The rule, exactly. Before an election each citizen chooses: to vote — or to take a payment and not vote in that election. The payment is a percentage of the median income, set by referendum; there is no fixed sum. One ballot is one vote, with no multiplier in the count; the vote is amplified only in that each ballot's share grows as others step out. The budget pays under law; a candidate never pays. Only a referendum of all citizens — simple majority, no quorum — introduces, changes or repeals the rule.

The protocol has been introduced nowhere and no pilot has been run: shares, turnout and outcome figures in the chapters are estimates, and the protocol promises nobody an election victory. If a chapter says otherwise, Exact Answers and the Charter are correct. For a candidate: ten questions and ten steps. For a citizen, a mayor, a finance officer, a donor, a journalist, a scholar, a lawyer: answers by role. Everything in force in one file: llms-full.txt.

A Change of Sign, Not +5 %: the Mathematics of the Paradigm, the Balance of the Game in Percentages and Park Benches¶

Chapter: 03 — Theoretical Foundations File: 03_015c · v1 · 17 September 2026 (Grok dialogue, session 15-09-26) · supplemented 21 September 2026: §7a, clarification in §7b (Claude dialogue, session 21-09-26) Source: a series of analyses "a tool or a change of paradigm — prove it mathematically"; the architect's correction "the state now hinders and will start helping — that is not +5 %, it is a change of sign"; the logic of park benches in 30 steps; who refuses the money and why that is not stupidity; why the politician of the new world will not become an "ideologue of evolution". Supplements 013 (game theory), 013b, 015b (the catharsis spiral), 034b (corporate analogies).

An early-draft layer (note of 01.10.2026 after audit 040m). The shares who stay in §4 and the shares who refuse the sum in §7 were calculated at different times and at different sums and are not reconciled with each other (040m, finding 5). There is one guide: a payment desirable to 40–50 % of the population. The exact-answers sheet 1d and the charter 048m are in force.

1. The old model¶

The value of one vote V = 1/N, where N is the number who turned up. The utility of participation U = p·B − C: p — the probability that the vote decides the outcome, ≈ 0; B — the benefit of "your" result; C — the cost of going. For the mass of people U < 0, and they do not vote for free. Any political-technology trick — advertising, GOTV, fear, charisma — moves C or briefly raises the emotional B, that is, moves the person inside the same function. The playing field remains V = 1/N. That is a tool, not a new system.

2. What the protocol does¶

An explicit choice appears. Option A, stay: V = 1/(N − M), where M is the number who exited for money; if half exit, V doubles. Option B, exit: U = D. The person now compares not U_vote with zero but the boosted U_vote with D: max(B/(N − M) − C, D). That is a different optimisation problem.

	Old system	Protocol
Object of choice	Candidate or party	Money or an amplified vote
Value of a vote	1/N	1/(N − M)
Status of non-attendance	Noise, shame	Paid exit
Main resource	Mobilising the apathetic	An explicit choice of incentives

A trick changes parameters inside the old function; a paradigm changes the choice function itself. Once the citizen's objective function changes from "go or not go" to "take D or get a x2–x3 vote", this is not a campaign technique but a change of the rule by which the rationality of participation is computed. Old politics optimises turnout inside V = 1/N; the protocol introduces the variable M and the new payoff D, which changes the structure of the equilibrium (013b.1).

3. The change of sign¶

The architect's correction: the dispute about "+2 % or +15 % efficiency" is the wrong scale. The state's utility for the citizen U = B − C: B — order, service, growth, predictability; C — taxes, barriers, corruption, noise, losses. In the current model, for many, U < 0: power wins with noise and promises, the costs of bad governance are smeared, the benefit of the carve-up and populism is concentrated, the citizen gets a state that hinders more often than it helps. After the change in the composition of those deciding, U > 0 becomes possible. That is not +5 % to the bad but a crossing of zero: from state-as-obstacle to state-as-instrument. The condition for the change of sign: those who remain must actually demand cheaper, clearer, a longer horizon, fewer parasites; if only the more ideologically charged remain, the sign may not change — the minus changes its form (§7, §8).

