiv.org/html/2606.28710#S3)\)\. Affine functions commute with expectation: for any probability measureν\\nu,𝔼ν​\[uperc,ig​\(b\)\]=uperc,ig​\(𝔼ν​\[b\]\)\\mathbb\{E\}\_\{\\nu\}\[u\_\{\\mathrm\{perc\},i\}^\{g\}\(b\)\]=u\_\{\\mathrm\{perc\},i\}^\{g\}\(\\mathbb\{E\}\_\{\\nu\}\[b\]\)\.

The fast chain onbbis, under linearRRand linearDD, itself affine:bt\+1=\(1−Lb\)​bt\+ηb​Ag​σig​\(X\)\+ξtb^\{t\+1\}=\(1\-L\_\{b\}\)\\,b^\{t\}\+\\eta\_\{b\}A\_\{g\}\\sigma\_\{i\}^\{g\}\(X\)\+\\xi\_\{t\}, with linear relaxation rateLb:=ηb​Ag​σig​\(X\)\+μ​ϵL\_\{b\}:=\\eta\_\{b\}A\_\{g\}\\sigma\_\{i\}^\{g\}\(X\)\+\\mu\\epsilon\(so the Lipschitz constant of the deterministic part is\|1−Lb\|\|1\-L\_\{b\}\|,<1<1whenLb∈\(0,2\)L\_\{b\}\\in\(0,2\)\) and per\-interaction noiseξt\\xi\_\{t\}centered by hypothesis\. For the unconstrained affine chain, taking expectations under stationarity givesm=\(1−Lb\)​m\+ηb​Ag​σm=\(1\-L\_\{b\}\)\\,m\+\\eta\_\{b\}A\_\{g\}\\sigma, hencem=ηb​Ag​σ/Lb=b∗​\(X;θ\)m=\\eta\_\{b\}A\_\{g\}\\sigma/L\_\{b\}=b^\{\*\}\(X;\\theta\)with no bias from the noise\. On a boundedℬ\\mathcal\{B\}, where reflection or truncation could otherwise displace it, the boundary\-preserving stationary\-mean assumption of[Section˜4](https://arxiv.org/html/2606.28710#S4)secures𝔼ν\(⋅∣X,θ\)​\[b\]=b∗​\(X;θ\)\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\[b\]=b^\{\*\}\(X;\\theta\)directly; this identity, not the affine recursion as such, is what the rest of the proof uses\.

Composing:𝔼ν\(⋅∣X,θ\)​\[upercg​\(b\)\]=upercg​\(b∗​\(X;θ\)\)\\mathbb\{E\}\_\{\\nu\(\\cdot\\mid X,\\theta\)\}\[u\_\{\\mathrm\{perc\}\}^\{g\}\(b\)\]=u\_\{\\mathrm\{perc\}\}^\{g\}\(b^\{\*\}\(X;\\theta\)\)exactly\.

##### Threshold and altruism averaging are exact per binding\.

For eachgg, averaging over the threshold priorθig\\theta\_\{i\}^\{g\}and the independent altruism prior𝜹i\\boldsymbol\{\\delta\}\_\{i\},

U¯g​\(X\)=𝔼θ,𝜹​\[uperc,ig​\(b∗​\(X;θig\),𝜹i\)\]=𝔼𝜹​\[uperc,ig​\(𝔼θ​\[b∗​\(X;θig\)\],𝜹i\)\]=upercg​\(σ¯g​\(X\),𝜹¯\),\\bar\{U\}\_\{g\}\(X\)\\;=\\;\\mathbb\{E\}\_\{\\theta,\\boldsymbol\{\\delta\}\}\\bigl\[u\_\{\\mathrm\{perc\},i\}^\{g\}\(b^\{\*\}\(X;\\theta\_\{i\}^\{g\}\),\\boldsymbol\{\\delta\}\_\{i\}\)\\bigr\]\\;=\\;\\mathbb\{E\}\_\{\\boldsymbol\{\\delta\}\}\\\!\\bigl\[u\_\{\\mathrm\{perc\},i\}^\{g\}\\bigl\(\\mathbb\{E\}\_\{\\theta\}\[b^\{\*\}\(X;\\theta\_\{i\}^\{g\}\)\],\\boldsymbol\{\\delta\}\_\{i\}\\bigr\)\\bigr\]\\;=\\;u\_\{\\mathrm\{perc\}\}^\{g\}\\\!\\bigl\(\\bar\{\\sigma\}\_\{g\}\(X\),\\bar\{\\boldsymbol\{\\delta\}\}\\bigr\),where the second equality is the affineness ofupercgu\_\{\\mathrm\{perc\}\}^\{g\}inbb, and the third uses the sharp\-threshold identityb∗​\(X;θ\)=cg​1​\[X≥θ\]b^\{\*\}\(X;\\theta\)=c\_\{g\}\\,\\mathbb\{1\}\[X\\geq\\theta\]together with𝔼θ​\[𝟙​\[X≥θ\]\]=1−F¯g​\(X\)\\mathbb\{E\}\_\{\\theta\}\[\\mathbb\{1\}\[X\\geq\\theta\]\]=1\-\\bar\{F\}\_\{g\}\(X\)to giveσ¯g​\(X\)=cg​\(1−F¯g​\(X\)\)\\bar\{\\sigma\}\_\{g\}\(X\)=c\_\{g\}\(1\-\\bar\{F\}\_\{g\}\(X\)\), plus the linearity ofualt,ig=𝜹i⋅𝐰gu\_\{\\mathrm\{alt\},\\,i\}^\{g\}=\\boldsymbol\{\\delta\}\_\{i\}\\cdot\\mathbf\{w\}\_\{g\}in𝜹i\\boldsymbol\{\\delta\}\_\{i\}to give𝔼𝜹​\[ualt,ig\]=𝜹¯⋅𝐰g=Wg\\mathbb\{E\}\_\{\\boldsymbol\{\\delta\}\}\[u\_\{\\mathrm\{alt\},\\,i\}^\{g\}\]=\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\mathbf\{w\}\_\{g\}=W^\{g\}\. The differenceWG2−WG1=𝜹¯⋅Δ​𝐇¯=WW^\{G\_\{2\}\}\-W^\{G\_\{1\}\}=\\bar\{\\boldsymbol\{\\delta\}\}\\cdot\\overline\{\\Delta\\mathbf\{H\}\}=Wenters the payoff differential\.

