Observer/projective representation

### PDF page 21 Daniel J. Murray Revised September 2026 Second, the observer formalism is operational, not a theory of consciousness. A thermostat- controller, automated microscope, cell, animal, or human laboratory can instantiate an observer- world interface at different levels of description. Third, this paper does not prove that observers create an ontologically private universe. It proves the narrower structural statement that empirical state identity and the existence of an empirical law depend on the equivalence relation induced by the observer’s accessible futures. Fourth, exact predictive state reconstruction may be statistically difficult. Weakly separated histo- ries, sparse interventions, measurement error, and nonstationarity can make distinct states empiri- cally unidentifiable. Fifth, the projective theorem is conditional and classical. It is valuable here because it supplies an exact death condition, not because fractional-linear transformations are new. Sixth, the proposed HepG2 thresholds are preregistration choices rather than universal biological constants. Replication in other systems and laboratories is required before transporting them. Finally, the present paper reports no new biological outcomes. The synthetic unit test verifies the logic when ground truth is known; the HepG2 programme is a prospective attempt to make the central matched-present claim experimentally vulnerable. 8. Conclusion The observer is inside the world. It cannot carry the world’s complete history forward, so every empirical present is a compression. To call that compression a state is to make a precise wager: the differences that have been forgotten will never reappear in the future records accessible within the declared jurisdiction. The State-Law Descent Theorem makes the wager explicit. A law exists on a proposed representation if and only if the future law is constant on the histories that representation identifies. If matched presents separate under the same future intervention, the failure is prior to curve fitting. The representation erased something the future still needs, and no deterministic reparameterization of that representation can recover it. Richer observers can refine the state space by gaining new interventions, records, horizons, or resolu- tion. Exact geometries can then emerge from the relation between latent structure and observational loss; the scale-blind two-channel projective branch provides one deliberately narrow example with an exact cross-ratio death condition. The resulting grammar is simple. The world has histories. The observer makes distinctions. The future decides which distinctions must be preserved. Only then does a state exist on which a law can act. The present is what remains of history after every distinction the future still needs has been preserved. Compress history only as far as the future permits. Appendix A. Scalar composition is a different branch Because bounded observables often tempt geometric overinterpretation, it is useful to separate projective observation from associative scalar composition. Suppose a nondegenerate scalar interval 𝐼 carries a total, continuous, strictly increasing-in-each- argument, associative binary operation ⊕ with identity, satisfying the standard interval conditions of Aczél’s representation theorem [16]. Then there exists an increasing generator 𝑔, unique up to positive multiplication after the identity fixes its zero, such that ### PDF page 22 Daniel J. Murray Revised September 2026 𝑔(𝑥 ⊕ 𝑦) = 𝑔(𝑥) + 𝑔(𝑦). For the particular total operation on (−1, 1)with identity zero, one possible mechanism-selected generator is 𝑔(𝑥) =artanh(𝑥), yielding 𝑥 ⊕ 𝑦 = 𝑥 + 𝑦 1 + 𝑥𝑦. This fractional expression may look projective, but its logical origin is different. Here the coor- dinate is selected by the specified composition operation , with continuity, totality and strict monotonicity; in Theorem 2 the fractional-linear action is selected by quotienting a homoge- neous linear two-channel state by common scale . Neither branch should be inferred from boundedness alone. Appendix B. Higher-dimensional extension For an (𝑚 + 1)-channel homogeneous latent state 𝑧 with common scale unobserved, the natural observed state is a point of projective 𝑚-space. Any invertible linear latent update 𝑧′ = 𝑀 𝑧 induces a projective transformation. In an affine chart with coordinates 𝑞𝑖 = 𝑧 𝑖/𝑧𝑚+1, 𝑞′ 𝑖 = ∑ 𝑚 𝑗=1 𝑀𝑖𝑗𝑞𝑗 + 𝑀𝑖,𝑚+1 ∑ 𝑚 𝑗=1 𝑀𝑚+1,𝑗𝑞𝑗 + 𝑀𝑚+1,𝑚+1 . The simple one-dimensional cross-ratio fingerprint no longer characterizes the full action. Higher- dimensional tests require collinearity/incidence invariants, compatible homogeneous lifts, or other projective constraints. Consequently, failure of the one-dimensional projective branch should not automatically be interpreted as evidence against projective structure at higher dimension; it may instead be evidence that the observer compressed a higher-rank state too aggressively. Appendix C. Reproducibility and preregistration inventory The current submission contains no empirical biological outcomes. Supplementary File S1 (S1_Synthetic_Unit_Test.py) regenerates the synthetic results with seed 20260827. Data-bearing plots are generated deterministically from code. Conceptual schematics and the graphical abstract are rendered from explicit vector drawing instructions and contain no generated biological or research data. Before any confirmatory HepG2 acquisition, the following are to be frozen in a time-stamped prereg- istration: culture/passaging limits; sensor calibration rules; cell and well exclusion rules; technical- variance estimator; intervention doses and timing; train/test randomization; matching tolerance; point-map threshold; cross-ratio quadruple separation; equivalence margin; model formulas; leave- one-replicate-out folds; viability jurisdiction threshold; orthogonal biochemical endpoints; and the hierarchy by which downstream tests are interpreted only after upstream gates are evaluated.

