Observer predecessor

### PDF page 26 THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY Conclusion: The Observer and the World We do not stand outside nature when we describe it. We are parts of it, interrogating it with finite bodies, instruments, interventions and records. The first task of empirical science is therefore not to announce the world’s nouns from a view from nowhere. It is to determine which distinctions available at an interface remain consequential to the future. That gives the paper its first-principles order: physical access → predictive closure → state → composition → observation geometry → reconstruction by failure. A state is a compression of history. To call two histories the same state is to wager that no declared future experiment will ever need the distinction that was erased. A law is what remains well de- fined after that compression. A geometry records what the observational lens preserves. And when prediction fails, nature has returned a distinction the representation tried to forget. Within that larger grammar , Theorem 1 gives one exact and unusually transparent observer geome- try. If common scale is unavailable and the relevant homogeneous channels transform linearly, the empirical state is projective: linear homogeneous state ⟶ Projective Shadow. On the projective line, the finite law is fractional-linear , the continuous law is Riccati, and the cross- ratio is the invariant fingerprint. This structure is not implied by boundedness, saturation, or obser- vation in general. It is earned by a specific lens. That is why the same Möbius grammar can recur in relativity, optics, networks, inference and selection without implying that those sciences share a substance: they can share an observational structure while describing different things. The Empirical Grammar exists because nature need not remain in that branch. Predictive state can be stochastic. A scalar can fail to close. A point map can fail. Cross-ratios can fail. A hidden coordinate can matter only at a longer horizon. The correct response is not to protect the curve. It is to type the failure, enlarge the state when warranted, and demand that the new description predict futures it was not built to fit. The synthetic study shows that this logic can work under controlled conditions and can fail in the in- tended direction under process noise, misspecification, correlated error , uneven sampling and weak state separation. The HepG2 experiment places the same architecture at genuine empirical risk. Its strongest possible outcome is not a beautiful fit. It is the sequence earned fast closure → controlled failure → identified missing capacity → held-out recovery. Its cleanest negative outcomes are equally valuable: rejection of the Point-Map Gate, rejection of the Projective Fingerprint, failure of the False-Noun test, or failure of the proposed enlarged state to recover prediction. The framework earns credibility only if those outcomes are allowed to kill the corresponding branch. The same discipline changes how scientific disagreement should be approached. Two theories can- not be said to conflict cleanly while they use different observational lenses, different jurisdictions, different state compressions or different horizons. Once those are aligned, either the future laws co- incide - in which case the disagreement is representational within that jurisdiction - or they separate, in which case the dispute has become an experiment. And the observer itself is no longer absent from the picture. The mathematics does not require con- sciousness, but human knowledge is consciously lived and experimentally situated. A mature science should neither elevate that fact into metaphysical magic nor erase it in the name of objectivity. It should declare the channel, declare the jurisdiction, and ask what predictive distinctions survive. ### PDF page 27 THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY Nature does not owe science its preferred nouns. A noun earns the name *state* only when the future can no longer distinguish the histories it compresses. A geometry earns phys- ical meaning only when the lens that induces it has been identified and tested. A hidden dimension earns reality in the model only when restoring it recovers held-out law. We are inside the world we measure. Begin there. Earn state. Test the lens. Let failure return what was forgotten. Let the future decide. This is the intended meaning of a grammar of empirical science . A grammar does not dictate ev- ery sentence nature can speak. It specifies how valid descriptions are constructed, how apparently different descriptions can be recognized as equivalent, and how contradiction forces revision. The ambition of the programme is therefore not to impose one geometry on every science. It is to make ex- plicit the conditions under which state, geometry and disagreement become scientifically meaningful at all. If that standard survives contact with natural systems and unrelated scientific controversies, its reach will not come from claiming universality in advance. It will come from becoming difficult to do care- ful science without first asking the question on which this paper begins: What can this observer distinguish, and which of those distinctions does the future still remember? Appendix A The Empirical Grammar in Practice The Empirical Grammar places Theorem 1 inside the broader first-principles order of predictive clo- sure, observer geometry and failure-driven reconstruction. It is placed in the appendix to keep the main text theorem-centred. The order matters because each later structure is licensed only after the earlier one has survived. EMPIRICAL GRAMMAR Earn state • • use failure to reconstruct • let held-out prediction decide 1 2 3 4 RAW ACCESS actions + records JOINT PREDICTIVE REDUCTION EMPIRICAL STATE + action quotient SCALAR CLOSURE? REGULAR CLOSURE Aczel coordinate LAWFUL ERASURE semigroup / memory FALSE NOUN same present, different future POINT-MAP GATE deterministic next state? PROJECTIVE FINGERPRINT cross-ratio + rank LINEAR LIFT / GEOMETRY FAILURE TOMOGRAPHY reversal + cycle circulation STATE ENLARGEMENT add missing distinction if stochastic: use transition kernels, not point geometry HELD-OUT FUTURE PREDICTION select the smallest adequate architecture new instruments, policies and longer horizons refine access test lens Figure 4 : The Empirical Grammar . Physical access is reduced to predictive state and effective jurisdiction before coordinates or geometry are assigned. Scalar closure, the Point-Map Gate, the Projective Fingerprint, and projective rank are empirical tests. Failure feeds back through state enlargement, and every branch returns to held-out future prediction. Step 1 Declare the physical access layer List what the apparatus can actually do and record, together with the background assumptions needed to treat those actions and records as reproducible. The repertoire is theory-laden; the latent ### PDF page 28 THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY state geometry is not assumed merely because the apparatus has coordinates. Step 2 Learn predictive state and effective jurisdiction jointly Estimate future-law features, reconstruct the coarsest supported history classes, and quotient inter- ventions that are future-law indistinguishable. Include adaptive policies as probes when useful; the policy-closure proposition guarantees that the same predictive state notion applies. Step 3 State the horizon, resolution and estimator Report the future-test family, predictive metric, uncertainty method, clustering/model-selection rule and held-out split. Approximate closeness is not an equivalence relation. Step 4 Test proposed state variables If matched proposed states produce different future laws, the variable is not sufficient. Enlarge the state before assigning geometry. Step 5 Test scalar sequential closure • no single-valued composition: Case I, state failure; • associative but nonregular/non-cancellative composition: Case II, lawful erasure/semigroup; • continuous strictly monotone closure: Case III, reconstruct the Aczél natural coordinate 𝑟 = 𝜙(𝑥). Step 6 Pass the Point-Map Gate Estimate the transition kernel 𝐾𝑢(𝑠, ⋅). If next predictive state retains non-degenerate stochastic spread beyond reconstruction error , reject deterministic point geometry for that representation. Model the kernel or enlarge state instead. Step 7 Test the Projective Fingerprint Rather Than Fitting It For a deterministic one-dimensional branch, test cross-ratio preservation on unused quadruples. A pass licenses a Möbius representation and an abstract two-channel homogeneous lift. A failure leaves a non-projective regular branch unless state adequacy itself has failed. Step 8 Select projective dimension under penalty When richer predictive coordinates exist, compare ℙ𝑑−1 candidates only up to a predeclared 𝑑max. Choose the smallest dimension supported by held-out prediction or a declared complexity penalty. Do not use unbounded rank as an unfalsifiable repair . Step 9 Search for physical homogeneous channels An abstract lift does not identify its physical coordinates. Ask whether independently measurable domain variables transform linearly/homogeneously in the required way. Failure to find them is a physical-lift failure, not a mathematical contradiction. Step 10 Earn smooth calculus Use Lie brackets only after the reconstructed predictive-law family has an identifiable smooth finite- dimensional constant-rank realization. Otherwise use discrete graph cycles and nonparametric pre- dictive tests. Step 11 Use Failure Tomography Where smoothness holds, test the 𝑎𝑏 reversal law and iterated commutators. At finite amplitude or without smoothness, test graph-cycle circulation. Candidate new coordinates are hypotheses gener- ated by the defect, not automatically real dimensions. Step 12 Demand held-out recovery Freeze the enlarged architecture and predict future records under histories/policies that were not used to construct it. A missing variable earns state status only if it improves held-out predictive law after complexity is accounted for .

