> # ⛔ RETIRED — 2026-09-23 > > **Do not quote any figure in this document.** It is kept as a > change-management record, not as a reference. > > The current PAR sheet is > **`aggregator/par_sheets_v13/par_sheet_shared_core_v13.md` (v13.1)**, with > per-game sheets beside it. > > This file is dated 2026-09-15 and describes the game as it was before the > 2026-09-21 progressive tier overhaul and the pool-seed accounting fix. Its > headline numbers are wrong by a wide margin: it quotes a 96 scale of 0.9427 > (now 0.842), a tier ladder with Mini and Major fixed (now Mini and Minor), a > random progressive trigger of 1 in 50,000 (now 1 in 20,000), and a > Cash-on-Reels ladder of 13 values averaging 5.32× per cell (now 3 values > averaging 1.4928×). The RTP figures it reports were produced by a simulator > that did not count pool seed refills, so they understate true cost by roughly > 25 points. > > **What was salvaged from it before retirement**, re-verified against the code > rather than copied across: > - Cash-on-Reels, including the player-facing guarantee that printed values > are the values awarded → v13.1 §5a.1, with the current distribution > - Pool-tier proportional payout, the reference bet, Must-Hit-By, and award > atomicity → v13.1 §6.5 > > Its §2 RNG description was already correct — it named the production CSPRNG > where the v12c sheet had described the simulator's Mersenne Twister — and > that correction is carried in v13.1 §2.1–2.5. --- # Refund Rush — PAR Sheet (Math Compliance Document) **Game:** Refund Rush **Version:** 3.0.0 (v3 math overhaul — cull + ways + volatility pivot) **Document date:** 2026-05-08 **Math design:** Oki Studios **Format:** This sheet is intended for submission to GLI (Gaming Laboratories International), iTech Labs, or equivalent test houses. --- > **Current math note (2026-05-08).** This sheet reflects the current family > math: 243-ways scoring, Direct Deposit hold-and-respin, Bonus Season free > spins, four jackpot tiers, and two readable random > events. Earlier prototype features were retired before this submission pack. > > **1B-spin validation receipts are pinned** in `docs/sim_receipts/`: > > | Target | PAYTABLE_SCALE | 1B measured RTP | Receipt | > |--------|----------------|------------------|---------| > | 88 | 0.8423 | 88.0347% | `sim_88_1B_final.txt` | > | 92 | 0.8920 | 91.9985% | `sim_92_1B_final.txt` | > | 96 | 0.9427 | 95.9966% | `sim_96_1B_final.txt` | --- ## 1. Game summary | Property | Value | | ------------------------- | -------------------------------------- | | Reels | 5 | | Rows visible | 3 (5×3 grid) | | Scoring | **All-ways (3^5 = 243 ways)** — v3; lines retired | | Bet range (minor units) | 100 – 2,500,000 ($1 – $25,000) | | Default bet (UI) | 100 ($1.00 total bet — no per-line div in ways) | | Maximum win | 10,000× non-progressive cap; Minor/Grand pools uncapped until hit | | RTP — default | 95.9966% measured at the 96 target | | RTP — configurable | 88.0347 / 91.9985 / 95.9966 measured | | Volatility | High (std dev/spin ≈ 9.2 — was 6.1 in v2) | | Hit frequency | **33.27%** | | Theme | Tax refund / personal-finance Americana | | Features | Direct Deposit + Bonus Season + 4 jackpot tiers + 2 random events (DD Alert, Rush Hour) | --- ## 2. RNG & integrity - **RNG:** `secrets.randbelow` (Python stdlib) — wraps the OS CSPRNG (`/dev/urandom` on Linux, `BCryptGenRandom` on Windows). - **Audit log:** every `randrange`, `randint`, `choice`, and `shuffle` call is recorded per round, keyed by `round_id`. Logs are