PART B · MATHEMATICAL PRINCIPLES

Mathematical Principles of the POSX Node Mode

The thirty-six-tier pricing is not a set of hand-picked lucky numbers; it is the solution of a constrained optimization problem. This part explains the theoretical grounds, formulas, and empirical literature behind the parameters from four directions — mechanism design, behavioral economics, operations research, and number theory — and lays out derivations that can be re-checked. All cited literature and case data come from verified primary sources in the project's research archive; wherever unpublished parameters are involved, concepts and methods lead, with exact values published progressively with each tier's formal announcement.

§1 · MECHANISM DESIGN

Mechanism Design: Why a Pre-Committed Ladder

The first pricing decision is not a number but a mechanism: is the price set by rules published in advance, or at the issuer's discretion as market conditions move?

POSX chose the former: the thirty-six-tier structure of prices and allocations is published before the sale opens and locked into the contract, and the price path is not adjusted to market conditions; the increment band sits wholly inside the externally committed 4%–10% envelope. The grounds are two independent results from macroeconomics and operations research.

Theory · Time consistency

Kydland & Prescott (1977) proved that, facing participants with forward-looking expectations, period-by-period discretionary optimization is dominated overall by pre-committed rules — and the rules should be simple, public, and such that deviations are detectable. This result was the core contribution behind the 2004 Nobel Prize in economics. Mapped onto a sale: if buyers expect the issuer to soften the increments whenever sales stall, the rational move is collective waiting, and sales momentum destroys itself.

Theory · Strategic consumers

Su (2007, Management Science) proved that facing strategically waiting customers, prices should rise over time to suppress unproductive waiting; Aviv & Pazgal (2008, M&SOM) further proved that pre-announced pricing beats contingent pricing, with revenue losses of roughly 20% for misjudging buyers' strategic behavior; Xie & Shugan (2001, Marketing Science) gave the precondition for advance selling — future higher prices must be credible, and when they are, advance-selling profit can nearly double. Together the three point one way: an ascending ladder draws its entire force from its credibility.

Mechanism choice: P = (P₁, …, P₃₆) published and contract-locked at t = 0 ⇒ ∀t: E[future P] is known; waiting carries no information value

Successful precedents and the POSX implementation

Aethir (2024) ran its ladder through 53 pre-deployed pricing contracts, selling $65M in the first hour; XAI's 40-tier ladder switches price automatically on sell-out via smart contract; Filecoin (2017) pre-published a price function and completed a $205.8M sale within a month. All eight comparable sales verified in the research archive used pre-committed structures, without exception. POSX is congruent with them: the ladder is published before the sale opens and locked in contract, and any subsequent arrangements follow the pre-published rules and official announcements.

§2 · INCREMENT GLIDE

The Increment Curve: A Glide from High to Low

Tier-to-tier increments start at +8.56% at the first step (0.0888 ÷ 0.0818 − 1) and decline overall, staying inside the externally committed 4%–10% envelope throughout; meanwhile, the absolute step between adjacent tiers grows tier by tier, so the nominal price rise faced by later buyers never shrinks.

Illustrative — not per-tier actual values

This shape is fixed jointly by three independent grounds: loss aversion from behavioral economics, the dual-entitlement principle from fairness research, and a cost accounting based on an elementary inequality.

Theory · Loss aversion, front-loaded

Kahneman & Tversky (1979) proved that losses carry more psychological weight than equal gains; meta-analysis puts the mean loss-aversion coefficient at 1.8–2.1 (Brown et al., 2021). “Missing this tier” is a computable loss, and its deterrent power is worth most at the opening — hence the largest percentage increment sits first, while the tier-by-tier growth of the absolute step guarantees that the nominal rise never shrinks in the later stretch.

Theory · The fairness constraint

The telephone surveys of Kahneman, Knetsch & Thaler (1986) showed that price rises which exploit demand shifts are judged unfair by 82% of respondents, while pre-announced, rule-based increases are not so judged. Declining percentage increments further lighten later participants' relative burden — the direction fairness research marks as benign.