velops a first-principles grammar for what follows from that fact. Relative to a declared access layer , histories are identified when every admissible future test assigns them the same future law; the resulting predictive quotient is the coarsest state sufficient for that jurisdiction. Raw interventions can be reduced by the same criterion. State is therefore earned from surviving predictive distinctions before coordinates or geometry are assigned. Within this architecture, Theorem 1 isolates an exact observation geometry. When common scale is unavailable and a homogeneous two-channel state transforms linearly, the observable ratio or nor- malized contrast necessarily transforms fractionally linearly and preserves cross-ratio. Conversely, ment. A reproducible synthetic unit test demonstrates quotient recovery, projective identification, controlled hidden-state failure, robustness under noise and misspecification, and held-out recovery. A quantitatively preregisterable HepG2/Grx1-roGFP2 protocol supplies a prospective natural-system test. The larger methodological implication is that scientific disagreement is well posed only after the ob- servational lens, intervention jurisdiction, predictive state, horizon, and future records have been made common. The paper does not claim that all observation is projective or that consciousness gen- erates physical law. It claims something more elementary: we do not stand outside nature when we describe it. A scientific state is a compression of history; a geometry records what a lens preserves; and failed prediction is evidence that the compression erased a distinction the future can still reveal. Claim architecture and evidential status

Layer Claim Status What can de- feat it Predictive closure Histories with identical future laws define the minimal predictive state relative to declared access established con- struction / con- stitutive decision archi- tecture prospective failure to iden- tify or recover predictive state under declared gates Synthetic study HepG2 redox A fast projective jurisdiction may fail when slower NADPH/NRF2 ca- pacity becomes predictive prospective hy- pothesis any prereg- istered gate failure every distinction between them has become irrelevant to every declared future test. If the future later separates them, the promise was false and the state was too small. In this sense a state is surviving predictive history: a compression of the past justified only by closure of the future. Geometry is downstream of that compression. One particularly important lens discards common scale. Suppose a latent description contains two nonzero homogeneous channels 𝑧 = ( 𝑧1 𝑧2

trary compression of a linear system is not generally projective. Theorem 1 applies when the obser- vation map is homogeneous or scale-blind in the stated sense. The power of the claim comes from this narrowness: when those hypotheses are earned, the geometry is not chosen. Corollary 1 The Projective Equivalence Principle Theorem 1 immediately explains why the same Möbius law can recur in sciences that share no phys- ical substrate. Corollary 1 (Projective Equivalence Principle). If two empirical domains independently

Γ, impedance Smith-chart Möbius maps [15,16] Binary inference two positive hypothesis weights posterior odds additive log-odds Two-allele fixed selection two type abundances allele-frequency odds additive log relative fitness [17] supplies linear homogeneous channels, and normalization removes common scale. The projective law is then the same representation-theoretic consequence. The catalogue therefore validates the typing of the mechanism , while the synthetic and prospective biological tests are where the frame- work itself takes discovery risk. A particularly useful comparison is binary inference versus fixed-fitness selection. Both obey new odds = multiplicative factor × old odds, so their log-odds are additive even though one factor is a likelihood ratio and the other a relative explained coincidence: scale-blind observation of linear homogeneous state necessarily casts a Projective Shadow. The cross-ratio is its operational fingerprint. Failure Tomography is the protocol for discovering predictive structure that lies beyond that shadow. Projectivity is not boundedness The observer-centered theorem also identifies an important limit. A bounded observable does not, merely by being bounded, inherit Möbius composition or hyperbolic geometry. Let

