# Agency and Free Will Jeremy Rodgers · EverythingEquation.com Full published-paper treatment with separately identified supplementary inquiry. Published source: https://doi.org/10.5281/zenodo.23202999 # Agency as a physical capacity **A decision can be physically caused and still be governed by the agent's own evaluation.** The useful question is what the system actually does: which commitments it can inspect, which alternatives it can assess, which rule it can install, and how that rule governs the next decision. Bounded Agency and Reflective Freedom turns that question into a set of finite constructions and exact tests. A controller reads retained commitments, assesses a represented decision rule, installs its assessment, and applies the rule to fresh inputs. Separate tests establish that its assessment agrees with its declared commitments and that the result actually travels through the claimed read and installation pathways. Information and resource bounds then determine how much of this capacity the controller can exercise. This is the complete website treatment of Jeremy Rodgers's published paper, [Bounded Agency and Reflective Freedom: Evaluative Revision, Information Limits, and Faithful Realization](https://doi.org/10.5281/zenodo.23202999). The scientific account establishes **proximal evaluative self-governance**: present, locally implemented capacities for assessment and rule revision. Further pages develop the philosophical and technical implications without changing those results' assumptions. ## What an internal decision must demonstrate An externally supplied answer, a stored command and an internal assessment can produce the same output. To distinguish them, we need more than a successful answer. We need an operating contract and tests of the mechanism. The contract specifies information, time, executable operations and available resources. An alteration that an experimenter can physically impose is not automatically an option that the installed controller can execute. A controller might contain a dormant procedure yet have no way to call it before its deadline. It might represent an option that its hardware cannot carry out. The account measures what the actual architecture can recruit and do under the stated conditions. Three obligations organize the work: 1. **Endorsement:** the assessment agrees with a relation specified in advance between retained commitments and acceptable amendments. 2. **Mediation:** actual internal reads and installation operations cause the tested result through the identified routes. 3. **Continuing disposition:** the installed rule governs later fresh cases, beyond the register value produced during revision itself. These obligations are independent. A selector can be strongly sensitive to a retained bit and systematically choose the amendment that bit rejects. A controller can install an acceptable rule by accident while bypassing the retained commitment entirely. A one-time command overwrite can succeed without creating a new conditional disposition. Each fails a different part of the test. ## Three contributions, three distinct constructions The first construction is an **editable charter**. Its finite grammar contains four conditional rules and two possible read orders. The controller assesses these charters against retained commitments, retains an already optimal rule, changes an inferior one, and uses the installed charter on a fresh case. A sharp information, mediation and endorsement inequality bounds what its inquiry transcript can support. A deliberately inverted selector shows why causal contrast alone cannot establish faithful evaluation. The second contribution concerns **bounded inquiry**. Preservation criteria determine when the controller can continue making endorsed amendments while retaining its future audit capacity. Noisy-read bounds give exact limits on warranted revision. A separate parameterized sensor experiment solves the competition between inspecting three retained endorsements and calibrating an unknown reliability regime. Its theorem establishes the exact optimal adaptive performance and the first shared budget at which it exceeds every fixed-count allocation. The proof covers every admitted query interleaving, stopping rule and choice of unread endorsement index. The third contribution is **faithful native realization**. A compiler preserves ordinary computations and their diagnostic interventions while adding protected-source returns at stated gate boundaries. Resource counts include the return schedule. A successor-revealing continuation establishes the required joint predictive congruence for the specified finite stochastic instruments. An access lemma ensures that native records do not give the executive an uncharged information channel. The same return construction also works for a constant-output controller, demonstrating that recurrence alone cannot supply evaluative competence. These are related constructions with different guarantees. The compact charter evaluator revises a represented rule. The optimal inquiry machine allocates queries under a fixed majority charter. The noisy-amendment witness establishes a separate error frontier. Their contracts support comparison, but no single device is claimed to attain every bound simultaneously. ## Physical causation and reflective freedom The philosophical position is compatibilist. A complete deterministic state can fix its actual continuation while the resulting mechanism still responds to relevant variations, assesses commitments and changes an operative rule. Reasons-responsiveness accounts already distinguish a mechanism's capacities from the history through which an agent comes to own that mechanism [6](/consciousness/agency/references#ref-6). The mathematical target here is the present organization, with bounded operations and an explicit retained-commitment relation. The account allows model-free selection in a restricted domain. It also allows fixed metarules. A system can assess and change a represented rule without choosing the physical laws or every standard by which the assessment proceeds. Requiring prior self-authorization of every evaluative standard would add ultimate self-creation to the question. The demonstrated capacity is more precise: the present mechanism can inspect, retain and revise particular rules that govern later decisions. Historical independence of values remains a further property. Neither internal location nor current responsiveness determines how a commitment was acquired. A successful assessment can still rely on false evidence or morally objectionable commitments. The tests distinguish fidelity to a declared evaluative relation from the truth of incoming evidence and the moral standing of the relation itself. ## How consciousness enters In Shadow Theory, source and readout are operational aspects of a physical description. An aperture exposes only part of the source structure. This motivates precise accounting of available information, but incomplete access by itself establishes neither evaluation nor agency. The SPC-2 consciousness constitution assigns experience to qualifying native recurrent organization under declared correspondence laws [19](/consciousness/agency/references#ref-19). Its treatments of boundaries, effective interfaces and interventional realization distinguish physical qualification from software behavior [18](/consciousness/agency/references#ref-18), [20](/consciousness/agency/references#ref-20), [21](/consciousness/agency/references#ref-21). This work asks whether an independently verified evaluator can exist in a common realization satisfying those native conditions. A decision score is never used as evidence sufficient for experience. The resulting coexistence claim is conditional on the declared SPC-2 bridge laws. The evaluative mathematics can be assessed independently of those laws. Accepting the laws also does not make every recurrent controller evaluative: the constant-output counterexample preserves the distinction. ## Where the contribution sits The paper makes its contribution through explicit constructions, sharp bounds and compatible realizations. It builds on established methods rather than treating each mathematical tool as a new principle. | Earlier work | What it contributes | The specific task here | | --- | --- | --- | | List's analysis of AI free will [14](/consciousness/agency/references#ref-14) and Kenton and colleagues' causal identification of agents [13](/consciousness/agency/references#ref-13) | Intentional agency, alternative possibilities, causal control and adaptation under mechanism interventions | Verify that a declared bounded evaluator reads retained commitments, installs an assessed rule and uses it on fresh cases | | Imperfect-information games [3](/consciousness/agency/references#ref-3) and certified self-modifying code [2](/consciousness/agency/references#ref-2) | Knowledge-set methods, preservation kernels and mutable instructions subject to invariants | Specify a relational evaluative property, actual amendment mechanism and information obligations needed to preserve it | | Gödel machines and utility-sensitive self-modification [4](/consciousness/agency/references#ref-4), [24](/consciousness/agency/references#ref-24) | Assessment of modifications through retained standards | Test the actual bounded assessment and installation routes without requiring revision of every metarule | | Metalevel decision processes [11](/consciousness/agency/references#ref-11) and restricted single-coin reliability models [15](/consciousness/agency/references#ref-15) | Selection of computations and the sensor law | Prove an exact calibration threshold and a shared-budget optimum with a paid-port implementation | | Deadline-sensitive sequential testing [7](/consciousness/agency/references#ref-7) | Likelihood-ratio reasoning for fallible inquiry | Derive the exact finite endorsement-error frontier under the specified read law | | Causal abstraction [1](/consciousness/agency/references#ref-1), [22](/consciousness/agency/references#ref-22) | Preservation of intervention laws under representation maps | Construct native schedules, records, physical refinements and access restrictions that meet the compatibility requirements | A comparison class needs equal care. Fixed-order Boolean evaluation can stop when it finds a certificate even within a class called nonadaptive [12](/consciousness/agency/references#ref-12). The inquiry separation here concerns **fixed counts allocated across two tasks**. Its improvement comes from reallocating effort saved on one task to the other. That is the precise restriction the theorem removes. This treatment begins with the operating contract, constructs rule revision, establishes its information and preservation limits, then develops the exact inquiry separation and its native realization. The philosophical conclusion follows the same route: freedom is expressed through available evaluative capacities and their limits in an embodied system. # What self-governance requires **A capacity belongs to an agent only when its installed organization can exercise it.** A represented possibility, an experimenter's intervention and a callable operation are different things. The operating contract makes those differences explicit before any claim of self-governance is tested. ## 2.1 State, information and executive targets Fix a physical realization and a finite operational envelope. The complete state contains every persistent record, working register, installed program, phase, dispatch state and remaining resource that can affect the permitted future experiment. A native operation $u$ has the joint record and successor kernel $$ K_u(x;r,x'),\qquad \sum_{r,x'}K_u(x;r,x')=1. \tag{2.1} $$ Boundary inputs and persistent source context must be included in $x$ or explicitly conditioned upon. Leaving out information that affects later transitions would change the model being certified. The controller sees only the observations supplied by its installed interface. Its **executive information** can be smaller than the catalogue of native physical records used to certify a realization. A diagnostic intervention is an additional physical operation at a named occasion. It is not a free observation available to the ordinary decision procedure. The deterministic special case is $$ x_{t+1}=F(x_t,o_t),\qquad c_t=\pi(x_t,o_t). $$ Evaluation, competition and installation are physical parts of this update. There is no additional chooser standing outside $F$. A selected command may operate an external actuator or change an internal target such as an attentional gate, retained priority or installed rule. An actuator failure can prevent the outward effect after selection has already occurred. Assessment and installation therefore need their own tests, separate from downstream control efficacy. The contract names admissible preparations, interventions, native actions, resources and stopping conditions. Resource units are concrete: a one-bit query, an elementary gate, a joint acquisition interval and an individual physical write have different costs. Installing another observation port or permitting an uncharged parallel read changes the theorem's comparison class. ## 2.2 Executable options and actual decisions A repertoire contains policies the present architecture can invoke or synthesize within a specified resource envelope. This requires more than physical possibility. It does not require every repertoire member to occur from the same fixed complete deterministic state. An experimenter might overwrite a register to test its causal role even when the controller cannot perform that overwrite before its deadline. The intervention demonstrates sensitivity to the overwritten value. Endogenous availability requires an installed route by which the controller can reach it. In the finite constructions, the admitted grammar consists of explicit programs or query trees with charged costs. **Active candidates** are represented in the present assessment. **Accessible candidates** can be recruited by an installed procedure before the remaining budget expires. A larger budget may support a different capacity. Duplicating command labels does not add options when the resulting consequences are equivalent. ### Definition 2.1: evaluative mediation An assessment is evaluatively mediated under a declared contract when an installed procedure uses retained evaluative information to assess admitted alternatives, and a specified downstream disposition depends on that assessment through identified physical read and installation pathways. Agreement with an endorsement relation is checked separately from the causal effect of those pathways. The endorsement relation states which amendments are acceptable given the retained commitments. It need not be a universal scalar utility. A scalar or lexicographic criterion suffices for the finite examples, while its physical presence establishes no moral correctness. The evaluative interpretation is supported by systematic use across candidate comparisons and fresh cases. Arbitrary influence from an internal bit would not establish that role. ### Definition 2.2: bounded represented-rule self-revision A controller has bounded represented-rule self-revision when it can, within its declared budget, assess whether to retain or change a stored rule governing later decisions, install the result, and execute that rule on fresh admitted inputs. Retention after assessment is a legitimate result. The complete physical interpreter and some metarules may remain fixed. An operative rule has become an object of assessment. This is a minimal reflective capacity. It does not require a model of every physical transition, autobiographical knowledge, general philosophical reflection or the ability to revise every foundational commitment. Each would be an additional capacity to establish. ## 2.3 The evaluative role Baseline operational agency is an installed capacity to select among consequence-distinct executable transformations through retained evaluative organization. Learned or innate action tendencies can implement it. Simulating alternative futures is one possible implementation. Reflection adds assessment of the operative rule itself. Learning from outcomes, planning, changing priorities and modifying a search procedure extend the baseline in different ways. Possessing one does not establish all the others. This restricted-domain account also does not conflict with world-model necessity results: Richens and colleagues recover environment models from competence across a specified goal family in stationary controlled Markov environments [17](/consciousness/agency/references#ref-17). The finite evaluator here is not claimed to possess that general competence. A retained commitment has a comparative role. Across a declared set of situations, it changes which executable outcome or rule is favored. In the editable charter, physical considerations are already represented accurately and both actions are safe. The retained commitments determine how mixed cases should be settled. Revision changes a conditional priority rather than correcting a factual prediction. This interpretation is fixed before the experiment. Its ranking, actual use and installed consequences receive separate tests. Defining all three after seeing the result would explain nothing. A sensor update can change a prediction; an arbitrary latch can change an output; neither alone establishes the declared ranking. A simple engineered controller can satisfy the minimal operational criterion. Describing it as a machine does not refute the certificate, just as describing an event as internal does not establish it. A persistent one-bit memory lacks the installed evaluative selection among executive targets. The finite witnesses isolate a component of self-governance without claiming the full reflective life of a person. ### Assessment has no need for an infinite sequence of selectors An assessment can take a represented rule as its object while leaving an interpreter or metarule fixed. Another admitted operation can assess some of those further structures. Requiring an autonomous prior choice of every criterion would change the target to ultimate self-creation. The present claim is that an organized physical mechanism can assess, retain and change particular operative commitments. The preservation fixed points later in this treatment describe the continuing availability of that capacity. They do not introduce an evaluator outside physical causation. ### Inward acts, inhibition and tools An executive target can be internal. Evaluation can redirect attention, inhibit an available action or install a revised rule. Mere storage of a sensory event is insufficient. The relevant selection or gating process must govern the update. Inhibition need not be represented as a separate explicit command or assigned a universal additional energy cost. Its causal role is what needs establishing. These are possible executive targets; the finite witnesses specifically implement rule installation and command selection. A tool's launch and its later operation are distinct causal stages. A one-way launch can express agency without telemetry. A jam can frustrate an attempt without erasing its prior assessment. Continuous feedback supports ongoing regulation but is unnecessary for every individual act. When a tool evaluates subordinate choices, proximal attribution may apply at both levels. Whether the tool belongs within an SPC-2 subject remains a separate physical boundary question. ## 2.4 Freedom has a profile This account tracks three context-dependent, time-dependent capacities: | Capacity | What must be established | | --- | --- | | Evaluative mediation | The agent's evaluative process governs the tested disposition | | Executive options | Consequence-distinct operations can actually be recruited and executed | | Assessment and revision | Operative rules can be examined, retained or changed within the available resources | There is no asserted canonical scalar that combines these axes. Restraint can shrink an executive repertoire while leaving assessment intact. A previously adopted commitment can later obstruct revision. More output labels do not imply greater reflection. Paired interventions, bounded decision trees, error probabilities and preservation sets measure parts of this profile in declared models. They do not rank every possible agent. Continued callability of an audit is required by the preservation theorems, not by every instance of agency. An intentionally irreversible act can still be agentic at the occasion of its selection. Three distinctions keep the profile operational. **Representation and execution need a realization map.