R12 Secret-Shared Causal Bootstrap No-Go
Status: rejected at invention gate 4. No CPU falsifier, neural fit, or GPU experiment is authorized from this construction.
1. Candidate
The proposed curriculum tried to make a persistent state path compulsory by
splitting a target across causally separated views. For a finite group G,
sample a uniform pad U independently of target Y and reveal
V = U^{-1} Y
only after the mechanism has committed a state from U and the source has
been deleted. The intended extension used a sequence of shares and a running
group product. Episode-private relabelings, conjugations, and hidden-state
interchanges were proposed to prevent a fixed local classifier from passing.
The construction does make memory causally necessary. It does not make reasoning necessary and it does not identify a new mechanism.
2. Tight information theorem
Let |G| = m. For every prior on Y, a uniform independent U gives
I(Y; U) = 0
I(Y; V) = 0
H(Y | U,V) = 0
Y = U V.
The first independence is immediate. For the second, for every y,v,
P(V=v | Y=y) = P(U = y v^{-1}) = 1/m.
The best single-share accuracy is max_y P(Y=y), not 1/m unless Y is
uniform.
Suppose a sequential mechanism reads U, commits state S, loses access to
U, then reads V and must return Y. Zero error on every pair requires at
least m distinguishable states. If u != u' produced the same state, then
the decoder given any fixed v would have to return both uv and u'v,
which differ by cancellation. Therefore
|S| >= m
B >= ceil(log2 m).
For uniform Y and error at most epsilon, Fano's inequality yields the tight
lower bound
I(U; S) >= log2(m) - h2(epsilon) - epsilon log2(m-1).
Retaining S=U and returning SV attains the exact bound. For an ordered
sequence of shares whose product is Y, a running product uses log2(m)
state bits independent of sequence length, while any T-1 shares reveal no
information about a uniform target.
This is a valid one-way communication and streaming-memory theorem. It is not a computational reasoning theorem.
3. Transition-mask variant also collapses
The less direct variant masked transitions rather than the final answer. Let
semantic state evolve as x_t = a_t x_(t-1), sample independent uniform pads
r_t, and expose
c_t = r_t a_t r_(t-1)^{-1}
z_t = r_t x_t.
Then the masked state obeys the ordinary recurrence
z_t = c_t z_(t-1).
For fixed semantic values, the map from (r_(t-1), r_t) to
(z_(t-1), c_t) is a bijection. Thus the previous masked state and current
masked transition are independent and uniform. A reset or state-free updater
cannot recover z_t above chance; an accurate updater must carry the same
Fano-bounded state information.
However, this is a time-dependent change of coordinates, or gauge transform, of the original group action. It is conjugate to the same residual automaton. The exact collapse test succeeds: the proposal is ordinary recurrence in masked coordinates.
4. Fatal non-identifiability
4.1 The answer-share task is decryption
V=U^{-1}Y contains a one-time-padded target. Combining the shares recovers an
encoded answer; it does not infer an answer from independent axioms or facts.
A matched model that receives both shares is exact with one group product.
4.2 The representation is not identified
For every bijection phi:G->G, the pair
S = phi(U)
D(S,V) = phi^{-1}(S) V
has identical behavior. Causal success therefore cannot select a privileged latent algebra, coordinate system, or semantic state.
4.3 Masked success gives no unmasked-transfer theorem
If masked and unmasked inputs are distinguishable, two models can agree on every masked training episode and behave arbitrarily differently on every unmasked input. No amount of masked accuracy or state mediation removes that extension ambiguity. Adding unmasked examples makes transfer an ordinary curriculum problem rather than a theorem.
4.4 Relabeling does not repair the gap
An arbitrary unseen episode relabeling is unidentifiable without a supplied operation table or demonstrations. Supplying that support changes the problem to episode-level task inference or meta-learning. Consistent group conjugation is an automorphism; in abelian groups it changes nothing, and in nonabelian groups canonical recovery requires either the conjugating element or another ordinary deconjugation step.
4.5 Serialization can reintroduce shortcuts
The secrecy statement covers the mathematical shares only. Prompt length, format, group choice, output frequency, mask reuse, RNG coupling, and query metadata can leak the target. Every serialized benchmark would require a separate whole-view audit.
5. Equivalence and prior-art boundary
- Two shares are perfect secret sharing / a group one-time pad in Shannon's perfect-secrecy framework.
- A running partial product is the minimal
m-state Cayley automaton and ordinary recurrence. - Hidden-state counterfactual swaps are interchange intervention training.
- Episode relabeling with support is meta-learning or task adaptation.
- Masked-to-unmasked staging is curriculum or transfer learning.
Relevant primary sources:
- C. E. Shannon, Communication Theory of Secrecy Systems (1949): https://onlinelibrary.wiley.com/doi/10.1002/j.1538-7305.1949.tb00928.x
- Geiger et al., Inducing Causal Structure for Interpretable Neural Networks (ICML 2022): https://proceedings.mlr.press/v162/geiger22a.html
- Finn, Abbeel, and Levine, Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks (ICML 2017): https://proceedings.mlr.press/v70/finn17a.html
The useful project-level delta is only a balanced causal-memory diagnostic: it can certify that a retained channel carries required source information. It cannot certify deduction, transferable axioms, intelligent context compression, or a new state primitive.
6. Decision
Reject Secret-Shared Causal Bootstrap and its transition-mask variant as R12 mechanisms. The exact collapse test already resolves the question, so a CPU neural falsifier would only demonstrate that a recurrent model can learn group multiplication. Preserve the theorem as a future causal-memory control, but do not train Shohin on it and do not claim masked decryption as reasoning.