From observer theorem to empirical grammar The universal kernel and the exact branch. The Projective Shadow is not the universal layer of the framework. **Predictive closure is.** Every application first asks which histories and interventions remain distinguishable by future law. Projective geometry enters only after a sufficient state has been earned, a point map is licensed, and scale-blind homogeneous linear structure survives testing. The Observation Projection Theorem begins with a state on which transformations act. Empirical science does not generally begin with such a state already identified. Before asking whether a state transforms projectively, one must determine what the state is, what interventions are effectively dis- tinct, whether one scalar closes, and whether the next predictive state is sufficiently deterministic for a point transformation to be meaningful. That is why the rest of this paper has the form of a grammar rather than a universal projective model. The theorem supplies the observer geometry for one sharply defined branch. The grammar deter- mines when an empirical system is entitled to enter that branch and what to do when it does not. The order is physical access → predictive distinction → state → closure → Point-Map Gate → geometry. When a proposed compression fails, the future itself identifies the problem: same recorded present + different future ⇒ the representation forgot something predictive . That is the False-Noun Criterion. Under the appropriate smoothness conditions, reversal defects probe Lie-bracket directions; at finite amplitude, cycle circulation tests global scalar integrability. Together these constitute Failure Tomography. No inferred variable is accepted merely because it repairs a fit: the enlarged state must improve prediction on data that were not used to construct it. The workflow is therefore not the primary discovery. It is the operational consequence of the observer theorem: Earn state. Test the lens. Use failure to reconstruct. Let held-out prediction decide. The full twelve-step implementation and decision diagram are collected in Appendix A, so that the theorem remains visually primary in the main argument. What the Grammar Adds Beyond Causal States and PSRs The predictive-state layer is established work. Causal states, epsilon-transducers, and predictive-state representations construct future-sufficient state from observations and controlled action-observation sequences [1-3]. CSSR and spectral PSR methods provide concrete learning procedures under explicit assumptions [24,25,26]. The Empirical Grammar is not a replacement estimator . It is a decision layer placed around those estimators. here What state is sufficient for future prediction? Core problem; established Uses the learned state as the starting object, not as the conclusion

geometry be fitted? Not a canonical PSR step Point-Map Gate requires the next predictive state to be sufficiently concentrated around a point map What observational condition selects projective geometry? Not supplied by predictive-state reconstruction Observation Projection Theorem: scale-blind access to homogeneous linear state forces the standard construction Projective Fingerprint: held-out Möbius prediction plus cross-ratio preservation What happens when one scalar is inadequate? State splitting/model refinement is complexity-penalized before the final test The distinction is therefore precise. Predictive-state methods answer what must be remembered for prediction. The additional grammar asks what operations are effectively distinct, whether a point geometry is licensed, which geometry is forced by the observer’s lens, and what struc- tured experiment should follow when a proposed representation fails . The projective mathe- matics itself is classical; the contribution is the observer-world typing and the sequence of empirical gates that makes that branch falsifiable. The forward algebra of Theorem 1 remains intentionally elementary. Scientific content resides in identifying scale-blindness as the observational condition that forces projective geometry and in turn- ing cross-ratio preservation into a held-out test rather than a coordinate choice. Where State Begins: Actions, Records, and the Birth of the Quotient Before jurisdiction: the raw operational repertoire The main apparent circularity in an operational reconstruction is immediate. Predictive state is de- fined relative to a family of admissible future tests, but how can those tests be specified without already knowing the world’s state space and causal structure? The answer is to distinguish an operational repertoire from an effective jurisdiction. Let 𝒰0 be a raw set of intervention labels corresponding to procedures the apparatus can physically some operations executable and some records measurable; even choosing a pulse amplitude rather than a waveform is a modeling decision. The claim here is therefore relative, not foundational in the sense of escaping all prior theory: once a physical access layer has been built, predictive state can be reconstructed without additionally assuming the state geometry that the reconstruction is meant to test. New instruments change that access layer and can refine the resulting state. A finite history is written ℎ𝑡 = (𝑢1, 𝑦1; … ; 𝑢𝑡, 𝑦𝑡),

when 𝑃 (𝐴 ∣ ℎ, 𝑢1∶𝑘) = 𝑃 (𝐴 ∣ ℎ′, 𝑢1∶𝑘) for every admissible future sequence and every future record event. The predictive state is 𝑆(ℎ) = [ℎ]∼. This is the controlled predictive-state or causal-state construction [1-3]. Intervention equivalence Two raw interventions 𝑢, 𝑣 ∈ 𝒰0 are predictively equivalent, written 𝑢 ≈ 𝑣, if replacing one by the other never changes the law of the complete future record, for any history and any later continuation. Explicitly, for every history ℎ, continuation 𝑤2∶𝑘, and future record event 𝐴, 𝑃 (𝐴 ∣ ℎ, 𝑢, 𝑤2∶𝑘) = 𝑃 (𝐴 ∣ ℎ, 𝑣, 𝑤2∶𝑘). The effective jurisdiction is the quotient 𝒥⋆ = 𝒰0/ ≈ . Thus physically different buttons that are predictively indistinguishable occupy the same operational action class. Minimal joint predictive reduction Relative to the declared access layer and horizon, the history and intervention quotients are the coarsest representations preserving all future-law distinctions. If a history representation 𝑅 is suf- ficient, 𝑅(ℎ) = 𝑅(ℎ′) ⇒ ℎ ∼ ℎ′, then its fibers refine predictive equivalence and there is a unique map on im 𝑅 such that 𝑆 = 𝑓 ∘ 𝑅. The same factorization holds for any sufficient intervention representation and 𝒥⋆. The direction matters: a richer representation may retain distinctions prediction does not need, but every sufficient

