# UD minimal pre-event response source theorem v0.1

14 October 2026 UTC. Development — no registry entry. Constructor: Codex.
Status: EXACT CONDITIONAL FINITE ALGEBRA / NATIVE SOURCE OPEN. Physical promotion: 0.
Independent adversarial proof review of this new result remains pending.

## 1. Source and domain

Source anchors: C0963, `declared_source_variables`, `exact_response_form`, `minimum_missing_object`; C0957, theorem and saturated-prior corollary; C0884 §§2, 7, 9; October 13 synthetic polarity audit E04 §§2–5. These source-text snapshots are preserved in the v0.3 dependency register with hashes. The attached v0.39 ledger, Q165–Q173, supplies fixed-carrier support and flux, not a response source law. The attached governance CRL-2 subdivision v1.0 is a proposal; its development lane and promotion distinctions are retained. None of these imported labels establishes independent review.

On three pair addresses fix binary prospective integrity T and latent-presence L, constructed independently before endpoint exposure. Define omega=T AND L and I=T AND NOT L. A response q=1−Theta is admitted iff omega≤q≤T coordinatewise. I is the intact-null potential mask; h is the actual idle-open response mask. They are not the same object. Prospective T and L are source outputs from a frozen pre-event mechanism, not retrospectively inferred post-event measurements.

## 2. Exact response-fiber theorem

Every admitted q has a unique representation q=omega+h with binary h≤I. Conversely every such h yields an admitted q. Proof: omega≤q makes h=q−omega nonnegative; q≤T restricts h to T=1,L=0; omega and h have disjoint support. The reverse follows coordinatewise.

Write i=|I|. The response fiber has exactly 2^i masks. With independently sourced idle count j, its cardinality is binomial(i,j), 0≤j≤i. For i=0,1,2,3 the full cardinalities are 1,2,4,8. There are 64 possible (T,L) pairs. Their i histogram is (27,27,9,1), by (3+x)^3. The sum of response fiber sizes is 27+54+36+8=125, matching the prior admissible-state count. This is a finite-domain enumeration, not a measured frequency distribution.

## 3. Minimal budget and address information

Let the admitted prior state be transported into the current pair frame. Write q−=omega−+h− and q+=omega++h+. For a supplied prospective omega+ and independently sourced j=|h+|,

muTheta=|q−|−|omega+|−j=ind Fomega+|h−|−j.

Thus the possible budgets form the contiguous integer interval [|q−|−|omega+|−i, |q−|−|omega+|]. Every integer occurs, with binomial(i,j) response addresses. A budget receipt needs j, rather than the whole idle mask. An address receipt generally needs the actual h+; j alone leaves binomial(i,j) possibilities. The exact worst-case distinguishability capacities are log2(i+1) bits for the count and i bits for the full mask; a fixed-length binary encoding needs ceil(log2(i+1)) bits for the count. These are finite labeling requirements, not physical entropy costs.

For i>0 both h=000 and h=I are admissible and permutation-equivariant. They share the same (T,L) but differ in response budget by i. Therefore admission and pair relabeling do not uniquely select POC-1 or another response law. This underidentification statement is on the declared binary domain; it does not assert that both alternatives are physically realized by one native apparatus.

## 4. Restricted symmetry theorem

For a deterministic selector using only (T,L), equivariance under pair permutations requires its output to be fixed by every permutation stabilizing (T,L). All i intact-null addresses have the same (T,L)=(1,0), so their permutations lie in that stabilizer. The selected idle mask must be constant across those addresses: h=0 or h=I. For i≥2, such a reduced deterministic selector cannot choose an intermediate count 0<j<i. The branch between empty and full may depend on the invariant profile; symmetry does not select it. Additional pre-event state, burdens, history or an admitted tie-break can break this stabilizer, but must be separately sourced. This theorem does not cover a selector using that additional information.

A stochastic kernel invariant under the same intact-null stabilizer has one probability p_j per cardinality orbit j=0,...,i. Every individual mask of size j has probability p_j/binomial(i,j); p_j≥0 and sum p_j=1. Conversely these formulas define every invariant kernel. There are i unconstrained probability degrees of freedom. Symmetry alone does not determine them or native clock rates. Conditioning on a supplied j gives a uniform invariant distribution on that orbit; it does not derive which j occurs.

## 5. Pre-event source contract

A complete prospective packet must declare: complete pre-state and its update index; old Theta,T,L; old-to-current frame permutation and source receipt; prospective output T,L generated from that pre-state; actual h or the count j with declared address policy; the frozen source construction, version/hash, event/null behavior, collision/tie behavior, feedback and renewal semantics; input seal, prediction freeze and endpoint-release chronology. Prior Theta and omega must both be transported before directed-address readout.

Net-budget-only admission can omit h addresses if it independently sources j. It cannot then claim a unique directed address except when binomial(i,j)=1. Full lineage provenance still additionally needs continuation/predecessor, merger and source/renewal records. A finite index difference does not supply these records.

The included reference receipt validator covers the full-mask route. It checks bits, prior admissibility, permutation convention, residual support, source-artifact SHA-256, binding of receipt fields, ordinal index and declared custody chronology. It rejects endpoint exposure, inconsistent source bytes and missing native program declarations. It does not execute a native law, authenticate timestamps, prove causal independence, establish artifact truth or certify empirical scoring eligibility. A structurally complete native-candidate packet still returns NATIVE_VALIDATION_PENDING__NO_SCORE.

No supplied specimen is native data. The example is explicitly synthetic: a declared pre-event construction supplies all masks and a candidate h=0. Another lawful h can give a different output on the same reduced variables. This tests the conditional readout, not model selection or a frozen POC-1 sidecar.

## 6. What closes and what remains open

Closed in the development lane: finite response parameterization, exact ambiguity/cardinality, count-versus-address distinction and restricted symmetry classification. The source obligation is reduced to independently deriving prospective T,L and an idle-response law/kernel on I, plus the frame and event/null chronology.

OPEN: native viability generation; native idle-response selection or probability weights; full pre-state and renewal law; native frame receipt; source-authenticated timing/custody; independent adversarial review; lawful target-hidden physical episode and scorer custody. Physical promotion: 0. The original primary sidecar and its missing-byte boundary are unchanged.

The user's “reviewed; continue next steps” acknowledges the prior presentation. It is not recorded as a theorem-specific adversarial finding, custodial promotion or independent empirical result. No original scientific registry status is changed.