Chicken Step Games: Conditional Risk, Rounded Payouts and a Toy Verifier

A step game asks whether to continue from a completed position or take the current offered payout. That differs from watching a shared flight, but a chicken theme does not identify one probability law or verifier. Chicken Train, Crossfire Chicken X5000 and Chicken Crash are separate products. We retain the useful conditional-step mathematics, correct the payout ladder and provide a reproducible synthetic ledger. No live-player ranking, decompilation or calibrated provider experiment is claimed.

Separate the named products

100HP's official catalogue identifies Chicken Train as a railway-crossing title. JackTop's provider portfolio separately lists Crossfire Chicken X5000. Galaxsys advertises Chicken Crash with 98% RTP and configurable difficulty. These primary descriptions already contradict treating the genre as one universally fixed 97% engine. They do not establish the selected operator configuration, every lane's probability, a common hash mapping or a particular disconnect remedy. Read the actual title, mode and installed rules before comparing decisions or payments.

Conditional survival comes first

Let q_i be the conditional probability of surviving step i given the surviving history, selected action and relevant state. By the probability chain rule, surviving n steps has probability P_n equal to the product of those conditional q_i values. This product does not require unconditional independence. If the steps are independently identical with constant survival q, it simplifies to q^n. A different difficulty, revealed state or action can change a conditional probability, so constant risk is an additional assumption rather than a consequence of the game format.

For an adaptive policy, combine the probabilities of compatible histories; do not multiply values taken from different state paths as if they described one path.

The Markov property allows transitions to depend on the current state and action. In symbols, a transition conditioned on the full history can reduce to P(S_next | S_current, A_current). It does not reduce automatically to the unconditional P(S_next), and it does not prove identical lane risks. A lane-history theory needs evidence under the actual rules. We have not established that every lane is identical or that observing a shared board always conveys zero information. A repeatable input pattern alone is not evidence of a predictive edge.

A complete educational ladder

Choose an independent toy step-survival probability q=0.822, so hazard is 0.178. Take rho=0.97 as an entry-return factor. Before rounding, set the gross payout M_n=rho/q^n for each positive target step. Then P_n M_n=rho and net expectation of a unit entry stake is -0.03. Step zero is separate: not entering preserves the unit and does not incur that model edge. No extra stake is charged on each intermediate step in this illustration.

At step ten, q^10 is approximately 0.140837457, not 0.1390. The ideal multiplier is about 6.887372, not 6.98. The table's actual toy payout floors the ideal multiplier to cents, so step ten pays 6.88 and its expected gross is about 0.968961706. Other steps also differ slightly from exactly 0.97 after rounding. The 97% factor is therefore not silently reapplied as a new loss at every lane, and the rounded right column is not incorrectly labelled invariant.

Decide one more step using the current offer

In the unrounded ladder, once step n has been reached, the conditional expected gross of advancing once is q M_(n+1)=M_n. That is a comparison with the current offered cashout, not a new unconditional edge calculation from entry. Losing the next step forfeits the current offer, so the distribution changes even when that conditional mean is equal. Personal limits can justify stopping without a claim that the next step is mathematically guaranteed to lose.

Rounding can make that local comparison slightly positive or negative. In our cent-floor ladder, step four offers 2.12 and step five 2.58 per original unit; 0.822×2.58-2.12=+0.00076. On an original stake of 10, the conditional expected difference is +0.0076. Both fixed exit targets still have negative expectation from entry. A tiny local rounding difference does not remove the entry cost, establish a provider exploit or support the earlier claim of a universal negative continuation penalty.

Reproduce a funded synthetic session set

Generate 10000 independent toy rounds with public seed 20261024. Each round has ten underlying Bernoulli opportunities, all generated in advance; the achieved prefix stops at the first failure. Four policies exit at steps 1, 3, 5 or 10. They share each underlying round, pay one unit once on entry and use the cent-floor ladder. Each policy starts with 10000 units, sufficient to fund all 10000 entries even on all-loss outcomes. There are no deposits, fees, progressions or additional stakes between steps.

The table is this seeded sample, not actual 100HP, JackTop or Galaxsys telemetry. Every payout and final cash value reconciles with integer-cent accounting. An independent implementation regenerated all 10000 bit strings and checked 40000 settlement rows. Maximum loss runs were also checked by grouping outcomes. Realised profit is not forced to equal the theoretical edge times realised turnover. Observing a longest loss run of 53 in this sample is not a guarantee or a forecast of another file.