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY representation can be compressed to the predictive quotient. The construction is canonical relative to the physical access layer , which remains theory-laden and may change when new instruments are built. Adaptive policies do not require a second state definition For measurable non-anticipating policies, open-loop predictive equivalence is already sufficient. If two histories induce identical future-record laws for every fixed action sequence through horizon 𝑇 , then induction over the record prefix gives identical laws under every policy whose next action depends only on the observable past. Fixed sequences are degenerate policies, so the converse is immediate. Adaptive experimentation therefore changes which distinctions are efficiently dis- covered, not the definition of predictive state. Jurisdiction refinement New instruments enlarge the executable repertoire. If 𝒰1 ⊆ 𝒰2, equivalence under the richer reper- toire implies equivalence under the poorer one, giving a canonical surjection 𝑆𝒰2 . Thus scientific state spaces form a refinement system rather than a fixed ontology: new actions or records can split an old predictive class without making it wrong in its former jurisdiction. Learning predictive state and jurisdiction from finite data Exact predictive equivalence is a population object. With finite data, choose a distance 𝐷 between conditional future laws and define finite-horizon predictive and intervention pseudometrics. Be- cause 𝜖-closeness is not transitive, finite-resolution state is a model-selection problem, not an exact quotient produced by thresholding noisy pairwise distances. This connects directly to probabilistic bisimulation metrics, CSSR, and spectral PSR learning [21,22,24,25,26]. Algorithm 1. Confidence-aware predictive reconstruction 1. Predeclare the horizon, probe policies, observation resolution, estimator , and reconstruc- tion/validation/test splits. 2. Estimate conditional future laws for sampled histories and probes, with uncertainty. 3. Cluster by a diameter-controlled rule such as complete linkage rather than single-link chaining. 6. Select new probes for expected information gain, then freeze the architecture before the final held-out test. Under a finite separated-state model, uniform consistency of the predictive-law estimates and a threshold lying strictly between within-state estimation error and the minimum between-state separation are sufficient for complete-link recovery of the true partition with probability tend- ing to one. This is a standard margin argument, not a universal sample-complexity theorem. Continuous-state, weakly separated, long-memory, and adaptively collected processes require the around those estimators. State adequacy and the false-noun criterion Scientific practice rarely manipulates the predictive quotient directly. It proposes variables 𝐿 ∶ ℋ → 𝒳, such as dose, temperature, concentration, velocity, genotype frequency, disease stage or a vector of biomarkers. State adequacy criterion

the measurement has forgotten a distinction the future can still reveal. The remedy is structural, not rhetorical: enlarge the representation to (𝐿, 𝑍), or to a richer object, until future prediction closes. The statement state = surviving predictive history is therefore not an ontological claim about what reality is made of. It is the canonical answer to a modeling question: which distinctions from the past remain necessary for the declared future tests? When One Number Is Enough: The Closure Test Sequential composition and the closure trichotomy laws, the scalar is not a sufficient state. Case II: lawful scalar closure outside the regular additive class. A single-valued associative law exists but cancellativity, strict monotonicity or another regularity hypothesis fails. Idempotent, absorbing and max/min-like semigroups are examples. Such laws can erase distinctions irreversibly while remaining lawful. Case III: regular ordered scalar closure. If 𝐹 is continuous and strictly monotone in each argu- ment on an interval, an additive natural coordinate exists.

of all physical theories. Aczél representation for regular closure (established) Under the Case III hypotheses there exists a continuous strictly monotone 𝜙 such that 𝜙(𝐹 (𝑥, 𝑦)) = 𝜙(𝑥) + 𝜙(𝑦). Equivalently, 𝐹 (𝑥, 𝑦) = 𝜙−1(𝜙(𝑥) + 𝜙(𝑦)). Thus additivity can be the natural coordinate representation of lawful composition rather than a tion family preserves. Earning the Right to Use Geometry: The Point-Map Gate The predictive quotient is generally stochastic. After an action 𝑢, the next predictive state may branch because the next record is random. Let 𝐾𝑢(𝑠, 𝐵) = 𝑃 (𝑆𝑡+1 ∈ 𝐵 ∣ 𝑆 𝑡 = 𝑠, 𝑢𝑡 = 𝑢) be the induced transition kernel on predictive states.

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY Deterministically controlled predictive systems A jurisdiction is deterministically controlled at the chosen sampling event if for every effective action 𝑢 and state 𝑠 there exists a point map 𝑓𝑢 such that 𝐾𝑢(𝑠, ⋅) = 𝛿𝑓𝑢 (𝑠). Only under this condition, or a declared approximation to it, is it legitimate to treat an intervention action should yield next-state dispersion compatible with the declared observation/reconstruction error . If a non-degenerate transition kernel remains after accounting for that uncertainty, a deter- ministic projective point map is rejected. One must then model 𝐾𝑢 directly, or find a richer predictive state in which the transition becomes deterministic; applying a Möbius fit to a conditional mean is not a substitute for this gate. No universal numerical Point-Map threshold is implied by the theory. In any application the gate is a domain-specific equivalence test : its margin must be fixed before confirmatory testing from HepG2 tests, not a constant of the grammar . This distinction separates two questions that are often conflated: predictive sufficiency is compatible with stochasticity; point geometry is not automatic. The Projective Shadow: Proof, Fingerprint, and Higher-Rank Gener- alization Proof and generalization of Theorem 1 Theorem 1 was stated at the outset because it is the mathematical centre of the paper . This section

𝜂𝑑 ∶ 𝑆 → ℙ 𝑑−1 and action matrices 𝑀𝑢,𝑑 acting projectively on that embedding. Fit each candidate on reconstruc- tion data and score it on held-out future laws with a proper predictive score. Select the smallest 𝑑 whose held-out performance is statistically indistinguishable from the best candidate, or equivalently use a predeclared MDL/BIC-type complexity penalty when its likelihood assumptions are appropriate. Two safeguards are essential. 1. Predeclare 𝑑max. Failure of every candidate up to 𝑑max rejects the finite-rank projective family within the searched jurisdiction rather than licensing unlimited rank inflation. 2. Demand identifiable coordinates. A scalar record does not by itself identify an arbitrary ℙ𝑑−1 state. Increasing 𝑑 requires additional predictive coordinates - measured variables, delay coordi- nates, or other independently validated features - sufficient to distinguish the higher-dimensional rays. The cross-ratio test is therefore the especially sharp 𝑑 = 2 case, not a promise that any failed scalar law can be rescued by adding enough projective channels. In dimensions 𝑑 > 2, the invariant struc-