The Observer and the World: Predictive state, lawful forgetting, and the geometry of empirical reality Daniel John Murray Revised September 2026 Article type: Full Length Article (theoretical methodology / biological hypothesis) Author: Daniel John Murray Independent Researcher, Melbourne, Victoria, Australia ORCID: 0009-0005-1794-5945 getting. We prove a State-Law Descent Theorem: a future law exists on a proposed representation if and only if the future kernel is constant on its history fibres. It follows that matched presents with different futures falsify statehood, and no reparameterization, nonlinear fit, spline, or neural network using only the failed representation can restore information already erased. Richer experimental jurisdictions can only refine the exact predictive quotient, formalizing the observer-dependence of empirical state without implying observer-created reality. A narrow projective branch is then de- veloped as an exact death condition: scale-blind observation of a homogeneous linear two-channel state forces a Möbius map and cross-ratio preservation. A frozen-seed synthetic test verifies failure and recovery, and a prospective HepG2 redox protocol makes the matched-present criterion experi- mentally killable. The central proposal is that scientific laws are defined only after lawful forgetting: compress history only as far as the future permits. Keywords: predictive state; biological information; observer; state reconstruction; projective ge- ometry; redox homeostasis 1. The observer is inside the world A measurement is not a state. Science does not encounter nature from outside nature. An observer - whether a cell, an organism,

interrogated. The empirical world available to that observer is therefore not a pre-labelled list of states delivered from a view from nowhere. It is organized by the distinctions that the observer-world interface can make and by whether those distinctions continue to matter to future prediction. This statement is not an appeal to subjectivism and does not require consciousness. The world need not depend on the observer for its existence. The narrower claim is operational and testable: empirical state identity depends on which histories remain distinguishable by the future experiments available to the observer. A richer sensor, a new intervention, a longer prediction horizon, or a finer record can split histories that were previously indistinguishable. The observer is therefore central to its empirical state space because the observer’s physical access helps define the equivalence relation by which different histories count as the same present. The consequence is easy to miss because scientific language converts histories into nouns. We say while retaining different enzyme capacities, transcriptional programs, damage burdens, reserves, or other hidden variables. If the same subsequent intervention separates their future distributions, then the recorded present omitted information predictive of the conditional future. The observer did not discover two equal states; it constructed an over-coarse present. Figure 1: The observer-world problem and matched-present refutation. The observer is a physical subsystem within the world, coupled to it through recording, intervention, and memory. Distinct histories may converge to the same measured present x. If the same future intervention u then produces different future laws, x is not a predictive state. Adding a missing predictive coordinate r can split the false equivalence class and restore closure. This paper develops the formal and experimental consequences of that observation. Its main claim is not that biology is projective, that one geometry governs living systems, or that observers create