Subtracting the two per\-binding identities reproduces[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25):

U¯Δ​\(X\)\\displaystyle\\bar\{U\}\_\{\\Delta\}\(X\)=U¯G2​\(X\)−U¯G1​\(X\)\\displaystyle=\\bar\{U\}\_\{G\_\{2\}\}\(X\)\-\\bar\{U\}\_\{G\_\{1\}\}\(X\)=upercG2​\(σ¯G2​\(X\),𝜹¯\)−upercG1​\(σ¯G1​\(X\),𝜹¯\)\\displaystyle=u\_\{\\mathrm\{perc\}\}^\{G\_\{2\}\}\(\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\),\\bar\{\\boldsymbol\{\\delta\}\}\)\-u\_\{\\mathrm\{perc\}\}^\{G\_\{1\}\}\(\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\),\\bar\{\\boldsymbol\{\\delta\}\}\)=\(uactualG2−uactualG1\)\+W−AG2−M\+AG2​σ¯G2​\(X\)\+AG1​σ¯G1​\(X\),\\displaystyle=\(u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\)\+W\-A\_\{G\_\{2\}\}\-M\+A\_\{G\_\{2\}\}\\,\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)\+A\_\{G\_\{1\}\}\\,\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\),identically\.

##### Inheritance of the basin existence conditions\.

BecauseU¯Δ\\bar\{U\}\_\{\\Delta\}equals the eq:3 form pointwise, each basin existence condition of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)transfers as a consequence of facts aboutσ¯G2,σ¯G1\\bar\{\\sigma\}\_\{G\_\{2\}\},\\bar\{\\sigma\}\_\{G\_\{1\}\}and the coefficientsAG2,AG1A\_\{G\_\{2\}\},A\_\{G\_\{1\}\}\.

*Monotonicity\.*Bothσ¯G2​\(X\)\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)andσ¯G1​\(X\)\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\)are non\-decreasing inXX, because the prior tailF¯g\\bar\{F\}\_\{g\}is non\-increasing at each binding\. The coefficientsAG2,AG1≥0A\_\{G\_\{2\}\},A\_\{G\_\{1\}\}\\geq 0\. The composition is therefore non\-decreasing inXX, andU¯Δ\\bar\{U\}\_\{\\Delta\}inherits the monotonicity pointwise\.

*Endpoint inversion\.*Writinguw:=\(uactualG2−uactualG1\)\+Wu\_\{w\}:=\(u\_\{\\mathrm\{actual\}\}^\{G\_\{2\}\}\-u\_\{\\mathrm\{actual\}\}^\{G\_\{1\}\}\)\+W, atX=0X=0both bindings haveσ¯g​\(0\)=0\\bar\{\\sigma\}\_\{g\}\(0\)=0, so

U¯Δ​\(0\)=uw−AG2−M\.\\bar\{U\}\_\{\\Delta\}\(0\)=u\_\{w\}\-A\_\{G\_\{2\}\}\-M\.\(44\)AtX=1X=1,σ¯g​\(1\)=cg​\(1−σgsat\)\\bar\{\\sigma\}\_\{g\}\(1\)=c\_\{g\}\(1\-\\sigma\_\{g\}^\{\\mathrm\{sat\}\}\)per binding, so

U¯Δ​\(1\)=uw\+AG1​σ¯G1​\(1\)−AG2​\(1−σ¯G2​\(1\)\)−M\.\\bar\{U\}\_\{\\Delta\}\(1\)=u\_\{w\}\+A\_\{G\_\{1\}\}\\,\\bar\{\\sigma\}\_\{G\_\{1\}\}\(1\)\-A\_\{G\_\{2\}\}\\,\(1\-\\bar\{\\sigma\}\_\{G\_\{2\}\}\(1\)\)\-M\.\(45\)
*Centro\-symmetric pairing \(CSP\)\.*For eachX∈\[0,1/2\]X\\in\[0,1/2\], direct substitution into the[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25)form gives the pair\-sum

U¯Δ​\(X\)\+U¯Δ​\(1−X\)=2​\(uw−AG2−M\)\+AG2​\[σ¯G2​\(X\)\+σ¯G2​\(1−X\)\]\+AG1​\[σ¯G1​\(X\)\+σ¯G1​\(1−X\)\],\\bar\{U\}\_\{\\Delta\}\(X\)\+\\bar\{U\}\_\{\\Delta\}\(1\-X\)=2\(u\_\{w\}\-A\_\{G\_\{2\}\}\-M\)\+A\_\{G\_\{2\}\}\\bigl\[\\bar\{\\sigma\}\_\{G\_\{2\}\}\(X\)\+\\bar\{\\sigma\}\_\{G\_\{2\}\}\(1\-X\)\\bigr\]\+A\_\{G\_\{1\}\}\\bigl\[\\bar\{\\sigma\}\_\{G\_\{1\}\}\(X\)\+\\bar\{\\sigma\}\_\{G\_\{1\}\}\(1\-X\)\\bigr\],\(46\)identical value\-by\-value to the corresponding pair\-sum on the[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25)expression\. A non\-negative pair\-sum on the pointwise form is therefore a non\-negative pair\-sum onU¯Δ\\bar\{U\}\_\{\\Delta\}, and strict positivity at any point in\[0,1/2\]\[0,1/2\]transfers as strict positivity at the same point\. ∎