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY The operationally complete branch here is therefore 𝑑 = 2 . Higher rank is tested only when ad- ditional predictive coordinates are independently identifiable and the predeclared complexity rule supports them. Continuous projectivization and Riccati dynamics Let ̇ 𝑧 = 𝐴(𝑡)𝑧, 𝐴(𝑡) = (𝑎 𝑏

to the simulator but withheld from the reconstruction. The workflow is required to recover a quo- tient, identify a projective law out of sample, fail when a hidden channel is introduced, diagnose that failure, and restore prediction only after the missing coordinate is admitted. Stage A recover state and effective action classes Six distinct latent rays were each instantiated at three different common scales, producing 18 raw histories. Future-law features were estimated under three probe actions. Because common scale is predictively irrelevant in the two-channel system, diameter-controlled clustering recovered exactly six predictive states from the 18 histories. The largest within-state feature distance was 0.0016, while the smallest between-state distance was 0.246. The raw action repertoire also contained two physically distinct matrices, 𝐴 and 𝐴⋆ = 2.35𝐴. They produce identical projective transformations because the common matrix scale cancels. Their esti- mated intervention distance was 0.0006, compared with at least 0.056 for every non-equivalent action CR(𝑞1, 𝑞2; 𝑞3, 𝑞4) ∣∣ was 0.000434, with 95th percentile 0.003566. This is the operational fingerprint; the Möbius fit is then the corresponding predictive representation. Stage C Same Shadow, Different Futures A third positive channel 𝑧3 was introduced and coupled into the next-step evolution of 𝑧1, 𝑧2, while the observer retained only 𝑥 = 𝑧1 − 𝑧2 𝑧3/(𝑧1 + 𝑧2). The best scalar Möbius model deteriorated to held-out RMSE 0.0416. When 𝜌 was admitted as an additional predictive coordinate, the appropriate two-coordinate fractional-linear model reduced held-out RMSE to 0.000271, a factor of 153.4. A complexity-penalized comparison strongly favored the enlarged model in this synthetic setting (ΔBIC ≈ 4195relative to the scalar model).

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY Δ𝑥 ≈ 𝜅𝑎𝑏, with 𝑅2 = 0.99995 . The fitted coefficient was -0.3771, close to the differential prediction -0.3856 from the commutator at the initial state; the remaining difference is higher-order in the finite ampli- tudes. −0.2 0.0 0.2 0.4 estimated future under probe A 0.4 estimated future under probe B 18 raw histories -> 6 predictive states A Predictive quotient is recovered 0.4 0.6 Möbius prediction RMSE = 0.0003 median |Δ log CR| = 0.0004 B Cross-ratio fingerprint -> held-out Möbius law −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 0.4 0.6 augmented-state prediction augmented RMSE = 0.0003 153.4x error reduction D State enlargement restores prediction A A* B C A A* B reversal: R2=1.000 Synthetic reconstruction: quotient, fingerprint, failure, recovery Figure 2: End-to-end synthetic unit test. (A) Eighteen raw histories collapse to six predictive states; the inset shows that two physically different but projectively identical interventions also collapse. (B) The cross-ratio fingerprint and a frozen Möbius law succeed out of sample. (C) Adding an unobserved third channel creates the defining false-state signature: the same present scalar supports different futures. (D) Adding the missing coordinate restores held-out prediction; the inset independently verifies leading reversal-bracket scaling. Table 3: Synthetic unit-test summary. Each stage is frozen before its held-out score is evaluated. Stage What the procedure must recover Held-out / separation metric Predictive quotient 18 raw histories → 6 predictive states max within-state distance 0.0016; min between-state 0.246 Action quotient 4 raw interventions → effective action classes 𝑑(𝐴, 𝐴⋆) = 0.0006; nearest non-equivalent pair ≥ 0.056 4195 Failure Tomography reversal defect vs. 𝑎𝑏 𝑅2 = 0.99995 ; fitted 𝜅 = −0.3771, local prediction - 0.3856 The simulation is not evidence that nature is projective. It is stronger in a different sense: it verifies that the proposed procedure can recover a quotient, identify a projective branch, reject an insuf-

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY ficient scalar , detect a local order defect and recover prediction after state enlargement. The exact script, random seed and generated tables accompany the manuscript. Stage E Robustness under noise, misspecification, and uneven sampling A decision procedure that works only in its generating model is not useful. The frozen simulation was therefore subjected to four additional stress tests. The point-map stress test used 5,000 latent states BIC∗ = 𝑛 log(RSS/𝑛) + 𝑘log 𝑛, where 𝑛 is the held-out sample size and 𝑘 the fitted parameter count. It is used only as a transparent penalty for otherwise indistinguishable predictive fits. The full script and CSV outputs are supplied in the supplement ( robustness_stress_tests.py). Stress test Result Interpretation Latent process noise in the two-channel system The quotient is not dependent on balanced sampling when predictive separation remains large The correlated-error stress test also constructs 299 pairs whose observed current coordinates match within 0.02 while their hidden capacities lie in opposite quartiles. Their median absolute future di- vergence is 0.0916. Thus the False-Noun Criterion survives substantial correlation between measure- ment error in the visible and hidden coordinates. A fifth stress test attacks the quotient itself rather than the projective branch. Six true predictive states were compressed toward one another while observation noise and sampling were held fixed (six his- tories per state, 40 probe replicates per action, observation SD 0.012). Identifiability was summarized by the separation margin 𝑚sep = 𝑑between ful: the gates fail in the intended direction. Process noise eventually blocks deterministic point geom- etry; flexible scalar misspecification cannot hide a missing state dimension; a correct enlarged state restores prediction; and a larger projective model is penalized when it adds no predictive value.