the representation must be shown to preserve the future law of the declared experiment. Predictive-state and causal-state constructions already contain the mathematical kernel of this idea: histories are grouped when they induce the same conditional futures [1-3]. The contribution here is to turn that kernel into a falsification-first audit for biological measurement and then show how exact geometric fingerprints can be deployed only after statehood has been licensed. The resulting order is world histories → observer-world distinctions → predictive state → law → coordinate geometry . A large fraction of modelling practice reverses the middle of this sequence by taking a convenient measurement as the state and immediately fitting a transition law. The present framework inserts the missing question: is there a single future law on that representation at all? sentation a state, how that claim can be killed, and how failed statehood can be used to reconstruct omitted biological information. 2. Lawful forgetting: predictive state before predictive law 2.1 Observer, jurisdiction, histories, and futures Let ℋ be the admissible histories of a world-system up to a declared present time 𝑡0. For the empirical question at hand, define an observer operationally by a jurisdiction 𝒥𝒪 = (ℋ, 𝒰, 𝒴, 𝑇 , Π), where 𝒰 is the set of future interventions available to the observer, 𝒴 the future record channel, 𝑇 the prediction horizon, and Π the admissible experimental policies and environmental regime. This tuple is not intended as an ontology of observers; it is the minimum declaration needed to make an empirical state claim testable. For each history ℎ ∈ ℋ and intervention program 𝑢 ∈ 𝒰, let 𝐾𝑢(ℎ, 𝐵) = 𝑃 (𝑌+ ∈ 𝐵 ∣ ℎ, 𝑢) ditional laws are defined only almost surely; claims then apply on a common declared support and to fixed versions, not to arbitrary null-history assignments. Interventions are externally specified policies with a stable deployment rule. Equality of history-conditioned future laws is a predictive claim; interpreting a contrast as the causal effect of preparation additionally requires an identified intervention design. Conditioning on a post-treatment biomarker may induce selection even when preparation was randomized.

Daniel J. Murray Revised September 2026 Definition 1. Predictive equivalence and empirical state Histories ℎ1, ℎ2 ∈ ℋ are predictively equivalent for observer 𝒪 in jurisdiction 𝒥𝒪 when ℎ1 ∼𝒪 ℎ2 ⟺ 𝐾 𝑢(ℎ1, 𝐵) = 𝐾𝑢(ℎ2, 𝐵) for every admissible 𝑢 and every future-record event 𝐵. The exact empirical state is the equivalence class 𝑆𝒪(ℎ) = [ℎ]∼𝒪 , 𝒮 𝒪 = ℋ/ ∼𝒪 . This is the controlled predictive quotient. It says exactly which aspects of history may be forgotten without changing any future law in the declared jurisdiction. The word state is therefore earned by future sufficiency, not bestowed by measurement convenience. 2.2 The State-Law Descent Theorem Let 𝑥 ∶ ℋ → 𝑋 be any proposed present representation: a scalar biomarker, a vector of measure- 𝑥(ℎ1) = 𝑥(ℎ2) ⟹ 𝐾 𝑢(ℎ1, 𝐵) = 𝐾𝑢(ℎ2, 𝐵) for every 𝑢 and 𝐵; 3. equality of 𝑥 implies predictive equivalence:

but 𝑃 (𝑌+ ∣ ℎ 1, 𝑢) ≠ 𝑃 (𝑌+ ∣ ℎ 2, 𝑢), then 𝑥 is not an exact predictive state in that jurisdiction and no single-valued future law on 𝑥 can represent both histories. This is the primary empirical death condition of the paper. The experiment does not ask whether one chosen curve is wrong. It asks whether the proposed present can support any law of the declared form. Corollary 2. No-reparameterization rescue Suppose 𝑥 fails Corollary 1. Then for every deterministic transformation 𝑓 ∶ 𝑋 → 𝑍 , the represen- This simple result blocks a common modelling escape. A polynomial, spline, saturation transform, kernel machine, or neural network can make the map from available information to output arbitrarily flexible, but it cannot recover a predictive distinction that was already destroyed before the model received its input. F unctional complexity cannot repair representational information loss. 2.3 Minimality: the predictive quotient is the lawful limit of compression The predictive quotient is sufficient by construction. It is also minimal in the sense relevant here.

