Accept rule and the game
Accept rule
Stage 1 then Stage 2 on the same signed epoch. A definition, not a theorem.
- An epoch is accepted only when both stages pass on that same signed epoch.
- Stage 1 is this room’s posterior. Quality gate and matching gate.
- No printed assumption is attached to the accept rule. α does not enter it.
DTS-Forge
The faker wins if a forged payload is accepted and written into the epoch root.
- Advantage is the chance the faker wins that game.
- For n rooms: every room must clear its own posterior and its own five classes.
- Success at one room is not granted at another.
Assumptions
A1–A5. They sit on the theorems and the conjecture, not on the accept rule.
- Hash hardness, per-room Stage 1 parameter α, and no shared cheat across rooms.
- These ifs attach to the many-room product and to nothing else.
- By themselves they accept nothing.
Bayesian stack · Theorem A · Five AND Conjunction · Theorem B
Bayesian stack
Per-room posterior. Independent Stage 1 tests. Not the n-room bound.
- Stage 1 is this room’s posterior. Quality gate and matching gate.
- Each node runs stacked Bayesian tests. The recipe is omitted.
- The n-room product is Theorem A. It is the next box.
Theorem A
Many rooms. Joint Stage 1 at most αⁿ under printed ifs.
- If each room’s Stage 1 chance is at most α and the rooms do not share one cheat, n rooms is at most αⁿ.
- That is the bound on the Bayesian stack. It is not the accept rule.
- That bound is not the Gardner Conjecture.
Five AND Conjunction
Physical AND cryptographic AND logistical AND statistical AND temporal.
- Stage 2 is the five-class conjunction on the same signed epoch.
- A miss on any one class is not an accept.
- Stage 2 alone is not the accept rule.
Theorem B
Modeled full sum under printed ifs. Does not enlarge Theorem A.
- The leftover term is query-bounded. Stage 2 leftover is in the sum.
- α is a parameter. This is a statement under those ifs.
- It is not a proof of the Gardner Conjecture.
Open claim, lemmas, and layers
Gardner Conjecture
Open. Theorem A and Theorem B do not close it.
- The claim is that advantage tends to vanish as hash strength, time depth, and honest rooms grow.
- It remains an open conjecture in the field.
- The printed lemmas do not upgrade it to a theorem.
Four lemmas
English in the paper. Lean source is not in the paper (available on request).
- Five-class AND. Rooms stack to αⁿ. Hash / replay. Fixed-guess distinguishability.
- Each is given in English first and by file second.
- Lean source is not in the paper (available on request).
Layers
Physics, information, crypto, logistics, Bayesian stack, time.
- The accept rule is not a sixth layer in that table.
- The Bayesian stack is the dominant term at large n.
- Reed 2ⁿ is economic only.
Instantiation and evidence
Instantiation
May 2026 production entropy validator. Not the live Sentry pin.
- Stacked Bayesian inference over camera, mic, motion, and cross-modal checks.
- Exact likelihoods and the accept threshold are omitted. Trade secret.
- The live Sentry deployment is a separate system.
Evidence
26 May Demonstrated, one node, Stage 1. 11 Sep lab experiment, Stage 2.
- 26 May 2026: five scripted forgeries rejected, genuine accepted. Stage 1 only.
- 11 September 2026: 47 cases, 45 held, 2 controls. Lab experiment. Not Demonstrated.
- Neither result proves the Gardner Conjecture.