Verification needs an actual mapped protocol

A SHA-256 hex digest has 64 hexadecimal characters. Four hex characters encode 16 bits; eight encode 32 bits. A 32-byte digest contains eight complete four-byte words, not unlimited step values. A real verifier must define byte order, input encoding, indexing, additional blocks, rejection rules and how a number becomes a hazard or action outcome. A seed commitment match alone does not compare the displayed board. Our existing generic crash verifier does not implement the three named chicken games.

For an explicitly separate educational example, the downloadable reference uses HMAC-SHA256 with the UTF-8 message client:nonce:block. It reads eight unsigned big-endian 32-bit words per digest and increments the block counter for more. To avoid modulo bias when mapping to 500 values, reject words at or above floor(2^32/500)×500, then reduce accepted words modulo 500. Values 0 through 410 survive, and 411 through 499 fail. This is our specified toy mapping, not a reverse-engineered commercial algorithm.

The public synthetic vector uses server crashmath_chicken_demo_server, client crashmath_chicken_demo_client and nonce 7. Its ten mapped values require two digest blocks. The first value 434 fails, so a playable prefix would end immediately; later values test the stream mapping rather than assert a chicken traversed ten steps. Python and an independent Node implementation agree on full digests, commitment and every mapped value. Boundary rejection and invalid nonce cases were checked separately.

Replay the data and the toy comparison

Download step_models.py and the synthetic step paths. Save them beside this Python 3 block. It regenerates the paths, checks the stored records and compares both a matching and deliberately changed mapped result. A missing expected result remains not compared. The example makes no casino connection, uses no private seed and places no wager. The published commitment is an illustrative expected value, not proof of an actual earlier casino publication.

from step_models import simulate, paytable, educational_sequence
from pathlib import Path
import csv
import json
import logging

data = simulate()
with Path("chicken_step_paths.csv").open(encoding="utf-8",newline="") as stream:
    stored = [{"round":int(row["round"]),"bits":row["bits"],
               "reached":int(row["reached"])} for row in csv.DictReader(stream)]
assert stored == data["paths"]
vector = educational_sequence("crashmath_chicken_demo_server",
                              "crashmath_chicken_demo_client",7)
expected_commitment = "216c239b0afaf4372316c213f0e809efccdd6787557dfc62f0b3a42a54ea4d09"
assert vector["commitment"] == expected_commitment
recorded = [434, 227, 263, 27, 88, 143, 314, 372, 239, 188]
changed = list(recorded)
changed[0] = (changed[0]+1)%500
checks = []
for claim in (recorded,changed,None):
    status = "not-compared" if claim is None else (
        "match" if claim == vector["values"] else "mismatch")
    checks.append(status)
assert checks == ["match","mismatch","not-compared"]
logging.basicConfig(level=logging.INFO,format="%(message)s")
logging.info(json.dumps({"paytable":paytable(),
    "syntheticResults":data["results"],"toyVector":vector,"checks":checks}))

Compare the decision space without universal claims

A step ladder can offer fewer exit amounts than a fine-grained target field, but quantisation alone does not prove higher variance. For a binary gross payout g with success probability p, variance per unit is p(1-p)g² regardless of theme. Compare the actual available goals and probabilities. A game that safely waits at a completed step may reduce urgency there; it does not establish immunity to network failure, accepted-move delay or automatic progression in another mode.

Confirm whether the chosen title offers different actions, difficulty levels, multiple bets, a timer or a disconnect rule. Observations of another player's board may or may not refer to the same hidden state. No guarantee of equal lanes, no adaptive behaviour or an earlier complete board commitment is inferred merely from a fairness label. Our separate Aviator protocol is also preserved rather than replaced with this toy HMAC stream.

You may choose an affordable stake, a preferred exit or no further step. There is no mandatory 1% stake, fixed training session or extra deposit in this guide. The EV calculator and direct partner registration remain optional routes. Use recorded accepted actions and actual rules to evaluate a product, keeping a model calculation separate from a verified provider claim.

Primary product descriptions checked 24 October 2026: 100HP catalogue, JackTop portfolio, Galaxsys Chicken Crash.