Structure before circuit tricks

Recoverable history on reachable states

When can a computation recover its branch decisions without a full history stack?

b=\operatorname{bit}_{j-1}(g_v^\prime q^{-1}\bmod 2^j)

What this technique preserves

Recover a branch only after a global reachable-state invariant makes the post-step state distinguish the alternatives. Compile and clear the decoder separately.

Hypotheses and hidden contracts

Pinned initialization and exact signed recurrence
Full invariant and decoder dependencies; odd modulus and valuation conditions are not disposable
Physical register refinement, phase/cleanup and actual gate costs remain separate

Mathematical proof mechanism

This is an authored reusable derivation guide, not a claim that the full family has been source-assimilated.

Audit Appendix A and existing upstream proofs, preserving their actual hypotheses.
Check invariant preservation and low-bit decoding as distinct suppliers.
Connect the decoded bit to inverse control and ancilla cleanup in the actual circuit; then count costs.

Exact Lean substrates

No local transport theorem is bound to this record. Do not infer formal truth from its position in the atlas.

Do not cross this boundary

Proposed local intake; existing upstream proof names are not new ASPBE certificates, and recurrence correctness is not a whole quantum circuit theorem.

Reusable technical lemma card

tl-reachable-branch-recovery obligation

Planned / uncertified technical lemma. This card is a retrieval obligation, not a local Lean certificate or a completed source route.

Use a proved reachable-state invariant to recover a prior branch from the updated state; supply the gate-level inverse/cleanup bridge separately.

Exact card declarations

No local transport theorem is bound to this record. Do not infer formal truth from its position in the atlas.

Planned consumers (not verified reuse)

Uncompute: digesting discovered reversible circuits

Required mechanisms

Finite matrices, norms and registers Charged access and finite-precision compilation

Failure modes

Applying the decoder to arbitrary unreachable states.
Reading a branch bit correctly but failing to clear its temporary register.
Counting external Lean proofs as local compilation or whole-circuit resource closure.

Next bounded action: Audit full upstream Recovery.lean and its imports, then compile an isolated Lean4.33.0 adaptation with explicit provenance before local admission.

uncomputereachable invariantsource-first

Card source ledger

Discovering state-of-the-art quantum circuits with large language models

primary-text-checked v1: design principles; Methods Eqs. (10)-(22); Appendix A, A.1-A.8 (PDF pp. 44-46). Other case studies remain a queued source audit.

Priority intake for reversible computation and proof digestion. First batch audits existing branch-recovery and DQI proof interfaces; full paper and circuit/resource closure remain open.

uncompute: pinned author circuit, evaluator and Lean artifacts

primary-text-checked Commit 64b5a13aa249f1360846657392e6c66817b26d0f; lean/divsteps/README.md, lean/dqi/README.md; selected Recovery.lean and Complete.lean theorem interfaces; not yet a whole-module audit or local build.

Observed pins: divsteps Lean/Mathlib4.32.2; DQI Lean/Mathlib4.33.1. Apache-2.0 repository; divsteps also retains MIT safegcd-bounds attribution. Upstream source and claimed builds are not ASPBE acceptance.

Download technical card JSON

Related transports

Source and prior-art ledger

Discovering state-of-the-art quantum circuits with large language models

primary-text-checked v1: design principles; Methods Eqs. (10)-(22); Appendix A, A.1-A.8 (PDF pp. 44-46). Other case studies remain a queued source audit.

Priority intake for reversible computation and proof digestion. First batch audits existing branch-recovery and DQI proof interfaces; full paper and circuit/resource closure remain open.

uncompute: pinned author circuit, evaluator and Lean artifacts

Copy mathematical mechanism as LaTeX

% Authored mechanism lesson; not a new theorem certificate.
\section*{Recoverable history on reachable states}
When can a computation recover its branch decisions without a full history stack?
\[
b=\operatorname{bit}_{j-1}(g_v^\prime q^{-1}\bmod 2^j)
\]
Recover a branch only after a global reachable-state invariant makes the post-step state distinguish the alternatives. Compile and clear the decoder separately.
\paragraph{Hypotheses and contracts.}
\begin{enumerate}
\item Pinned initialization and exact signed recurrence
\item Full invariant and decoder dependencies; odd modulus and valuation conditions are not disposable
\item Physical register refinement, phase/cleanup and actual gate costs remain separate
\end{enumerate}
\paragraph{Mathematical proof mechanism.}
This is a reusable derivation guide; exact certified scope is given by the linked Lean signatures.
\begin{enumerate}
\item Audit Appendix A and existing upstream proofs, preserving their actual hypotheses.
\item Check invariant preservation and low-bit decoding as distinct suppliers.
\item Connect the decoded bit to inverse control and ancilla cleanup in the actual circuit; then count costs.
\end{enumerate}
\paragraph{Boundary.} Proposed local intake; existing upstream proof names are not new ASPBE certificates, and recurrence correctness is not a whole quantum circuit theorem.

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