### B\.3Proof of[Proposition˜B\.2](https://arxiv.org/html/2606.28710#A2.Thmtheorem2)

A Kramers\-style barrier\-crossing analysis on the Moran–Fermi birth–death chain of[Section˜5\.2](https://arxiv.org/html/2606.28710#S5.SS2), paralleling the stochastic\-stability and escape\-cost analysis for two\-strategy logit\-noise chains\[[16](https://arxiv.org/html/2606.28710#bib.bib13), §6–7\]and the analogous treatment of best response with mutation inEllison \[[6](https://arxiv.org/html/2606.28710#bib.bib6)\]\. The waiting\-time scaling instantiates this logit\-noise escape\-cost analysis for the prior\-induced gradientU¯Δ\\bar\{U\}\_\{\\Delta\}, recovering the classical mutation\-selection results ofFoster and Young \[[7](https://arxiv.org/html/2606.28710#bib.bib7)\]andKandoriet al\.\[[11](https://arxiv.org/html/2606.28710#bib.bib10)\]for2×22\\times 2coordination games as the uniform\-mutation special case\. What is specific to the present setting is finite\-population and non\-asymptotic\.[Equation˜42](https://arxiv.org/html/2606.28710#A2.E42)bounds the barrier sum from the Lipschitz Riemann estimate and the running\-maximum location alone, without the small\-noise limit those results rest on\.

##### Step 1: Mean first\-passage on the discrete chain\.

For the slow chain onj∈\{0,1,…,Nc\}j\\in\\\{0,1,\\ldots,N\_\{c\}\\\}with up\- and down\-switch probabilitiesT\+​\(j\),T−​\(j\)T^\{\+\}\(j\),T^\{\-\}\(j\)of[Section˜5\.2](https://arxiv.org/html/2606.28710#S5.SS2), the leading\-order mean\-first\-passage approximation for a birth–death chain gives the expected time to reachj=Ncj=N\_\{c\}starting fromj=1j=1as

𝔼​\[τfix\]=∑k=1Nc−11T\+​\(k\)​∏j=1kT−​\(j\)T\+​\(j\)⋅\(1\+o​\(1\)\),\\mathbb\{E\}\[\\tau\_\{\\mathrm\{fix\}\}\]\\;=\\;\\sum\_\{k=1\}^\{N\_\{c\}\-1\}\\frac\{1\}\{T^\{\+\}\(k\)\}\\prod\_\{j=1\}^\{k\}\\frac\{T^\{\-\}\(j\)\}\{T^\{\+\}\(j\)\}\\cdot\\bigl\(1\+o\(1\)\\bigr\),where the\(1\+o​\(1\)\)\(1\+o\(1\)\)factor absorbs the inner geometric correction\. Substituting the Fermi\-rate ratioT−​\(j\)/T\+​\(j\)=exp⁡\(−U¯Δ​\(j/Nc\)/ηs\)T^\{\-\}\(j\)/T^\{\+\}\(j\)=\\exp\(\-\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)/\\eta\_\{s\}\)of[Equation˜26](https://arxiv.org/html/2606.28710#S5.E26)yields

𝔼​\[τfix\]=∑k=1Nc−11T\+​\(k\)​exp⁡\(−1ηs​∑j=1kU¯Δ​\(j/Nc\)\)⋅\(1\+o​\(1\)\),\\mathbb\{E\}\[\\tau\_\{\\mathrm\{fix\}\}\]\\;=\\;\\sum\_\{k=1\}^\{N\_\{c\}\-1\}\\frac\{1\}\{T^\{\+\}\(k\)\}\\exp\\\!\\left\(\-\\frac\{1\}\{\\eta\_\{s\}\}\\sum\_\{j=1\}^\{k\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\\right\)\\cdot\\bigl\(1\+o\(1\)\\bigr\),carrying only the\(1\+o​\(1\)\)\(1\+o\(1\)\)correction noted above, the large\-NcN\_\{c\}reduction of the inner geometric series near the barrier\.

##### Step 2: Discrete and continuous quasi\-potential\.

Define the discrete quasi\-potential as the running sum of selection gradients,

VNc​\(k/Nc\):=−1Nc​∑j=1kU¯Δ​\(j/Nc\)\.V\_\{N\_\{c\}\}\(k/N\_\{c\}\)\\;:=\\;\-\\frac\{1\}\{N\_\{c\}\}\\sum\_\{j=1\}^\{k\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)\.Under the Lipschitz bound onU¯Δ\\bar\{U\}\_\{\\Delta\}over\[0,1\]\[0,1\]\(following from the bounded density of the threshold prior;[Equation˜22](https://arxiv.org/html/2606.28710#S5.E22)\), the right Riemann\-sum error isO​\(1/Nc\)O\(1/N\_\{c\}\), soVNcV\_\{N\_\{c\}\}converges uniformly to its continuous limit

V​\(X\):=−∫0XU¯Δ​\(Y\)​𝑑Y,VNc​\(k/Nc\)=V​\(k/Nc\)\+O​\(1/Nc\)\.V\(X\)\\;:=\\;\-\\int\_\{0\}^\{X\}\\bar\{U\}\_\{\\Delta\}\(Y\)\\,dY,\\qquad V\_\{N\_\{c\}\}\(k/N\_\{c\}\)\\;=\\;V\(k/N\_\{c\}\)\+O\(1/N\_\{c\}\)\.WithV​\(0\)=0V\(0\)=0andV′​\(X\)=−U¯Δ​\(X\)V^\{\\prime\}\(X\)=\-\\bar\{U\}\_\{\\Delta\}\(X\), monotonicity together with endpoint inversion givesVVstrictly increasing on\(0,X∗\)\(0,X^\{\\ast\}\), attaining its maximum atX=X∗X=X^\{\\ast\}, and strictly decreasing on\(X∗,1\)\(X^\{\\ast\},1\)\(the interiorX∗X^\{\\ast\}is the basin boundary of[Theorem˜5\.4](https://arxiv.org/html/2606.28710#S5.Thmtheorem4)\)\. The basin depth is the barrier height,

Δ​V=V​\(X∗\)−V​\(0\)\>0\.\\Delta V\\;=\\;V\(X^\{\\ast\}\)\-V\(0\)\\;\>\\;0\.