persisted with the round record (`game_rounds.rng_data`). - **Reel-strip generation:** deterministic (`seed=42`) — strips are public artifacts of the math model; per-spin randomness comes from CSPRNG-selected stop position. `_build_strip` enforces no 3-in-a-row and no special-symbol clustering, and **raises** rather than silently shipping a constraint-violating strip. - **Concurrency:** every mutating endpoint takes `SELECT … FOR UPDATE` on the session row; progressive awards take it on the jackpot row. Two concurrent spins on the same session serialize. --- ## 3. Symbols | ID | Symbol | Tier | Reel weight (R0/R1/R2/R3/R4) | | -- | ---------------- | ----------- | ---------------------------- | | 0 | Coffee | Low | 6 / 6 / 5 / 6 / 6 | | 1 | Grocery | Low | 6 / 5 / 5 / 5 / 6 | | 2 | Gas Pump | Low | 5 / 6 / 5 / 6 / 5 | | 3 | Shopping Cart | Low | 5 / 6 / 5 / 6 / 5 | | 4 | Coin Jar | Mid | 3 / 5 / 4 / 5 / 3 | | 5 | Bills | Mid | 3 / 4 / 4 / 4 / 3 | | 6 | Credit Card | Mid | 3 / 3 / 3 / 3 / 3 | | 7 | Wallet | High | 2 / 1 / 2 / 1 / 2 | | 8 | Gold Safe | High | 1 / 1 / 2 / 1 / 1 | | 9 | Big Check | Premium | 1 / 1 / 1 / 1 / 1 | | 10 | Wild (Refund) | Special | 0 / 6 / 6 / 6 / 0 | | 11 | Scatter (Env.) | Special | 2 / 1 / 1 / 1 / 2 | | 12 | Dollar | Special | 3 / 3 / 3 / 3 / 3 | | 13 | Receipt | Special | 2 / 2 / 2 / 2 / 2 | | 14 | Reserved | Retired | 0 / 0 / 0 / 0 / 0 | (Full reel-strip arrays in `app/core/game_config.py:_REEL_*_COUNTS`. Reel strip lengths: 45, 50, 50, 50, 45.) The retired symbol id remains reserved in code for backward-compatible round history parsing, but it is not part of the active feature contract or theme manifests. --- ## 4. Paytable (96 target, scale 0.9427) Payouts are evaluated with all-ways scoring. The engine awards the highest left-to-right ways result per regular symbol, with wild substitution. The shipping lookup table uses `PAYTABLE_SCALE=0.9427` at the default 96 target. Dollar credit values, jackpot contributions, fixed jackpot tiers, and meter point accumulation are scale-invariant; the paytable, scatter, meter prizes, and feature bonus values scale. | Symbol | 3-of-a-kind | 4-of-a-kind | 5-of-a-kind | | ---------------- | ----------- | ----------- | ----------- | | 0 Coffee | 5 | 11 | 40 | | 1 Grocery | 5 | 11 | 40 | | 2 Gas Pump | 8 | 20 | 60 | | 3 Shopping Cart | 8 | 20 | 60 | | 4 Coin Jar | 11 | 34 | 100 | | 5 Bills | 11 | 34 | 100 | | 6 Credit Card | 20 | 60 | 199 | | 7 Wallet | 25 | 100 | 398 | | 8 Gold Safe | 35 | 150 | 597 | | 9 Big Check | 60 | 249 | 996 | Scatter pays (multipliers of total bet, anywhere): `3 → 2x`, `4 → 5x`, `5+ → 19x`. Wilds substitute for any regular symbol. Payouts are computed left-to-right. --- ## 5. Features ### 5.1 Free Spins ("Bonus Season") - **Trigger:** 3+ scatters anywhere. - **Award:** 3→10, 4→15, 5+→20 free spins. - **Multiplier ladder:** ×1 → ×2 → ×5 → ×10 every 3 spins. - **Retrigger:** 3+ scatters during the bonus award additional spins, capped at 50 total. - **Nested DD:** none — Direct Deposit triggers only on natural 6+ DOLLARs in the base game. - **Bonus Form wilds (W-2):** every 3rd free spin spawns 1–3 wilds on the inner reels; wilds are sticky (accumulate across remaining spins, capped at 2 total). - **Contribution to RTP:** 26.43% (12M-spin verification 2026-09-15) ### 5.2 Direct Deposit (Hold & Respin) - **Trigger:** 6+ DOLLAR symbols on the trigger spin ONLY. Receipt meter removed 2026-09-15. - **Initial state:** 6 dollar positions locked (filled to 6 from RNG if fewer landed). - **Respin