Return to the full study

KEDDEH SYSTEMS · K-WP17 · E2

Observer, Foreground, and Residue in Cosmological Inference

A falsifiable KEX wrapper around established cosmological likelihoods

Abstract

This paper asks whether a source-preserving, observer-conditioned framework can support a discriminating cosmological inference rather than merely a different description of unexplained data. The July source proposes a residual state-space extension around an established ΛCDM forward model. This edition preserves its equations and develops the meanings, units, identifiability requirements and experimental controls needed to use them. It distinguishes a coordinate origin from a physical origin, an observation from a fitted parameter, and retained memory from an arbitrary residual. New exact toy calculations illustrate the state update, an observational no-residue null, latent-scale ambiguity and foreground confounding. They also show why the source's κ=0 null alternative needs initial-state and process-noise conditions supplied with its application. These arithmetic results are not cosmological measurements. The proposed physical evaluation requires baseline reproduction, controlled injections, invariant observables, reserved-data prediction and a justified complexity comparison. The current literal .1 origin designation is preserved without numerical coercion or interpretation as a cosmological parameter.

1. Question, research lineage and scope

Cosmological measurements reach an observer through instruments, foregrounds and a physical environment. Inferring source properties requires a model of those transformations. The Keddeh research question is whether retaining their identities and dependencies can make a proposed additional mechanism reproducible and testable. The paper's immediate contribution is a model-and-evidence contract; a physical residue becomes a scientific claim only through its specified observable consequences. [1]

The original WP-17 isolates this branch from wider KEX and BRAINK research. It proposes a baseline-plus-residue model and explicitly does not claim a new force, matter component, expansion law or replacement for general relativity. Its wake-shell and recursive-field interpretations remain research hypotheses in this paper. A complete evaluation would attach the exact originating derivation and the conditions under which it maps into the chosen astronomical observable.

A chart's coordinate origin is not automatically a physical origin. The current author designation .1 names a shared referent and is a literal nonnumeric label. It does not denote 0.1, a fitted initial condition, a probability or a displacement of the cosmological coordinate system. A conventional numerical zero may occur in a data vector or null hypothesis without becoming that native designation. Nor does an unobserved object or relation thereby become nonexistent. [2, 3]

These distinctions protect the research in both directions. A numerical fit cannot by itself establish the origin ontology, and an ontological proposal cannot by itself determine a likelihood. This edition therefore keeps the native ordered observer/observed/context framework, the conventional statistical comparison and a possible physical mechanism separately inspectable.

2. Original equations and a declared statistical interpretation

The original three equations are retained unchanged:

y_t=g(θ_ΛCDM)+Fα+M_oβ+Br_t+ε_t (1)

r_{t+1}=λr_t+κu_t+η_t (2)

H_0: B=0 or κ=0; H_1: residue term enabled (3)

The source declares these symbols local to this paper unless explicitly imported from Series Paper 01. Equation (1) is the proposed observable decomposition; (2) is its retained-state update; (3) describes the proposed comparison. This edition does not merge their symbols with identically named quantities elsewhere in the corpus. In particular, B here maps residue into an observation and is not automatically a boundary-separation operator from another passage. [1]

The following is an editorial statistical interpretation for evaluation, not a newly asserted native operator. Let y_t be an m-component observation vector. Let g(θ_ΛCDM) be the selected baseline prediction in precisely that observable space. F has m rows and p columns and maps p foreground coefficients α. M_o has m rows and q columns and maps q observer/instrument coefficients β. For the scalar-state case used below, r_t is a scalar and B is an m-component response column. A vector-state generalisation needs its dimensions and transition operator declared separately.

All terms added to y_t must have the same units by component. If residue has unit U_r and an observed component has unit U_y, that component of B has unit U_y/U_r. In the scalar update, λ is dimensionless per declared step, κ has unit U_r/U_u, u_t has unit U_u, and η_t has unit U_r. Measurement-noise covariance has units U_y,i U_y,j; scalar process-noise variance has unit U_r². A fitted coefficient without these roles is not yet a physical mechanism.

The index t must be defined. It may label acquisition order in a declared experiment; it is not automatically cosmological time, redshift, sky position or multipole. A spatial or frequency-indexed dataset does not gain a temporal memory law merely by renumbering its samples. If an application uses irregular sampling, it must specify the applicable transition for those intervals instead of silently assuming a constant per-step λ.