Daniel J. Murray Revised September 2026 Proposition 1. Minimal exact predictive state Let 𝑅 ∶ ℋ → ℛ be any representation sufficient for every future law in the same jurisdiction. Then 𝑅(ℎ1) = 𝑅(ℎ2) ⟹ 𝑆 𝒪(ℎ1) = 𝑆𝒪(ℎ2). Thus every exact sufficient representation refines the predictive quotient; it may retain irrelevant distinctions, but it cannot identify histories that the declared future distinguishes. Proof. If 𝑅(ℎ1) = 𝑅(ℎ2)and future law factors through 𝑅, then all conditional future laws agree under every admissible intervention. Definition 1 gives ℎ1 ∼𝒪 ℎ2. □ This gives a precise meaning to lawful forgetting. Compression is scientifically legitimate exactly to the point at which further identification would merge histories with different futures. 2.4 Jurisdiction-dependent statehood Statehood is incomplete without a jurisdiction. A variable may close prediction over minutes and fail over hours; may close under passive observation and fail once a new perturbation becomes available; or may close at one measurement resolution and split at another. Proposition 2. Jurisdiction refinement Let 𝒥1 and 𝒥2 be two jurisdictions on the same admitted history class such that every future test 𝒮𝒪 = ℋ/ ∼𝒪 . If two observers possess different experimental jurisdictions and those jurisdictions induce different predictive equivalence relations, then they possess different empirical partitions of the same history space. This is the precise sense in which the observer is central to its empirical universe: not because the observer creates the underlying world, but because empirical identity - what counts as the same present - is defined by the distinctions its physical coupling to the world can test.

The decisive experiment is almost embarrassingly simple. Prepare two systems through different histories ℎ1 and ℎ2. Match them at the declared present on candidate representation 𝑥 within a preregistered tolerance. Apply the same future intervention 𝑢. Compare the future distributions. If the distributions remain equivalent to the declared resolution, the proposed compression survives that test. If they diverge beyond the tolerance allowed by measurement and process noise, 𝑥 fails as state for that jurisdiction. The world has revealed that the observer compressed too far. In finite biological data, equality is replaced by a declared matching relation 𝑑𝑋(𝑥1, 𝑥2) ≤ 𝜖𝑥 and predictive equality by an equivalence margin or calibrated discrepancy metric on future records. These tolerances belong to the empirical jurisdiction. Failure to reject divergence alone licenses no equivalence. Use three outcomes: a simultaneous confidence lower bound above the declared material margin establishes separation; an upper bound below it certifies equivalence for the tested battery; otherwise the result is unresolved. Matching and measurement uncertainty must be trans- bound. 3.2 The Point-Map Gate Many modelling claims are stronger than predictive sufficiency: they assume a deterministic or low-noise transition from a present point to a future point. Before any such geometric branch is tested, the conditional relation must pass a Point-Map Gate . For a fixed intervention 𝐴, 𝑥′ = 𝐹 𝐴(𝑥) + 𝜀 must have residual dispersion compatible with independently characterized uncertainty, and his- tory/preparation labels must not retain predictive information after conditioning on 𝑥. If matched presents occupy distinct future branches, a deterministic point map is not merely a bad fit; it is undefined on the proposed state. This gate is therefore logically prior to curve selection. 3.3 State enlargement is accepted only by held-out recovery Suppose scalar or low-dimensional 𝑥 fails. Let 𝑟 be a candidate variable representing a historical