##### Step 3: Finite log\-ceiling\.

The Karlin–Taylor terms are

Pk=∏j=1kT−​\(j\)/T\+​\(j\)=exp⁡\(Nc​VNc​\(k/Nc\)/ηs\),P\_\{k\}\\;=\\;\\prod\_\{j=1\}^\{k\}T^\{\-\}\(j\)/T^\{\+\}\(j\)\\;=\\;\\exp\\bigl\(N\_\{c\}\\,V\_\{N\_\{c\}\}\(k/N\_\{c\}\)/\\eta\_\{s\}\\bigr\),using∑j=1kU¯Δ​\(j/Nc\)=−Nc​VNc​\(k/Nc\)\\sum\_\{j=1\}^\{k\}\\bar\{U\}\_\{\\Delta\}\(j/N\_\{c\}\)=\-N\_\{c\}\\,V\_\{N\_\{c\}\}\(k/N\_\{c\}\)and the Fermi\-rate ratio of[Equation˜26](https://arxiv.org/html/2606.28710#S5.E26)\. Bounding the sum by its largest term times the number of terms,

∑k=1Nc−1Pk≤\(Nc−1\)​max1≤k≤Nc−1⁡exp⁡\(Nc​VNc​\(k/Nc\)/ηs\)\.\\sum\_\{k=1\}^\{N\_\{c\}\-1\}P\_\{k\}\\;\\leq\\;\(N\_\{c\}\-1\)\\,\\max\_\{1\\leq k\\leq N\_\{c\}\-1\}\\exp\\\!\\bigl\(N\_\{c\}\\,V\_\{N\_\{c\}\}\(k/N\_\{c\}\)/\\eta\_\{s\}\\bigr\)\.The running maximum is controlled by the continuous barrier:VNc​\(k/Nc\)≤V​\(X∗\)\+L/Nc=Δ​V\+L/NcV\_\{N\_\{c\}\}\(k/N\_\{c\}\)\\leq V\(X^\{\\ast\}\)\+L/N\_\{c\}=\\Delta V\+L/N\_\{c\}, sinceVVattains its maximumΔ​V\\Delta VatX∗X^\{\\ast\}\(Step 2\) and the right\-Riemann error is bounded byL/NcL/N\_\{c\}withLLthe Lipschitz constant ofU¯Δ\\bar\{U\}\_\{\\Delta\}\(riemann\_sum\_error\_lipschitz\)\. Taking logarithms,

log​∑k=1Nc−1Pk≤log⁡\(Nc−1\)\+Ncηs​\(Δ​V\+LNc\)=Ncηs​Δ​V\+Lηs\+log⁡\(Nc−1\),\\log\\\!\\sum\_\{k=1\}^\{N\_\{c\}\-1\}P\_\{k\}\\;\\leq\\;\\log\(N\_\{c\}\-1\)\+\\frac\{N\_\{c\}\}\{\\eta\_\{s\}\}\\Bigl\(\\Delta V\+\\frac\{L\}\{N\_\{c\}\}\\Bigr\)\\;=\\;\\frac\{N\_\{c\}\}\{\\eta\_\{s\}\}\\,\\Delta V\+\\frac\{L\}\{\\eta\_\{s\}\}\+\\log\(N\_\{c\}\-1\),which is[Equation˜42](https://arxiv.org/html/2606.28710#A2.E42)\. This bound uses only the Lipschitz constant and the location of the running maximum atX∗X^\{\\ast\}, not theC1C^\{1\}or strict\-crossing hypotheses; it is the only part of this proposition the achievability constraint of[Section˜5\.3\.2](https://arxiv.org/html/2606.28710#S5.SS3.SSS2)consumes, and it is the form mechanized asfixation\_log\_rate\_cdepth\_le\.

##### Step 4: Discrete Laplace approximation\.

The inner sum in Step 1 equalsNc​VNc​\(k/Nc\)N\_\{c\}V\_\{N\_\{c\}\}\(k/N\_\{c\}\), so the product term isexp⁡\(Nc​VNc​\(k/Nc\)/ηs\)\\exp\(N\_\{c\}V\_\{N\_\{c\}\}\(k/N\_\{c\}\)/\\eta\_\{s\}\)\. AsNc→∞N\_\{c\}\\to\\infty, the outer sum is dominated by terms near the barrierk/Nc=X∗k/N\_\{c\}=X^\{\\ast\}\. ExpandingVVaroundX∗X^\{\\ast\},

V​\(X\)=Δ​V\+12​V′′​\(X∗\)​\(X−X∗\)2\+O​\(\(X−X∗\)3\),V\(X\)\\;=\\;\\Delta V\+\\tfrac\{1\}\{2\}V^\{\\prime\\prime\}\(X^\{\\ast\}\)\(X\-X^\{\\ast\}\)^\{2\}\+O\\bigl\(\(X\-X^\{\\ast\}\)^\{3\}\\bigr\),withV′′​\(X∗\)=−U¯Δ′​\(X∗\)<0V^\{\\prime\\prime\}\(X^\{\\ast\}\)=\-\\bar\{U\}\_\{\\Delta\}^\{\\prime\}\(X^\{\\ast\}\)<0\. Applying the discrete Laplace method at grid spacing1/Nc1/N\_\{c\}against a continuous Gaussian of widthηs/\(Nc​\|V′′​\(X∗\)\|\)\\sqrt\{\\eta\_\{s\}/\(N\_\{c\}\|V^\{\\prime\\prime\}\(X^\{\\ast\}\)\|\)\}gives