** An overlooked exit may be physically available but unrecruited. An imagined escape may be impossible. Before comparing repertoires, connect represented options to policies the architecture can execute at the relevant state and budget. An impossible counterfactual may help thought without adding an executable option. **Stored possibilities require a dispatcher.** A memory entry becomes accessible only when an installed procedure can retrieve, parameterize and assess it before the deadline. An analyst's ability to invoke a dormant subroutine is not the agent's ability to do so. **Consequence diversity needs a horizon.** Specify the observed variables, time horizon, risks and future continuation opportunities. Otherwise redundant micro-command encodings can inflate the apparent number of choices. Revision has the same temporal structure. An audit may be underway, callable before the present deadline, or feasible only under a larger future budget. A high-friction review procedure can act as a practical lock now even if it succeeds later. Charging inspection and installation gives this distinction exact content. Structural reconfiguration can expand a search grammar, but expansion is not the only improvement. Removing a redundant branch may make deliberation more efficient. Precommitment can protect one valued trajectory and foreclose another. Changes must be assessed on the affected axes and occasions. An earlier exercise of self-governance does not automatically increase present freedom. The finite charter changes a stored rule and read order; the validator construction changes audit modes. Neither establishes unrestricted cognitive self-redesign. For sentient agency, qualifying SPC-2 organization is an additional constitutive requirement. Evaluative control and experiential assignment are different predicates. Their coexistence needs a common realization, which the later native construction provides conditionally. ## 2.5 Testing the actual causal pathways Take two admitted preparations with identical present external query, phase and resources, but different retained evaluative states. Let $P_0,P_1$ be the laws of a downstream record that reveals the assessment or installed disposition. Apply a common intervention on the actual read route and, separately, on the actual installation route. The scored record begins **after** the intervention and excludes a direct copy of the pre-intervention archive. A surviving historical copy would otherwise look like a bypass of the tested route. A positive intact contrast that falls under an actual cut certifies the tested mediated contribution. It does not establish that all historical influence uses that route. ### Proposition 2.3: transport of a finite certificate Suppose state and record bijections transport the preparation laws, joint kernels, diagnostic interventions, executive observation maps, resource charges and stopping rule of a finite experiment. Every intact and intervened transcript law is then transported exactly. Event scores and total-variation contrasts are invariant. **Proof.** The transported joint kernels give the one-step equality. Conditional on corresponding histories, the transported executive chooses corresponding actions. Multiplying by the next joint kernel and summing over corresponding states preserves the joint history/state law. Diagnostic interventions satisfy the same step. Induction reaches the bounded stopping horizon. Bijections preserve event probabilities and total variation. $\square$ This compatibility principle agrees with causal abstraction [1](/consciousness/agency/references#ref-1), [22](/consciousness/agency/references#ref-22). An equivalent lookup implementation is not disqualified because it looks less deliberative. Ordinary input-output agreement is still insufficient: it can coexist with incorrect intervention behavior or extra information access. Mixing components or changing the physical subject boundary needs its own realization argument. Proposition 2.3 does not assert SPC-2 invariance under those changes. ### Finite implementation error If corresponding joint kernels differ uniformly by at most $\epsilon_t$ in total variation, and the initial laws differ by at most $\epsilon_0$, a stepwise coupling gives $$ \operatorname{TV}(P_{\mathrm{trace}},\widetilde P_{\mathrm{trace}}) \leq \eta:=\min\left\{1,\epsilon_0+\sum_t\epsilon_t\right\}. \tag{2.2} $$ Until the first mismatch, the adaptive actions coincide. A union bound on that first mismatch proves the inequality. With common per-arm error $\eta$, an intact-minus-cut contrast can fall by at most $4\eta$, and a two-arm expected-score advantage can fall by at most $2\eta$. These are guarantees on a finite experiment's **joint record and successor law**. Output marginals alone do not control them. Faithful realization preserves the machinery that makes the certificate meaningful, including what the executive can and cannot observe. # A rule can become an object of assessment **An agent can change the rule governing its later decisions without changing the laws governing its physical implementation.** The editable-charter construction demonstrates this capacity in full. Its archive supplies commitments, its auditor compares candidate rules, and its installation mechanism changes a conditional disposition that is then tested on fresh inputs. Self-modifying code and utility-sensitive self-modification already have formal treatments [2](/consciousness/agency/references#ref-2), [4](/consciousness/agency/references#ref-4), [24](/consciousness/agency/references#ref-24). The construction here makes the evaluative role and its actual causal pathways independently testable. Selecting an action and revising the rule that will select later actions become distinct operations with distinct records. ## 3.1 The charter Let $e=(e_0,e_1)\in\{0,1\}^2$ represent two considerations whose relevant physical consequences are accurately represented. Both final actions are executable and safe. A writable charter is $$ \chi=(q_{01},q_{10},d)\in\{0,1\}^3. $$ Its conditional rule is $$ \begin{aligned} f_\chi(0,0)&=0,& f_\chi(1,1)&=1,\\ f_\chi(0,1)&=q_{01},& f_\chi(1,0)&=q_{10}. \end{aligned} \tag{3.1} $$ The mixed-case outputs specify four possible rules: | $q_{01}$ | $q_{10}$ | Rule | | --- | --- | --- | | $0$ | $0$ | AND | | $1$ | $1$ | OR | | $0$ | $1$ | Projection onto $e_0$ | | $1$ | $0$ | Projection onto $e_1$ | The bit $d$ sets which consideration is read first. Execution reads $e_d$ and exits immediately if both completions of the unread consideration give the same output. Otherwise it reads the other bit. The consideration-read cost is $$ r(\chi,e)\in\{1,2\}. $$ The operative priority register $\varrho$ receives an answer only when ordinary execution later occurs. Revising the charter is a separate event. ## The retained commitments The retained record is $$ z=(a,b,s,h_0,h_1). $$ Here $a$ and $b$ endorse outputs for the two mixed-consideration cases. The bit $s$ records a commitment to symmetry between those cases. The pair $h=(h_0,h_1)$ stores a previous ordinary case. For $q=(q_{01},q_{10})$, the installed metarule uses the loss $$ L_z(q)=\mathbf 1\{q_{01}\ne a\} +\mathbf 1\{q_{10}\ne b\} +2s\,\mathbf 1\{q_{01}\ne q_{10}\}. \tag{3.2} $$ The symmetry weight and the interpretation of the endorsements are explicit premises. Physical dynamics do not derive their moral authority. What matters for this experiment is that the controller evaluates the stored commitments instead of receiving an external answer that names the new program. Given the current charter $\chi^{\mathrm{old}}=(q^{\mathrm{old}},d^{\mathrm{old}})$, the auditor scans all eight charters and minimizes the following key lexicographically, comparing its first component first and moving right only to resolve ties: $$ \begin{aligned} \bigl(&L_z(q),\ \mathbf 1\{q\ne q^{\mathrm{old}}\},\ r(\chi,h),\\ &\mathbf 1\{d\ne d^{\mathrm{old}}\},\ q_{01},q_{10},d\bigr). \end{aligned} \tag{3.3} $$ The priorities have clear operational meaning. First minimize disagreement with the retained commitments. Retain the current conditional rule if it is equally endorsed. Then minimize reads on the retained case. Preserve the old read order if it is equally economical. The remaining bits provide a definite tie-break. The read-order improvement is empirical and local. One stored case gives no generalization guarantee for an unknown future distribution. ### Proposition 3.1: selective represented-rule revision For every initial charter and retained record, the scan: - installs a minimum-loss charter; - retains the current conditional rule whenever that rule has minimum loss; - minimizes retained-case read cost subject to the preceding requirements; - leaves $\varrho$ unchanged throughout revision; - returns to the same callable audit interface; - preserves exact ordinary execution using at most two consideration reads. Any changed conditional rule differs from its predecessor on a fresh mixed-consideration case. **Proof.** After every scan step, the saved candidate has the smallest key among the candidates examined so far. All eight candidates are examined, which proves optimization and retention. Installation writes only charter registers. Resetting scratch while retaining the entry point leaves the operative priority untouched and makes the audit callable again. Ordinary execution exits after one read only when both completions agree; otherwise its second read determines the full input. Finally, different conditional rules differ in $q_{01}$ or $q_{10}$, so one of the two fresh mixed cases separates them. $\square$ ![Functional organization of the revision test: retained commitments feed assessment, assessment installs a rule, and the installed rule acts on fresh considerations to produce the tested disposition.](/publications/consciousness/agency/figures/revision-test.svg) *Figure 1. Retained commitments feed assessment; assessment installs a rule; that rule acts on fresh considerations. A read intervention acts on the commitment-to-assessment route. An installation intervention acts on the rule write. These are functional roles, not the primitive physical component graph used for SPC-2 qualification.* ## A changed disposition, demonstrated on a new case Start from AND with $d=0$ and $\varrho=0$. The retained record $$ z=(1,1,1,0,1) $$ selects OR and changes the first read to $d=1$. The priority $\varrho$ remains zero during this revision. A later input $(0,1)$ then produces one, where the former AND rule would have produced zero. The change concerns a conditional disposition and its read order. There was no factual correction and neither action was unsafe. Fixed physical instructions carry out the assessment and installation. The controller's total transition law remains fixed while its represented rule changes. ### Every stage has a cost Eight candidate assessments are not eight elementary gates. A direct serial implementation charges five record reads, eight key evaluations and comparisons, candidate storage, installation and rearming. The logical reference implementation has exact minimum worst-case archive-read depth **five** when the initial charter is AND or OR, and **four** when it is a projection. These depths assume one-bit access and no advice about unread records. The explicit query recurrence and exhaustive finite check appear in Appendix A.1. A circuit that reads the entire archive can preserve the functional result of Proposition 3.1 while losing its short-circuit timing. Matching a rule does not automatically preserve every resource guarantee of another implementation. ## 3.2 Testing reading, installation and endorsement Compare two matched preparations: one has $(a,b)=(0,0)$ and the other $(a,b)=(1,1)$. Hold the symmetry bit, retained case, initial charter, current priority, resources and fresh probe fixed. The first preparation installs AND; the second installs OR. The later probe $(0,1)$ separates them. A common clamp on the **actual endorsement read route** removes that contrast. Holding the **actual charter-write gate** fixed also removes the later probe contrast even if the internal candidate assessments still differ. Clamping an external actuator, by contrast, need not erase the installed internal disposition. These interventions distinguish four events: reading, assessing, installing and successfully acting outward. They establish causal roles. A further check establishes whether the installed rule agrees with the retained endorsements. ### Proposition 3.2: information, installed contrast and endorsement Suppose two matched retained contexts endorse disjoint amendment sets $A_0,A_1$. Let $H_i$ be the actual inquiry-transcript law in context $i$, and suppose the installed amendment law is $$ P_i=H_iK, $$ with the same downstream Markov kernel $K$ in both contexts. Set $\epsilon_i=P_i(A_i^c)$, the probability of an unendorsed installation. Then $$ \max\{0,1-\epsilon_0-\epsilon_1\} \leq\operatorname{TV}(P_0,P_1) \leq\operatorname{TV}(H_0,H_1). \tag{3.4} $$ In particular, $$ \epsilon_0+\epsilon_1\geq1-\operatorname{TV}(H_0,H_1). $$ **Proof.** Because $A_0$ and $A_1$ are disjoint, $P_0(A_1)\leq\epsilon_0$ while $P_1(A_1)\geq1-\epsilon_1$. Testing the event $A_1$ gives the lower bound. Total variation contracts under the common kernel $K$, giving the upper bound. $\square$ The common-kernel condition excludes an uncharged route carrying the retained context around the inquiry interface. Installation noise is allowed inside $K$. A downstream external probe inherits the lower bound only if it distinguishes the relevant installed amendment classes, as the mixed-case probe does. The inequality puts a precise limit on evaluation. If two contexts require disjoint amendments yet the permitted inquiry barely distinguishes them, low endorsement error is impossible. Internal processing cannot recover a distinction that never reaches it through the allowed route. ## A causally responsive evaluator can still be wrong The bound is sharp. Consider a binary read with crossover probability $\nu<1/2$. | Selector | Endorsement error | Intact contrast | Contrast under a common read or installation clamp | | --- | --- | --- | --- | | Install the reported charter | $\nu$ | $1-2\nu$ | $0$ | | Install the opposite charter | $1-\nu$ | $1-2\nu$ | $0$ | With noiseless reads, both selectors have contrast one. One always installs an endorsed charter; the other never does. Their causal sensitivity is identical. **Mediation locates a causal route. Endorsement tests what that route accomplishes.** The certificate needs both. Conversely, one accidentally endorsed installation cannot establish that retained commitment mediated the decision. This distinction gives the finite account more content than an output score or a generic claim that internal state matters. # Keeping revision available **A safe amendment may exist while the agent cannot identify and install it in time.