The source writes a fixed g, F and M_o. Whether an actual data product requires changing response blocks is an application decision to record explicitly. A time-dependent or channel-dependent extension should receive its own versioned equation. Environment is implicit in the selected forward and response models here; that placement does not establish that environmental contributions are absent.

3. Baseline and provenance contract

A reproducible comparison first defines what “the baseline” means. Planck's cosmological-parameter analysis provides a specific established comparator: its methodology states the theoretical model, priors, spectra, likelihoods and calibration treatment. Its distinction between the measured acoustic-scale information and model-conditioned late-Universe inference is particularly relevant. This reference is used as a documented method, not as a claim about the latest cosmological constraints. [4, §§2.1–2.2, 3.1]

Before enabling residue, retain the selected data release, data product, mask, channel selections, calibration, nuisance parameters, likelihood implementation and version, prior definitions, numerical settings and random seeds where used. Record the target published analysis and tolerances in advance. Reproducing a plot approximately is not a substitute for specifying which likelihood and parameterisation produced it.

Baseline parity should be assessed with the residue contribution disabled while keeping the selected original nuisance model intact. If a published likelihood already accounts for a foreground, calibration or observer correction, adding Fα or M_oβ again can double count it. The wrapper must identify whether it preserves an existing block, replaces a specific block or adds a separately justified contribution. Those alternatives define different experiments.

The provenance formulation uses the proof-bearing state relation defined in WP-01 §2, equation (3), under the mapping conditions in §5. Its identity, payload, observer, environment, path, rule and proof fields bind an observation to its construction; its anchor, adjacency, orientation and commit-order conditions constrain the translation. Here the cosmological data and inferred parameters belong to declared payloads, not to an untyped substitution of native state labels. The July anchor notation remains versioned separately from the current .1 designation. [7]

The current retained-source architecture can support this audit by binding a result to the actual inputs and transformations. That engineering capability does not establish that the full cosmology pipeline has already been implemented. A digest confirms that a carrier matches a retained version; an independently reproduced likelihood establishes a different, scientific-computational property. [1, 5]

4. Derivation of the residue and null conditions

This formulation uses the recurrence defined in WP-18 §2, equation (2), under its §1 requirement that every term have a type, unit, dimension and update law. WP-18 writes R_{t+1}=λR_t+κ·residual_t+η_t. Applied here, r_t occupies that retained-state role, while u_t denotes the explicitly declared driver used in WP-17. This is a structural correspondence; identifying u_t with a particular physical residual requires the application's source-derived mapping. [8, §§1–2]

For constant scalar λ, repeated substitution in equation (2) gives the editorial expansion

r_t = λ^t r_0 + Σ(j=0 to t−1) λ^(t−1−j)(κu_j+η_j), for integer t≥1.

This expression uses conventional multiplication and powers within the source's state-space comparison. It is not a rule for native predecessor-dependent relational powers. It shows the three possible contributions to retained state: the initial state, driven input and process disturbance.

An observational no-residue null requires Br_t to vanish in the predicted observable and, in a stochastic model, requires the induced residue contribution to the observation distribution to vanish. Setting B=0 is a sufficient way to obtain that null while leaving the declared baseline unchanged.

The original alternative κ=0 removes the driven term but does not, by that fact alone, remove λ^t r_0 or the propagated η terms. It is a sufficient zero-state construction if the application also imposes r_0=0 and η_t=0 at every step. Other sufficient conditions may be established by a cited source, such as a residue lying entirely in the observation map's null space. Those conditions must accompany the null definition rather than remain implicit.

This is a conditional interpretation issue in the displayed local model, not a finding that the entire source corpus is inconsistent. Equation (3) is preserved. The inspected WP-01 and WP-18 texts supply the state and recurrence dependencies above but do not impose r_0=0 and η_t=0 for this null. If an application imports those conditions from another defined relation, it must name that source and apply them. For the local comparison specified here, B=0 supplies the unambiguous comparator. Section 6 provides counterexamples to treating κ=0 alone as sufficient.

A likelihood-level null also needs consistent nuisance treatment and priors. Disabling a mean contribution while retaining an additional residue covariance is not the same observational model as removing the residue entirely. The exact distribution used under each hypothesis should therefore be recorded, not inferred from a label such as “off.”