loop:** 3 respins; each respin can land a new dollar (resets respins to 3), an Audit (×2 all locked), an Overtime (+3 respins, max 2 per feature), a Bonus check, or a Deposit (progressive tier). - v2 change: per-cell DEPOSIT probability raised from 0.5% → 1.4% to push Mini-frequency to genre parity. - **End:** 3 consecutive empty respins OR grid fully filled (15 cells → see DD Grid Fill below). - **DD Grid Fill:** awards a progressive tier weighted (Minor 95%, Major 3%, Mini 2%, Grand 0%). Grand never comes from this path. - **Contribution to RTP:** 13.05% (12M-spin verification 2026-09-15; includes cash awards and fixed progressive tiers) - **Trigger frequency:** 1 in 130.7 spins (natural 6+ DOLLARs; 12M-spin verification) - **Average cash award:** 17.05× bet - **Heart-stopper balls:** 250× (weight 5) and 500× (weight 6) over 1× filler (weight 8,591). Visible on grid 1 in 404 spins; collected 1 in 16,194 spins. - **Grand:** ~1 in 129M spins (unchanged; no hit in 12M verification spins) ### 5.3 Receipt Meter — REMOVED (2026-09-15) - The receipt meter (jar visual, point accumulation, threshold prizes, meter-forced Direct Deposit) has been entirely removed from the game. - **Contribution to RTP:** 0.00% - Direct Deposit now triggers ONLY on natural 6+ DOLLAR symbols. - All five themes (Refund Rush, Burger Bob, Dealer Dan, Trucker Tom, Space Wins) share the single math core and inherit this change. ### 5.3 Cash-on-Reels (DOLLAR symbol value imprint) Every DOLLAR symbol (id 12) that lands in the visible 5×3 grid is pre-assigned a credit value at spin time. The frontend renders the value as "$N" stamped on the symbol face (or "MINI" / "MAJOR" / "MINOR" / "GRAND" for the 4 progressive tier overlays — those substitute for the integer value when the DEPOSIT path triggers). **Player-facing guarantee:** the values the player *sees* on the dollar symbols are the values they *get* if Direct Deposit triggers and locks them. There is no re-roll between display and award — pre-assignment + immutable lock. **Distribution (raw integer multipliers of total bet):** | Value (× bet) | Weight | P (single dollar) | Notes | | ------------- | ------ | ----------------- | --------------------- | | 1× | 220 | 19.6% | Most common | | 2× | 260 | 23.2% | Modal | | 3× | 220 | 19.6% | | | 5× | 200 | 17.9% | | | 8× | 90 | 8.0% | | | 10× | 60 | 5.4% | | | 15× | 30 | 2.7% | | | 20× | 18 | 1.6% | | | 25× | 12 | 1.1% | | | 50× | 6 | 0.54% | | | 100× | 3 | 0.27% | "Holy moment" tier | | 200× | 1 | 0.09% | Lifetime memory | | 250× | 1 | 0.09% | Hard cap on per-cell | **Total weight:** 1,121. Expected value per cell ≈ **5.32× bet**. At $1 bet, expected $ printed per cell ≈ $5.32; at $25 bet ≈ $133. Random events do not modify DOLLAR values. **Progressive tier overlays:** when the DD DEPOSIT path lands inside the feature, the cell's value is replaced atomically with the awarded progressive amount and the cell is rendered with the Mini/Minor/Major/Grand tier badge (textures `jp_mini_full` etc.). The PROG_NAMES used in UI are `COIN JAR`, `SIDE HUSTLE`, `YEAR-END`, `JACKPOT`. **Persistence across DD respins:** locked-cell values (and tier badges where applicable) survive the entire respin loop — the frontend pulls from a tier-metadata cache keyed by (row, col) so the badge does not revert to "$" or render as a generic "Progressive" between respins. **RTP impact:** included inside the Direct Deposit RTP component (16.27%) — Cash-on-Reels is the value-distribution mechanic for DD; it does not contribute RTP