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY When the Shadow Fails: Tomography of Forgotten Dimensions When does a predictive quotient become a smooth manifold? History quotients require no topology; Lie brackets do. A smooth calculus is licensed only when the reconstructed family of future laws admits a locally injective finite-dimensional 𝐶𝑟 parameteriza- tion with constant rank and the usual embedding regularity. Under those standard conditions, the predictive-law image inherits a smooth manifold structure and smoothly parameterized interven- tions define vector fields on it. If the predictive image is discrete, singular , stratified, fractal, or discontinuous, the bracket approxi- mation is not used. The finite graph-cycle tests below remain meaningful without inventing a tangent space. Smoothness is therefore an empirical branch condition, not an assumption hidden in the word “history.” Local failure tomography: reversal and Lie brackets Assume now that the smooth predictive-realization conditions above apply and that two small inter- ventions are represented by flows generated by vector fields 𝑋𝐴 and 𝑋𝐵. Write 𝐴𝑎 = 𝑒𝑎𝑋𝐴 , 𝐵 𝑏 = 𝑒𝑏𝑋𝐵 . The Baker-Campbell-Hausdorff expansion gives log(𝐴𝑎𝐵𝑏) = 𝑎𝑋𝐴 + 𝑏𝑋𝐵 + 𝑎𝑏 Iterated commutator sequences probe 𝑋𝑖, [𝑋 𝑖, 𝑋𝑗], [𝑋 𝑖, [𝑋𝑗, 𝑋𝑘]], … and geometric control theory relates the rank of this Lie closure to local accessibility. In a predictive- state experiment this becomes a tomography programme: vary the order and amplitude of interven- tions and determine how many independent directions the future can distinguish. Global failure tomography: graph cycles and integrability Smooth local analysis is not always available. Finite history contrasts provide a complementary test.

The word tomography is reserved here for a defect that is turned into a predeclared reconstruction loop rather than merely reported as a residual. 1. Earn smoothness before brackets. Fit local predictive-law coordinates on training data and re- quire stable finite dimension, local injectivity and approximately constant rank under resampling. If those conditions fail, do not infer tangent directions; use the discrete cycle route. 2. Run a local reversal scan. Choose perturbation families 𝐴𝑎, 𝐵𝑏, randomize 𝐴𝐵versus 𝐵𝐴, and repeat on a shrinking amplitude grid. Regress Δ𝑌 /(𝑎𝑏)against amplitude. Retain a bracket di- state graph, then test a basis of independent cycle sums. Circulation compatible with zero leaves a scalar node potential admissible; persistent circulation rejects global scalar integrability. 5. Demand predictive recovery. Domain variables suggested by the defect are added on training data, frozen, and accepted as state only if they improve held-out future laws after complexity control. Failure Tomography is therefore a diagnostic-to-reconstruction protocol. A nonzero bracket or cycle defect is evidence that a proposed representation is structurally incomplete; it is not by itself a unique A Quantitative Biological Trial: HepG2 Glutathione Redox Across Two Jurisdictions The prospective biological test is now specified as a registered-report-style protocol rather than a narrative programme. Its purpose is to place three claims independently at risk: (i) a short-window Projective Shadow exists for glutathione redox, (ii) the same scalar becomes insufficient when reduc- tive/adaptive capacity changes, and (iii) measured enlargement of state restores held-out prediction. System, sensor, and pre-registered windows The primary system is HepG2 human hepatoma cells (ATCC HB-8065) stably expressing cytoso- lic Grx1-roGFP2, a genetically encoded probe that equilibrates rapidly with the glutathione redox couple and supports second-to-minute live-cell measurements [27,28]. HepG2 is chosen because an

State recovery succeeds only if held-out RMSE improves ≥30% and ΔBIC ≥ 10 Prospective HepG2 / Grx1-roGFP2 test: two jurisdictions, four predeclared gates Figure 3: Pre-registered biological test. The fast jurisdiction tests the Point-Map Gate and Projective Finger- print under a fixed 50 𝜇M H2O2 pulse. A G6PDi-1 arm creates a matched-present hidden-capacity challenge. The 4-24 h TBH/NRF2 arm tests whether adding NADPH and NRF2-capacity coordinates restores held-out prediction. Thresholds shown here are fixed before confirmatory acquisition. Primary fast-window test: the Projective Fingerprint For the sensor signal, define a calibrated bounded redox coordinate 𝑥 ∈ (−1, 1) from the Grx1- roGFP2 oxidation fraction; in parallel destructive wells quantify GSH, GSSG, and NADP +/NADPH by targeted LC-MS or validated enzymatic assays. The physical two-channel hypothesis is written in reducing-equivalent units as 𝑅 = [ GSH], 𝑂 = 2[ GSSG], 𝑥 = 𝑅 − 𝑂 𝑅 + 𝑂 . The Projective Fingerprint is not tested by taking four successive time points on one trajectory. It the best deterministic map must satisfy 𝑠point ≤ 0.020. If this gate fails, the deterministic projective hypothesis is rejected for that jurisdiction and cross-ratio fitting is not used to rescue it. If the gate passes, two co-primary projective criteria are evaluated: 1. Held-out Möbius accuracy: replicate-level held-out RMSE in 𝑥 must be ≤ 0.030.

The one-sided 95% upper confidence bound for the mean replicate-level 𝐷CR must be <0.05. This corresponds to a predeclared approximately 5% multiplicative tolerance in the cross-ratio finger- print. This 0.05 margin is likewise a domain-specific preregistered equivalence margin, not a universal definition of projectivity. The projective branch is called supported only if the Point-Map Gate and both co-primary criteria pass. Failure of any gate is reported as failure, not repaired by increasing model rank after seeing the final test data. absolute difference in replicate-level mean 𝑥 at 10 min after challenge must exceed 0.08, with the 95% confidence interval excluding zero. This test is deliberately directional only in magnitude; the framework predicts separation, not which pretreatment must have the larger signed response. Adaptive-window challenge: NRF2-dependent capacity For the 4-24 h jurisdiction, scrambled-control and 70 nM NRF2-siRNA HepG2 cultures [31] receive 200 𝜇M TBH and are measured at 4, 8, and 24 h, matching an established HepG2 oxidative-stress exposure schedule [29]. Parallel wells quantify: • viability, with a predeclared requirement of ≥ 90% of vehicle for the trajectory to remain in the intended non-lethal jurisdiction. The enlarged predictive state is specified before fitting as 𝑆enlarged = (𝑥, NADP+/NADPH, 𝐶 NRF2), where 𝐶NRF2 is the first standardized principal component of the four predeclared NRF2 target tran- scripts, with the loading vector estimated on training replicates only. Success requires leave-one-biological-replicate-out held-out prediction to improve by at least 30%in RMSE relative to the scalar 𝑥-only model, together with ΔBIC ≥ 10 in favor of the enlarged state. If the enlarged state does not meet both criteria, the proposed recovery is rejected even if individual coef- ficients are significant.

