1 Scope#

This paper is a grounding, not a physical theory. It inherits Graphic Equalisation’s position that the claims are about information — distinction, frame, admissibility, realisation — of which physical theory is one embodiment, and that a physical result bears on an information-level claim when it settles an informational fact. Everything below is at that level. The strong claims are structural: that there is a c, a ℏ, an uncertainty principle, a Landauer floor, a second law, and that special and general relativity reconcile as one change-economy. To these the quantum arc adds collapse as frame-relative realisation, entanglement as a superframe dimension, and — the one place a number is reached — the Born rule, recovered given contingent rule-content rather than derived as a necessity, with the uniqueness supplied by cited theorems. The weak part is otherwise anything carrying a number: no field equations, no quantitative metric, no exact Lorentz factor, no dimensionless constant, and no theory of quantum gravity. Each grounding is a promoted constraint exposed to falsification, in the manner of Graphic Equalisation’s collapse bridge, not a result asserted from within.

The minimal-universe argument that licenses the contingency verdicts below — a rock is a universe, and the minimal universe does without time, the gauge group, the constants and the dimensions, so none can be compulsory — together with the run/record axis, free will, personal identity, and death, is developed in the mind companion (its Section 2 and Proposition 2.1), which treats philosophy of mind as applied physics of the run.

2 c as the change-to-time rate#

Graphic Equalisation establishes that time is not a substrate dimension but is inferred by a frame, whose unit is a cycle — one traversal of a closed rule-loop, of no fixed length. Once time is the inferred count of cycles, the invariant speed c has a natural reading, consistent with the missing-denominator result (Graphic Equalisation’s Remark 13.5: there is no rate in substrate-time, since there is no substrate-time to divide by). c is the fixed rate at which change is realised — per unit of the substrate’s throughput, a cycle consuming a variable amount of it — from which a frame’s time is inferred. So c is prior to time: the conversion between change-count and inferred duration, which is exactly what a constant relating the temporal and spatial parts of an interval is. That is runner c, the first of the three things the word names here, and the separation has to come before anything is claimed of any of them.

A value other than 1 is a cross-frame quantity: the familiar number appears only when one frame is compared with another — change-throughput against a spatial metric, seconds against metres — which is where the continuous lives, in the maps between frames rather than in a frame’s own discrete states. One can give time a number at all only by comparing frames; in its own frame c is the unit, and its particular value (299792458 in SI) is an artefact of the comparison, not a fact about the substrate. And c is a limit because a signal is a sequence of changes and none can be realised faster than the frame’s throughput.

It matters what is and is not being stipulated here, because the nearest neighbouring account stipulates precisely what this one does not. Discrete-substrate derivations of relativity typically assign a metric quantity and obtain c-constancy from the assignment: in the Wolfram model, constancy of the speed of light is, in the author’s own words, “enforced axiomatically by our definition of the edge lengths in causal graphs” (Gorard, 2020a). Nothing is assigned here. A cycle is a closure event, not a length: how much substrate throughput one traversal consumes depends jointly on the rules and on the graph those rules are applying to, so there is no fixed unit available to be set. A frame reads its own c as 1 because it counts its own cycles and possesses no in-frame length against which to compare them — the unit is unavailable, not chosen.

The difference is not verbal, because it is where the empirical content sits. A stipulated edge length is constant by construction and can therefore predict nothing further. A cycle whose length varies with rules and graph predicts a great deal: that concentrating change-load lengthens cycles and slows clocks, that mass is throughput reserved for internal change, and that the whole of Section 3 follows as consequences rather than as further postulates. The honest residual is that this does not make c-invariance an empirical prediction — a frame counting its own cycles will read 1 whatever the substrate does — and the paper does not claim it as one. What it claims is that the invariance costs nothing to assume because it is a fact about self-measurement, and that the variation in cycle length, which is not a fact about self-measurement, carries the physics.

3 The change-budget: speed, dilation, mass, aperture#

The budget is the substrate’s throughput per cycle, and c is its unit rather than its size (Definition 2.1); nothing below turns on the number, only on the fixity. That fixed throughput splits between spatial motion and internal change (proper time) — in squares, the two shares summing to one, which is the constant-magnitude four-velocity below — and a speed is the share spent on motion. At rest the whole budget is internal change and the clock ticks fully; increasing speed moves budget to motion, leaving less for internal change, so the clock slows — time dilation as budget re-allocation — and at c the whole budget is motion and no proper time remains. Mass is what holds the internal share above zero at every finite speed: a massive graph’s motion share approaches one and never reaches it, so it asymptotes to c and the energy to approach diverges, while a massless one has no internal share at all and is at c, with no proper time.

Because observation is funded by internal-change throughput, approaching c coarsens a frame’s aperture. The transformation between frames in relative motion — time dilation and length contraction — is an aperture transformation between different budget-splits, and each frame’s aperture self-calibrates so that its own c still reads 1. The Lorentz factor is the aperture change that holds c = 1 across the split. (The constant-magnitude four-velocity of Minkowski geometry (Minkowski, 1908) is this budget-split stated as a metric fact rather than as a mechanism.)