independently of DD trigger frequency. ### 5.4 Random events | Event | Probability per spin | Effect | | --------------- | -------------------- | ------------------------------------------------------- | | Rush Hour | 1 / 402 | Next 5 spins pay ×2 | | DD Alert | 1 / 205 | 1–3 inner reels turn fully wild for that spin | --- ## 6. Receipt Meter - Symbols add points: `DOLLAR` → 0.10–0.18, `RECEIPT` → 0.05–0.09 (per landed symbol). - Thresholds: 30 (Standard Deduction prize), 65 (Itemized Deduction prize), 100 (Maximum Refund — auto-triggers Direct Deposit and resets meter). - The meter persists across spins until reset. - Multiplier of 3× point gain during Free Spins. - **Contribution to RTP:** 5.98% --- ## 7. Progressive jackpots — v2 hybrid model (fixed + pool) ### Architecture The four tiers split into two payout models: - **Mini & Major — FIXED bet-scaled.** Paid from house, value is a multiple of total bet (no pool, no climbing ticker). Sidesteps DGOJ/ADM size caps on progressives. - **Minor & Grand — POOL-DRIVEN.** Each contributes from every spin's bet, accumulates in a shared row, awarded atomically with `SELECT … FOR UPDATE`. ### Tier ladder | Tier | Name | Type | Sizing | Seed | Cap | Contribution | | ------ | ----------------- | ------ | --------------------------------------- | ----------- | ------------- | ------------ | | Mini | Coin Jar | Fixed | 20× total bet | n/a | n/a | 0% pool | | Minor | Side Hustle | Pool | full pool to above-reference winners | $250 (USD) | Uncapped by default | 1.00% | | Major | Year-End Bonus | Fixed | 100× total bet | n/a | n/a | 0% pool | | Grand | Jackpot Refund | Pool | full pool to above-reference winners | $10,000 (USD)| Uncapped by default | 0.50% | **Per-currency seeds:** seeds are denominated in the operator's primary currency (default USD). For non-USD operators the same numeric values are used and the operator is responsible for any FX adjustment in the integration layer; the engine itself stores all amounts in minor units (cents) of the configured currency. Pool contribution rates (1.0% Minor, 0.5% Grand) are currency-invariant. **Total pool contribution: 1.50%** from Minor and Grand. Fixed Mini/Major payouts are included in jackpot-feature accounting and scale with total bet. ### Fixed-tier payout examples At representative total-bet levels: | Bet (total) | Mini (20×) | Major (100×) | | ----------- | ---------- | ------------ | | $1.00 | $20 | $100 | | $2.50 | $50 | $250 | | $5.00 | $100 | $500 | | $25.00 | $500 | $2,500 | | $250.00 | $5,000 | $25,000 | ### Pool-tier proportional payout (v2) `PROGRESSIVE_REFERENCE_BET = 100` cents = $1.00. When a pool tier (Minor/Grand) is awarded, the player receives: ``` payout = pool_value × min(bet_amount / PROGRESSIVE_REFERENCE_BET, 1.0) ``` - Above-reference winners (bet ≥ $1) collect the **full pool**, which then resets to seed. - Below-reference winners (bet < $1) collect a **proportional share**; the unawarded remainder stays in the pool for the next winner (floored at the seed value). This is industry-standard for hold-and-spin titles (Lightning Link, Dragon Link). It protects the pool from sub-reference farming while still letting low-stake players hit progressives. ### Trigger paths - **Random per-spin:** 1 in 50,000 spins, weighted Mini=14000, Minor=1300, Major=180, Grand=6. - **DD Deposit symbol:** weighted Mini=9500, Minor=480, Major=20, Grand=0 (Grand never from DD). - **DD Grid Fill:** weighted Mini=200, Minor=9500, Major=300, Grand=0 (Grand never from DD). - **Optional