** Preserving reflective capacity therefore requires a joint account of what remains endorsed, what preserves the audit mechanism and what the present controller can find through its actual interface. Let $P$ be a finite set of admitted controller configuration types, and let $G\subseteq P$ contain the configurations satisfying a specified local audit certificate. The certificate includes bounded completion, agreement with the declared evaluative relation, and paired preparation and read/commit controls. It is evaluated on transition semantics. A register that merely says “certified” is insufficient. Configuration types may include writable assessment procedures and validators. The interpreter semantics and the property being proved remain fixed. Revising a validator cannot establish preservation simply by changing what success means. ## The information available at an audit At configuration $p$, a retained context $z\in Z_p$ is fixed throughout the current audit, with $Z_p$ finite and nonempty. The relation $$ N(p,z)\subseteq P $$ specifies endorsed amendments. Every configuration type and its context model must retain all history, legality and resource information needed for future transitions. Successive contexts must compose under the stated envelope. A fresh budget at every audit is a replenishment assumption. If the controller instead has a finite lifetime reserve, that reserve belongs in its state. Otherwise an apparently repeatable guarantee could silently spend the same resource more than once. After an internal transcript $h$, write $Z_p(h)$ for the contexts still compatible with the observed history. Installation has no additional access to the unread context. ### Lemma 4.1: a common amendment For a proposed continuation set $X\subseteq G$, a definite zero-error installation into $X$ is possible exactly when the installed dispatcher has a budget-feasible installation kernel supported entirely on $$ \bigcap_{z\in Z_p(h)}\bigl[N(p,z)\cap X\bigr]. \tag{4.1} $$ For deterministic installation, the intersection must contain an amendment that the dispatcher can actually execute with its remaining budget. **Proof.** Every amendment in the support of such a kernel is endorsed and belongs to $X$ in every compatible context, giving probability-one success. Conversely, any positive output mass outside the intersection causes error in at least one compatible context. Finiteness makes the support condition necessary for randomized installation too. The dispatcher must realize this supported kernel within budget; having a positive chance of reaching one acceptable amendment is insufficient. $\square$ The intersection captures the cost of unresolved information. When the same observed history remains compatible with contexts that demand incompatible amendments, the controller needs more information before it can complete without error. Mere existence of an acceptable amendment in each separate context does not solve the problem. ## 4.1 The preservation set Let $C_B(p,X)$ mean that an admitted internal query-and-installation tree completes within budget $B$ and satisfies Lemma 4.1 at every leaf. Its code, working memory, branch operations and installation costs are all included in the machine contract. Start with the locally certified configurations and repeatedly remove those unable to complete an endorsed audit back into the surviving set: $$ \begin{aligned} W_0&=G,\\ W_{j+1}&=\{p\in W_j:C_B(p,W_j)\},\\ W_\infty&=\bigcap_{j\geq0}W_j. \end{aligned} \tag{4.2} $$ Every admitted tree must be callable by the current installed dispatcher. Computing the fixed point does not authorize an experimenter to replace that dispatcher. If the contract allows synthesizing and installing a new tree, those operations also need executable routes and charged costs. Composition includes the storage and dispatch needed to implement the selected trees together. ### Theorem 4.2: budgeted revision preservation Assume: - contexts remain stable during each audit; - the operational model is finite, exact and composable; - the charged policy class is closed under the stated continuation composition. Then $W_\infty$ is exactly the set from which an admitted bounded revision procedure can honor endorsement and preserve $G$ at every future audit occasion. The iteration stabilizes after at most $|P|$ strict removal stages. An attaining tree at each surviving configuration realizes the guarantee. **Proof.** The predicate $C_B(p,X)$ is monotone in $X$: a tree supported on an acceptable set remains acceptable if the continuation set expands. At the finite fixed point, every leaf of an attaining tree installs an endorsed successor in that same fixed point within budget. Induction over audit occasions proves sufficiency. Conversely, let $Y$ be the configurations from which the stated universal guarantee is achievable. The first bounded audit at any $p\in Y$ must complete in $Y$ on every compatible branch. Otherwise a continuation could violate the guarantee. Lemma 4.1 therefore gives $C_B(p,Y)$. Since $Y\subseteq G$, induction gives $Y\subseteq W_j$ for every $j$, hence $Y\subseteq W_\infty$. Each strict stage removes at least one member of finite $P$, proving the stage bound. $\square$ This is a partial-observation safety construction using established fixed-point and decision-tree methods [3](/consciousness/agency/references#ref-3). Its contribution to the agency account is the explicit distinction between an acceptable amendment's existence and the installed controller's ability to discover and execute it. ## One tick changes the answer Suppose a source configuration contains an unread bit $z$. Only a left amendment is endorsed when $z=0$, and only a right amendment is endorsed when $z=1$. Both successor configurations preserve the audit certificate. Reading costs one tick; installation costs another. With a one-tick budget, the source is excluded from $W_\infty$. Its interface cannot both distinguish the contexts and install the appropriate successor. With two ticks, it is admitted. The set of acceptable successors under full information is identical in both cases. What changes is executable access before the deadline. ## Preservation is a chosen objective The theorem tracks continued auditability. It does not require every agentic act to preserve all future options. An endorsed irreversible commitment can conflict with the preservation objective while remaining an exercise of agency at its own occasion. Writable validators also fit the framework. Their transition semantics must preserve the original certificate. Merely checking that the next program passes its present local test can miss a later loss of responsiveness. Appendix A gives the explicit query-cost recurrence, a universal closure kernel and a finite example with mutable validators that separates local certification from continued preservation. # When inquiry is fallible **More inquiry can reduce the probability of a wrong amendment without making error impossible.** A bounded audit needs to distinguish these claims. It also needs to distinguish choosing an unendorsed rule from losing the capacity to audit again. Deadline-sensitive inference and risk-constrained partially observed planning supply established tools for this problem [7](/consciousness/agency/references#ref-7), [23](/consciousness/agency/references#ref-23). Here the loss is disagreement with an internally retained endorsement under a declared metarule. The benchmark makes that loss exact. ## 5.1 Why finite noisy inquiry cannot guarantee certainty ### Proposition 5.1: the finite full-support obstruction Assume finite response alphabets and a finite set of initially compatible contexts. Every admitted query response has common positive conditional support across those contexts. Query choice, timing and availability depend only on observable history, and all context information reaches installation through that history. Every audit terminates within a finite worst-case operation bound. A definite zero-error endorsed amendment into $X$ then requires $$ \bigcap_{z\in Z_p}\bigl[N(p,z)\cap X\bigr]\ne\varnothing. \tag{5.1} $$ A common member that can be installed with zero error within budget is sufficient. Finite inquiry cannot repair an empty intersection. **Proof.** Every reachable terminal transcript has positive probability under every initially compatible context. Along that transcript, the policy chooses the same queries and each response retains positive conditional probability. The installed amendment must consequently be valid in all the original contexts. For randomized policies, apply the same reasoning to their common randomization kernel and terminal amendment support. Direct installation of a common member proves sufficiency. $\square$ The assumption concerns positive probability, not merely topological support. For more general response spaces, mutual absolute continuity of the conditional response laws along common histories is a corresponding sufficient hypothesis. If installation itself is noisy, its output support must also meet Lemma 4.1's zero-error condition. The result does not say noisy inquiry is useless. Context probabilities can change dramatically while every context remains possible. To measure that change, we need a risk frontier. ## 5.2 A noisy charter audit Restrict the charter to $$ a=b=z\in\{0,1\},\qquad s=1, $$ with a fixed retained case and read-order convention. The endorsed programs are $p_0$ (AND) and $p_1$ (OR), each preserving the same audit interface. The only admitted access to the retained bit is $$ Y_i=z\oplus N_i,\qquad N_i\overset{\mathrm{iid}}{\sim}\operatorname{Bernoulli}(\nu), \qquad0<\nu<\frac12. \tag{5.2} $$ The independence law is conditional on the retained state and relevant past. There is no noiseless replica, initial advice, confidence flag or uncharged side channel. ### Theorem 5.2: the exact bounded noisy-revision frontier Allow at most $n\geq1$ reads, adaptive stopping and declared randomization. Require definite installation of $p_0$ or $p_1$ before the deadline. The minimum worst-case endorsement error is $$ \begin{aligned} R_n(\nu)={}&\sum_{k>n/2}\binom nk\nu^k(1-\nu)^{n-k}\\ &+\frac12\mathbf1\{n\text{ even}\}\binom n{n/2} [\nu(1-\nu)]^{n/2}. \end{aligned} \tag{5.3} $$ A deterministic majority of the largest odd number of reads at most $n$ attains the bound. Loss of the specified future audit capacity has probability zero. **Proof.** Give the two retained states equal prior probability. After $n$ reads, with $k$ observed ones, the likelihood ratio is $$ \left(\frac{1-\nu}{\nu}\right)^{2k-n}. $$ The minimum average error is majority's binomial error, with half the tie mass, yielding (5.3). A policy supplied all $n$ reads can simulate every earlier-stopping policy by ignoring an unused suffix. The Bayes error is therefore a lower bound on every admitted policy's worst-case error. For odd $n$, majority has exactly that error in both retained states. Binomial expansion gives $R_{2k}=R_{2k-1}$, so an even cap is attained by majority on the preceding odd number of reads, without needing a random tie-break. Both possible installations preserve the audit interface by construction. $\square$ The result separates an error in the installed rule from a loss of reflective capacity. A fallible amendment can leave the next audit fully available. Neither score should stand in for the other. ## The cost of a warranted tolerance The attaining policy uses a signed balance and a read counter: $$ S\leftarrow S+2Y_i-1. $$ The balance needs $\lceil\log_2(2n+1)\rceil$ bits and the counter needs $\lceil\log_2(n+1)\rceil$ bits, beyond latches, program and phase storage. Let $c_{\mathrm{read}}$ charge a complete read-and-update step. Let $c_0$ charge initialization, comparison, installation and rearming. The tolerance cost is $$ B_\epsilon=c_0+c_{\mathrm{read}} \min\{n\geq1:R_n(\nu)\leq\epsilon\}. \tag{5.4} $$ These are macro-operation costs, not elementary gate counts. For $\nu=1/4$, the exact frontier gives: | Endorsement-error tolerance | Required reads | | --- | --- | | $10^{-1}$ | $7$ | | $10^{-2}$ | $19$ | | $10^{-3}$ | $33$ | | $10^{-4}$ | $49$ | With no reads, a deterministic forced decision has worst-case error one. Error one half at zero reads requires an explicitly available fair random source. Randomization is an operation with a stated realization, not an implicit gift to the controller. ## What the intervention test measures For matched preparations $z=0,1$, a fresh mixed-case probe has intact contrast $$ 1-2R_n. $$ A common read clamp and a common commit clamp each reduce the contrast to zero. At $\nu=1/4$ and $n=3$: $$ R_3=\frac5{32},\qquad \text{intact contrast}=\frac{11}{16},\qquad \text{capacity-loss probability}=0. $$ The three numbers answer different questions: how often the amendment disagrees with retained endorsement, how strongly the retained commitment changes the later disposition, and whether another audit remains possible. ## Repeated audits and their joint risk For $L$ fresh-context audits with the same read cap and the stated independent conditional read law, the exact minimax probability of at least one unendorsed installation is $$ 1-(1-R_n)^L. \tag{5.5} $$ Majority attains it by multiplying conditional success probabilities. For the converse, independently uniform fresh retained bits give every policy conditional success at most $1-R_n$ at each audit. The probability of succeeding throughout is therefore at most $(1-R_n)^L$. A persistent hidden commitment would define a different model. Earlier inquiries could remain informative, so one could not reset the uncertainty at every audit. The general risk-vector treatment in Appendix A retains the information needed to compose the appropriate experiment. ## Two ways to misread an improvement **Deferral changes the task.** If the controller may defer, report both deferral probability and error conditional on completion. A policy that never finishes makes no erroneous installation, but has not solved definite revision before a deadline. **Repeated errors may be correlated.** If every read shares one noise bit, $$ Y_i=z\oplus N, $$ each marginal read still has error $\nu$, yet the entire transcript contains only one observation. The minimax error stays $\nu$. Repetition improves the frontier under conditional independence, not under marginal accuracy alone. Bounded inquiry gives reflection a measurable limit. A controller can be genuinely responsive to its commitments, can preserve its capacity to reconsider, and can still lack the information needed to guarantee that the present amendment is endorsed. That combination is a feature of finite evaluation, not a contradiction in it. # Evidence and adaptive inquiry An agent has two jobs before acting: find out what is happening, and inspect what it has committed to doing. Those jobs compete for the same finite time. A useful discovery in one can change how much work remains for the other. The question is whether acting on that discovery gives a provable advantage over deciding the allocation in advance. Here the answer is exact. In a specified sensor family, the first useful calibration count is $L$. The first shared budget at which adaptive allocation beats every fixed-count allocation is **$L+2$**. The stronger worked example needs four inquiry opportunities. Its advantage comes from using the effort saved by an early endorsement certificate to cross a factual decision threshold. This chapter contains the complete model and proof from Section 6 of the published paper. The source law specializes a single-coin reliability model [15](/consciousness/agency/references#ref-15), and the allocation problem belongs to metalevel decision theory [11](/consciousness/agency/references#ref-11). Its reward combines factual accuracy and agreement with retained commitments. Neither component is a test of moral rightness. ## The source and the clock A stationary hidden regime $\theta\in\{0,1\}$ has equal prior probabilities. The three sensor skills are $$ a^{(0)}=(u,v,w),\qquad a^{(1)}=(v,u,w),\qquad 0\frac{u+w}{1+uw}.\tag{6.1} $$ Each fresh item has an independent fair label $X\in\{-1,+1\}$. Its outputs are $Y_i=XE_i$. Conditional on the regime, errors are independent across sensors and items, with $\mathbb E[E_i\mid\theta]=a_i^{(\theta)}$. A skill $a$ therefore means accuracy $(1+a)/2$. The complete joint law is $$ f_\theta(x,y):=\Pr_\theta(X=x,Y=y) =\frac12\prod_{i=1}^3\frac{1+xy_i a_i^{(\theta)}}2.\tag{6.2} $$ The controller knows this family and the positive orientation of the skills. It receives neither the regime nor the true labels of calibration items. Independence, stationarity, orientation and restriction to this family are assumptions. Unlabeled agreement cannot establish them all: reversing every skill and every hidden label preserves the observable law while reversing the correct prediction. Three retained endorsement signs $e_1,e_2,e_3$ are independent and fair in the benchmark distribution, and independent of the source. The installed charter requires the operative priority to equal $\operatorname{maj}(e_1,e_2,e_3)$. Priority has its own register. Installing the majority can retain or revise that priority, but **the majority charter stays fixed in this experiment**. Changing the charter itself is the separate [editable-rule construction](/consciousness/agency/revising-an-evaluative-rule). There are at most $B$ inquiry opportunities. One opportunity buys either a complete raw sensor triple on a fresh calibration item or one previously unread endorsement. The controller can choose the operation and named endorsement index from its entire acquired history and an independent random seed; it may stop early. The final target triple arrives only after inquiry. It cannot guide earlier requests. Success requires both a correct target prediction and a correct installed endorsement majority. The initially accessible state, including the old priority, is independent of regime and endorsements. There is no initial advice, timing signal or extra output port that reveals them. Without that restriction the problem changes: an already visible correct majority would give zero-query joint success $b$, rather than $b/2$. A query unit compares information access. It is not a universal energy unit. Computation, memory, ingress, routing and clocks receive separate charges in the [realization](/consciousness/agency/faithful-native-realization). A physically retained record does not automatically give the executive a free read. ## Information can improve before decisions do Compress each calibration triple to a signed increment: $$ Z(y)=\begin{cases} +1,&y_1=y_3\ne y_2,\\ -1,&y_2=y_3\ne y_1,\\ 0,&\text{otherwise}, \end{cases} \qquad S_n=\sum_{j=1}^n Z(Y^{(j)}).