post hoc. • Cell-state heterogeneity. Cell-cycle state, baseline metabolic capacity and viability may remain predictive even at matched visible redox coordinate. If they explain held-out divergence better than the proposed NADPH/NRF2 variables, the intended enlargement is rejected or expanded un- der a new preregistration. • Damage-induced jurisdiction change. A perturbation that causes irreversible damage, loss of viability or discontinuous transcriptional state change may leave the proposed smooth jurisdiction rather than merely expose a hidden coordinate. Such observations are typed as jurisdiction failure, not forced into the projective branch. State recovery also requires orthogonal validation. Any recovered capacity coordinate must predict not only future Grx1-roGFP2 trajectories but independent biochemical readouts already specified in the design, including NADP+/NADPH and GSH/GSSG measurements. The same training/held-out split is used for both classes of endpoint. Power, multiplicity, and preregistration power for a paired standardized effect 𝑑 = 1.0. These assumptions are planning alternatives, not empirical estimates of the expected HepG2 effect . The quoted power therefore characterizes the preregistered design conditional on those alternatives. A blinded technical run-in may estimate assay variance but may not change the hypothesis directions, margins, time windows, effect targets, or held-out analysis after confirmatory data are opened. The gate order is hierarchical, which controls interpretation without post hoc multiplicity fishing: Point-Map Gate -> Projective Fingerprint -> False-Noun test -> state-enlargement test. All exclu- sions, image-quality thresholds, cell tracking rules, cross-ratio quadruple separation criteria, model formulas, random seeds, and held-out folds are to be preregistered before confirmatory acquisition. Decisive outcomes The experiment has four clean outcomes: 1. Fast success, long failure, successful recovery: the strongest confirmation of the jurisdiction- dependent grammar . 2. Fast projective failure: rejects the proposed two-channel linear lift even before adaptive capac- ity is invoked. 3. Scalar failure without recovery: confirms that the redox scalar is insufficient but falsifies the proposed NADPH/NRF2 enlargement. 4. No scalar failure on extension: narrows the claim by showing that the redox coordinate re- mains sufficient over the tested adaptive jurisdiction. No outcome is labeled a success merely because a more complicated model can be fitted. The biolog- ical result is accepted only through predeclared held-out prediction. This protocol is grounded in established Grx1-roGFP2 redox sensing [27,28], a published HepG2 oxidative-stress time course [29], cell-active G6PD inhibition [30], HepG2 NRF2 knockdown [31], and independent HepG2 NRF2 timing data [32]. It is nevertheless a prospective test : none of those studies tested the Observation Projection Theorem or the complete failure-and-recovery sequence proposed here.

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY Where the Present Result Ends The strongest claims in this paper are intentionally asymmetric. The predictive quotient is a repre- sentation construction relative to stated future laws. The Observation Projection Theorem is exact in its two-channel scale-blind linear branch. The Empirical Grammar is a methodological architecture whose external validity remains to be established prospectively. Four boundaries matter most. First, finite-sample identifiability can fail before geometry is reached; the weak-separation stress test makes that boundary explicit. Second, jurisdiction is physical and revisable : new instruments or longer horizons can refine the state. Third, the simple cross-ratio fingerprint is special to ℙ1; higher-dimensional projective identification requires richer observables and invariants. Fourth, the glutathione programme is a prospective test. No synthetic success substitutes for a natural system surviving predeclared gates. These limits are not retreats from the programme. They are the points at which the programme can be wrong. Scientific Disagreement After the Lens Is Declared A consequence of predictive closure is a simple criterion for when two scientific descriptions gen- uinely disagree empirically. Let two models 𝑇𝐴, 𝑇𝐵 be expressed on a common observational juris- diction 𝒥, common predictive state 𝑆, common record space and horizon. Define 𝑇𝐴 ≡𝒥,𝐻 𝑇𝐵 ⟺ 𝑃 𝐴(𝑌1∶𝑘 ∣ 𝑠, 𝑢1∶𝑘) = 𝑃𝐵(𝑌1∶𝑘 ∣ 𝑠, 𝑢1∶𝑘) for every admissible 𝑠, intervention sequence and 𝑘 ≤ 𝐻. If this equality holds, the two descriptions do not make an empirical disagreement inside that jurisdiction, however different their coordinates or interpretation may be. If it fails, a discriminating future experiment exists in principle within the This suggests a discipline for scientific controversy. Before deciding which camp is wrong, ask whether the apparent conflict is instead a difference of lens, jurisdiction, state compression , aggregation level, or only then dynamics. Different coordinates can describe the same predictive law. Different horizons can make different state variables sufficient. A genuine dynamics conflict begins only when the same state, lens and intervention imply different future records. This is a programme rather than a theorem that every historical dispute will dissolve. Its proposed standard is nevertheless sharp: a scientific disagreement is not fully posed until both sides have formalize. In a science of consciousness, first-person reports or structured experiential records and third-person neural or behavioral records can be treated as distinct observational channels within a joint predictive problem. The question becomes: what state closes their joint future law under intervention? Consciousness is returned to the map without being made a mystical cause. The frame- work neither claims that consciousness creates physical law nor treats the observer as an ex- ternal contamination to be removed. It places every mode of access - including first-person access when scientifically operationalized - inside the same demand for predictive sufficiency, explicit jurisdiction and held-out discrimination.