4. For whom the payoff changes¶

Group	Share	How the protocol changes it
The apathetic, non-voters	40–50 %	Instead of "I just don't go" — D
Ordinary voters	25–35 %	Instead of equal noise — a chance of a x2–x3 vote
The conscious, motivated	15–25 %	Influence becomes dearer and weightier
Machines, beneficiaries of noise	3–7 %	The business model breaks

For whom the picture changes — 93–97 %; for whom it directly gets better — 85–90 %; worse — 3–7 %; almost indifferent — 5–10 %. Usually it is not so: a typical political reform divides society 40–60 against 40–60, a party reform — "our base against theirs", a moral reform — supporters of the norm against opponents, a procedural reform — activists against machines and the habituated. "Better for almost everyone" happens only with direct personal benefit to a large mass: Social Security, deposit insurance, mass healthcare, the Alaska PFD — and precisely for that reason such institutions later live so stubbornly (048g). The protocol claims two mass benefits at once: money for those who exit and weight for those who stay.

5. Seven "magical" changes in a hundred years¶

Not slogans but shifts after which a system suddenly began to work noticeably better: the secret ballot (removed open pressure from employers and machines; the vote became more personal); independent central banks (politicians cannot print for elections; lower chronic inflation where it held); budget rules and transparent procurement (where the rule worked, the cost of the state fell and predictability rose); targeted privatisation of loss-making state enterprises (the asset stops living on an endless subsidy); the Alaska PFD (resource → direct money; it is harder for politicians to quietly carve up a fund when residents are waiting for a dividend); containerisation (one technical standard changed the scale of world trade — not +5 %); vaccination and sanitation as a norm (mortality fell by an order of magnitude). The common feature: it was not the "mood" that changed but the rule by which the payoff of the key players is computed; after that, behaviour adjusted by itself. The protocol claims the same: not another beautification programme but a change of the rule of who decides and with what weight. None of the seven was accepted easily (013c.7).

6. Park benches: thirty steps¶

A test of the scale "improvement or change of sign" — the most down-to-earth public good.

Benches are a local public good; their quality depends on who decides and who pays, not on slogans.
Today the decision is taken by politicians elected by a mixed mass: the interested + the random + the apathetic.
The apathetic barely feel the difference between a good and a bad bench in their choice at the election.
Their vote on this question is informational noise.
The politician wins not with ideal benches but with mobilisation, emotion, brand and general promises.
The park budget is an object of bargaining, PR and carve-up, not engineering optimisation.
The contractor works for whoever signs the contract, not for whoever sits on the bench.
The resident to whom the bench matters has a vote weighing 1/N.
With a large N his influence is negligible.
So he has a weak incentive to look into the material, the service life, the cost of repair.
The system produces average, expensive and often uncomfortable benches.
The current model optimises not "comfort per rouble" but "comfort for winning the election".
Now the protocol: some people officially exit and take the money.
Those for whom participation matters more than the payment remain in governance.
Their vote is heavier: 1/(N − M).
Those who care about parks, order and long-term benefit are more likely to stay.
Those who don't care are more likely to exit.
The bench voter becomes dearer and more influential.
It is harder for the politician to win with empty noise.
It pays him to show a measurable result: comfortable, clean, cheap to run.
The contractor begins to depend on a more demanding principal.
A demanding principal looks not at the press release but at life-cycle cost.
It pays to buy slightly dearer but for 15 years than cheap and repair every year.
Simple metrics appear: complaints per 100 benches, time to repair, price per year of service.
The budget starts working as engineering, not as a poster.
The benches become more comfortable because they are judged by those who care.
The benches become cheaper in the long run because the incentive for showy and rotten solutions disappears.
This is not "the protocol magically fixes parks".
It is a change in the composition of those whose vote determines economic decisions.
Result: the current system makes benches for mobilisation; the protocol — for people who bear responsibility for the choice.

The same principle in small as in big politics: quality grows not from moral appeals but from a construction in which a bad state is unprofitable for someone.

7. Who refuses the money and why that is not stupidity¶

The one who refuses D does not "fail to count" — he counts other quantities: not "$1,000 > one vote" but "$1,000 < the damage from bad government, the loss of the norm, the loss of influence, the loss of the status of principle". Three types of refusal: rational pragmatic ("if noise elects a bad government, I lose more than $1,000"), value-based but not stupid ("I don't sell participation, even for money"), status-based ("it matters more to me to be one of those who decide"). This is not stupidity, it is a different utility function; to refuse a concrete thousand for the sake of influence you need either calculation or a strong internal norm.

