# Projection-Relative Truth and Moving Boundaries **Status:** Open philosophical and mathematical research. **Edition:** English editorial reconstruction preserving competing formulations. The notation is a set of candidate formalizations, not an established theory. No undisclosed mathematical system is inferred or reproduced. ## 1. A problem of coordinates before a problem of priority Are being, knowledge, and mind independent layers, or different projections of a structure that does not privilege their separation? A candidate notation introduces a structure X and projections π_O(X), π_E(X), and π_M(X). This is not the claim that mind creates everything, that matter creates everything, or that all three are identical. It asks whether questions of priority depend partly on the representation used to ask them. The simple hierarchy O → E → M is inadequate if a mind also determines which distinctions it uses and changes its surroundings through action. Even a reciprocal causal diagram may be insufficient for the stronger hypothesis: different apparent causal orders could be projections of the same transformation. A hierarchy H(r) could depend on reference r. A >_p B and B >_q A need not assert opposite priorities in an identical context. ## 2. Scale changes what counts as an object A category can remain stable over some resolutions and cease to be the useful local description at others. A person, cells, molecules, and fields need not be rival descriptions at one scale. Write C_ε(x) for a description at resolution ε. A category might behave like an equivalence class [x]_ε within a limited range of resolutions. The proposed recurrence of object, knowledge, and representation inside each other is called “fractal” in a broad sense. It does not yet establish geometric self-similarity, a fractal dimension, or a literal tree. A higher node may mean another zoom level rather than a more fundamental reality. ## 3. Local truth without unrestricted relativism The candidate notation T(φ | r) makes the reference explicit. T(φ | r₁) and T(φ | r₂) can differ without constituting a contradiction under a shared interpretation. A stronger possibility is that no single global truth assignment combines all local descriptions. This is not permission to accept every proposition. It creates an obligation to define local validity, overlap, translation, and incompatibility. What evidence would distinguish a genuinely obstructed global description from incomplete observations or a poorly chosen vocabulary? ## 4. Overlaps, transitions, and time Suppose local domains U_i are related by transitions g_ij. One candidate compatibility condition is: g_jk ∘ g_ij = g_ik Transitions among local descriptions might be read, in different projections, as time, causal continuity, information updating, or compatibility of physical states. **Editorial mathematical qualification:** Calling these objects a sheaf would impose specific gluing axioms. Compatible local sections over a cover in a sheaf glue. A claim of compatible locals without a global section must therefore specify what fails: compatibility, the chosen cover, the relevant object, or another hypothesis. “No global truth” does not follow merely from writing transition maps. This qualification leaves the intended local/global problem open while preventing a metaphor from silently becoming a theorem. ## 5. Thin intersections and the distinction between zero and one A candidate coherent domain is C = ∩_λ U_λ. Under a specified measure, μ(C) might be zero while C remains nonempty. The phrase “zero from outside, one from within” admits several readings: different measures, changed support, observer selection, or a realized-membership indicator. They are not interchangeable. If P(C)=0, ordinary division does not define P(C | C). A construction for conditioning on a null event is needed. Alternatively, 1_C(ω)=1 states that a particular ω belongs to C; it does not turn its prior probability into one. See [Measure, Actuality, and Internal Consistency](measure-actuality-and-consistency.md). ## 6. Indeterminacy is not a final foundation A sequence such as nothingness → indeterminacy → coherence → being → mind would reinstall an absolute hierarchy. The stronger proposal makes even “what is fundamental?” projection-dependent. Indeterminacy must also be distinguished from ignorance. Ignorance can mean that a determinate answer exists but is unknown. Indeterminacy may mean that a final reference has not been fixed, or that fixing it changes the domain. ## 7. Objectifying indeterminacy moves the boundary Naming an indeterminate domain gives it a name, a boundary, and an identity. That determination can produce a new exterior: U → D(U) + U′ U′ → D(U′) + U″ The research motif is not merely infinite repetition. Objectification may change the reference, semantic space, or boundary of the system. A candidate operator Φ(X,r) = (X′,r′) could express such a change. A