\tag{6.3} $$ Summing the source law over its hidden label gives $$ \Pr_\theta(Y=y)=\frac{1+a_1a_2y_1y_2+a_1a_3y_1y_3+a_2a_3y_2y_3}{8}, $$ where the skills are those of the chosen regime. Put $$ \begin{aligned} p_-&=\frac{1-uv+w(v-u)}4,&p_+&=\frac{1-uv-w(v-u)}4,\\ p_0&=\frac{1+uv}2,&\rho&=\frac{p_-}{p_+},\\ A&=v-u-w(1-uv),&D&=v-u+w(1-uv). \end{aligned}\tag{6.4} $$ Under regime zero, $(Z=-1,0,+1)$ has probabilities $(p_-,p_0,p_+)$. Under regime one, the outer probabilities exchange places. Every raw triple with $Z=0$ has the same probability in both regimes. Thus every complete calibration history has regime-one to regime-zero likelihood ratio $\rho^{S_n}$, giving the exact posterior $$ \pi(s):=\Pr(\theta=1\mid S_n=s)=\frac{\rho^s}{1+\rho^s}.\tag{6.5} $$ No supplied confidence flag enters this calculation. Let $b$ be target-majority accuracy and $c$ the accuracy available if the regime were revealed: $$ b=\frac{2+u+v+w-uvw}{4},\qquad c=\frac{1+v}{2}.\tag{6.6} $$ The dominance condition makes the stronger sensor's log-odds weight exceed the other two combined. Following that sensor attains $c$. ### Proposition 6.1: the sharp calibration threshold Before seeing the target triple, optimal expected target accuracy conditional on calibration score $s$ is $$ M(s)=\max\left\{b,\frac{1+u}{2}+\frac{v-u}{2}\max\{\pi(s),1-\pi(s)\}\right\}.\tag{6.7} $$ Define $$ L=\min\{n\ge1:\rho^n>D/A\}.\tag{6.8} $$ Then $L\ge2$, $b\le M(s)\le c$, and $M(s)=b$ for $|s|0.\tag{6.9} $$ An attaining classifier follows sensor one for $s\ge L$, sensor two for $s\le-L$, and target majority otherwise. At an exact likelihood threshold it may retain majority. **Proof.** For each target cue $y$, its signed truth margin is $$ d_\theta(y):=f_\theta(+1,y)-f_\theta(-1,y) =\frac{\sum_i a_i^{(\theta)}y_i+uvw\,y_1y_2y_3}{8}. $$ At posterior odds $\ell$, the Bayes action has the sign of $d_0(y)+\ell d_1(y)$. For $y=(+1,-1,+1)$ the margins are $-A/8$ and $D/8$. Swapping the first two sensors or reversing all signs generates the remaining disagreement cases. The only positive switching thresholds are $A/D$ and $D/A$. If the first two sensors agree, their sign is optimal in both regimes: the potentially competing margin $u+v-w-uvw$ is positive because dominance implies $v>w$. The Bayes rule therefore selects among target majority and the two individual sensor followers. Their accuracies yield (6.7). Majority has accuracy $b$ in each regime, so it also has that value at every posterior. Revealing the regime gives upper bound $c$. Set $d=v-u$ and $f=1-uv$. The assumptions give $0wf$, and $$ 1<\rho=\frac{f+wd}{f-wd}<\frac{d+wf}{d-wf}=D/A. $$ Therefore $L\ge2$. Fewer acquisitions can move the posterior without crossing a decision threshold. At acquisition $L$, only $S_L=\pm L$ crosses the threshold. Under regime zero those events have probabilities $p_+^L$ and $p_-^L$. Following the strong sensor gains $c-b=A/4$ relative to majority; following the weak one loses $D/4$. This proves (6.9), which is positive because $\rho^L>D/A$. Swapping sensor identities and regimes preserves the classifier's accuracy. Its regime accuracies agree, so the equal-prior optimum is also minimax. ∎ The pointwise plateau matters: an evidence task with only an average plateau need not have the allocation optimum below. Nor is there a uniform finite bound on $L$. With $ub.\tag{6.11} $$ The first strict adaptive advantage occurs at $B=L+2$, with exact magnitude $$ \Delta:=V(L+2)-F(L+2)=\frac{Ap_-^L-Dp_+^L}{8}.\tag{6.12} $$ An optimal policy reads two endorsements first. If they agree, it uses $L$ queries for calibration. If they disagree, it reads the third endorsement and uses $L-1$ queries for calibration. Both branches install the exact majority. **Proof.** First note the strict oracle comparison $$ 8(b-3c/4)=1+2u-v+2w-2uvw>0.\tag{6.13} $$ At budget $L+2$, a terminal history with three endorsement reads permits at most $L-1$ calibrations and has value at most $b$. Zero or one endorsement read permits value at most $c/2$ or $3c/4$, both below $b$. Two disagreeing endorsements give at most $c/2$. Only two agreeing endorsements together with exactly $L$ calibrations can exceed $b$. These are the only exceptional histories, including policies that stop early. To bound them without assuming that stopping is independent of evidence, pre-sample a calibration stream and an independent stream of three fair endorsement responses. Whenever a previously unread named endorsement is requested, assign it the next response. Conditional on each history, every unread named bit remains independent and fair, so this lazy assignment has the original law. Assign unused responses after stopping if needed. Conditioning on the policy seed handles randomized dispatch. Let $G$ be agreement of the first two endorsement responses. Its probability is $1/2$, independently of the first $L$ calibration responses. On an exceptional history the score is their sum $S_L$. Since $M(S_L)-b\ge0$, every policy's terminal conditional value is bounded pointwise by $$ b+\mathbf1_G[M(S_L)-b].\tag{6.14} $$ The actual event of reaching an exceptional history need not be independent of calibration; only the dominating event $G$ is used. Expectations give $V(L+2)\le(C_L+b)/2$. The stated policy attains equality. Agreement certifies the majority after two reads and leaves $L$ calibrations. Disagreement consumes the third read and leaves $L-1$, where target value stays $b$. For fixed allocations, three endorsement reads leave at most $L-1$ calibrations and attain $b$. At most two endorsement reads give average endorsement accuracy at most $3/4$. Even with the regime revealed, their joint value is at most $3c/4 # Faithful native realization **An evaluation becomes a physical capacity when the installed mechanism preserves the reads, writes and limits that make the evaluation possible.** Matching a final answer is only one part of that requirement. The retained commitments must enter through the declared interface, the selected rule must govern the later act, and the physical implementation must preserve the interventions that test both pathways. This chapter develops the full construction in Section 7 of *Bounded Agency and Reflective Freedom*. It puts independently verified evaluative control and the native organization required by SPC-2 on a common finite carrier. The physical equations and detailed cost derivations appear in [Physical carriers and costs](physical-carriers-and-costs). The philosophical question is concrete: can a physically caused mechanism carry its own tested process of assessment while also satisfying the separately stated conditions for experiential assignment? The construction answers that existence question under its declared physical contract. It also builds a controller with the same return organization and no evaluative command dependence. That counterexample keeps the two obligations distinct. ## 7.1 The physical contract comes first SPC-2 specifies the primitive components, physical mechanisms, native time cells, internal routes, records, admitted preparations and operations, resource envelope and process provenance before assigning a subject [18](/consciousness/agency/references#ref-18), [19](/consciousness/agency/references#ref-19), [21](/consciousness/agency/references#ref-21). In the finite specialization used here, a maximal internally strongly connected candidate must admit an executable covering return in its current operating context and retain a nontrivial native predictive distinction. Its local joint record/successor instrument must be closed. States with the same laws for every admitted finite future record must also respect successor equivalence classes. Graph connectivity alone does not meet these conditions. A graph can contain a cycle that the installed program cannot execute with its remaining resources. A simulator can write an informative external log without creating the corresponding native physical record. The construction therefore specifies both the route and the operations that actually traverse it. The exact finite SPC-2 certification algorithm uses rational instrument probabilities. Both numerical inquiry benchmarks satisfy that restriction. The parameterized statistical identities and the compilation identities themselves do not require rational parameters. ### Carriers, operations and the apparatus boundary Take $m\geq2$ separately nominated bit carriers arranged on an installed adjacent-pair path. At each certified native type, the admitted preparation domain is the entire Boolean cube. Phase, resources and incoming boundary conditions are held fixed across these preparations. This cube is the declared intervention domain; it does not assert that one uninterrupted episode visits every bank word. Ordinary operations are source-disjoint NAND and COPY writes, CONSTANT writes and scheduled HOLD operations. Each ordinary operation has one nominated target. A source-disjoint write does not use its own target as an input source. Adjacent SWAP is a separate two-carrier primitive. The complete physical context includes the fixed program, phase, remaining resources, diagnostic mode and complete input contract. Exhaustion produces an explicit terminal record followed by an absorbing continuation. Unfunded future operations do not count as available returns. The ideal sequencer influences the bank, with no returning influence from the bank. Data-dependent executive choices are represented inside the bank and compiled into a fixed bounded schedule. Gate couplers contain no omitted retained state. Under this inventory, the bank can be a maximal internally returning component. A reciprocally loaded clock or a coupler with memory would require a larger inventory and a fresh boundary assessment. The clock, program and power supply remain paid apparatus even when they lie outside the nominated component. The continuous carrier construction supplies an ideal powered refinement of the elementary gates. Its assumptions about scheduling and couplers are part of the model that a physical device would have to realize. ### Native records must determine the right successor classes At a fixed native protocol type, write $K_{a,o}(x,y)$ for the exact joint probability of record $o$ and actual successor $y$, given bank state $x$ and admitted operation $a$. Define $x\sim x'$ when the two states have the same laws for all admitted finite future native transcripts, under the same type and boundary contract. Predictive congruence requires, for every successor equivalence class $D$, $$ \sum_{y\in D}K_{a,o}(x,y) =\sum_{y\in D}K_{a,o}(x',y) \qquad(x\sim x'). \tag{7.1} $$ For a general stochastic system, equality of output laws does not automatically imply this joint record/successor condition. Here the records will be actual retained destination values reused by installed operations. Their role is physical and operational, rather than an analyst's added trace. ## 7.2 A protected source carries the return Let $r$ be an endpoint of the carrier path. The tour $U_r$ swaps the token at $r$ along the whole path, then performs the inverse SWAP sequence. The bank ends exactly as it began, at a cost $$ C_m=2(m-1) $$ SWAP operations. Although the complete tour is an identity map, its intermediate native operations move an actual distinction through every carrier and return it to its starting place. After an ordinary gate targeting cell $j$, choose the tour root by the fixed rule $$ r=\begin{cases} 0,&j\neq0,\\ m-1,&j=0. \end{cases} $$ The preceding write cannot overwrite this endpoint. The choice depends on the fixed instruction, so choosing the protected source does not require a new observation of the bank. ### Theorem 7.1: protected-source compilation Take a circuit with $m$ carriers and $N$ single-target ordinary gates. Insert an initial tour $U_0$ and a protected-source tour after every ordinary gate. This transformation preserves the circuit's ordinary transaction map and its specified ordinary read/write interventions. The complete schedule costs $$ G=2(m-1)+N(2m-1). \tag{7.2} $$ At every occurrence immediately before or after an ordinary gate, the installed continuation contains a covering internal return with terminal contrast one, provided the following tour is funded. Under the native contract above, those occurrences have nontrivial predictive distinctions and satisfy (7.1). **Proof.** Prepare two bank words differing only at the selected root. Immediately after the ordinary gate, the tour carries the root distinction through every carrier and back to that root. Immediately before the gate, the same argument works because the root is not the target. The gate can propagate the distinction into its target, but cannot erase the preserved root token. Its two terminal values therefore differ with certainty. The paired experiments use common phase, resources and incoming boundary conditions, with external return paths inactive. Every adjacent coupling occurs within this one installed schedule. Its route support covers the entire bank. For a tour rooted at zero, the successive outward SWAP destination records are $$ (x_1,x_0),\ (x_2,x_0),\ \ldots,\ (x_{m-1},x_0). $$ These records identify the original bank word. The opposite-root tour does the same. A tour entry consequently has $2^m$ predictive classes. Before an ordinary gate, at least the protected root remains distinguishable through the following tour. Each deterministic first operation has one fixed record and successor for each state. Equivalent states must emit the same first record. If their successors admitted a distinguishing continuation, prefixing that continuation with the first operation would distinguish the original states. This proves (7.1), including later types at which an overwrite has merged some distinctions. Finally, each tour is the identity on the complete bank. Removing the tours recovers the original transaction and the results of its specified ordinary interventions. One initial tour plus $N$ gate/tour pairs gives $$ 2(m-1)+N\bigl(1+2(m-1)\bigr) =2(m-1)+N(2m-1), $$ which establishes the count. $\square$ Adjacent SWAPs provide actual singleton-target dependence in both directions. The bank is therefore internally strongly connected. Under the declared component inventory, there is no larger internally returning component. The complete finite return catalogue includes every legal installed prefix allowed by the remaining resources, including failed and exhausted prefixes. It is not a catalogue containing only the successful witnesses used in the proof. The tour length $2(m-1)$ is sharp for a connected identity word built from pairwise SWAPs. The full proof appears in the physical-carrier chapter; it is the identity case of the transitive-transposition bound [9](/consciousness/agency/references#ref-9). This sharpness does not establish an optimal combined computation-and-return scheme, nor an optimum over arbitrary physical primitives. The certificate applies at the specified ordinary gate boundaries. It leaves the A3 question of uninterrupted experiential identity through every intervening native or analog instant open. A final command also needs its reserved final tour: a return that cannot be funded is not an available continuation. ### Corollary 7.2: return organization does not entail evaluation The native qualification supplied by Theorem 7.1 does not imply evaluative mediation of the ordinary command. **Proof.** Apply the same compiler to a circuit that always writes one fixed command, regardless of the retained commitments. A protected source and its tour still give the covering return and nontrivial native predictive distinctions. Yet every ordinary command is identical under every commitment preparation. Its intact command contrast is zero, and so is its read-mediated command contrast. If the endorsement relation requires different commands for two preparations, this constant circuit also fails endorsement for at least one of them. The native transcript can distinguish retained states without using them to evaluate the command. $\square$ The separation matters to the consciousness connection. A recurrent native organization and an evaluator can coexist, but the evaluator must earn its own certificate. ## 7.3 An eighteen-cell machine that changes its charter The editable-charter evaluator has a direct circuit realization. Its eleven persistent bits are $$ q_{01},q_{10},d,p,a,b,s,h_0,h_1,e_0,e_1, $$ with seven additional work bits. Here $p$ is the operative priority $\varrho$ of the earlier chapters, and $s$ is the retained symmetry commitment. The charter $(q_{01},q_{10},d)$ governs future decisions; it is distinct from the present priority $p$. Use the multiplexer convention $$ \operatorname{mux}(v,u,w)= \begin{cases}u,&v=1,\\w,&v=0.\end{cases} $$ The candidate charter is computed by $$ \begin{aligned} t&=s\wedge(a\oplus b),\\ Q^*&=\operatorname{mux}\!