The reconstruction cycle is 𝑆1 fails ⟹ 𝑆 2 = (𝑆1, 𝑟2) ⟹ retest descent and held-out closure . If 𝑆2 fails, the added coordinate is not preserved out of loyalty to the hypothesis. The failure becomes a new constraint on what the observer must measure next. 3.4 Failure tomography by order and path Matched-present divergence establishes insufficiency but may not localize it. Two optional diagnos- tics can expose structure in the omitted state. the fully closed scalar maps 𝐴(𝑥) = 𝑥/2and 𝐵(𝑥) = (𝑥 + 1)/2have 𝐵(𝐴(𝑥)) − 𝐴(𝐵(𝑥)) = 1/4. Reversal rejects an order-blind increment representation; a separate matched-present test is needed to reject scalar predictive sufficiency. A complementary finite test requires no differentiability. If measured pairwise contrasts satisfy a scalar node-potential model Δ𝑖𝑗 = 𝑉 𝑗 − 𝑉𝑖, then every directed cycle 𝐶 must satisfy ematically elementary. Its role is methodological: once a point-valued state has passed the prior gates, one independently motivated latent architecture can be converted into an exact, parameter- free death condition. If the invariant fails, the branch dies. That rejection is information, not a failure of the framework.

ferent latent systems can induce the same projective map. Third, stochastic biology is not made deterministic by notation. A broad, history-independent stochastic kernel can be perfectly predictively sufficient while rejecting a deterministic point map. Residual history dependence is a different failure: it rejects predictive sufficiency of the chosen representation. Neither result should be inferred from the other. The projective theorem is therefore a kill-switch, not a worldview. Its purpose is to show how an observer-aware state audit can generate exact falsifiers once the representational prerequisites have been earned. 5. Frozen-seed synthetic unit test The workflow should fail and recover in the intended direction before it is proposed for biology. Supplementary File S1 therefore implements a deterministic synthetic unit test with random seed

Within finite-width bins of current 𝑥, states at opposite extremes of the hidden coordinate had median next-state divergence 0.200. This statistic is descriptive: the selected pairs are not exactly matched and it is not an optimized lower bound over all scalar predictors. Exact insufficiency follows independently from the known model. Writing 𝑞 = 𝑧 1/𝑧2 and 𝜌 = 𝑧 3/𝑧2 gives 𝑞′ = 1.10𝑞 + 0.17 + 0.24𝜌 0.06𝑞 + 0.93 + 0.04𝜌, 𝜕𝑞 ′ 𝜕𝜌 = 0.2164 − 0.0296𝑞 (0.06𝑞 + 0.93 + 0.04𝜌)2 . At 𝑞 = 1 , changing 𝜌 changes the future while leaving the present 𝑥 = 0 exactly fixed. This explicit collision excludes every deterministic scalar predictor of the exact future, not only a fitted Möbius family. The missing homogeneous coordinate was then admitted as 𝜌 = 𝑧3 𝑧2

0.00170, a 38.1-fold recovery relative to the scalar model. The point is not that an added parameter always helps; it is that a coordinate that restores the latent projective state restores out-of-sample prediction after a failure created specifically by omitted state. Figure 2: Frozen-seed synthetic reconstruction. (A) A two-channel scale-blind linear system is accurately predicted by a scalar Möbius law. (B) Introducing an unobserved third channel makes the scalar future multi-valued and increases held-out error. (C) Adding the missing projective coordinate restores prediction. Seed 20260827; code in Supplementary File S1. 5.3 Reversal diagnostic Finally, two noncommuting 3 × 3 generators were applied in opposite orders over small amplitude pairs. The reversal defect was fit to Δ𝑥 = 𝜅𝑎𝑏. The fitted coefficient was 𝜅 = −0.13530 , compared with the differential commutator prediction −0.13711, and the zero-intercept scaling fit achieved 𝑅2 = 0.999983 . This verifies that the local failure diagnostic behaves as derived when the ground truth is known. 5.4 What the simulation establishes tive. It establishes three narrower facts: the code recovers the correct projective branch when its assumptions are true; omitted state produces the intended matched-present/different-future failure; and admitting the missing coordinate restores held-out prediction. Biological support must come from an independent experiment in which the latent truth is not supplied to the algorithm. 6. Prospective biological test: HepG2 glutathione redox 6.1 Biological rationale Glutathione redox is a useful stress test for state reconstruction because it combines a fast chemical pool with slower resource and transcriptional capacities. The genetically encoded Grx1-roGFP2 probe reports the local glutathione redox potential in real time [9,10]. HepG2 cells have a published