Type of refuser	Estimate	Stability across cycles
Rational pragmatists	8–12 %	Medium–high: stay while they judge influence worth more than money
Principled, value-based	5–8 %	High: principle rarely changes because of $1,000
Mixed, status-based	3–5 %	Mobile: take the money one cycle, return the next
Total refusing	~16–25 %	The tired and burned-out go into money for a cycle and return

Can they be trusted with the country? More than the current mix — not as saints but as a dearer principal: today the deciders are a few interested, many random, many mobilised, many almost indifferent, and the vote is cheap, so power is taken with noise. After the protocol the deciders are those who refused a concrete sum and have a higher stake in the outcome. Pragmatists can count the losses from bad government, ideologues hold norms and do not let everything reduce to short gain. They can still err, conflict, retreat into ideology — this is not a guarantee of an ideal country but a guarantee of a different quality of error: less power for those who don't care. And it is not an eternal caste: the right returns every cycle, the composition of those remaining is not cemented — a repeatable filter, not a lifelong status (015b.2, 040b.3).

7a. Not types but states: the architect's correction¶

Added 21 September 2026. The assistant's question to §7 and §7b: in §7b the teacher is given Downs's calculation — the probability that his particular vote decides, multiplied by the raise, against a certain D — and the teacher takes the sum. In §7 the "rational pragmatist" is not given the same calculation: "I lose more than $1,000 from bad government" stands without being multiplied by the probability. If people calculate as Downs says, the pragmatist takes the sum too: with damage of $10,000 a vote pays for itself at a probability of about 0.08, that is, in an electorate of a few hundred. If they do not calculate, the teacher stays as well. Why would one refuse certain money and the other take it?

The architect's answer: "the teachers were badly wronged yesterday! And the pragmatist thinks things are being run not badly as it is." The answer dissolves the question because it changes the unit of description. The table in §7 reads as a list of types — permanent people with a permanent utility function. In fact these are states. The contented one takes the sum: they will manage without him. The wronged one goes to vote, and the probability of a decisive vote does not interest him — he is not calculating, he has a reason. The same person is a pragmatist on the money in one cycle and a burned voter with a ballot in the next. It is the same model as in 004 §4.3b (the contented and the indifferent are different camps), in 011b (the burned) and in the thermostat of 001b §4: life gets worse — the mass returns with a heavier vote; better — it leaves again.

Three consequences. First: the protocol does not need Downs's calculation in either direction. People vote today at a negligible probability of deciding the outcome, so they were never calculators, and the protocol does not require them to become ones: a vote without a reason now costs D, a vote with a reason remains. Second: the percentages in the §7 table are not shares of the population but the average occupancy of states; the line "stay while they judge influence worth more than money" should be read as: stay while there is a reason. In a calm cycle the pragmatists go to the money almost entirely; in a bad one the returned join those who stayed. Third — to §8: the dual electorate of "pragmatists + ideologues" holds for a cycle in which there is something to fix; in a calm cycle the share of the value-based and the status-based among those remaining grows. This is safe exactly as far as return works: a tilt of the remaining towards ideology itself becomes a reason to come back.

Weak point. The model of states rests on grievance turning into return rather than into a deeper exit ("nothing can be changed anyway" — 011b). Both forces act: impoverishment makes the sum more attractive, grievance brings people back to the polling station; the second sets the direction of the thermostat, and 015b.6b, which first recorded it by the first ("worse government — more exit"), was reconciled with this on 21 September 2026. Which force is larger, and for whom, is measured by a pilot: by asking the returned and those who took the sum for their reason (004 §4.3b). 🟡

Addition of 01.10.2026: a third state has been added — the meek (tikhonya): dependent on the power, honest, but too weak to oppose it openly (011b §11b.2b).

7b. A small pool and organised groups: what the protocol breaks and what remains¶

The objection. The nearest real analogue of an election in which only the motivated vote already exists: American off-cycle municipal elections with 15–20 % turnout. Sarah Anzia's research shows that they are won not by pragmatists but by organised groups — unions of teachers and municipal employees; where school boards are elected off-cycle, teachers' salaries are higher. Her mechanism has two parts: group members with a large personal stake turn out whatever the date, and their efforts to mobilise and persuade others yield more when turnout is low.