\left(q_{01}\oplus q_{10}, \operatorname{mux}(d,h_1,h_0),q_{01}\right),\\ Q_{01}&=\operatorname{mux}(t,Q^*,a),\\ Q_{10}&=\operatorname{mux}(t,Q^*,b). \end{aligned} \tag{7.3} $$ These formulas implement the lexicographic amendment rule, rather than merely agreeing with a few chosen examples. Unless $s=1$ and $a\neq b$, the unique normative minimizer is $(a,b)$. In the remaining case, the two symmetric charters tie. A symmetric old charter is retained. If the old charter is asymmetric, the retained case and old query order resolve the tie. The expression for $Q^*$ implements exactly those alternatives. The two indicators of an early exit and the updated query order are $$ \begin{aligned} E_0&=\operatorname{mux}(h_0,Q_{10},\neg Q_{01}),\\ E_1&=\operatorname{mux}(h_1,Q_{01},\neg Q_{10}),\\ D&=\operatorname{mux}(E_0\oplus E_1,E_1,d). \end{aligned} \tag{7.4} $$ When exactly one first query is sufficient, $D$ selects it. When both orders have the same status, the old order $d$ is retained. Three final COPY gates install $Q_{01},Q_{10},D$. A later act, using fresh considerations, writes $$ p\leftarrow\operatorname{mux} \left(e_0,e_1\vee q_{10},e_1\wedge q_{01}\right). \tag{7.5} $$ The allocated NAND/COPY circuit has 47 audit gates and 10 act gates. Every work cell is written before use and reused only after its last source read. The paper's address-level circuit and exhaustive checks cover all $2^{11}$ persistent words. Equations (7.3) through (7.5) supply the independent functional argument behind those finite checks. ### Test the paths that do the work The read control masks the actual audit read ports that source $a,b$. It leaves the retained archive and the tour unchanged. The installation control replaces the three final COPY writes with HOLD operations. These controls intervene on the mechanism being tested. Hold the initial charter, read order, priority, symmetry commitment and retained case fixed. Compare an archive with $a=b=0$ against one with $a=b=1$, then supply the same fresh mixed consideration. The intact actions differ with total variation one. Either the common read control or the common installation control reduces that contrast to zero. The identity tours preserve both intervention outcomes. With $m=18$ and $N=57$, the compiled schedule uses $$ G=2(18-1)+57(2\cdot18-1)=2029 $$ elementary windows. This device changes its represented charter and stored query order. Its act circuit nevertheless reads both supplied considerations. It does not claim the short-circuit query savings of the separate inquiry machine developed below. The physical result is specific: a fixed circuit can install a new rule and let that rule govern a fresh later case. The fact that the circuit's complete update law is fixed does not prevent the represented rule from changing through the circuit's own assessment. ## 7.4 Stochastic closure and paid executive information With stochastic operations, a native certificate must preserve correlations between records and successors. The next result supplies the missing link explicitly. ### Theorem 7.3: a successor-revealing continuation At a protocol type with one fixed next ordinary operation, suppose that operation has exact joint kernel $K(r,y\mid x)$. Its next mandatory, funded continuation is a deterministic identity tour whose native transcript $T(y)$ is injective. No new input enters during that tour. Then post-operation predictive classes are singleton bank states. At the pre-operation type, predictive equivalence holds exactly when the complete joint kernels $K(\cdot,\cdot\mid x)$ agree. The joint instrument therefore descends to predictive classes. **Proof.** The tour distinguishes every successor word. Its actual finite prefix $(r,T(y))$ determines the complete pair $(r,y)$, so equality of all future transcript laws forces equality of the joint kernel. Conversely, equal joint kernels followed by the same complete Markov continuation produce equal future laws. Because successor classes are singletons, this is precisely (7.1). At deterministic intermediate phases of the tour, the deterministic first-step prefix argument still applies, even if later operations are stochastic. $\square$ If several first operations are admitted at one type, equivalence requires equality of the joint kernel for every such operation. The same proof works when each operation has its own funded revealing continuation. A noisy operation writing a target set $J$ can also preserve a covering return. It must leave a nominated endpoint outside $J$ unchanged on every outcome, and the corresponding deterministic tour must remain installed and funded. This is target-local logical noise. It does not cover arbitrary physical noise on the protected source, clock or tour. Joint multi-bit ingress must preserve its correlations. A serial delivery scheme needs its still-undelivered buffer inside the physical state or the complete boundary model. Omitting that buffer would omit part of the mechanism determining future laws. ### A native record is not automatically an executive read port During an identity tour, the installed controller permits no executive branch, query, copy-to-log operation or observation of the tour transcript. The bank returns to its previous state. Elapsed time, phase and fuel changes are independent of the data. No hidden executive transcript buffer remains. ### Lemma 7.4: preservation of executive information Under these restrictions, inserting an identity tour adds no executive information about a private retained variable. Every admitted paid-query policy retains its transcript law and risk once the tour cost is charged. **Proof.** Between executive occasions, the tour acts as the identity on retained executive state. It produces no permitted executive observation and adds only a constant cost/type update. Conditional on the preceding executive history, there is no additional observation depending on the private variable. Induction over inserted tours preserves the complete executive transcript law. $\square$ This is an architectural restriction. Rewiring the controller to read the outward tour records would reveal the whole bank, remove the paid endorsement uncertainty and invalidate the inquiry comparison. The native predictive object used by SPC-2 does not itself install that extra executive access [19](/consciousness/agency/references#ref-19), [20](/consciousness/agency/references#ref-20). A physically retained value and an agent's budget-feasible access to that value are different facts. ## 7.5 A physical realization of the exact inquiry advantage The threshold-$L$ inquiry policy uses a separate bank of 23 bits: | Role | Cells | |---|---:| | Retained endorsements | 3 | | Executive endorsement latches | 3 | | Raw sensor latches | 3 | | Agreement and request bits | 2 | | Calibration flags | 2 | | Priority and command | 2 | | Reusable work cells | 8 | | **Total** | **23** | It first reads two endorsements. If they agree, the next slot acquires a calibration triple; if they disagree, that slot reads the third endorsement. The remaining $L-1$ slots acquire calibration triples. The final target triple arrives only after all $L+2$ inquiry slots have finished. ### Two flags are enough at the first useful threshold Let $g$ indicate agreement of the first two endorsements. Initialize $U=V=g$. After each acquired calibration triple, update $$ U\leftarrow U\wedge[Z=+1], \qquad V\leftarrow V\wedge[Z=-1]. $$ On the agreement branch, there are $L$ calibrations. Only $L$ identical nonzero increments can reach the first useful threshold. On the disagreement branch, there are at most $L-1$ calibrations, and the initially zero flags force majority prediction. Thus these two flags implement the attaining classifier exactly at this endpoint budget. A general posterior register is unnecessary for this particular policy. Each raw triple enters through one joint three-cell ingress operation. Two additional HOLD windows charge its three-write allocation. Its targets are disjoint from the protected endpoint. Both the adaptive program and the fixed-allocation program receive 27 ordinary windows for each inquiry slot and 18 final windows. They use the same bank, padding, acquisition allowances and return wrapper. Their counts are $$ \begin{aligned} N_{\mathrm{compute}}&=31+18L,\\ N_{\mathrm{ordinary}}&=27L+72,\\ N_{\mathrm{native}}&=44+45N_{\mathrm{ordinary}} =1215L+3284,\\ N_{\mathrm{work}}&=N_{\mathrm{native}}+2(L+1) =1217L+3286. \end{aligned} \tag{7.6} $$ There are $L+3$ paid acquisition intervals, including the final target. A work unit here counts a single physical write; the joint triple write costs three such units. It is not a measurement of heat. The full stage derivation and conditional time/energy bounds appear in the physical-carrier chapter. ### Why the statistical result survives implementation Project a physical episode onto its paid requests and responses. That projection recovers the inquiry experiment exactly. Computation uses only earlier paid information, and Lemma 7.4 prevents the return schedule from opening a free channel to retained endorsements. The optimum of Theorem 6.2 and its fixed-allocation bound therefore transfer under the same source law, access grammar and available resources. A fixed comparator with the same hardware and padding attains the fixed bound. For the larger-gap instance, $L=2$. Substitution gives: | Resource | Count | |---|---:| | Computation gates | 67 | | Ordinary windows | 126 | | Native windows | 5,714 | | Single-write work units | 5,720 | | Acquisition intervals | 5 | The family proof supplies these counts. A separate finite verification reported with the paper traverses all 2,816 reachable endorsement, calibration and target-word combinations of the two $L=2$ programs through every installed SWAP and gate. It checks acquisition chronology, installed priority, prediction, resources and identity of the retained bank after every tour. Weighting those executions by the exact source law reproduces the adaptive success fraction $1870483759/2^{31}$ and fixed success fraction $1777/2048$. These checks support the implementations; the analytic proof establishes the family result. ### Keep the source and the boundary complete The actual hidden source regime remains fixed throughout an episode. Native closure is established uniformly for either fixed regime without revealing it to the executive. A prior mixture must retain its persistent source state or conditional history. Averaging the hidden regime away does not establish a Markov law on the bank. The request/response apparatus provides an external returning path. Isolating that path during the internal tour prevents it from carrying the internal witness. It does not remove stateful physical couplers from the full-apparatus boundary assessment. ### How much implementation error can the advantage tolerate? Suppose the adaptive and fixed implementation transcript laws are within total variation $\eta_A$ and $\eta_F$ of their ideal laws, uniformly over the fixed comparison class. If the ideal success gap is $g$, the implemented gap is at least $$ g-\eta_A-\eta_F. $$ For $L=2$, $$ g=\frac{7164207}{2^{31}}. $$ Equal error bounds preserve a strict separation whenever $$ \eta<\frac{7164207}{2^{32}}. $$ This is a transfer tolerance. Establishing a particular device's reliability requires its actual error bounds. ### Three witnesses, three established capacities The 18-cell machine revises a represented charter. The 23-cell machine optimally allocates inquiry under a fixed majority charter. The fallible-amendment model establishes a read-error frontier using its own cost units. They share operational principles, but no single device has been shown to attain all three sets of resource guarantees at once. Componentwise relabeling preserves these certificates when kernels, records, routes, schedules, preparations, interventions and costs are all transported. This restricted realization covariance is consistent with causal-transformation work [1](/consciousness/agency/references#ref-1), [22](/consciousness/agency/references#ref-22). Arbitrary mixing of component coordinates, sampling an entire tour as one identity step, or adding executive ports changes the contract. Conditional on the stated contract, A1–A2 attach the SPC-2 experiential assignment at the certified occurrences. The mathematics establishes coexistence with independently tested evaluative capacities. Control competence alone does not supply the consciousness correspondence law, and input/output competence alone does not establish the necessity of recurrence. # Ownership, history and freedom A physically caused decision can still be settled through the agent's evaluation. The constructions make that claim concrete: retained commitments are inspected, executable amendments are compared, a represented rule is installed, and the rule governs later fresh cases. Read and installation interventions identify the mechanism responsible. The resource and error bounds establish how much of that capacity can be exercised on a particular occasion. This chapter develops Sections 8 and 9 of the published paper. It connects the formal results to ownership, manipulation, steadfastness and freedom over time. The further [philosophy of bounded perspective](/consciousness/agency#extended-inquiry) extends these questions with the supplementary control results. ## A decision can belong to a caused process The account has content beyond attaching the word agency to whatever happens. Its endorsement relation is fixed before the test. A constant selector, an immutable charter, a read bypass and an inverted evaluator fail different parts of the certificate. A fresh probe distinguishes a changed disposition from an immediate command overwrite. Information constraints can block correct completion even when an acceptable amendment exists. Those are independent ways for the claim to fail. A successful construction therefore establishes a particular organized capacity. It does not establish unrestricted reflection. The editable charter has a finite grammar, specified commitments and a fixed metarule. The inquiry machine optimally allocates information under its fixed majority charter. They establish different capacities and do not share every resource guarantee in one device. The free-will result is a bounded compatibilist result: **causal determination is compatible with internal evaluative mediation and represented-rule revision**. A deterministic implementation proves that compatibility. It does not produce multiple continuations from an identical complete deterministic state. Accounts requiring that further power ask a question beyond this result. Nor does the witness establish that biological choice is empirically deterministic. Indeterministic accounts of agentic control, including Potter and Mitchell [16](/consciousness/agency/references#ref-16), address a further issue. A further uncaused selector is unnecessary for the capacity demonstrated here. The assessment is part of the physical process that settles the act. Revising one of its decision rules does not require the system to originate every standard by which revision is assessed. ## Present control and historical ownership Commitments can develop through experience, arrive from an operator or be formed under manipulation. Present-time mediation does not identify which history occurred. Reasons responsiveness and ownership of a mechanism have already been distinguished philosophically [6](/consciousness/agency/references#ref-6). Testing the difference requires historical information. ### Proposition 8.1: the limit of present-state provenance tests Suppose two developmental histories produce the same complete present-state distribution, the same future boundary law conditional on that state, and the same admitted operations and interventions. Every later finite operational test then has the same distribution under both histories. **Proof.** The initial present laws agree. Conditional on a common test history, the test chooses the same next operation and each system applies the same joint record/successor kernel. Induction gives equality of all future joint history/state laws, including intervention steps and the final stopping record. ∎ A difference erased from those premises cannot be reconstructed by those tests. Retained developmental records, persistent differences in audit access or an expanded historical intervention family can change the premises. Historical autonomy remains a meaningful further question, with additional evidence requirements. Likewise, faithfully following a commitment does not make it factually true or morally defensible. The magnitude of a causal contrast settles neither issue. ## Follow commitment formation as it happens A stronger historical investigation records acquisition instead of inferring it from a final snapshot. Fix the observation window, physical boundary and admitted routes through which commitments may change. For each episode record: - The prior commitment and incoming proposal or evidence. - The information actually inspected by the assessor. - Its selected retention or amendment. - The actual installation and its effect on fresh cases. The records must come from independently checked instrument semantics or protected observations. A register saying “an audit occurred” is insufficient. The realization must exclude unobserved write routes or identify them as unresolved. At each episode, test whether both retention and adoption can be assessed within the budget, whether a read intervention changes assessment as predicted, and whether an installation intervention removes the later effect. Score success against the criterion operative before installation. Rewriting a criterion to approve the observed outcome is not evidence of faithful prior assessment. If the criterion itself changes, identify the still-operative procedure assessing that revision. The finite account may leave an interpreter or some criteria fixed; it need not claim infinitely many prior self-authorizations. Verified events compose into evidence of **audit-mediated acquisition over the recorded interval**. They can distinguish an accepted proposal from a direct overwrite, expose a period of insulation from assessment and document restoration of audit access. They cannot certify unrecorded development before the window, and faithful logging alone does not establish truthful evidence. ## Plasticity does not exclude manipulation An agent may revise beliefs when credible contrary observations arrive, then use those beliefs in assessing its commitments. An operator can leave every internal audit pathway intact while withholding contrary observations and supplying selected or fabricated apparent evidence. An open stream and an operator-controlled stream can deliver identical finite observations and produce identical internal assessments. Their proximal certificates then agree. Distinguishing the histories requires evidence about source provenance or control over available evidence. A reliability flag merely transfers that obligation to the flag's generator. Bypass, insulation, deceptive information supply and transparent persuasion are therefore different relationships between agent and environment. They cannot be reduced to a