adaptation dominates; • a capacity challenge in which NADPH supply is changed while current redox is matched; • an adaptive jurisdiction in which NRF2-dependent capacity should become predictively relevant. The experiment is prospective. None of the cited studies tested the projective fingerprint or the complete failure-and-recovery sequence described below. 6.2 System, replication, and quality control The primary system is authenticated HepG2 human hepatoma cells (ATCC HB-8065) stably ex- pressing cytosolic Grx1-roGFP2. Cells are routinely screened for mycoplasma. Sensor oxidation is prespecified survivor analysis is explicitly conditional and accompanied by the full-cohort outcome. A blinded technical run-in may estimate assay variance and verify that the proposed doses remain in a nonlethal dynamic range. It may not alter confirmatory hypotheses, equivalence margins, time windows, model formulas, or outcome thresholds after confirmatory acquisition begins. 6.3 Fast jurisdiction: test the point map before the geometry The fast jurisdiction is fixed as -2 to +20 min around a 5-min 50 µM H2O2 pulse, with Grx1- roGFP2 imaging every 15 s. The proposed starting dose is already represented in the GSE39291

Daniel J. Murray Revised September 2026 Prospective HepG2 redox protocol Separate preparation, measured-state matching, and future assessment FAST ARM Baseline 2 to 0 min Held-out improvement plus residual-history equivalence Figure 3: Prospective HepG2 experiment. The fast jurisdiction tests the point-map gate and projective fingerprint. A G6PD/NADPH perturbation creates a matched-present hidden-capacity challenge. The 4-24 h TBH/NRF2 arm tests whether measured capacity variables restore held-out prediction.

but the live-cell projective test is performed on the calibrated sensor coordinate. Neither numerical identity nor a fractional-linear relationship between the sensor coordinate and this biochemical contrast is assumed. The projective hypothesis concerns the explicitly declared measured coordinate. For the transition test, 𝑥𝑖 is the mean over the final 60 s before the pulse and 𝑥′ 𝑖 the mean over the final 60 s of the 5-min pulse. Within each biological replicate, 70% of tracked cells are assigned to model calibration using a frozen pseudorandom split; the remaining 30% are held out. standard deviation around the prespecified deterministic model class must satisfy 𝑠point ≤ 0.020 . For every preregistered history contrast, a simultaneous upper confidence bound on the abso- lute residual-history effect in future 𝑥, including a justified matching/measurement discrepancy allowance, must also be below the operational margin 0.020. The distributional discrepancy and test family must be frozen if a full-law claim is made; a mean-only contrast certifies only that feature. Lack of significance is unresolved. Statistical calibration uses independent culture days, retaining the nested dependence of cells and wells. If this gate fails, the deterministic projective hypothesis is rejected for the fast jurisdiction. Cross-ratio analysis is not used to rescue it. If the gate passes, two co-primary projective criteria are evaluated. Held-out Möbius accuracy . A simultaneous one-sided upper confidence bound on mean replicate- level held-out RMSE in 𝑥 must be CR(𝑥1, 𝑥2; 𝑥3, 𝑥4)∣∣ . The calibrated one-sided upper confidence bound for the mean replicate-level 𝐷CR must be below the preregistered equivalence margin 0.05 . The 0.05 margin is an assay-level decision threshold, not a universal definition of projectivity. A mean of replicate medians does not test every quadruple; success licenses only this finite fingerprint

margin. The bound 𝑏 requires a supported transport/control model, not the matching tolerance 0.03 alone. If 𝑏 cannot be justified, report a sensitivity analysis and an unresolved closure decision. This is a predictive contrast conditional on measured matching; attribution specifically to NADPH requires controls for off-target effects and post-treatment selection. The prediction is about separation, not its sign. If matched current redox values produce different futures under the same challenge, current redox is insufficient state in the capacity-perturbed jurisdiction. 6.5 Adaptive jurisdiction: test a prespecified enlarged state The adaptive jurisdiction is measured at 4, 8, and 24 h after 200 µM TBH, matching the exposure defeats transport of that smooth model; the trajectory remains in the full randomized-cohort report. Conditioning on survival creates a separate estimand and cannot establish full-cohort sufficiency. The enlarged predictive state is fixed before fitting as 𝑆enlarged = (𝑥, NADP+ NADPH , 𝐶NRF2) ,