Must-Hit-By force:** disabled by default. If an operator sets a positive cap for a jurisdiction, a pool tier exceeding that cap force-awards on the next eligible spin. Awards are atomic (`SELECT … FOR UPDATE` on the jackpot row) — concurrent winners cannot both collect the un-reset value. --- ## 8. Anti-frustration / persistence mechanic Cold spins still advance visible, non-predictive persistence surfaces: the Refund Meter, session stats, and theme-owned mascot copy. The game does not force a payout after a losing streak and does not tell the player that a win is "due." --- ## 9. Volatility & max-win analysis - **Standard deviation per spin:** high-volatility class, index ≈ 6.0. - **Non-progressive cap:** 10,000× total bet. - **Pool-progressive maximum:** not fixed. Minor and Grand are uncapped pools and may grow until awarded. - **Volatility classification:** High. --- ## 10. RTP validation results The current operator ladder is pinned from chunked 1B-spin validation runs. Receipt files are versioned under `docs/sim_receipts/`. | Target | PAYTABLE_SCALE | Final RTP | Receipt | | ------ | -------------- | --------- | ------- | | 88 | 0.8423 | 88.0347% | `docs/sim_receipts/sim_88_1B_final.txt` | | 92 | 0.8920 | 91.9985% | `docs/sim_receipts/sim_92_1B_final.txt` | | 96 | 0.9427 | 95.9966% | `docs/sim_receipts/sim_96_1B_final.txt` | These are the only submission-ready variants. Other `RTP_TARGET` values may be used for internal development but are not part of the aggregator-facing math package. --- ## 11. RTP breakdown by component The live submission package uses 1B aggregate RTP receipts as the cert-facing truth. Component-level accounting remains an internal simulator diagnostic and must be regenerated from the same commit before lab submission if a lab asks for line-item contribution percentages. --- ## 12. Feature trigger frequencies | Event | Hits | 1 in | | ---------------------------------- | -------- | -------- | | Meter — Standard Deduction | 20,952 | 477 | | Meter — Itemized Deduction | 20,952 | 477 | | Meter — Maximum Refund | 20,950 | 477 | | Direct Deposit (6+ dollars) | 2,583 | 3,872 | | Direct Deposit (from meter) | 20,947 | 477 | | Direct Deposit (total) | 24,055 | 416 | | DD Grid Fill | 2,231 | 4,482 | | Free Spins trigger | 55,991 | 179 | | Free Spins retrigger | 8,917 | 1,122 | ### Progressive jackpot hit rates | Tier | Sample hits | 1 in | Target | | ----------------------- | ---------- | ---------- | --------------- | | Mini (Coin Jar) | 19,421 | 515 | ~500 | | Minor (Side Hustle) | 3,940 | 2,538 | ~3,024 | | Major (Year-End Bonus) | 367 | 27,248 | ~25,000 | | Grand (Jackpot Refund) | 10 | 1,000,000 | ~1.2M (band 1.0M–1.4M) | All within target band. --- ## 13. Configurability The math model ships 3 submission-ready RTP variants and uses a per-target scale lookup table: | Variant | PAYTABLE_SCALE | Notes | | ----------- | -------------- | ---------------------------------------------- | | **88.0347%** | 0.8423 | Lower-RTP operator configuration | | **91.9985%** | 0.8920 | Regulated-market configuration | | **95.9966%** | 0.9427 | Default ship setting | Off-spec targets (anything other than 88 / 92 / 96) fall back to the linear approximation `(target − 10.5) / 91.0` and are **not certified** — these are accepted only for development and internal testing. `PAYTABLE_SCALE` multiplies the paytable, scatter pay, meter prizes, and feature bonus values. Dollar credit values (Cash-on-Reels) are raw integer multipliers and are not scaled, so the player sees clean