stronger candidate makes Φ itself state-dependent. Neither notation supplies its required definitions or proves that one operator explains all the examples. ## 8. Self-reference, simultaneous tension, and revision A temporal sequence true → false → true is one possible model of self-revision. Another representation might retain two incompatible commitments simultaneously. A third might abandon the original true/false coordinates. These are distinct possibilities. Contradiction in one interpretation, context-indexed disagreement, and alternation over time must not be collapsed into one phenomenon. The suggested role of contradiction is generative: a failure of the current representation might motivate a larger representation space. Whether it must do so is an open question. ## 9. Truth as an orbit A self-representing system might follow: X_(t+1) = F(X_t, P_Xt(X_t)) Representing itself changes its state, so its old self-description may no longer be complete. A fixed point is not assured. An orbit, cycle, attractor, or branching family could be a more useful object of study. The possibility that time is a projection of this failure to close remains a philosophical hypothesis. It is not a physical result. ## 10. Absorbing the meta-language Moving to a meta-language need not provide a permanent outside. Once the meta-language is described, it can itself become an object within an expanded system. The recurrent motif is: incorporating the boundary changes the boundary. This can produce a hierarchy, but a linear ladder is not mandatory. Cycles, partial references, and changing contexts remain legitimate alternatives. The appropriate mathematics must be discovered through explicit examples. The assertion “there is no absolute outside” must itself face the same scrutiny. Treating it as an unquestionable final truth would exempt the proposal from its own claim. ## 11. Mind-like and cosmological readings A global organization can have properties not evident in one component. This motivates possible readings of coherence as mind-like, or of a whole structure as having a mental and a physical projection. These are optional interpretations. They establish neither cosmic consciousness nor an actual Boltzmann-brain scenario. They also do not require the ontology–epistemology–mind triplet to be the final coordinate system. Another intelligence could make different distinctions. ## 12. Two legitimate mathematical directions One direction searches for invariants preserved across changing projections: I(P_r(X)) = I(P_s(X)) Another subjects every proposed invariant and meta-level to the same process of recontextualization. The second is more radical but also harder to constrain. Neither should be silently substituted for the other. A useful model must state which relationships remain fixed, and what counts as a meaningful comparison if those relationships can change. ## 13. Comparisons to investigate Diagonalization, reflection, revision semantics, paraconsistent logic, indefinite extensibility, contextual semantics, local/global constructions, and non-well-founded structures are comparison directions. Their appearance here is not a literature review or an originality claim. The task is to determine whether the recurring motifs reduce to known constructions, combine them in a new way, or require a different object. Theorems about formal systems do not automatically prove metaphysical claims. ## 14. Conditions for a substantive research program A model that admits every result explains too little. The research program should produce an explicit invariant, obstruction, impossibility result, or necessary expansion rule. Questions that separate candidate models include: - Does truth have a fixed codomain? - Does representation change only the described state, or also the semantics? - Is the exterior a complement, an undefined region, a modality, or an open-ended operation? - Is contradiction necessary for expansion? - Is time primitive or derived? - Is a fractal structure actually measurable? - Does a fixed point preserve the distinctions under investigation? - Does “self-refutation” concern syntax, semantics, proof, or changing reference? **New editorial challenge:** Construct two small models sharing the vocabulary but disagreeing on a concrete outcome. If no observation or formal property separates them, the proposed unification may still be a language for inquiry rather than one theory. ## 15. Relation to the broader problem-space Projection-dependent hierarchy, scale-sensitive categories, reflexive boundaries, and internal consistency can be traversed together. Their connection does not force them into one global account. The strongest open question is whether one minimally specified, self-referential transformation can generate several of these structures while excluding others. A coherent negative result would be useful: it would identify where the problem-space branches.