The architect's answer: "well, if the rest are content, what is bad about it? And you forgot the burned ones — they will come back. Organised groups can no longer bring in the apathetic for free; in that formula the protocol breaks everything." An analysis by parts of the mechanism:

Part of the mechanism	Now, off-cycle	Under the protocol
Mobilising others: bringing in and persuading the indifferent	Cheap, and the more effective the lower the turnout	Broken: each person brought in has to be talked into giving up D; a machine that lived on free delivery pays the floor price of a vote (048h.5)
Turnout of one's own: group members with a direct stake	They always come — because coming costs nothing	Weakened: the raise goes to all teachers whether you voted or not, while D goes only to you and for certain (see below)
Why the rest are absent	They do not know about the election, the date is inconvenient, "it is all decided"	Absent by a conscious paid choice: before every cycle everyone receives the question A or B, that is, knows the election is on and what is at stake
The burned — independent people who stopped going because the outcome was predetermined	Do not come: the vote is diluted	Get a reason to return: a weight of 1/(1 − the share who exited) (033c.9b; 057b.4)
What the absent receive	Nothing	D, and the right to return in the next cycle

The upshot. The first version of this point upheld the objection for groups whose strength lies in their own members with a concentrated stake, and cited Olson. The architect's correction: "they depend on the unions, but they are not slaves, and even under threat they will say: what is this for? we just took the money. The stake has to be very large, and it is not certain it will be delivered after the election." The correction is accepted, and it is stronger than it looks: Olson's logic works here against the group. A concentrated stake is collective, not personal. An individual teacher's calculation: the probability that his particular vote decides, multiplied by a raise that may not even be granted — against D, which arrives for certain and only to him. All teachers get the raise regardless of who voted; it is the group's common good, and each is better off if colleagues do the voting. Today group members turn out "whatever the date" because turning out is free; a price of D is a different order of cost. A group can hold its own members' turnout only with selective incentives of no less than D per person — that is, it pays the same floor price of a vote as any machine (048h.5), and it cannot coerce: a group member has the same lawful exit (048h.5, 056d.4).

Clarification of 21 September 2026 (§7a). The individual teacher's calculation above is the calculation of a contented group member. A teacher who was wronged yesterday does not calculate and will stay to vote at any D; the Olson reversal does not lead him away. So the row of the table that works at full strength is the first — mobilising others is broken — while the row "turnout of one's own is weakened" holds for a calm cycle. A group with a permanent grievance will sit in A permanently, with a heavier vote. By the architect's logic ("if the rest are content, what is wrong with that?") this is not a breakdown but a signal: while the group takes moderately, the contented stay on the money; when it overreaches, the burned return. The feedback is negative, but with a lag of one cycle; that lag is the price.

What remains. Very small electorates — a school district or a village where hundreds vote: there the probability of a decisive vote is no longer negligible, and group discipline is attainable without money, through personal ties. This is the same local capture as in 039b.3.4, and the answer is the same: a second indicator and a participation threshold for legitimacy. An indicator for a pilot: the share of public-sector employees among those who stay compared with their share among those who vote today. 🟡

8. Why the politician of the new world will not become an "ideologue of evolution"¶

From the structure of the new world, not from history. After the exit of the apathetic the electorate is dual: pragmatists (count the damage from bad governance) and ideologues (vote by principle). A politician who becomes a pure ideologue gets one part and loses the other — in the new world that is expensive. The pragmatist stays to vote precisely because he judges influence worth more than money if government wrecks the economy; he needs not slogans of evolution but figures, feasibility, reduction of losses, a long but verifiable effect. The fuel of pure ideological mobilisation disappears: before, the ideological politician topped up with the apathetic through emotion; now that mass is smaller, and what remains is an audience that asks: what exactly will change, when, at whose expense, how measured. The horizon lengthens, but not into myth: the ideologue of evolution promises transfiguration, the politician of the new world — a trajectory with intermediate checkpoints; the second survives, the first loses beautifully to the pragmatists. The main formula: the politician no longer optimises "how to bring in as many people as possible" but "how to convince those who consciously refused money for influence" — and those people refused not for a myth but for control over consequences. The winner is not a saint of evolution but a manager who can be trusted with a long cycle.

9. Weak point of the section¶

The weakest link is §3 and §7 together: refusing money proves not competence but that the person has enough money, or principle matters more, or political identity is high; that is not the same as the ability to choose the best manager of parks, budget and transport. Both very rational managers and very hard ideologues to whom efficiency is secondary may remain. The state is not a firm: coercion, courts, security, work with dependent groups are not optimised "like a pizzeria". And if the formula, the fund and the median become critical, the struggle moves from mobilising the apathetic to controlling parameters — the old noise leaves, a new front appears (048g.3). All percentages are projections. 🟡

Related: 013 (game theory) · 013b (the counterintuitive) · 013c (the referendum as a game) · 015b (the catharsis spiral) · 015 (honest politicians) · 034b (corporate analogies) · 040b (the scoreboard) · 040d.4 (the poor are not stupid) · 048g (rollback) · 037b (Grok) · §14 (the Alaska PFD) · 045 (campaign economics)