It is to determine which distinctions available at an interface remain consequential to the future. That gives the paper its first-principles order: physical access → predictive closure → state → composition → observation geometry → reconstruction by failure. A state is a compression of history. To call two histories the same state is to wager that no declared future experiment will ever need the distinction that was erased. A law is what remains well de- fined after that compression. A geometry records what the observational lens preserves. And when prediction fails, nature has returned a distinction the representation tried to forget. Within that larger grammar , Theorem 1 gives one exact and unusually transparent observer geome- try. If common scale is unavailable and the relevant homogeneous channels transform linearly, the empirical state is projective: linear homogeneous state ⟶ Projective Shadow. in relativity, optics, networks, inference and selection without implying that those sciences share a substance: they can share an observational structure while describing different things. The Empirical Grammar exists because nature need not remain in that branch. Predictive state can be stochastic. A scalar can fail to close. A point map can fail. Cross-ratios can fail. A hidden coordinate can matter only at a longer horizon. The correct response is not to protect the curve. It is to type the failure, enlarge the state when warranted, and demand that the new description predict futures it was not built to fit. The synthetic study shows that this logic can work under controlled conditions and can fail in the in- tended direction under process noise, misspecification, correlated error , uneven sampling and weak state separation. The HepG2 experiment places the same architecture at genuine empirical risk. Its Its cleanest negative outcomes are equally valuable: rejection of the Point-Map Gate, rejection of the Projective Fingerprint, failure of the False-Noun test, or failure of the proposed enlarged state to recover prediction. The framework earns credibility only if those outcomes are allowed to kill the corresponding branch. The same discipline changes how scientific disagreement should be approached. Two theories can- not be said to conflict cleanly while they use different observational lenses, different jurisdictions, different state compressions or different horizons. Once those are aligned, either the future laws co- sciousness, but human knowledge is consciously lived and experimentally situated. A mature science should neither elevate that fact into metaphysical magic nor erase it in the name of objectivity. It should declare the channel, declare the jurisdiction, and ask what predictive distinctions survive.

What can this observer distinguish, and which of those distinctions does the future still remember? Appendix A The Empirical Grammar in Practice The Empirical Grammar places Theorem 1 inside the broader first-principles order of predictive clo- sure, observer geometry and failure-driven reconstruction. It is placed in the appendix to keep the main text theorem-centred. The order matters because each later structure is licensed only after the earlier one has survived. EMPIRICAL GRAMMAR Earn state • • use failure to reconstruct • let held-out prediction decide 1 2 3 4 RAW ACCESS actions + records JOINT PREDICTIVE REDUCTION EMPIRICAL STATE + action quotient SCALAR if stochastic: use transition kernels, not point geometry HELD-OUT FUTURE PREDICTION select the smallest adequate architecture new instruments, policies and longer horizons refine access test lens Figure 4 : The Empirical Grammar . Physical access is reduced to predictive state and effective jurisdiction before coordinates or geometry are assigned. Scalar closure, the Point-Map Gate, the Projective Fingerprint, and projective rank are empirical tests. Failure feeds back through state enlargement, and every branch returns to held-out future prediction. Step 1 Declare the physical access layer List what the apparatus can actually do and record, together with the background assumptions needed to treat those actions and records as reproducible. The repertoire is theory-laden; the latent

THE OBSERVER AND THE WORLD DANIEL JOHN MURRAY state geometry is not assumed merely because the apparatus has coordinates. Step 2 Learn predictive state and effective jurisdiction jointly Estimate future-law features, reconstruct the coarsest supported history classes, and quotient inter- ventions that are future-law indistinguishable. Include adaptive policies as probes when useful; the policy-closure proposition guarantees that the same predictive state notion applies. Step 3 State the horizon, resolution and estimator Report the future-test family, predictive metric, uncertainty method, clustering/model-selection rule and held-out split. Approximate closeness is not an equivalence relation. Step 4 Test proposed state variables If matched proposed states produce different future laws, the variable is not sufficient. Enlarge the state before assigning geometry. • continuous strictly monotone closure: Case III, reconstruct the Aczél natural coordinate 𝑟 = 𝜙(𝑥). Step 6 Pass the Point-Map Gate Estimate the transition kernel 𝐾𝑢(𝑠, ⋅). If next predictive state retains non-degenerate stochastic spread beyond reconstruction error , reject deterministic point geometry for that representation. Model the kernel or enlarge state instead. Step 7 Test the Projective Fingerprint Rather Than Fitting It For a deterministic one-dimensional branch, test cross-ratio preservation on unused quadruples. A a non-projective regular branch unless state adequacy itself has failed. Step 8 Select projective dimension under penalty When richer predictive coordinates exist, compare ℙ𝑑−1 candidates only up to a predeclared 𝑑max. Choose the smallest dimension supported by held-out prediction or a declared complexity penalty. Do not use unbounded rank as an unfalsifiable repair . Step 9 Search for physical homogeneous channels An abstract lift does not identify its physical coordinates. Ask whether independently measurable domain variables transform linearly/homogeneously in the required way. Failure to find them is a physical-lift failure, not a mathematical contradiction. Step 10 Earn smooth calculus Use Lie brackets only after the reconstructed predictive-law family has an identifiable smooth finite- dimensional constant-rank realization. Otherwise use discrete graph cycles and nonparametric pre- dictive tests. Step 11 Use Failure Tomography Where smoothness holds, test the 𝑎𝑏 reversal law and iterated commutators. At finite amplitude or without smoothness, test graph-cycle circulation. Candidate new coordinates are hypotheses gener- ated by the defect, not automatically real dimensions. Step 12 Demand held-out recovery Freeze the enlarged architecture and predict future records under histories/policies that were not used to construct it. A missing variable earns state status only if it improves held-out predictive law after complexity is accounted for .