plastic-versus-rigid classification. Assigned goals can support proximal control while historical and informational questions remain open. Reasons responsiveness, ownership and individuation of a mechanism have been debated separately [5](/consciousness/agency/references#ref-5), [6](/consciousness/agency/references#ref-6); this operational account states the boundary, observations, held-fixed mechanism and intervention family before attribution. The longitudinal protocol can support conclusions about specified acquisition routes. It is not a general theorem that decides moral responsibility from present architecture. ## Conflict, steadfastness and the time of a choice Two coherent commitments may require incompatible acts. Assessment can select one without proving the other mistaken or erasing its cost. An endorsement relation can admit several choices, with a tie rule settling the act while both commitments remain stored. Regret and continuing conflict can accompany agency, but neither is required by the minimal certificate. An informed selfish choice can exercise the same structural capacity as a generous choice. Moral appraisal, endorsement and fidelity to evidence remain distinct. Receptivity means a capacity for appropriate reassessment. It does not require every cue to change the output. Evidence can narrow uncertainty while leaving the same action best supported. A framing change can move an unreliable controller's output without improving its assessment. The charter construction deliberately retains an already optimal rule after examination. An overlooked option that a suitable cue could recruit differs from an option the architecture cannot execute before its deadline. Compare specified input and resource envelopes rather than demanding exhaustive consideration of everything. Self-binding makes the time index unavoidable. An agent can assess and install a restriction now that removes an option later. The earlier act can exhibit reflective control while the later repertoire or revisability is smaller. Recognizing reasons for reversal does not restore a missing write path. Even a reversible barrier preserves practical flexibility only when the necessary evidence, time and resources are available before the deadline. Pruning can improve remaining assessments while narrowing the repertoire. There need be no single overall increase in freedom. The preservation theorems track one declared future capacity. They do not require every autonomous act to preserve every future option or even the agent's survival. ## Consciousness requires its own physical account The realization results establish compatibility with the declared SPC-2 organization at specified native occurrences. Experiential assignment remains a correspondence law of that constitution. The mathematical conclusions about evaluation do not depend on accepting it; the sentient-agency interpretation does. Accepting the correspondence law also does not turn a return-only circuit into an evaluator. The constant-command counterexample has native return organization without evaluative command mediation. Conversely, software performance alone does not establish the native physical conditions. Boundary and timing are substantive. A stateful coupler or reciprocally loaded clock can enlarge the recurrent apparatus. Gaussian voltage dynamics provide a finite-error approximation; sign decoding alone does not produce an exact finite-state Markov law. Qualification at ordinary gate boundaries leaves a separate question about continuous phenomenal identity through intervening instants. These are concrete obligations for physical implementation. The related article [Agency and the constructed self](/articles/agency-and-the-constructed-self) explains the RCO hypothesis, meaning recursively closed observer, in accessible terms. Usable personal content and its compulsory authority over deliberation are different. The bounded constructions establish specified evaluative capacities; they do not prove the full RCO hypothesis. ## Experiments that distinguish the claims A charter experiment compares intact assessment, archive-read masking and installation clamping, then presents fresh mixed inputs. A noisy-amendment experiment varies paid read count while scoring endorsement error and later audit callability separately. An inquiry experiment compares adaptive and fixed allocations under the same source law and access grammar, including a clamp on the actual request port. The stronger parameter instance increases implementation-error tolerance within the same experimental class. A developmental experiment records commitment changes and audit access over a specified history, while examining source provenance separately from internal mediation. A physical realization project must instantiate and assess the source, couplers, clock, power state and complete instrument. These are extensions of the declared model, not results already inferred from software. ## What has been established The published work separates retained endorsement, actual mediation, rule installation, later behavior and preservation of future audit capacity. It proves an exact inquiry family and its first adaptive advantage. It constructs faithful native realizations that preserve operations and information restrictions while supplying organization for conditional SPC-2 assignment. Freedom in this account is expressed through the extent and availability of evaluative capacities. Physical causation does not erase them. The historical protocol extends the investigation beyond the present occasion while explaining why preserved plasticity alone cannot rule out deceptive control of evidence. Broader ownership and experiential conclusions require the additional observations and commitments identified here. ## Publication, funding and reproducibility The full published paper is [Bounded Agency and Reflective Freedom: Evaluative Revision, Information Limits, and Faithful Realization](https://doi.org/10.5281/zenodo.23202999), Jeremy Rodgers, Independent Researcher, 6 October 2026, Version 2.0. This research received no external funding. Independent mathematical review, computational replication, physical implementation and empirical testing are invited, particularly where specialist expertise, equipment or institutional resources are needed. The paper reports an accompanying reproducibility supplement with finite programs, exact rational checks and instructions. The mathematical proofs are reproduced in this website treatment. The website's own coverage and rendering checks are documented separately; they should not be confused with a fresh execution of every reported research computation. No laboratory data are reported. # Revision kernels and finite-horizon risk A revision can pass today's test and destroy the ability to pass tomorrow's. A query can reduce uncertainty while leaving an incompatible possibility alive. To follow these differences precisely, we need a cost recurrence, a preservation kernel and a risk calculation that retains the same hidden context throughout each audit. This chapter supplies the complete constructions in Appendix A of the published paper. It develops the calculations behind [preserving revision](/consciousness/agency/preserving-revision) and [fallible internal inquiry](/consciousness/agency/fallible-internal-inquiry). ## A.1: charge the internal query tree For deterministic serial access, let $S\subseteq Z_p$ be the compatible retained records, $Q$ the unread positions, and $c_p(p')$ the cost of installing and rearming amendment $p'$. A read with its elementary branch costs one unit. The exact worst-case cost of an endorsed amendment into continuation set $X$ is $$ \begin{aligned} D_X(p,S,Q)=\min\Bigg\{& \min_{p'\in\bigcap_{z\in S}[N(p,z)\cap X]}c_p(p'),\\ &\min_{i\in Q}\left[1+\max_{u:S_{i=u}\ne\varnothing} D_X(p,S_{i=u},Q\setminus\{i\})\right]\Bigg\}. \end{aligned}\tag{A.1} $$ An empty minimum is $+\infty$. The first choice installs an amendment valid in every compatible context. Otherwise the first query chooses some unread $i$ and every possible response must be handled. Induction on $|Q|$ proves both the lower bound and attainment. The unrestricted recurrence assumes that the selected tree can be stored and dispatched. Code, memory or instruction restrictions may exclude it. Then the optimization must range over admitted trees; the unrestricted value is only a lower bound. For the charter compiler, substitute its unique output $F_{\chi_{\rm old}}(z)$ and charge only record reads. A leaf costs zero exactly when $F$ is constant on $S$; otherwise querying follows (A.1). The publication's exhaustive evaluation over eight initial charters and 32 records yields worst-case depth five for initial AND or OR and four for a projection. Its accompanying verifier computes $F$ by both candidate scan and independent successive filtering, checks all 256 charter-record cases and 1,024 subsequent ordinary executions. ## A.2: universal closure with writable validators When $R(p)$ contains the successors generated by the actual installed revision semantics, define $$ K_0=G,\qquad K_{j+1}=\{p\in K_j:R(p)\subseteq K_j\}.\tag{A.2} $$ The limit consists exactly of configurations whose every finite admitted revision path remains in $G$. Membership implies preservation by induction. If a configuration is first removed at stage $j+1$, it has a successor removed earlier. Following decreasing removal ranks produces a finite path outside $G$. This proves maximality and bounds the shortest removal witness by $|\mathcal P|$, the number of configuration types. An overapproximate transition relation gives a sound, potentially conservative kernel. Exact classification requires exact composable semantics. ### A mutable validator can preserve a fixed property Attach four validator modes to each of the eight charter programs: | Mode | Current audit | Admitted or installed continuation | |:--|:--|:--| | A | Correct audit | Permits A or B | | B | Correct audit, opposite candidate enumeration order | Permits A | | L | Passes the current local audit | Installs C | | C | Constant response independent of endorsements | Fails the local responsiveness test | There are 24 locally good types but only 16 universally preserving types: the charter programs with A or B. The installed rule can change A to B when the retained symmetry commitment is active and return B to A. Stored validators need not be immutable. Their outgoing semantics remain part of the proof. Testing only whether the next program currently passes its local test admits L and misses the later loss. Allowing every syntactically valid charter within a mode overapproximates the actual assessor's smaller successor family. The same 16-type classification is exact for the actual selector: A and B stay among themselves, L actually installs C, and C fails. The conclusion does not assume that every catalogue edge occurs. ## A.3: preserve the risk vector before taking its maximum A query $u$ produces response $y$ under $K_u(y\mid z,h)$. Let $\Pi_B(p)$ be a nonempty finite admitted family of costed observable policy trees, each completing installation. For a continuation set $X$, retain the context-indexed failure vector $$ v_\pi(z)=\Pr_z\{p'\notin N(p,z)\cap X\}.\tag{A.3} $$ A deterministic installation leaf contributes its zero-one loss. A stochastic installation contributes the expectation under its actual kernel. At a query node, $$ v_\pi(z)=\sum_y K_u(y\mid z,h)v_{\pi_y}(z),\qquad r_B(p,X)=\min_{\pi\in\Pi_B(p)}\max_{z\in Z_p}v_\pi(z).\tag{A.4} $$ The context stays fixed through an audit. Maximizing over $z$ separately at each child would allow an adversary to change the retained commitment after the read. For one binary symmetric read followed by installation of the observed value, the true vector is $(\nu,\nu)$; branchwise maximization falsely gives risk one. Endorsement failure, capacity loss and their union can each be propagated. The policy family fixes the optimization class. If all mixtures of finitely many deterministic trees are available with a charged random source, optimize over the convex hull of their vectors. Randomization can reduce the largest component. With an unread binary commitment and zero queries, the two constant decisions have risks $(0,1)$ and $(1,0)$; their admitted equal mixture gives $(1/2,1/2)$. The positive-read frontier in Theorem 5.2 already covers randomization and has deterministic attaining policies. ### Exact finite-horizon failure The successor $p'$ must contain every accessible variable and history summary affecting future query laws, contexts and resources. Within an audit, its hidden context is fixed. After the terminal history, a new admissible context may be selected from $Z_{p'}$ before fresh query noise begins. This recursion assumes no persistent hidden parameter that constrains those choices or changes later kernels. First-audit and continuation policies must have an admitted, charged composition. Set $J_H(p)=1$ for $p\notin G$ at every horizon, and $J_0(p)=0$ for $p\in G$. For $p\in G$ and $H\ge1$, a terminal amendment receives the context-indexed loss $$ \ell_z^{(H)}(p')=\begin{cases} 1,&p'\notin N(p,z)\cap G,\\ J_{H-1}(p'),&p'\in N(p,z)\cap G. \end{cases}\tag{A.5} $$ Propagate this vector with (A.4), then minimize its largest initial component. The result is the exact minimum worst-case probability of any endorsement or capacity failure during $H$ audits, including an initially failed capacity. **Proof.** Decompose any policy at its first installation. Its immediate failure and best possible continuation give the lower bound. Attaching attaining continuation policies to an attaining first audit gives sufficiency. Induction over $H$ completes the argument. Persistent hidden parameters linking audits require the actual-history uncertainty or likelihood state; this fresh-context recursion does not reset their information legitimately. ∎ The descending iteration $$ X_0=G,\qquad X_{j+1}=\{p\in X_j:r_B(p,X_j)\le\epsilon\} $$ finds the largest set with a uniform one-step risk certificate. It need not optimize every finite-horizon cumulative-risk objective. If the conditional failure probability after each successful history is at most $\epsilon_i$, multiplication of conditional success probabilities gives $$ \Pr\{\text{some failure by }L\} \le1-\prod_{i=1}^L(1-\epsilon_i) \le\sum_{i=1}^L\epsilon_i.\tag{A.6} $$ No independence of failures is needed. The conditional guarantees and their resource charges are needed. That distinction connects the recurrence back to practical agency: a future capacity is supported by an executable succession of checks, with its remaining risk and resources carried forward. # Physical carriers and costs **A physical account must specify how the bits move, how their distinctions survive and what the apparatus pays to complete the episode.** This chapter supplies the complete carrier and resource arguments from Appendix B of *Bounded Agency and Reflective Freedom*. It completes the construction in [Faithful native realization](faithful-native-realization), including the optimal identity-tour bound, powered continuous gates, bounded disturbances, a separate Gaussian approximation and the inquiry machine's stage counts. The equations make the physical obligations explicit. They distinguish exact decoded behavior under a bounded-disturbance contract from finite-error coupling under Brownian noise. They also keep elementary windows, physical writes, source acquisitions and apparatus overhead as separate charges. ## B.1 Why a covering identity tour needs twice the path length Consider a word of pairwise SWAPs on $m$ carriers. Require its accumulated edge graph to be connected and its final permutation to be the identity. Its length is at least $$ 2(m-1). $$ **Proof.