Π(1+gᵢ) ≤ (1+ḡ)³⁵, with equality iff gᵢ is constant; ln(P₃₆ ⁄ P₁) = Σ ln(1+gᵢ) = ln 12.22

Mathematics · The glide's cost is computable

By the arithmetic–geometric mean inequality: with the mean fixed, a constant increment maximizes the first-to-last multiple, and any varying increment necessarily yields a smaller one. The system trades less than 1% of the multiple for opening intensity and late-stage fairness — a cost accepted after being computed, not an unexamined default. For reference: Filecoin's price function price = max($1, raised ⁄ $40M) is mathematically exactly “percentage increments high first, low later” (with equal absolute steps, relative increments decline as the price rises); CARV's measured tier gaps likewise converged from about 16% early on to about 12.5% mid-sale.

§3 · ALLOCATION MODEL

The Allocation Model: Load Flattening under Dual Constraints

Per-tier allocations are generated by a one-parameter family of geometric weights: given the price table and the two total targets, the allocation's “evenness” is solved uniquely from an identity — there is no room for manual tuning.

Illustrative — not per-tier actual values

wᵢ = cᵢ · r^(i−1) (c₁, c₂ are start-up coefficients); r is solved by bisection from Σwᵢ Pᵢ ⁄ Σwᵢ = total value ÷ total quantity (the left-hand side is strictly monotonic in r)

Theory · Congruent with revenue management

The structure is congruent with airline revenue management: low-fare classes are capped and high-fare classes protected — “closing a low-fare class is conceptually a price increase” (INFORMS history of revenue management). Nested protection and the EMSR heuristics (Belobaba; McGill & van Ryzin, 1999) supplied the operations-research foundation for multi-class capped pricing, a system credited with roughly $1.4B in incremental revenue for American Airlines over three years.

Theory · The goal gradient

The field experiments of Kivetz, Urminsky & Zheng (2006, JMR) proved that purchase frequency rises as a goal draws near. Compressing each tier's sale size into a narrow band turns every tier into a completable small goal: thirty-six approach-and-complete cycles in place of a few insurmountable mountains.

Implementation and verification

Peak single-tier load is below the 5% internal-control ceiling and within the verified peer range (Aethir mean about 1.9%; 0G mean about 3.1%); the quartiles of full-sale progress fall at Tiers 9, 18, and 26 — a nearly even cadence. As per-tier allocations are published progressively, the narrow load band and the peak share can be recomputed and re-checked tier by tier.

§4 · EXACT BALANCING

Exact Balancing: A Diophantine Solution for Two Totals

The total allocation of 980,000,000 tokens and the full-sale scale must both hold exactly at the same time. With every bin's value constrained to the auspicious-number lattice, this is a system of two linear integer equations, and number theory guarantees its solvability.

With two free bins a, b: Δa + Δb = ΔQ; Δa·p_a + Δb·p_b = ΔV ⇒ Δa = (ΔV − p_b·ΔQ) ⁄ (p_a − p_b); an integer solution requires (p_a − p_b) | (ΔV − p_b·ΔQ)

Solution order: first pick values greedily on the lattice and balance in one dimension (quantity exact); then sweep the bin pairs for one satisfying the divisibility condition of the two-bin correction; where no pair solves, bring in a third bin for congruence adjustment. The final solution landed on bins 10 and 15 (called the Gate of Life and the Gate of Scenery within the system), with corrections inside a ±8% window of the original allocation curve, leaving the load shape intact.

Why it is worth it

An integer total is a commitment any third party can verify on-chain in a single step. Compared with a rounded approximation, the exactness of 980,000,000 reduces the cost of an audit to one summation and makes the total a contract constant rather than marketing rhetoric.

Verification

Σ Qᵢ = 980,000,000 (a 36-term sum); the full-sale-scale identity Σ QᵢPᵢ can be independently verified the same way in a spreadsheet once the full table of parameters is formally published.

§5 · SIMULATION & ROBUSTNESS

Simulation Robustness: Quantified Evidence from Mechanism Comparison

Before locking in the ladder's shape, the project ran Monte Carlo comparisons of four increment mechanisms (geometric-Brownian market regimes × discount-elastic demand × a strategic-waiting penalty, 200–400 runs per grid cell).

Illustrative — not per-tier actual values

Under the baseline regime, the median comparison of the committed ladder against discretionary market-tracking shows that the latter buys its higher completion rate with an average price about three-tenths lower and about seven-tenths more tokens sold — its advantage is, in substance, a disguised discount (model output, not a prediction).