tive baseline and future sampling use distinct randomized parallel wells with explicit hierarchical uncertainty. Model assessment uses leave-one-biological-replicate-out prediction. Relative to the scalar 𝑥-only model, the enlarged model first qualifies as predictively improved only if it achieves both ≥ 30% reduction in held-out RMSE and ΔBIC ≥ 10 in favor of the enlarged representation. A significant coefficient without held-out improvement does not count as state recovery. Neither improvement nor BIC is an equivalence test. Operational recov- ery additionally requires simultaneous upper bounds on residual-history mean-feature discrepancies below 0.020 on held-out common support in the enlarged state, and prespecified absolute prediction- error criteria. Unsupported regions remain unresolved. A claim about complete future laws requires a separately calibrated law-separating discrepancy and cannot be obtained from RMSE alone. 6.6 Orthogonal validation and confounders The proposed two-channel fast state and NADPH/NRF2 enlargement are hypotheses, not unique biochemical decompositions. Four confounder classes are therefore explicit. Sensor artifacts. Calibration drift, photobleaching, motion/segmentation error, sensor saturation, and compartmental heterogeneity are quantified from technical controls. Unexplained variance cannot be subtracted post hoc merely to pass the Point-Map Gate. Cell-state heterogeneity . Cell-cycle state, morphology, baseline metabolic capacity, and viability may remain predictive at matched 𝑥. If a prespecified nuisance variable explains the held-out divergence better than NADPH/NRF2 capacity, the proposed enlargement fails and a new state hypothesis is required. Perturbation off-targets. G6PDi-1 has known metabolomic effects beyond a single idealized flux coordinate [13]. Target-engagement measurements are therefore necessary but not sufficient for mechanistic attribution. The false-state result only requires that matched redox states differ in future response; attribution to one molecular mechanism remains a second question. change may move the cells outside the regime in which smooth projective or adaptive dynamics were proposed. Such observations reject that jurisdiction rather than being absorbed as outliers. State recovery also requires orthogonal prediction. The enlarged coordinate must improve predic- tion not only of future Grx1-roGFP2 trajectories but also of at least one independent biochemical endpoint (NADP+/NADPH or GSH/GSSG) using the same training/held-out partition. The bio- chemical validation endpoint is measured at a future time distinct from its baseline state input; re-predicting an input is not orthogonal validation.

The experiment is constructed so that negative results are informative. • F ast projective success; capacity failure; enlarged-state recovery: supports the full jurisdiction-dependent hypothesis. • F ast projective failure: rejects the optional scale-blind two-channel linear branch before adaptive capacity is invoked. • Scalar failure without enlarged-state recovery: shows that current redox is insufficient but falsifies the prespecified NADPH/NRF2 reconstruction. • Certified scalar equivalence under capacity/adaptive extension: licenses the tested compression for the specified features, histories and margins. • Neither material separation nor equivalence certified: remains unresolved; a nonsignif- icant result does not establish sufficiency.

The phrase observer-dependent is often heard as a claim about subjectivity, consciousness, or observer-created reality. None is required here. An observer is simply a physically specified sub- system with a particular set of interventions and records. Given those capacities, the predictive quotient is objective: either two histories induce the same accessible future laws or they do not. What depends on the observer is the empirical partition of histories. A spectrometer, a fluorescence microscope, and a naked eye can inhabit the same world while supporting different empirical state spaces because they can distinguish different histories. Proposition 2 makes this refinement explicit. present, the observer has written those histories as one empirical narrative. A matched-present future challenge asks whether the world accepts that narrative. Divergence means that a discarded distinction remained predictive even though the measured present omitted it. Causal attribution requires the additional identification assumptions stated above. The claim is therefore stronger than “history matters. ” Some past distinctions can become irrelevant to the declared future; no universal microscopic memory premise is needed. The operational question is which historical distinctions must still be represented now for the declared future law to exist? The predictive quotient answers that question without requiring the observer to reconstruct every microscopic event. 7.3 State before law changes modelling logic Theorem 1 separates two kinds of failure that are often conflated. A functional failure occurs when a valid state supports a law but the chosen functional family is adding functional flexibility alone cannot solve the problem. This distinction matters increasingly as biological modelling adopts high-capacity machine-learning systems. Predictive power can improve dramatically with flexible models, but high capacity does not remove the requirement that the input representation contain the predictive distinctions needed by the future task. A model can interpolate complicated structure in what it sees; it cannot reconstruct distinctions that are identical in its input but associated with different conditional future laws. 7.4 Relation to predictive-state and organizational approaches The predictive-equivalence construction is not claimed as new. Computational mechanics, epsilon-transducers, predictive-state representations, bisimulation, and related controlled-process