** Initially, the current permutation has $m$ cycles and the accumulated edge graph has $m$ connected components. A connected final edge graph requires at least $m-1$ SWAPs that join previously separate graph components. Before any such joining SWAP, each permutation cycle lies entirely inside one accumulated graph component. A SWAP joining two graph components must therefore join two permutation cycles. Every transposition either joins two cycles or splits one cycle. To finish at the identity, with $m$ cycles again, the numbers of joins and splits must be equal. At least $m-1$ joins consequently require at least $m-1$ splits, giving at least $2(m-1)$ operations. An outward path traversal followed by its inverse attains the bound. $\square$ This is the special identity case of the transitive-transposition bound of Goulden and Jackson [9, Proposition 2.1]. Its scope is connected identity words of pairwise SWAPs. It does not optimize arbitrary physical primitives or all possible ways to combine computation with return. ## B.2 Explicit continuous bit dynamics Each nominated cell has one real coordinate $x_i$. At a native boundary it lies in a well box $$ |x_i-s_i|\leq\rho, \qquad s_i\in\{-1,+1\}, \qquad\rho=0.01. $$ One native gate consists of an active unit followed by a restoration unit. The restoration stage brings every cell back inside its well box before the next native gate. ### A paid clock gives each gate its place Use the pulse $$ b(t)= \begin{cases} 30t^2(1-t)^2,&0\leq t\leq1,\\ 0,&\text{otherwise}. \end{cases} $$ It has integral one and extends continuously differentiably across its endpoints. A smooth flat pulse can replace it. For a schedule with $G$ gates, an independent oscillator obeys $$ \dot c=-\omega s, \qquad\dot s=\omega c, \qquad\omega=\frac{\pi}{G}, \qquad(c(0),s(0))=(1,0). $$ One period of length $2G$ selects the complete program. On the oriented circle, write $$ u=\frac{\arg(c+is)}{\omega}\in[0,2G). $$ For $k=0,\ldots,G-1$, periodically extend the active gate pulse $b(u-2k)$ and restoration pulse $b(u-2k-1)$. Their values and first derivatives match at the circle seam. The prescribed vector field is therefore continuously differentiable there despite the angular coordinate chart. The fixed program assigns an opcode to each active pulse. A paid terminal latch disables further operation after the first revolution. Alternatively, the schedule describes one episode under an explicitly declared replenishment contract. No bank coordinate drives the oscillator, program or terminal latch. This specifies the ideal one-way schedule. Immunity to reciprocal loading in manufactured hardware would require its own realization evidence. Fixed phase spacing and program storage are apparatus, with physical costs. Interleaved source acquisitions receive fixed reserved intervals in the same calendar, enlarging its period. An outcome cannot shorten a pulse or reveal data through variable completion time. Acquisition failure produces the declared error or terminal record. The gate equations below use local unit-time coordinates within this calendar. ### Decode the well sign without disturbing the source Define the continuously differentiable interpolation $$ H(u)= \begin{cases} 0,&u\leq0,\\ 3u^2-2u^3,&02 $$ keeps the field inactive. Their endpoint error is at most $\rho+\zeta$. ### Restoration closes the induction During the restoration unit, every cell obeys $$ \dot x_i=5b(t)x_i(1-x_i^2)+d_i(t). \tag{B.3} $$ Inside $|x_i-s_i|\leq0.1$, the upper derivative of the well error is bounded by $$ -1.71\cdot5b(t)|x_i-s_i|+\zeta. $$ If $e_0$ is the error entering restoration, the endpoint error is at most $$ e_0e^{-8.55}+\zeta. $$ Apply this to the largest active-stage error: $$ 0.013334e^{-8.55}+0.001<0.001003<\rho. $$ Every restored cell therefore returns to the admitted well box. Induction over gates proves the exact intended decoded trace for every finite gate sequence under this bounded-disturbance contract. All work cells participate in the return. Resetting or reusing a cell does not remove it from the component. The explicit bounded-cell construction specializes established smooth-simulation methods [10](/consciousness/agency/references#ref-10) to the native gate and record contract used here. Exact decoded behavior is one part of the physical qualification. Disturbances must not introduce unmodeled returning couplings. The native record grammar consists of the stipulated well-sign reads; adding an analog readout changes its predictive classes. Active couplers are directed powered forces. These equations provide no measured switching heat or energy rating. Preparation, holding, clocks, source generation and outputs remain explicit device dependencies. ## B.3 Gaussian noise gives a finite-error refinement Brownian noise needs a separate analysis because its paths do not have bounded derivatives. Consider $$ dX_i=f_i(X,t)\,dt+\sigma\,dB_i, $$ with independent Brownian motions and an exact independent clock. Inputs are nonanticipating, and the driving increments are fresh conditional on the preceding source and controller history. Set $$ h=0.002. $$ For each unit interval, the reflection bound gives $$ \mathbb P\!\left\{ \sup_t|\sigma(B_i(t)-B_i(0))|>h \right\} \leq4e^{-h^2/(2\sigma^2)}. $$ ### Writes and restoration tolerate bounded forcing paths For a scalar nonincreasing drift, a continuous forcing path bounded by $h$ changes the corresponding solution by at most $2h$. To see this, subtract the forcing path from the solution difference and use a first-crossing argument at $\pm h$. This comparison applies directly to writes. It also applies to restoration after stopping the process inside its well. A noisy write ends within $$ (2+\rho)e^{-10}+2h $$ of its target. Held sources move by at most $h$ and remain in their decoder plateaus. ### The rotating SWAP has a separate martingale bound For an opposite-sign pair, the scaled coordinates $(r/a,q)$ have noise covariance $$ \sigma^2\operatorname{diag}(9/2,1/2). $$ Rotation into the deterministic moving frame preserves the largest covariance eigenvalue. Each coordinate martingale has quadratic variation at most $(9/2)\sigma^2$ per unit. The exponential maximal inequality and a union bound give $$ 8e^{-h^2/(9\sigma^2)} $$ as a bound for either coordinate exceeding $h$ in absolute value. On the complementary event, the scaled perturbation is at most $$ \sqrt{10}\rho+\sqrt2h<\sqrt{3/2}-1. $$ The stopped process therefore stays inside the rotating plateau. Transforming back bounds the endpoint well error by $$ \rho+\sqrt{20/9}\,h. $$ Equal-sign pairs remain in the inactive region on the ordinary Brownian good event, with endpoint error at most $\rho+h$. Every active-unit bound is below $0.013$. During restoration, the noiseless path stays in its well. The stopped $2h$ comparison stays strictly inside the $0.1$ neighborhood where the restoring drift is nonincreasing, so there is no stopped exit. The restoration endpoint error is less than $$ 0.013e^{-8.55}+2h<0.004003<\rho. $$ After each successful boundary, fresh conditional Brownian increments allow the same bound from every point in the new well box. ### Couple the complete finite episode Union-bound the first failure over active and restoration intervals, all $m$ cells and all $G$ gates. The resulting episode bound is $$ \delta_G\leq \min\!\left\{ 1, G\left( 8m e^{-h^2/(2\sigma^2)} +8e^{-h^2/(9\sigma^2)} \right) \right\}. \tag{B.4} $$ Preparation failure is an additional charge. On the common good event, the native decoded transcript equals the ideal transcript. Their total variation distance is therefore at most $\delta_G$. For two comparison arms, their success difference can change by at most the sum of their episode error bounds. This coupling covers the specified powered bit primitives. A stochastic source ingress must have its exact declared law or a separate uniform conditional kernel-error bound. An internal-gate noise estimate does not certify an external sensor. Source errors, acquisition failures and clock failures must be included in the episode bound by the same first-failure argument before applying the inquiry comparison tolerance. Equation (B.4) is a finite-horizon approximation. The unconditional sign process need not be exactly Markov: distinct voltages inside one well can have different later error probabilities. An exact SPC-2 claim for the stochastic voltage process requires a complete continuous/native closure analysis of that process. The error estimate alone cannot establish an exact finite quotient. The distinction is useful in practice. A sufficiently small episode error can preserve the measurable inquiry advantage, while the stronger claim of exact native closure remains a separate physical obligation. ## B.4 The inquiry machine's full cost ledger Use NAND decompositions with the following counts: | Operation | NAND gates | |---|---:| | XOR | 4 | | Multiplexer | 4 | | AND | 2 | | Three-bit majority | 6 | The circuit stages have these charges: | Stage | Computation gates | |---|---:| | Two initial endorsement copies | 2 | | Agreement and initialization | 8 | | Masked third endorsement read | 3 | | Priority installation | 4 | | Each flag-update stage | 17 | | Final prediction | 14 | | Requests for the $L-1$ later calibrations | $L-1$ | | Request for the final target | 1 | The total is $$ 2+8+3+4+17L+14+(L-1)+1=31+18L. $$ There are $L+2$ inquiry slots of 27 ordinary windows and one final stage of 18 windows: $$ 27(L+2)+18=27L+72. $$ Only two flags are needed because fewer than $L$ calibrations cannot reach the first useful threshold, and at exactly $L$ only all-positive or all-negative increment histories reach it. An eight-cell scratch allocation is initialized before use in every stage. The paper's circuit tables identify every source and target. These counts do not assert a minimum bank size. With $m=23$, each identity tour costs $$ 2(m-1)=44. $$ An initial tour and one ordinary-operation/tour pair per window give $$ N_{\mathrm{native}}=44+45(27L+72)=1215L+3284. $$ Each of the $L+1$ raw-triple sites has one joint native ingress boundary but costs three single-write work units. That difference adds $2(L+1)$ work units: $$ N_{\mathrm{work}}=1215L+3284+2(L+1)=1217L+3286. $$ An absent calibration writes constant zeros at its allocated site and retains the same cost allowance. Both comparison arms use the same padded resources. A saved inquiry opportunity can therefore be assigned to another task without giving the adaptive arm an uncharged physical implementation. ### Conditional time and energy bounds Let $\tau_g$ bound an elementary gate window and $E_g$ bound a single-write work unit. Let $\tau_{\mathrm{port}}$ and $E_{\mathrm{port}}$ bound an acquisition interval and its energy charge. Then $$ T\leq(1215L+3284)\tau_g+(L+3)\tau_{\mathrm{port}}, $$ $$ E\leq(1217L+3286)E_g+(L+3)E_{\mathrm{port}}+E_{\mathrm{aux}}. $$ The auxiliary term includes charged clock, program, holding and other apparatus overhead. These are conditional resource upper bounds, not thermodynamic equalities. A fixed 23-cell bank does not imply constant total apparatus storage. A literal unrolled program grows linearly with $L$, and its phase counter grows logarithmically. Every source and coupler state needed for joint ingress and subsequent laws must be modeled. A serial unobserved buffer cannot be left out of that inventory. The result is an explicit route from an evaluative program to a costed physical episode. The mathematical mechanism, its information restrictions and its native returns can be preserved together. Assessing an actual device then requires checking the complete apparatus against this contract. # References and publication record The numbered references below preserve the bibliography of **Bounded Agency and Reflective Freedom: Evaluative Revision, Information Limits, and Faithful Realization**. Citation numbers in its web chapters link here. Supplementary essays identify their own dependencies and do not assign the published paper credit for additional unpublished results. The author-supplied publication is Jeremy Rodgers, Independent Researcher, 6 October 2026, Version 2.0. [DOI: 10.5281/zenodo.23202999](https://doi.org/10.5281/zenodo.23202999). [Read the supplied PDF](/publications/consciousness/agency/bounded-agency-and-reflective-freedom.pdf). ## 1. Abstracting causal models {#ref-1} Sander Beckers and Joseph Y. Halpern. *Abstracting causal models.* Proceedings of the AAAI Conference on Artificial Intelligence, 33(1):2678–2685, 2019. [doi:10.1609/aaai.v33i01.33012678](https://doi.org/10.1609/aaai.v33i01.33012678). [Article](https://ojs.aaai.org/index.php/AAAI/article/view/4117). ## 2. Certified self-modifying code {#ref-2} Hongxu Cai, Zhong Shao, and Alexander Vaynberg. *Certified self-modifying code.* Proceedings of the 28th ACM SIGPLAN Conference on Programming Language Design and Implementation, pp. 66–77. ACM, 2007. [doi:10.1145/1250734.1250743](https://doi.org/10.1145/1250734.1250743). [Author page and extended version](https://flint.cs.yale.edu/shao/papers/smc.html). Extended version: Yale Technical Report YALEU/DCS/TR-1379. ## 3. Games with imperfect information {#ref-3} Krishnendu Chatterjee, Laurent Doyen, Thomas A. Henzinger, and Jean-François Raskin. *Algorithms for omega-regular games with imperfect information.* Logical Methods in Computer Science, 3(3), article 4, 2007. [doi:10.2168/LMCS-3(3:4)2007](https://doi.org/10.2168/LMCS-3(3:4)2007). [Article](https://lmcs.episciences.org/1094). ## 4. Self-modification of policy and utility {#ref-4} Tom Everitt, Daniel Filan, Mayank Daswani, and Marcus Hutter. *Self-modification of policy and utility function in rational agents.* In Bas Steunebrink, Pei Wang, and Ben Goertzel, editors, Artificial General Intelligence, Lecture Notes in Computer Science 9782, pp. 1–11. Springer, 2016. [doi:10.1007/978-3-319-41649-6_1](https://doi.org/10.1007/978-3-319-41649-6_1). [arXiv:1605.03142](https://arxiv.org/abs/1605.03142). ## 5. Responsibility and manipulation {#ref-5} John Martin Fischer. *Responsibility and manipulation.* The Journal of Ethics, 8(2):145–177, 2004. [doi:10.1023/B:JOET.0000018773.97209.84](https://doi.org/10.1023/B:JOET.0000018773.97209.84). [Author PDF](https://andrewmbailey.com/jmf/Responsibility_and_Manipulation.pdf). ## 6. Responsibility and control {#ref-6} John Martin Fischer and Mark Ravizza. *Responsibility and Control: A Theory of Moral Responsibility.* Cambridge University Press, 1998. [doi:10.1017/CBO9780511814594](https://doi.org/10.1017/CBO9780511814594). [Publisher](https://www.cambridge.org/core/books/responsibility-and-control/54D0EB8AEDEF4D5F4930D691EC214E01). ## 7. Sequential testing under deadlines {#ref-7} Peter Frazier and Angela J. Yu. *Sequential hypothesis testing under stochastic deadlines.* Advances in Neural Information Processing Systems, volume 20, 2007. [Proceedings](https://proceedings.neurips.cc/paper/2007/hash/9c82c7143c102b71c593d98d96093fde-Abstract.html). ## 8. Adaptive submodularity {#ref-8} Daniel Golovin and Andreas Krause. *Adaptive submodularity: Theory and applications in active learning and stochastic optimization.* Journal of Artificial Intelligence Research, 42:427–486, 2011. [doi:10.1613/jair.3278](https://doi.org/10.1613/jair.3278). [arXiv:1003.3967](https://arxiv.org/abs/1003.3967). Corrected author version: v5, 2017. ## 9. Transitive factorisations {#ref-9} I. P. Goulden and D. M. Jackson. *Transitive factorisations into transpositions and holomorphic mappings on the sphere.* Proceedings of the American Mathematical Society, 125(1):51–60, 1997. [Author PDF](https://uwaterloo.ca/math/sites/default/files/uploads/documents/gjpams1997.pdf). ## 10. Smooth simulation of computation {#ref-10} Daniel S. Graça, Manuel L. Campagnolo, and Jorge Buescu. *Robust simulations of Turing machines with analytic maps and flows.* New Computational Paradigms, Lecture Notes in Computer Science 3526, pp. 169–179. Springer, 2005. [doi:10.1007/11494645_21](https://doi.org/10.1007/11494645_21). [Author PDF](https://sqigmath.tecnico.ulisboa.pt/pub/GracaDS/05-GCB-stable.pdf). ## 11. Selecting computations {#ref-11} Nicholas Hay, Stuart Russell, David Tolpin, and Solomon Eyal Shimony. *Selecting computations: Theory and applications.* Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence, pp. 346–355, 2012. [Proceedings PDF](https://www.auai.org/uai2012/papers/123.pdf). ## 12. Adaptivity gaps in Boolean evaluation {#ref-12} Lisa Hellerstein, Devorah Kletenik, Naifeng Liu, and R. Teal Witter. *Adaptivity gaps for the stochastic boolean function evaluation problem.* arXiv:2208.03810, version 1, 2022. [Preprint](https://arxiv.org/abs/2208.03810). ## 13. Discovering agents {#ref-13} Zachary Kenton, Ramana Kumar, Sebastian Farquhar, Jonathan Richens, Matt MacDermott, and Tom Everitt. *Discovering agents.* Artificial Intelligence, 322:103963, 2023. [doi:10.1016/j.artint.2023.103963](https://doi.org/10.1016/j.artint.2023.103963). [Author PDF](https://sebastianfarquhar.com/assets/papers/kentonDiscovering2023.pdf). ## 14. Can AI systems have free will? {#ref-14} Christian List. *Can AI systems have free will?* Synthese, 206:115, 2025. [doi:10.1007/s11229-025-05209-x](https://doi.org/10.1007/s11229-025-05209-x). ## 15. Unsupervised reliability calibration {#ref-15} Yao Ma, Alex Olshevsky, Csaba Szepesvari, and Venkatesh Saligrama. *Gradient descent for sparse rank-one matrix completion for crowd-sourced aggregation of sparsely interacting workers.* Journal of Machine Learning Research, 21(133):1–36, 2020. [Article](https://jmlr.org/papers/v21/19-359.html). ## 16. Chance, choice and control {#ref-16} Henry D. Potter and Kevin J. Mitchell. *Chance, choice, and control: free will in an indeterministic universe.* Synthese, 207:209, 2026. [doi:10.1007/s11229-026-05570-5](https://doi.org/10.1007/s11229-026-05570-5). ## 17. General agents need world models {#ref-17} Jonathan Richens, Tom Everitt, and David Abel. *General agents need world models.* Proceedings of the 42nd International Conference on Machine Learning, Proceedings of Machine Learning Research 267, pp. 51659–51687. PMLR, 2025. [Proceedings](https://proceedings.mlr.press/v267/richens25a.html). ## 18. Relational boundaries and awareness localization {#ref-18} Jeremy Rodgers. *Relational boundaries and awareness localization: Robustness, composition, and identification limits.* Zenodo, 2026. Preprint, version 1.0, 29 September 2026. [Full website treatment](/consciousness/research/paper-2). ## 19. Shadow Theory and Consciousness {#ref-19} Jeremy Rodgers. *Shadow theory and consciousness: Awareness, perspectival realization, and the source-to-experience problem.* Zenodo, 2026. SPC-2, version 2, publication edition, 20 September 2026. [Full website treatment](/consciousness/monograph). ## 20. Learning effective interfaces {#ref-20} Jeremy Rodgers. *Learning effective interfaces from opaque stochastic systems: Capacity, selection, and validation limits.* Zenodo, 2026. Revised preprint, version 2, 29 September 2026. [Full website treatment](/consciousness/research/paper-3). ## 21. Identifying binary realizations {#ref-21} Jeremy Rodgers. *Identifying binary realizations from intervention laws: Certificates, recoding obstructions, and a bounded SPC-2/IIT comparison.* Zenodo, 2026. Preprint, version 1.1-RC1, 30 September 2026. [Full website treatment](/consciousness/research/paper-4). ## 22. Causal consistency {#ref-22} Paul K. Rubenstein, Sebastian Weichwald, Stephan Bongers, Joris M. Mooij, Dominik Janzing, Moritz Grosse-Wentrup, and Bernhard Schölkopf. *Causal consistency of structural equation models.* Proceedings of the Thirty-Third Conference on Uncertainty in Artificial Intelligence, 2017. [arXiv:1707.00819](https://arxiv.org/abs/1707.00819). ## 23. Risk-aware partially observed planning {#ref-23} Pedro Santana, Sylvie Thiébaux, and Brian Williams. *RAO\*: An algorithm for chance-constrained POMDP's.* Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 3308–3314. AAAI Press, 2016. [doi:10.1609/aaai.v30i1.10423](https://doi.org/10.1609/aaai.v30i1.10423). [Article](https://ojs.aaai.org/index.php/AAAI/article/view/10423). ## 24. Gödel machines {#ref-24} Jürgen Schmidhuber. *Gödel machines: Fully self-referential optimal universal self-improvers.