In this article
- A failed attempt can contain real partial work
- The controlled-prefix structure
- The recovered instrument
- Why the full inverse matters
- A simple partial-gate example
- What recovery costs
- Why this is useful for a dual-time program
Undoing rejected partial work
A failed completion attempt can be coherently undone by reversing its actual source-controlled work prefix. The recovery restores an unknown payload under explicit inverse-access, memory, and control assumptions.
01A failed attempt can contain real partial work
A completion attempt may stop after changing the payload but before reaching its accepted region. If the payload is unknown, simply resetting the apparatus or repeating the final operation can corrupt it. The draft constructs a coherent recovery that reverses the actual source-dependent partial work on the failure branch.
This gives a concrete operational meaning to unfinished inner generation in SMM/DTT. A failed attempt need not become a public completed record, but its physical effects still exist and must be accounted for. Recovery is a dynamical operation with resources, not a deletion of an inconvenient branch.
02The controlled-prefix structure
Let G be the known unitary describing partial work as a function of source state. After an attempt the joint state has the form G(|ξs⟩C⊗|ψ⟩D). The acceptance projector P acts on the source, commutes with G, and selects a region where G equals the completed payload operation V.
Coherently copy the acceptance bit into a fresh flag. Apply G† only when that flag indicates failure. The successful branch retains V|ψ⟩, while the failed branch loses its partial work and returns to |ψ⟩. The spent source states remain as accounted auxiliary output.
P|ξs⟩⊗V|ψ⟩⊗|1⟩+(I−P)|ξs⟩⊗|ψ⟩⊗|0⟩
03The recovered instrument
The successful and failed source components are orthogonal. Tracing them therefore yields a flagged channel with accepted payload VρV† and failed payload exactly ρ. The success weight p is the norm squared of P|ξs⟩.
This identity holds for mixed inputs and for payloads entangled with a passive reference. The failed operation is the identity channel on the payload, not merely a restoration of one selected probability distribution. Entangled-state failure tomography would therefore be a stringent operational test.
R(ρ)=p|1⟩⟨1|⊗VρV†+(1−p)|0⟩⟨0|⊗ρ
04Why the full inverse matters
Applying V† on failure generally does not work because the failed branch has not applied V. It may contain a coherent superposition of several different prefixes. The correct inverse is G†, controlled on the actual source-dependent prefix structure.
The recovery also has to occur before an irreversible work readout discards relevant coherence. If an inaccessible environment has recorded intermediate information, the inverse on the retained apparatus need not restore the unknown input. An arbitrary heralded channel with the same successful branch need not admit this recovery.
05A simple partial-gate example
Suppose a source has two rejected positions: at one the payload is unchanged, and at the other a gate U has been applied. A coherent rejected source can therefore be correlated with |ψ⟩ and U|ψ⟩. Applying a fixed U† to the payload repairs the second component but changes the first to U†|ψ⟩.
A source-controlled inverse applies identity at the first position and U† at the second, restoring |ψ⟩ in both. This elementary example captures the general prefix recovery. It also shows why source memory and coherent control are essential rather than optional implementation details.
06What recovery costs
The enlarged device requires a fresh flag, access to the prefix inverse, coherent conditional control, retained source information, and a later export or readout. Failed source carriers are not magically restored to their original preparation. They remain spent and cannot be reused as fresh attempts without an additional reset process.
If several attempts are compiled into an autonomous Hamiltonian, the compilation must include these gates and their memory. The source may be autonomous after initialization while still requiring a substantial prepared resource. Autonomy does not mean that preparation, memory capacity, and energy costs vanish.
07Why this is useful for a dual-time program
Recoverable failure separates logical completion from physical activity. After several failed attempts and one success, the final payload can equal one application of V, while the apparatus has performed much more work. A net activity record that is uncomputed during recovery does not measure the full elapsed effort.
SMM/DTT should therefore distinguish successful generative activity, actual device runtime, and later outer reading. The recovery theorem makes reliable finite completion possible under explicit assumptions. It does not derive a universal clock law or permit failed work to be excluded from the physical resource budget.
KEEP THE CONNECTION
Key ideas
- Unknown-input retrying needs recovery of the failure channel.
- The source-dependent inverse prefix restores the payload, including reference entanglement.
- Spent sources, flags, coherent control, and runtime remain physical resources.
Scope of the argument. Recoverability theorem for a coherent factorized implementation with inverse access; it does not apply to arbitrary heralded channels.
Sources & further reading
- quantum_relativity_temporal_generation_v10.pdfWorking draft, pp. 26–29
Source editions and catalogue entries
Working manuscript
Quantum relativity and temporal generation — working draft v10
Working manuscript. Article-level references identify the consulted sections and the scope of the interpretation. No public URL was supplied for this exact manuscript version.
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