∑k=1Nc−1exp⁡\(Nc​VNc​\(k/Nc\)ηs\)∼2​π​Nc​ηs\|V′′​\(X∗\)\|⋅exp⁡\(Nc​Δ​Vηs\)\.\\sum\_\{k=1\}^\{N\_\{c\}\-1\}\\exp\\\!\\left\(\\frac\{N\_\{c\}V\_\{N\_\{c\}\}\(k/N\_\{c\}\)\}\{\\eta\_\{s\}\}\\right\)\\;\\sim\\;\\sqrt\{\\frac\{2\\pi N\_\{c\}\\,\\eta\_\{s\}\}\{\|V^\{\\prime\\prime\}\(X^\{\\ast\}\)\|\}\}\\cdot\\exp\\\!\\left\(\\frac\{N\_\{c\}\\Delta V\}\{\\eta\_\{s\}\}\\right\)\.TheT\+​\(k\)−1T^\{\+\}\(k\)^\{\-1\}prefactor in Step 1 isO​\(1\)O\(1\)near the barrier\. Combining,

𝔼​\[τfix\]∼Nc⋅exp⁡\(Δ​V​Nc/ηs\)\.\\mathbb\{E\}\[\\tau\_\{\\mathrm\{fix\}\}\]\\;\\sim\\;\\sqrt\{N\_\{c\}\}\\cdot\\exp\\\!\\left\(\\Delta V\\,N\_\{c\}/\\eta\_\{s\}\\right\)\.

##### Step 5: Selection\-intensity inheritance\.

The selection intensityβ=1/ηs\\beta=1/\\eta\_\{s\}enters the exponent inverse\-linearly through the Fermi\-rate ratio of[Equation˜26](https://arxiv.org/html/2606.28710#S5.E26), giving exponentNc​Δ​V/ηsN\_\{c\}\\Delta V/\\eta\_\{s\}\. Under the conventionηs=ηb\\eta\_\{s\}=\\eta\_\{b\}\([Section˜5\.2](https://arxiv.org/html/2606.28710#S5.SS2)\) this readsNc​Δ​V/ηbN\_\{c\}\\Delta V/\\eta\_\{b\}, so the accessibility constraint[Equation˜36](https://arxiv.org/html/2606.28710#S5.E36)is governed by the calibration\-update step size, which enters both as the comparison temperature and, throughρg\\rho\_\{g\}, in the barrier heightΔ​V\\Delta V\. ∎

## Appendix CThreshold\-prior shape class

The basin\-existence framework rests on five prior\-shape conditions, not on any specific parametric family:*normalization*, the prior is a probability measure on\[0,∞\)\[0,\\infty\)or\[0,1\]\[0,1\]with a density;*tail regularity*, its tailF¯i\\bar\{F\}\_\{i\}is continuous, monotone non\-increasing, and bounded Lipschitz on\[0,1\]\[0,1\];*endpoint\-low*and*endpoint\-high*of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)atF¯i​\(0\)\\bar\{F\}\_\{i\}\(0\)andF¯i​\(1\)\\bar\{F\}\_\{i\}\(1\); and*centro\-symmetric pairing*\(CSP\) at the prior’s centro\-symmetric maximum on\[0,1/2\]\[0,1/2\]\. Normalization and tail regularity are intrinsic to the prior; endpoint\-low, endpoint\-high, and centro\-symmetric pairing are the three conditions of[Definition˜5\.2](https://arxiv.org/html/2606.28710#S5.Thmtheorem2)read off at the values the prior supplies\. Four families discharge them in the Lean 4 development: Hill \(hillPrior,Priors/Hill\.lean\), Pareto Type I \(paretoPrior,Priors/Pareto\.lean\), Lomax / Pareto Type II \(lomaxPrior,Priors/Lomax\.lean\), and Fréchet / EV Type II \(frechetPrior,Priors/Frechet\.lean\)\. Each is a fully certifiedThresholdPriorinstance; normalization is handled inPriors/Normalization\.lean\(hill\_integral\_eq\_one,pareto\_integral\_eq\_one,lomax\_integral\_eq\_one,frechet\_integral\_eq\_one\)\.

The four tails and the location of the centro\-symmetric pair\-sum maximum on\[0,1/2\]\[0,1/2\]per family are as follows\. Pareto, tail\(xm/X\)α\(x\_\{m\}/X\)^\{\\alpha\}with modal density at the scalexmx\_\{m\}: the maximum is atX=xmX=x\_\{m\}, proved byparetoPairSum\_le\_at\_threshold\. Lomax, tail\(1\+X/λ\)−α\(1\+X/\\lambda\)^\{\-\\alpha\}with monotone\-decreasing density: the pair\-sum is antitone, so the maximum is atX=0X=0, proved bylomaxPairSum\_antitoneOn\. Hill, tailKn/\(Kn\+Xn\)K^\{n\}/\(K^\{n\}\+X^\{n\}\), and Fréchet, tail1−e−\(s/X\)α1\-e^\{\-\(s/X\)^\{\\alpha\}\}, are the right\-skew families: in Regime II the pair\-sum is monotone on\[0,1/2\]\[0,1/2\]\(hillPairSum\_monotoneOn,frechetPairSum\_monotoneOn\), so the maximum is atX=1/2X=1/2; in Regime III \(X∗<1/2X^\{\\ast\}<1/2\) it sits at an interior local maximizer, the critical point located byregimeIII\_interior\_critical\_hillandregimeIII\_interior\_critical\_frechet\. In every figure the globalWmaxW\_\{\\max\}on\[0,1/2\]\[0,1/2\]is then evaluated numerically, with the maximizer checked against a brute\-force grid scan before any diagram is drawn\.