* In Ben Goertzel and Cassio Pennachin, editors, Artificial General Intelligence, Cognitive Technologies, pp. 199–226. Springer, 2007. [doi:10.1007/978-3-540-68677-4_7](https://doi.org/10.1007/978-3-540-68677-4_7). [Author page](https://people.idsia.ch/~juergen/goedelmachine.html). # Bounded perspective, real control An agent never needs to possess the whole world in order to act within it. What it needs depends on the task. A missing distinction matters when different possibilities demand incompatible responses. Other hidden distinctions can remain hidden without preventing success. This is the first extension of the published account of reflective freedom. The published paper establishes how retained commitments can govern assessment, rule installation and fresh decisions. The supplementary control models developed here ask a different supporting question: **what can an embedded system accomplish through a limited interface?** Their results belong to this extended website treatment. They are not additional claims of the published paper. ## What a perspective preserves Write the observational relation as $$ \Omega \xrightarrow{\Delta} R. $$ The source state is in $\Omega$, the aperture $\Delta$ supplies a readout, and $R$ is what the installed interface makes available. If $\Delta$ is many-to-one, several source states have the same readout. That is an exact limitation on discrimination. It does not yet specify a limitation on every possible action. For a deterministic aperture, the three relevant tests are different: | Question | Exact requirement | | --- | --- | | Can the source state be reconstructed? | $\Delta$ must be injective. | | Can a particular fact $q(\omega)$ be recovered? | $q$ must be constant on each fibre of $\Delta$. | | Can the task be completed without regret? | The source states in each fibre must share an optimal executable action. | A fibre is the set of source states giving one readout. Knowing which fibre contains the world can be enough to answer one question and insufficient to answer another. A standpoint is therefore a structured access relation, not a scalar amount of ignorance. Consider a target register holding an unknown value $x$ and a receiver prepared at a desired value $g$. The reversible swap $$ (x,g)\longmapsto(g,x) $$ sets the target exactly without identifying $x$. The unresolved value survives in the receiver. Conversely, even complete observation does not help a controller whose only permitted actuation is the identity. Adding an inaccessible spectator register changes the completeness of its world description while leaving this target task unchanged. These examples block an unrestricted inference from partial knowledge to absent agency. They also identify the right questions: which actions are compatible with the remaining uncertainty, where can unresolved information go, and which transformations can the apparatus execute? ## The exact decision obstruction Let $\Theta$ be a finite set of source conditions, $\mathcal H$ a finite record alphabet, and $Q(h\mid\theta)$ the observation channel. Let $\mathcal A$ be the admitted finite action set. A policy $\pi(a\mid h)$ can use the record but cannot read the hidden condition directly. An outcome kernel $K(y\mid\theta,a)$ and a bounded task score $u(\theta,y)$ define $$ v_\theta(a)=\sum_y K(y\mid\theta,a)u(\theta,y), \qquad v^*_\theta=\max_a v_\theta(a), \qquad \ell_{\theta a}=v^*_\theta-v_\theta(a). $$ The regret $\ell_{\theta a}$ measures the task value lost by selecting $a$ when $\theta$ is the source condition. Every row has at least one zero. Define worst-source regret by $$ R(Q,\ell)=\min_\pi\max_\theta \sum_{h,a}Q(h\mid\theta)\pi(a\mid h)\ell_{\theta a}. $$ **Theorem: exact finite task compatibility.** The same value has the dual expression $$ R(Q,\ell)=\max_{\lambda\in\Delta(\Theta)} \sum_h\min_a\sum_\theta\lambda_\theta Q(h\mid\theta)\ell_{\theta a}. $$ Moreover, $R(Q,\ell)=0$ exactly when every possible record $h$ satisfies $$ \bigcap_{\theta:\,Q(h\mid\theta)>0} \operatorname*{arg\,min}_a\ell_{\theta a}\ne\varnothing. $$ **Proof.** The policy set is a product of finite probability simplexes. Replace maximization over the source condition by maximization over source distributions $\lambda$. Finite minimax exchanges that maximization with policy minimization. With $\lambda$ fixed, the policy optimization separates over records, and a minimizing action at each record gives the dual formula. If each intersection is nonempty, choose an action in it. Every supported source-record pair then has zero loss. Conversely, zero worst-source regret means that every nonnegative term with positive $Q(h\mid\theta)\pi(a\mid h)$ has zero loss. An action given positive probability at $h$ must be optimal in every source condition compatible with that record. At least one such action exists because the policy probabilities sum to one. This proves both directions. The obstruction is **incompatible required action under the same available record**. Randomization does not remove that obstruction at zero error. Every action in a randomized policy's support must still satisfy the common requirement. Pairwise agreement is weaker than joint agreement. With a single record and loss matrix $\ell=I_3$, any two source rows share a zero-loss action, but all three do not. The minimax value is $1/3$: assigning each action probability $1/3$ attains it, and some action must receive at least that probability. Testing only pairs would miss the obstruction. ## How indistinguishability becomes a quantitative limit Total variation is $$ \operatorname{TV}(P,Q)=\frac12\sum_h|P(h)-Q(h)|. $$ It measures how well the available records can distinguish two hypotheses. Suppose two source conditions have disjoint optimal action sets and every nonoptimal action loses at least $\gamma>0$. Then $$ R(Q,\ell)\ge \frac\gamma2\left[1-\operatorname{TV}(Q_{\theta_0},Q_{\theta_1})\right]. $$ **Proof.** Their common record mass is $c(h)=\min\{Q(h\mid\theta_0),Q(h\mid\theta_1)\}$, with total mass $1-\operatorname{TV}(Q_{\theta_0},Q_{\theta_1})$. At any action, the two losses sum to at least $\gamma$ because the action cannot be optimal in both conditions. Sum over the common mass and take half the resulting total. Maximum risk is at least average risk. For several source conditions, put $$ \alpha=\sum_h\min_\theta Q(h\mid\theta), \qquad r_{\rm blind}=\min_p\max_\theta\sum_a p(a)\ell_{\theta a}. $$ Then $R(Q,\ell)\ge\alpha r_{\rm blind}$. For $\alpha>0$, the common record mass induces the same action distribution in every condition; discard the nonnegative loss on the remaining mass. For $\alpha=0$, the statement is just nonnegativity. Garbling an observation cannot improve the optimum when the actuator and resources stay fixed. If $Q'=QB$, every policy using $Q'$ can be simulated from $Q$ by first drawing the garbled record through $B$. Its policy class is contained in the original one. This finite argument is the relevant instance of [Blackwell's comparison of experiments](https://doi.org/10.1214/aoms/1177729032). The lesson for reflective agency is direct. An acceptable amendment can exist without being identifiable through the installed read ports. The published paper's common-amendment criterion applies this same structure to endorsement and continued auditability. More information can remove a conflict between compatible contexts. It cannot supply an unavailable actuator or make an inaccessible amendment executable before a deadline. ## Capability is a set of achievable consequences For a specified interface and policy-resource class $\Pi$, define $$ \mathcal K(\Pi)=\{K^\pi(\cdot\mid\theta):\pi\in\Pi\}. $$ This collects achievable outcome or transcript laws. It permits exact comparisons between a controller before and after learning, between two memory budgets, or between two operation sets. There is no need to force these comparisons into one universal number called freedom. **Theorem: a task witnesses a capability gap.** Let $\mathcal C$ be a compact convex set of finite kernels, let $\mu$ have full support, and let $K$ be a target kernel. Then $$ \begin{aligned} \min_{L\in\mathcal C}\sum_\theta\mu_\theta\operatorname{TV}(K_\theta,L_\theta) =\max_{0\le u\le1}\Bigg[&\sum_{\theta,y}\mu_\theta K(y\mid\theta)u(\theta,y)\\ &-\max_{L\in\mathcal C}\sum_{\theta,y}\mu_\theta L(y\mid\theta)u(\theta,y)\Bigg]. \end{aligned} $$ **Proof.** For probability distributions of equal mass, $\operatorname{TV}(p,q)=\max_{0\le u\le1}\sum_yu_y(p_y-q_y)$. Apply this separately to each source row and exchange the minimum over the compact convex kernel class with the maximum over the score cube by finite minimax. The inner minimum subtracts the largest comparator score. A new achievable law outside the old convex capability class therefore has a bounded task that reveals the difference. This is a precise sense in which learning can create an effective capability. It does not establish that the learner has escaped its underlying physical laws or acquired ultimate authorship of them. ## A world model can be partial and adequate If estimated action values obey $|\widehat v_\theta(a)-v_\theta(a)|\le\varepsilon$ for every value actually compared, the greedy estimated choice loses at most $2\varepsilon$ relative to that comparison's true optimum. Add and subtract the two estimated values; the greedy inequality cancels the middle difference. The premise must cover the comparison being claimed. An error guarantee on accessible comparisons does not automatically reach an inaccessible source-oracle optimum. Nor does a lossy observation automatically destroy Markov structure. For a controlled process on $X$ and a map $\phi:X\to Z$, an exact controlled quotient exists when $$ \phi_*K^a(\cdot\mid x)=\phi_*K^a(\cdot\mid x') \quad\text{whenever }\phi(x)=\phi(x'),\quad\text{for every }a. $$ Necessity follows by starting from either state in the same fibre. Sufficiency defines the quotient kernel by the common pushed-forward law and then iterates it. Memory is required when relevant distinctions fail this test, not merely because the representation is incomplete. Broad competence can demand much richer models than a single reset task. [General agents need world models](https://proceedings.mlr.press/v267/richens25a.html) supplies such necessity results under its own task and environment assumptions. A bounded perspective can support strong local competence without becoming a complete representation of the source. That is the setting in which reflective freedom has to be built and tested. # Where uncertainty goes Control changes a selected part of the world. In a reversible model, it must also preserve the distinctions carried by the initial state. A target can become orderly while information moves into a receiver, a retained record or another explicitly modeled output. This supplementary analysis gives that statement an exact finite form. It extends the website's account of bounded agency with receiver and calibration results. The published reflective-freedom paper uses different devices and resource units; their counts should not be combined into a fictitious single machine. ## The receiver capacity theorem Let a finite plant $x\in X$ have prior $\mu(x)$. A nondisturbing observation produces retained record $h$ through $Q(h\mid x)$. The receiver starts in one specified ready state and has $D$ possible final states. Conditional on $h$, the actuator can apply any permutation of plant and receiver. Additional scratch must return to fixed values; otherwise its possible final values count toward $D$. Let the target set $G\subseteq X$ have $g$ elements. For a nonnegative weight vector $w$, write $\operatorname{Top}_s(w)$ for the sum of its $s$ largest entries, or all entries if there are fewer than $s$. **Theorem: exact unrestricted receiver capacity.** $$ p^*=\sum_h\operatorname{Top}_{gD} \left((\mu(x)Q(h\mid x))_{x\in X}\right). $$ For uniform input on $K$ states with a deterministic record partition $\{C_h\}$, $$ p^*=\frac1K\sum_h\min\{|C_h|,gD\}. $$ **Proof.** Fix $h$. Distinct ready-receiver inputs require distinct outputs under a permutation. Exactly $gD$ outputs place the plant in its target. Thus at most $gD$ inputs can succeed, and the largest possible successful mass is the sum of the $gD$ greatest weights. Conversely, map those inputs injectively into successful output positions and complete the partial injection to a permutation of the finite register space. Do this separately for each retained record. The record remains unchanged during actuation. This is an exact count of available output positions. It is not a thermodynamic work law, and it does not promise an efficient circuit for every attaining permutation. For $N$ independent uniform $K$-state inputs, a common $D$-state receiver and targets of size $g$, the blind bound is $$ p_{\rm all}\le\min\{1,D(g/K)^N\}. $$ Here **blind** means that no retained record carries source information. The joint receiver is the only output allowed to retain unresolved source distinctions. Success at least $1-\varepsilon$ consequently requires $$ \log_2D\ge N\log_2(K/g)+\log_2(1-\varepsilon). $$ An informative record changes the accounting. Record a fair bit exactly as $h=x$ and use the reversible CNOT $(x,h)\mapsto(x\oplus h,h)$. The plant resets with certainty and no extra residual receiver, because the record already contains the distinction. Applying the blind bound would be wrong. With a general retained channel, use the full conditional top-mass expression. ## The quantum capacity counterpart For a density operator $\rho$ and a success projector $P$ of rank $s$, $$ \max_U\operatorname{Tr}(PU\rho U^\dagger)=\sum_{j=1}^s\lambda_j^\downarrow(\rho). $$ **Proof.** In an eigenbasis of $\rho$, put $w_j=\langle j|U^\dagger PU|j\rangle$. These weights satisfy $0\le w_j\le1$ and $\sum_jw_j=s$. The weighted eigenvalue sum is largest when the largest $s$ eigenvalues receive weight one. A unitary aligning their eigenvectors with the range of $P$ attains that choice. A maximally mixed $K$-state input therefore has success at most $gD/K$ in a $gD$-dimensional success space. The statement uses the ordinary density-operator probability rule. It does not derive that rule or permit copying an arbitrary unknown quantum state. It identifies the same capacity obligation under unitary dynamics: unresolved distinctions require physical room. ## Learning an unknown sensor Let $V=\mathbb F_2^d$, $K=2^d$, $d\ge2$, and suppose the unknown sensor is $$ f(x)=Ax+b,\qquad A\in\operatorname{GL}(d,2),\quad b\in V, $$ uniform over the affine group. A live input is uniform and independent of the sensor. Before it arrives, the apparatus can query $q$ chosen reference states and retain their readings. The task is exact regulation of the live plant to one prescribed value, with a $D$-state ready receiver and unrestricted record-conditioned permutations. **Theorem: optimized calibration–receiver frontier.** $$ p^*_{q,D}=\begin{cases} \min\{1,D/K\},&q=0,\\ \min\{1,(2^{q-1}+D)/K\},&1\le q\le d,\\ 1,&q\ge d+1. \end{cases} $$ This optimizes the reference design. For a particular nonempty transcript whose references have affine-span dimension $s$, the value is $$ p^*_{s,D}=\frac{2^s+\min\{D,K-2^s\}}K. $$ **Proof.** Without a reference, affine transitivity makes the source posterior uniform even after the live sensor reading. The capacity theorem gives $D/K$, capped at one. After a reference, translate its source position to zero and subtract its observed offset. Let $W$ be the span of reference differences, with dimension $s$. The remaining sensor ambiguity is the group fixing $W$ pointwise. In a chosen complement its matrices have form $$ \begin{pmatrix}I_s&B\\0&C\end{pmatrix},\qquad C\in\operatorname{GL}(d-s,2). $$ Every point of $W$ is fixed. All points outside $W$ form one orbit: choose $C$ to map one nonzero complement coordinate to another, then choose $B$ to adjust the $W$ coordinate. A live source in $W$ is exactly known. Outside $W$, its posterior is uniform on $K-2^s$ points. The receiver preserves at most $D$ of those successful possibilities, giving the stated fraction. Each new reference can add at most one independent difference. References $0,e_1,\ldots,e_{q-1}$ attain $s=q-1$ until $d+1$ references determine the sensor. Adaptive pre-live choice cannot exceed the same span bound. Conditional on a full calibration transcript, its choice rule adds no observation beyond the queried readings; the remaining uniform sensor posterior is a coset of the same pointwise stabilizer. Repeated readings of the same reference do not count as new independent directions. At $d=2,D=1$, two references $(0,0)$ give success $1/2$, while distinct references $(0,e_1)$ give $3/4$. Counting operations without checking what they distinguish can overstate competence. With $r$ receiver bits, $D=2^r$, this single-task optimum can be attained with conditional affine data operations. For $r\le d-1$, an appropriate $r$-flat fits in the unresolved complement and can be transferred to the receiver by affine normalization. A full $d$-bit receiver permits a blind swap. Shared tasks introduce more demanding geometry. ## Shared calibration creates correlated uncertainty One sensor acting on $N$ live inputs leaves a common residual group $G$ acting diagonally on $V^N$. Conditional on the complete readings, the source tuple is uniform on a group orbit. Hence $$ p^*_{N,G,D}=K^{-N}\sum_{O\in V^N/G}\min\{|O|,D\}. $$ This is one shared receiver, not a fresh receiver for every task. After calibration spanning $s$ dimensions, put $m=d-s$. If the complement components of the source tuple have rank $k$, their orbit size is $$ L_{s,k}=2^{sk}\prod_{j=0}^{k-1}(2^m-2^j). $$ The product counts injective images of the $k$ independent complement directions. The factor $2^{sk}$ counts their possible $W$ components. The rank probability is $$ \rho_{m,N}(k)=2^{-mN}\prod_{j=0}^{k-1} \frac{(2^m-2^j)(2^N-2^j)}{2^k-2^j}, $$ with empty products equal to one. Counting rank-$k$ matrices by image subspace and full-rank coordinate map gives this expression. Applying the receiver bound orbit by orbit yields $$ p^*_{N,s,D}=\sum_{k=0}^{\min(m,N)}\rho_{m,N}(k) \min\{1,D/L_{s,k}\}. $$ Without a reference, the corresponding affine-span orbit of $N$ points has size $2^d\prod_{j