Three simulation findings

First, discretionary repricing wins on completion rate but issues 43%–77% more tokens for the same raise, compressing early participants' relative advantage — and it can be fully replicated by simply lowering the fixed increments; the flexibility itself creates no value. Second, deadline-forced price rises rank last on revenue in every regime: the price is pushed into a zone with no demand. Third, a conditional dynamic component triggered only by clear strong-demand signals under pre-published rules sells about 10% fewer tokens for the same revenue in strong markets and switches itself off in weak ones — the only directionally harmless dynamic term; whether and how POSX adopts such a component is governed by the formally published sale rules.

Literature sources of the behavioral parameters

The demand-side strategic-waiting penalty takes the 20%-scale lower bound from Aviv–Pazgal; the scarcity response follows Worchel et al. (1975) — scarcity from demand is most effective — and Aggarwal et al. (2011) — limited quantity beats limited time; same-screen option discipline follows Iyengar–Lepper (2000) and Chernev et al. (2015).

Boundary statement

The simulations rest on assumed parameters (demand level, elasticity, reflexivity coefficients); the absolute values are not predictions. The directional conclusions — discretion equals hidden discounting, time triggers are harmful, conditional components are neutral-to-favorable — remain stable under parameter perturbation.

§6 · DIGIT-LEVEL EVIDENCE

Digit-Level Evidence: Price Endings, Precise Prices, and Clustering

The choice of the specific digits also has its literature. Three independent empirical threads jointly support the four-decimal, auspicious-ending, non-round price shape.

Price endings (Chinese-speaking markets)

Simmons & Schindler (2003) found in statistics on Chinese price advertising a systematic preference for endings in 8 and avoidance of 4; Fortin et al. (2014) measured transaction-price gaps of +2.5% for 8-ending and −2.2% for 4-ending house numbers in Chinese-community neighborhoods; Shum et al. (2014) measured a premium of about 235 yuan per square meter for 8-ending floors in Chengdu; Bhattacharya et al. (2018) observed Taiwanese retail futures limit orders clustering significantly at 8-ending prices. Price endings are not superstition dressing — they are a market preference with transaction-price evidence.

The precise-price effect

Thomas, Simon & Kadiyali (2010): precise (non-round) list prices are judged smaller under uncertainty, and in real property transactions precise listings correspond to higher sale prices; Janiszewski & Uy (2008): adjustments away from precise anchors are smaller. Four-decimal pricing (0.0818 rather than 0.08) therefore wins at the anchoring level. The left-digit effect (Thomas & Morwitz, 2005) neutralizes naturally across the candidates — all of them begin with 0.08.

Clustering in crypto markets

Urquhart (2017) found that Bitcoin prices cluster significantly at round numbers; Quiroga-García et al. (2022) replicated the same clustering in Ether, Ripple, and Litecoin. Market participants' attention naturally moors at particular digit positions — the pricing deliberately places those mooring points at auspicious digits rather than arbitrary ones, an active use of the same phenomenon.

The POSX implementation

All prices in the table carry four decimals; except for the final tier's return to one (1.0000), prices end in 8 or 6; the published tier prices and displayed increments contain no digit 4 and none of the designated sensitive combinations, and later tiers can be checked continuously as they are published. The rulebook thereby satisfies the direction of both the precise-price and the price-ending literatures.

§7 · CASE BENCHMARKS

Case Benchmarks: Where POSX Sits in the Peer Coordinate System

The research archive verified the complete structures of six comparable sales. POSX's per-tier increment is below the node-sale mainstream (12%–15%), and its first-to-last multiple of 12.22× (= 1.0000 ÷ 0.0818) sits mid-range within the “completion band” (2.5×–25×).