These are three structurally distinct density shapes, monotone\-decreasing \(Lomax\), scale\-modal \(Pareto\), and right\-skew unimodal \(Hill, Fréchet\), so the four families exercise the centro\-symmetric\-pairing condition across the range of cases it must handle\.

The phase diagrams of[Section˜6\.1](https://arxiv.org/html/2606.28710#S6.SS1)express these conditions through three dimensionless boundary functions\. Dividing[Equation˜25](https://arxiv.org/html/2606.28710#S5.E25)byMMand writingσ¯g​\(X\)=cg​\(1−F¯g​\(X\)\)\\bar\{\\sigma\}\_\{g\}\(X\)=c\_\{g\}\(1\-\\bar\{F\}\_\{g\}\(X\)\)gives the dimensionless payoff differential

U¯Δ​\(X\)/M=w~−b−1\+b^​\(1−F¯G2​\(X\)\)\+a^​\(1−F¯G1​\(X\)\),\\bar\{U\}\_\{\\Delta\}\(X\)/M\\;=\\;\\tilde\{w\}\-b\-1\+\\hat\{b\}\\,\\bigl\(1\-\\bar\{F\}\_\{G\_\{2\}\}\(X\)\\bigr\)\+\\hat\{a\}\\,\\bigl\(1\-\\bar\{F\}\_\{G\_\{1\}\}\(X\)\\bigr\),\(47\)in the controls of[Appendix˜A](https://arxiv.org/html/2606.28710#A1), withw~:=d\+ud\\tilde\{w\}:=d\+u\_\{d\},a:=AG1/Ma:=A\_\{G\_\{1\}\}/M,b:=AG2/Mb:=A\_\{G\_\{2\}\}/M, and calibration\-rescaleda^:=a​cG1\\hat\{a\}:=a\\,c\_\{G\_\{1\}\},b^:=b​cG2\\hat\{b\}:=b\\,c\_\{G\_\{2\}\}\. Writinggi​\(X\):=F¯i​\(X\)\+F¯i​\(1−X\)g\_\{i\}\(X\):=\\bar\{F\}\_\{i\}\(X\)\+\\bar\{F\}\_\{i\}\(1\-X\)and the pair\-sum maximumWmax:=maxX∈\[0,1/2\]⁡\[b^​gG2​\(X\)\+a^​gG1​\(X\)\]W\_\{\\max\}:=\\max\_\{X\\in\[0,1/2\]\}\\bigl\[\\hat\{b\}\\,g\_\{G\_\{2\}\}\(X\)\+\\hat\{a\}\\,g\_\{G\_\{1\}\}\(X\)\\bigr\], the three conditions become

πlow:=w~−b−1,πhigh:=πlow\+b^​\(1−F¯G2​\(1\)\)\+a^​\(1−F¯G1​\(1\)\),πbind:=2​πlow\+2​\(a^\+b^\)−Wmax,\\pi\_\{\\mathrm\{low\}\}:=\\tilde\{w\}\-b\-1,\\quad\\pi\_\{\\mathrm\{high\}\}:=\\pi\_\{\\mathrm\{low\}\}\+\\hat\{b\}\\,\(1\-\\bar\{F\}\_\{G\_\{2\}\}\(1\)\)\+\\hat\{a\}\\,\(1\-\\bar\{F\}\_\{G\_\{1\}\}\(1\)\),\\quad\\pi\_\{\\mathrm\{bind\}\}:=2\\pi\_\{\\mathrm\{low\}\}\+2\(\\hat\{a\}\+\\hat\{b\}\)\-W\_\{\\max\},namely endpoint\-lowπlow<0\\pi\_\{\\mathrm\{low\}\}<0, endpoint\-highπhigh\>0\\pi\_\{\\mathrm\{high\}\}\>0, and centro\-symmetric pairingπbind≥0\\pi\_\{\\mathrm\{bind\}\}\\geq 0\(strict at someXX\)\. The diagrams fix the working pointcG1=cG2=1c\_\{G\_\{1\}\}=c\_\{G\_\{2\}\}=1\(soa^=a\\hat\{a\}=a,b^=b\\hat\{b\}=b\) andd=1d=1, and partition each scale plane by the zero level sets ofπhigh\\pi\_\{\\mathrm\{high\}\}andπbind\\pi\_\{\\mathrm\{bind\}\}for all four families\.

Table 2:Four threshold priors that satisfy the basin\-existence conditions, spanning monotone\-density \(Lomax, Pareto\) and unimodal\-density \(Hill, Fréchet\) structure\. Middle column: on the matched\-scale slice the tipping point is the single inverse\-tail valueX∗=F¯−1​\(v\)X^\{\\ast\}=\\bar\{F\}^\{\-1\}\(v\)withv=\(ud\+d−1\+a\)/\(a\+b\)v=\(u\_\{d\}\+d\-1\+a\)/\(a\+b\), defined whenv∈\(F¯​\(1\),1\)v\\in\(\\bar\{F\}\(1\),1\)so that a stable all\-G2G\_\{2\}state exists\. Right column: the Lean object certifying all five shape conditions, with the location of the centro\-symmetric pair\-sum maximum that each family’s centro\-symmetric\-pairing lemma supplies; the full declaration list and the normalization route are in[Appendix˜D](https://arxiv.org/html/2606.28710#A4)\. The shape index \(nnorα\\alpha\) is the architect’s awareness\-sharpness knob and the scale \(KK,xmx\_\{m\},λ\\lambda,ss\) is unit choice;[Figure˜2](https://arxiv.org/html/2606.28710#S6.F2)fixesα=2\\alpha=2to match Hilln=2n=2\.Table 3:Laundering thresholdudminu\_\{d\}^\{\\min\}across the four families, evaluated at the figure working point: a representative community \(median acceptance threshold0\.30\.3;a=0\.8a=0\.8,b=0\.3b=0\.3;cG1=cG2=1c\_\{G\_\{1\}\}=c\_\{G\_\{2\}\}=1;α=2\\alpha=2matched to Hilln=2n=2\) on theW=0W=0basin frontier \(d=0d=0\), whereudmin=1−a\+Wmax/2u\_\{d\}^\{\\min\}=1\-a\+W\_\{\\max\}/2\.WmaxW\_\{\\max\}is the pair\-sum maximum located by each family’s centro\-symmetric\-pairing maximizer \([Table˜2](https://arxiv.org/html/2606.28710#A3.T2)\);F¯​\(1\)\\bar\{F\}\(1\)is the shared endpoint\. The threshold is a field over the adoption plane, varying with the community \(along the matched\-scale diagonal it ranges over\[0\.75,1\.30\]\[0\.75,1\.30\]\); at a fixed community it varies little across the prior because at the frontier it rides onWmaxW\_\{\\max\}, set by the perception levers andF¯​\(1\)\\bar\{F\}\(1\)rather than the tail\. The invariant isudmin\>0u\_\{d\}^\{\\min\}\>0throughout: laundering requires the agent to be substantially, not marginally, more useful to the individual\.
## Appendix DMachine verification