5. Identifiability before model selection

WP-05 supplies the relevant comparison relation: one unknown source-plus-foreground mixture is underdetermined, whereas a specified second independent transform restores full rank. This application uses that relation under its known-response, conventional linear-model conditions. It motivates checking whether cosmological, foreground, observer and residue responses are genuinely distinguishable in the selected data. It does not prove that every multi-channel cosmology experiment is identifiable. [6, §2]

There are two immediate ambiguities. First, if a residue response duplicates an allowed foreground response over all measured components, their amplitudes can trade against each other. The data may constrain their total while leaving the separate labels unsupported. Foreground injection and recovery are therefore essential to the residue claim, rather than merely general robustness exercises.

Second, an unconstrained scalar latent state has a scale freedom. For any nonzero constant c, the replacements r'_t=c r_t, B'=B/c, κ'=cκ, η'_t=cη_t and r'_0=c r_0 preserve the same observable Br_t and the form of the update with unchanged λ. A model estimating all these quantities cannot claim a unique physical residue scale without a declared normalisation, known response or other independently justified restriction.

A physical interpretation must consequently state what fixes the residue units and scale, how the driver u_t is measured or constructed, which coefficients are known and which are inferred, and what observation could distinguish the mechanism from an existing foreground or calibration explanation. A prior can make an inference mathematically specified, but the publication should make clear which separation comes from data and which comes from the imposed prior.

WP-18 additionally supplies a memoryless λ=0 comparator in its experiment table. That comparator tests carry-over memory: it still permits a current driven or noise contribution. It is therefore distinct from the no-contribution B=0 null. WP-18 §2 also gives the local fixed-point stability condition ρ(J)<1. For the unforced scalar residue recurrence, the Jacobian is λ and attraction requires |λ|<1; extending that conclusion to coupled, forced or stochastic dynamics requires their declared model. [8]

A stable-looking recurrence is also not sufficient evidence of physics. If an experiment requires bounded long-run memory, it must establish that requirement for its admitted λ and input/disturbance model. A finite toy trajectory only tests the chosen finite case.

6. New editorial worked examples and executed results

All examples in this section are synthetic, scalar, conventional calculations. Their purpose is to expose the model's operation and boundaries. They are not a fit to cosmological observations.

Choose an arbitrary observation unit U_y and residue unit U_r. Let λ=1/2, κ=1 in compatible driver-to-residue units, B=3 U_y/U_r, r_0=0 and η_t=0. Use the four driver values 2, 0, 1, 0. Equation (2) then gives residue states

r_0,…,r_4 = 0, 2, 1, 1.5, 0.75 U_r.

For this toy alone, set the fixed baseline contribution to 10 U_y, foreground to 1 U_y and observer contribution to −0.5 U_y, with ε_t=0. The generated observations are

y_0,…,y_4 = 10.5, 16.5, 13.5, 15, 12.75 U_y.

These values result from substituting a declared state into a declared forward model. The exact fit is guaranteed by construction and supplies no evidence that such residue exists in nature. Setting B=0 returns 10.5 U_y at all five steps.

For the null-condition counterexample, choose κ=0, r_0=2 U_r, λ=1/2, η_t=0 and the same B. The residue contribution remains 6, 3, 1.5 and 0.75 U_y at the first four states. Alternatively, choose r_0=0 and η_0=1 U_r with κ=0; the next state contributes 3 U_y. Each example isolates one condition that must be controlled when interpreting the source's null alternative.

For the scale ambiguity, multiply every residue state in the first example by two and divide B by two, with the corresponding scaled driver coefficient. Every generated observation remains unchanged. For foreground confounding, two identical response columns (1,1,1) have stacked design rank one, although two separate amplitudes are being proposed.

On 18 October 2026, all six cases were executed in Python 3.12.14 using exact rational arithmetic, direct recursion and a rational rank calculation. The retained calculation receipt records six passed cases and zero failures. No Planck likelihood, astronomical data, posterior fit, held-out score or evidence ratio was calculated. The original series calculation-receipt identifier remains preserved in the original text; this new receipt is independently identified and does not claim to reproduce that archive.