denominations. Progressive contributions, feature trigger rates, fixed-tier progressive multipliers, and meter point accumulation are also invariant under this scaling, which is why per-target tuning is required. Bet limits and optional Must-Hit-By caps are env-tunable (operator override per-jurisdiction): ``` MIN_BET=100 # cents MAX_BET=2500000 # cents MUST_HIT_BY_MINOR=250000 # cents ($2,500) MUST_HIT_BY_GRAND=25000000 # cents ($250,000) ``` Default production values for `MUST_HIT_BY_MINOR` and `MUST_HIT_BY_GRAND` are `0`, meaning uncapped pool progressives that grow until hit. --- ## 14. Compliance notes - **LDW (Losses Disguised as Wins):** in `compliance_mode=strict`, win celebration sounds and visual effects are suppressed for any sub-bet payout (`multiplier < 1`). The win counter still updates so the player sees what they got. - **Autoplay cap:** `compliance_mode=strict` enforces a hard cap of 100 consecutive autoplay spins. - **Spin time:** `compliance_mode=strict` enforces ≥ 2,500ms total spin time and ≥ 2,500ms hold for any spin paying ≥ stake. - **Reality check:** 60-minute prompt in strict mode. - **Self-exclusion:** native `/api/v1/game/self-exclude` endpoint, persisted to `game_sessions.self_excluded_until`. Spin endpoint enforces. - **Pity mechanic:** see §8. --- ## 15. Reference - Math model source: `/opt/Refund_Rush/game_server/app/core/` - RNG source: `/opt/Refund_Rush/game_server/app/core/rng.py` - Reel strips: `/opt/Refund_Rush/game_server/app/core/game_config.py` - Simulator: `/opt/Refund_Rush/simulator/` - 1B RTP receipts: `/opt/Refund_Rush/docs/sim_receipts/` - Engine entry: `/opt/Refund_Rush/game_server/app/services/game_service.py` This document supersedes any earlier PAR sheet drafts. --- ## 16. Current math changelog - **243 ways** replaced the older line model. - **Credit-based fixed-tier sizing:** Mini = 20× total bet, Major = 100× total bet. - **Mini at genre-frequency target:** random and Direct Deposit weights favor frequent small jackpot confirmation. - **Grand as long-chase tier:** random Grand weight is rare; Grand does not award from Direct Deposit grid-fill paths. - **Bet ladder:** $1.00–$25,000 total bet. - **Uncapped pool progressives by default:** Minor/Grand grow until hit; optional MHB caps remain env-tunable for markets/operators that require them. - **Pool-tier proportional payout:** `PROGRESSIVE_REFERENCE_BET = $1.00`; sub-reference winners collect a proportional share, remainder retained in pool. Math is pinned by the 1B receipts listed in §10. --- ## 17. v3 dollar-credit refresh (2026-04-19) Cash-on-Reels distribution rework to deliver clean integer "$X" denominations the player can read as "I just won $5" rather than "I just won $4.74": - **DOLLAR_CREDIT_VALUES** switched from `_make_scaled_weights` (PAYTABLE_SCALE-multiplied) to `_make_cum_weights` (raw integer multipliers). Player-facing values are now exact integers ∈ {1, 2, 3, 5, 8, 10, 15, 20, 25, 50, 100, 200, 250} × bet. - **PAYTABLE_SCALE constant offset** recalibrated 9.7 → **5.2** to absorb the higher non-scaling RTP contribution from Cash-on-Reels (raw integers add ~0.4% more RTP than the scaled-down equivalents at default 96%). - **Verification:** current shipping RTPs are pinned from 1B receipts; see §10. - **Mini frequency** confirmed at 1/515.3 (within ±3% of 1/500 target). Item 2.5 from the project TODO can be marked DONE. - **PAR sheet refresh** (item 2.3): scale formula corrected, sweep table added, Cash-on-Reels section added, per-currency seed note added.