Illustrative — not per-tier actual values

Six comparable sales and POSX's structural coordinates
Project	Tiers	Per-tier increment	Structure	Outcome (verified basis)
Aethir 2024	53	~15% (inferred)	Pre-deployed pricing contracts · advances on sell-out	$65M in the first hour; voluntarily paused after 73,000+ nodes in 42 days
Sophon 2024	—	~12% (inferred)	Pre-published price table	$60M; 60.5% sold; unsold portion burned
CARV 2024	—	~12.5–16% (inferred)	Pre-published ladder	Over $30M in 4 days of public sale
0G 2024	32	~3% (mean, inferred)	Price-capped · per-tier limits	$28M+; 80,000 nodes
XAI 2023–	40+	15% (exact)	Contract-hardcoded · declining quantities	Perpetual ladder
Filecoin 2017	Continuous function	Declining as price rises	Pre-published price function	$205.8M; $135M in the first hour
POSX	36	From +8.56% at the first step, declining within the band (externally committed 4%–10% envelope)	Contract-locked · pre-published rules	First-to-last multiple 12.22×; peak single-tier load below the 5% internal-control ceiling

Notes on the figures

Except for XAI (exact values from its official table), each case's increment is inferred from two verified price points; parts of the Sophon and CARV data come from media reports, with the nature of each source annotated item by item in the research archive. POSX's unpublished tier parameters are published progressively with official announcements.

§8 · METHODS & REFERENCES

Methods, Limitations, and Reference Sources

Research workflow: parallel literature and case retrieval (every citation was actually opened and checked against the original during the research) → second-pass verification of key facts → simultaneous parameter solving → Monte Carlo mechanism comparison → numeric-rule value selection and balancing → full-table audit.

The principal sources are listed below by category, as textual records without external links.

Principal reference sources
Category	Source (author · year · finding)
Mechanism and commitment theory	Kydland & Prescott (1977, JPE) rules beat discretion; Su (2007, Management Science) rising prices suppress strategic waiting; Aviv & Pazgal (2008, M&SOM) pre-announced beats contingent pricing; Xie & Shugan (2001, Marketing Science) credibility conditions for advance selling
Behavioral economics	Kahneman & Tversky (1979) prospect theory; Brown et al. (2021) loss-aversion meta-analysis; Kahneman, Knetsch & Thaler (1986, AER) fairness and dual entitlement; Worchel et al. (1975) scarcity effects; Aggarwal et al. (2011) limited quantity beats limited time; Iyengar & Lepper (2000) choice overload; Chernev et al. (2015) moderator meta-analysis; Kivetz, Urminsky & Zheng (2006, JMR) the goal gradient
Revenue management	McGill & van Ryzin (1999) revenue-management survey; Belobaba: EMSR heuristics (MIT OCW lecture notes); INFORMS: history of revenue management and the American Airlines DINAMO case
Digit and price-ending evidence	Simmons & Schindler (2003) Chinese advertising price endings; Fortin et al. (2014) house-number price gaps; Shum et al. (2014) floor-number price gaps; Bhattacharya et al. (2018) trader price-ending behavior; Hirshleifer et al. (2018) lucky-number premiums; Thomas & Morwitz (2005) the left-digit effect; Thomas, Simon & Kadiyali (2010) the precise-price effect; Janiszewski & Uy (2008) precise anchoring; Urquhart (2017) Bitcoin clustering; Quiroga-García et al. (2022) crypto-asset clustering
Case sources (selected)	Aethir official documentation and sale data; the XAI pricing contract table; 0G sale notes and results; Filecoin sale economics (official PDF) and completion announcement; the Ethereum genesis sale; the Binance EDU subscription; the CoinList Casper case; the FTC dark-patterns report and UK CMA commitments (compliance baseline)

Limitations

Peer increments are mostly two-point inferences; the Monte Carlo conclusions are directionally robust but their absolute values depend on assumptions; behavioral effect sizes come from Western and East Asian consumer samples, and extrapolation to crypto participants carries uncertainty. This part is a methodological explainer, not investment advice; the numeric-culture reading is in Part A, and the two parts' statements stand independently of each other.

A · Digit Energetics of the POSX Node Mode

Boundary statement: Part A of this appendix is a cultural interpretive frame; Part B is a methodological and bibliographical explainer. Hexagram imagery, decans, digit roots, and the like do not constitute scientific conclusions; simulation and literature findings do not constitute predictions of, or commitments to, any price, value, or return; neither part constitutes investment advice. This page is for information only and provides no purchase or redirect functions; sale parameters follow the Node Sale Mechanism V1.0 and official announcements, and each tier's exact prices and allocation quantities are published progressively with the corresponding tier's formal announcement.