7. Proposed cosmological evaluation

The source's experimental programme can be executed as a sequence of non-substitutable gates:

1. Freeze the comparison. Select the observable and exact data product. Define its sampling, units, covariance or likelihood, response maps, priors, initial state and admitted updates. Bind every imported relation to its source section and conditions.
2. Reproduce the baseline. Disable observable residue using the declared sufficient null. Recover the selected published analysis within predeclared numerical and statistical tolerances. Failure blocks interpretation of an extension's apparent gain.
3. Inject controlled components. Generate synthetic datasets with known foreground and observer effects, including a no-residue case. Test false-positive residue attribution and parameter recovery. Then inject a declared residue to measure recovery and confusion under the same workflow.
4. Test equivalent representations. Transform data, forward response and uncertainty consistently under the admitted frame change. Compare the same physical observable after reconstruction. Merely relabelling coordinates must not create a detection.
5. Reserve genuinely predictive tests. Fit using the training portion and evaluate specified sky, frequency or other justified splits. Model correlations and shared calibration across the split. Do not treat dependent portions as independent evidence or refit the extension using the nominal test result.
6. Account for added flexibility. Preselect a justified likelihood-based complexity or evidence procedure and declare all estimated states and parameters, priors and relevant assumptions. The original source lists Δχ², WAIC/AIC/BIC and evidence as candidate approaches; they are not interchangeable votes. Training improvement alone does not determine acceptance.
7. Reproduce independently. Retain failed runs and rejected parameterisations as well as successes. A separate implementation should recover the reported comparison from the same declared inputs before a physical interpretation is promoted.

The planned acceptance result is a calibrated predictive gain that survives known foreground and observer explanations under a stated model comparison. A rejected residue model can still leave a useful reproducibility system. Conversely, successful record preservation cannot rescue a physical model that fails its discriminating tests.

8. Claims and conclusion

C-01: Coordinate classification. The coordinate origin and a proposed physical origin are different claims. Conventional coordinates and the literal .1 designation have separate roles.

C-02: Formalised, conditional hypothesis. The source's residue model is testable once its units, driver, dynamics, initial state, noise, observation map and null conditions are fixed. The editorial calculations establish selected arithmetic consequences and expose identifiable ambiguities.

C-03: Physical outcome. This edition establishes no cosmological anomaly, new component or improvement over ΛCDM. Its domain experiment remains the explicit programme above, rather than an implied completed result.

The decisive contribution of the paper is to connect a retained relational record to a comparison that could fail. The baseline, foreground, observer and residue must remain distinguishable both in the stored provenance and in the inference. A credible residue result would require data to prefer its specified predictions under that complete contract. Until then, the equations provide an inspectable research proposal and the editorial examples make its conditions concrete.

References and edition custody

[1] A. Keddeh. *Observer, Foreground, and Residue in Cosmological Inference*, KEX-WP-17, 19 July 2026, §§1–2, 5–7, claim and experiment tables. Source document.

[2] Aboudy Keddeh. Current author interpretation directions, retained in *Observation and Identity Review*, R3, current at 18 October 2026. Owner-supplied clarification; source identity is recorded in this edition's metadata.

[3] A. Keddeh. *Observer Relational Mathematics*, Publication Extension R1, 10 October 2026, especially §§2–6. Source document.

[4] Planck Collaboration. “Planck 2018 results. VI. Cosmological parameters.” *Astronomy & Astrophysics* 641, A6 (2020), §§2.1–2.2 and 3.1. DOI; author manuscript. Accessed 18 October 2026.

[5] Keddeh Systems. *From Observation to Governed Computation: A relational architecture study across measurement, interacting bodies, cosmology and quantum mapping*, KEDDEH-WP-RELATIONAL-ARCHITECTURE-001, R1, 18 October 2026, §§5.3 and 6. Source-based internal synthesis and evidence register.

[6] A. Keddeh. *Observer-Anchored Measurement Algebra*, KEX-WP-05, 19 July 2026, §2. Source document. Its conventional identifiability result is applied under the conditions stated in Section 5.

[7] A. Keddeh. *KEX Active-State Foundations*, KEX-WP-01, 19 July 2026, §2 equation (3) and §5. Source document.

[8] A. Keddeh. *Recursive Field Persistence and Memory-Bearing Systems*, KEX-WP-18, 19 July 2026, §§1–2 and experiment table, “Comparator.” Source document.

The original text extraction, its digest, source metadata, a change log and the new calculation receipt accompany E2. The original monograph and master-document citations remain in that unchanged extraction; citing them as lineage does not imply that E2 has freshly verified every passage in those works.

© 2026 Aboudy Keddeh | Publisher: Keddeh Systems | KEX-WP-17 | E2 | Internal research
Licence profile: KEDDEH-RESERVED-1.0. All rights reserved. No licence is granted by this document. Separate written permission governs reuse, redistribution and implementation, subject to applicable law. Third-party material retains its own rights. Authorship, copyright ownership and legal publishing capacity are separate records; no assignment of rights, patent, trademark or software licence is asserted.