3. use matched-present divergence as the primary refutation; 4. refuse reparameterization as a cure for erased information; 5. accept state enlargement only when it improves held-out prediction and certifies residual- history equivalence in its declared support; 6. test exact structural fingerprints only after the representation has passed the prior gates. This is also complementary to theories of biological organization. Organizational closure concerns how a living system sustains and constrains its own processes; predictive closure asks a different op- erational question: what representation is sufficient for the future accessible to a declared observer? A system can possess rich organizational closure while a chosen instrument observes a non-closing projection. Conversely, a predictive state can be sufficient for one narrow experimental jurisdiction without revealing the full ontology of the organism. 7.5 Empirical geometry can arise at the observer-world interface The projective branch illustrates why the observer cannot be added as an afterthought. If a latent process is homogeneous and linear but the observer loses common scale, the residual coordinate The same logic motivates searching for other exact fingerprints associated with other observational quotients. The broader research programme is therefore not “fit Möbius maps to biology,” but derive the strongest falsifiable structure that follows after the observer, the state claim, and the information loss have been declared. 7.6 Biological meaning of a successful HepG2 result The proposed redox experiment is intentionally narrower than the philosophical reach of the frame- work. A successful matched-present capacity experiment would not prove that every biological state is history-dependent, nor that the entire cell has been reconstructed. It would establish something more concrete: within a specified redox jurisdiction, one commonly used present coordinate was insufficient because a historical capacity distinction remained predictive after the coordinate had been matched. If the enlarged NADPH/NRF2 state then improved held-out prediction and passed the absolute residual-history equivalence criterion, the experiment would demonstrate operational recursive re- construction: the future would identify a distinction the observer had forgotten, and a new mea- surement would restore the law on a finer present. If the proposed enlargement failed, that failure would itself constrain the next candidate state. 7.7 Limitations and boundary of the claim Several boundaries are necessary to keep the central claim exact. First, predictive equivalence is always relative to accessible tests and finite resolution. Failure to distinguish histories does not establish absolute identity beyond the declared jurisdiction.

the narrower structural statement that empirical state identity and the existence of an empirical law depend on the equivalence relation induced by the observer’s accessible futures. Fourth, exact predictive state reconstruction may be statistically difficult. Weakly separated histo- ries, sparse interventions, measurement error, and nonstationarity can make distinct states empiri- cally unidentifiable. Fifth, the projective theorem is conditional and classical. It is valuable here because it supplies an exact death condition, not because fractional-linear transformations are new. Sixth, the proposed HepG2 thresholds are preregistration choices rather than universal biological constants. Replication in other systems and laboratories is required before transporting them. Finally, the present paper reports no new biological outcomes. The synthetic unit test verifies the logic when ground truth is known; the HepG2 programme is a prospective attempt to make the central matched-present claim experimentally vulnerable. 8. Conclusion The observer is inside the world. It cannot carry the world’s complete history forward, so every empirical present is a compression. To call that compression a state is to make a precise wager: the tion. Exact geometries can then emerge from the relation between latent structure and observational loss; the scale-blind two-channel projective branch provides one deliberately narrow example with an exact cross-ratio death condition. The resulting grammar is simple. The world has histories. The observer makes distinctions. The future decides which distinctions must be preserved. Only then does a state exist on which a law can act. The present is what remains of history after every distinction the future still needs has