A machine\-checked Lean 4 development againstmathlibaccompanies the analysis\. This section separates the results derived from first principles inside Lean from the hypotheses retained at the boundary with the analytic asymptotics that frame them\.

The development mechanizes the Moran–Fermi switch\-rate ratio and the Karlin–Taylor ratio identity \([Equations˜26](https://arxiv.org/html/2606.28710#S5.E26)and[28](https://arxiv.org/html/2606.28710#S5.E28)\); the fixation probabilityϕk\\phi\_\{k\}of[Equation˜27](https://arxiv.org/html/2606.28710#S5.E27), whose absorbing\-boundary harmonic recurrence is verified so thatρG2,G1=ϕ1\\rho\_\{G\_\{2\},G\_\{1\}\}=\\phi\_\{1\}andρG1,G2=1−ϕNc−1\\rho\_\{G\_\{1\},G\_\{2\}\}=1\-\\phi\_\{N\_\{c\}\-1\}are obtained from it rather than posited as closed forms; the within\-community ratio\-favoredness criterion \([Proposition˜5\.1](https://arxiv.org/html/2606.28710#S5.Thmtheorem1)\); the sufficiency of the structural conditions \([Proposition˜5\.3](https://arxiv.org/html/2606.28710#S5.Thmtheorem3)\), including the continuous\-to\-grid bridge and an explicit community\-size threshold above which the cumulative sum is strictly positive; the affine lifting mechanism and the outer threshold\-prior averagingσ¯g​\(X\)=cg​\(1−F¯g​\(X\)\)\\bar\{\\sigma\}\_\{g\}\(X\)=c\_\{g\}\\bigl\(1\-\\bar\{F\}\_\{g\}\(X\)\\bigr\)underlying[Theorem˜B\.1](https://arxiv.org/html/2606.28710#A2.Thmtheorem1); the finite\-population accessibility rate bounds \(the discrete\-barrier log\-rate form of[Proposition˜B\.2](https://arxiv.org/html/2606.28710#A2.Thmtheorem2)\); and[Lemma˜B\.3](https://arxiv.org/html/2606.28710#A2.Thmtheorem3)for the Hill density at generaln\>1n\>1in both its right\-skew form \(a\) and its centro\-symmetric monotonicity form \(b\), with the Regime III interior\-critical\-point lemma \([Lemma˜B\.4](https://arxiv.org/html/2606.28710#A2.Thmtheorem4)\) resting on the mechanized unimodality of the Hill density\.

Three hypotheses are retained at the boundary, indexed \(1\)–\(3\) to match the table below\. \(1\) Centro\-symmetric pairing \(CSP\) is retained at the model capstoneg2\_ratio\_favored\_model\. Monotonicity of the model differential is derived from the monotonicity ofσ¯g\\bar\{\\sigma\}\_\{g\}\(modelFdelta\_condA\), and endpoint inversion reduces to the explicit control inequalities[Equations˜29](https://arxiv.org/html/2606.28710#S5.E29)and[30](https://arxiv.org/html/2606.28710#S5.E30)\. CSP is not derivable universally from the primitive parameters; it is hypothesised at the capstone\. Per\-family sufficient frontiers for CSP are mechanised via the centro\-symmetric\-pairing lemmas of[Appendix˜C](https://arxiv.org/html/2606.28710#A3), which exhibit the three structurally distinct cases CSP turns on\. \(2\) The abstract capstoneaccessible\_and\_favoredis stated for aU¯Δ\\bar\{U\}\_\{\\Delta\}carrying continuity, a Lipschitz bound, and threshold\-root regularity atX∗X^\{\\ast\}\. The model differential is shown continuous on\[0,1\]\[0,1\]viamodelFdelta\_continuousOn; the global Lipschitz constant and the threshold\-root regularity enter the abstract chain rather than being re\-derived for the model\. \(3\) The discrete\-LaplaceNc\\sqrt\{N\_\{c\}\}prefactor of[Equation˜43](https://arxiv.org/html/2606.28710#A2.E43)is hypothesised inprop54\_logas the classical leading\-order form\. The proved accessibility chain,fixation\_log\_rate\_cdepth\_leand its downstream consequences, does not consume it\.

Table 4:Correspondence between manuscript claims and the Lean 4 development\. “Fully mechanized” marks results proved from first principles; “assumption\-level” marks results carrying an explicit analytic hypothesis in their statement\. The three retained hypotheses are CSP at the model capstone, model\-level accessibility composition, and the KramersNc\\sqrt\{N\_\{c\}\}prefactor stated in classical leading\-order form and not consumed downstream\.The development issorry\-free and introduces no custom axioms, depending only on the standard Lean/mathlibfoundationspropext,Classical\.choice, andQuot\.sound\. The Lean sources and the phase\-diagram scripts are available at[https://github\.com/dlewissandy/two\-genie\-scripts](https://github.com/dlewissandy/two-genie-scripts)\(commitebbd849\), built with Leanv4\.29\.0andmathlibv4\.29\.0;lake buildre\-checks the development, and the repositoryreadmemaps each paper result to its Lean declaration\. The Lean declarations named in this paper reside underformalization/, the figure scripts underfigures/\.

## References

- \[1\]Z\. Alalawi, P\. Bova, T\. Cimpeanu,et al\.\(2024\)Trust AI regulation? Discerning users are vital to build trust and effective AI regulation\.arXiv preprint\.Note:arXiv:2403\.09510Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p8.1)\.
- \[2\]D\. Amodei, C\. Olah, J\. Steinhardt, P\. Christiano, J\. Schulman, and D\. Mané\(2016\)Concrete problems in AI safety\.arXiv preprint\.Note:arXiv:1606\.06565Cited by:[§2\.1](https://arxiv.org/html/2606.28710#S2.SS1.p1.2)\.
- \[3\]R\. Axelrod\(1984\)The evolution of cooperation\.Basic Books\.Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p1.1)\.
- \[4\]P\. Bova, A\. Di Stefano, and T\. A\. Han\(2024\)Both eyes open: vigilant incentives help auditors improve AI safety\.Journal of Physics: Complexity5\(2\),pp\. 025009\.External Links:[Document](https://dx.doi.org/10.1088/2632-072X/ad424c)Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p8.1)\.
- \[5\]A\. Buscemi, D\. Proverbio, P\. Bova,et al\.\(2025\)Do LLMs trust AI regulation? Emerging behaviour of game\-theoretic LLM agents\.arXiv preprint\.Note:arXiv:2504\.08640Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p8.1)\.
- \[6\]G\. Ellison\(2000\)Basins of attraction, long run stochastic stability, and the speed of step\-by\-step evolution\.Review of Economic Studies67\(1\),pp\. 17–45\.Cited by:[§B\.3](https://arxiv.org/html/2606.28710#A2.SS3.p1.2),[Proposition B\.2](https://arxiv.org/html/2606.28710#A2.Thmtheorem2.p1.16.16)\.
- \[7\]D\. Foster and H\. P\. Young\(1990\)Stochastic evolutionary game dynamics\.Theoretical Population Biology38\(2\),pp\. 219–232\.Cited by:[§B\.3](https://arxiv.org/html/2606.28710#A2.SS3.p1.2)\.
- \[8\]M\. Granovetter\(1978\)Threshold models of collective behavior\.American Journal of Sociology83\(6\),pp\. 1420–1443\.Cited by:[§5\.1](https://arxiv.org/html/2606.28710#S5.SS1.p2.4)\.
- \[9\]T\. A\. Han, L\. M\. Pereira, F\. C\. Santos, and T\. Lenaerts\(2020\)To regulate or not: a social dynamics analysis of the race for AI supremacy\.arXiv preprint\.Note:arXiv:1907\.12393Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p8.1)\.
- \[10\]T\. A\. Han, C\. Perret, and S\. T\. Powers\(2021\)When to \(or not to\) trust intelligent machines: Insights from an evolutionary game theory analysis of trust in repeated games\.Cognitive Systems Research68,pp\. 111–124\.Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p8.1)\.
- \[11\]M\. Kandori, G\. J\. Mailath, and R\. Rob\(1993\)Learning, mutation, and long run equilibria in games\.Econometrica61\(1\),pp\. 29–56\.Cited by:[§B\.3](https://arxiv.org/html/2606.28710#A2.SS3.p1.2)\.
- \[12\]S\. Karlin and H\. M\. Taylor\(1975\)A first course in stochastic processes\.2nd edition,Academic Press,New York\.Cited by:[§5\.2](https://arxiv.org/html/2606.28710#S5.SS2.p2.11)\.
- \[13\]D\. Manheim and S\. Garrabrant\(2018\)Categorizing variants of Goodhart’s law\.arXiv preprint\.Note:arXiv:1803\.04585Cited by:[§2\.1](https://arxiv.org/html/2606.28710#S2.SS1.p1.2)\.
- \[14\]D\. Manheim, S\. Martin, M\. Bailey, M\. Samin, and R\. Gruetzemacher\(2024\)The necessity of AI audit standards boards\.arXiv preprint\.Note:arXiv:2404\.13060Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p8.1)\.
- \[15\]E\. Ostrom\(1990\)Governing the commons: the evolution of institutions for collective action\.Political Economy of Institutions and Decisions,Cambridge University Press,Cambridge\.Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p7.1)\.
- \[16\]W\. H\. Sandholm\(2010\)Stochastic evolutionary game dynamics: Foundations, deterministic approximation, and equilibrium selection\.InProceedings of Symposia in Applied Mathematics,Vol\.69\.Cited by:[§B\.3](https://arxiv.org/html/2606.28710#A2.SS3.p1.2)\.
- \[17\]T\. C\. Schelling\(1971\)Dynamic models of segregation\.Journal of Mathematical Sociology1\(2\),pp\. 143–186\.Cited by:[§1](https://arxiv.org/html/2606.28710#S1.p1.1)\.
- \[18\]I\. Shapira, G\. Benadè, and A\. D\. Procaccia\(2026\)How RLHF amplifies sycophancy\.arXiv preprint\.Note:arXiv:2602\.01002Cited by:[§2\.1](https://arxiv.org/html/2606.28710#S2.SS1.p1.2)\.
- \[19\]A\. Traulsen, M\. A\. Nowak, and J\. M\. Pacheco\(2006\)Stochastic dynamics of invasion and fixation\.Physical Review E74\(1\),pp\. 011909\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevE.74.011909)Cited by:[§5\.2](https://arxiv.org/html/2606.28710#S5.SS2.p1.5)\.