Both ends need stating, because the negative half is the smaller half. A frame learns a great deal here: the two-way speed, its isotropy, and the constraints that isotropy places on any anisotropic account are all in hand and tightly bounded by experiment. The sharpest item on that side is one-way after all, and it is worth spelling out because it reads at first like a counterexample. The absolute one-way speed is unavailable; the ratio of one-way propagation times through two media is not, and is measured as a matter of routine — a pulse sent along a fixed path through a medium arrives late against the same path in vacuum by an amount that is a real datum. That media slow light by large factors is itself routine: the group velocity in an ultracold gas has been reduced to seventeen metres per second and read off (Hau et al., 1999). The synchrony convention cancels: whatever offset 𝜀 assigns to the far end is a function of the separation, both signals cross the same separation, so the offset is common to them and absent from the difference (Anderson et al., 1998). The condition carrying that is along the same path — two one-way measurements in different directions do not share an offset under an anisotropic convention, and their ratio is convention-dependent again.

This is the closure criterion delivering the positive result rather than an exception to it, and getting all three cases from the same criterion is a check the account passes rather than a convention it adopts. And the impossibility is indexed to frames inside the structure being measured, which is Axiom −1 in its plainest form. That qualification has to be read as a real one and not as decoration. The framework supplies outsides routinely: a superframe is one, and Section 6’s reading of entanglement as a superframe dimension is load-bearing for the recovery of the Born rule two sections later. Not measurable in frame is therefore the whole of the claim, and it does not extend to not measurable.

4 Special and general relativity reconciled#

The two relativities are one mechanism, because the one fixed throughput budget — whose unit is c — can be drained two ways.

The equivalence principle is the seam. Accelerating (changing one’s motion) and standing in a gravitational field (change-load) draw on the budget identically, so inertial mass (resistance to a change of motion — the impulse relation of the momentum paper, its Remark 10.1, with mass as the exchange rate) and gravitational mass (change-load consuming throughput and slowing local time) are one quantity, “how much change this matter loads onto the ruleset,” measured two ways. General relativity assumes their equality and does not explain it; here they coincide by construction.

4.1 Relativity and collapse#

Relativity also reconciles with the collapse picture of quantum measurement, by the division of Section 2: the continuous in the maps between frames, the discrete in a frame’s own states. The continuous geometry of general relativity is a cross-frame map; the collapse of quantum measurement is the in-frame event; and c — the change-to-time rate — is the seam joining the discrete event to the continuous metric. The two are the map and the event of one change-substrate, not rival ontologies, and the standing incompatibility is the artefact of treating them as rivals. This is a claim about the shape of the reconciliation, not a derivation: that the field equations follow is not asserted, and no theory of quantum gravity is offered.

5 The measurement problem is frame-relative collapse#

Collapse in Graphic Equalisation is a local write that realises one continuation relative to the frame that interacts. It is not a physical process needing a mechanism — no dynamical collapse, no spontaneous localisation — but the in-frame realisation that occurs when frames couple. So there is no frame-independent “when”: each frame realises its own outcome upon interaction, and a frame that has not interacted holds no collapse. This is Axiom −1 and relational quantum mechanics (Rovelli, 1996) — facts indexed to interactions, not assigned from nowhere.

That there is no observer-independent fact of the matter is, in this literature, a theorem and not only a stance. Brukner (2018) shows that the existence of observer-independent facts, together with locality and freedom of choice, is inconsistent with quantum predictions in an extended Wigner’s-friend setting, and Frauchiger and Renner (2018) show that agents who use quantum theory to reason about other agents who use quantum theory can be brought to contradictory conclusions about the same run. Neither result is evidence for the change-substrate — they constrain every account equally — but they remove the option Proposition 5.1 declines to take, that of a single frame-independent record the friend’s and Wigner’s descriptions must both be answerable to. Axiom −1 forbids that record on structural grounds; these results forbid it on quantum-mechanical ones, given the auxiliary assumptions each names.

This dissolves the measurement problem — the when, the how, the who, and the definiteness. That outcomes are realised frame-relatively is the shape; how their probabilities are weighted is a separate question, taken up in Section 7.

6 Entanglement is dimensional access through frames#

An entangled pair is one system in the superframe, described by a joint dimension — the correlation — that is a first-class quantity of the superframe and null in each subframe individually (Graphic Equalisation’s Proposition 10.8, higher-frame first-class structure). Each subframe has dimensional access to it only through its own boundary, and only in part; the full dimension lives one level up, along the structural-depth axis rather than the spatial one.

That the subframe/superframe split is itself frame-relative is not only assumed here. Ali Ahmad et al. (2022) prove, in the quantum-reference-frame setting, that the decomposition of a composite system into subsystems is frame-dependent — what counts as “the parts” changes with the frame relative to which the whole is described. That result takes frame-covariance as its premise and derives the relativity of subsystems from it, so it does not establish the non-privilege of frames — that remains this framework’s own commitment, held as an axiom and not borrowed as a theorem. What it does establish is that the subframe carving inherits the non-privilege: no single decomposition into parts is preferred, so the joint dimension below is a claim about a decomposition, made from inside it, and not a claim about a preferred carving of the world.

Quantum mechanics is the standing demonstration that the in-frame description is not exhaustive, and it is an empirical one rather than an interpretive preference. Correlations violating Bell’s inequality are not functions of the separated subframes’ own states (Bell, 1964), which is precisely what this paper means by a dimension held at the superframe, and they have been exhibited without the loopholes that once allowed the conclusion to be resisted (Hensen et al., 2015). So structure beyond a subframe’s own description is not merely coherent in this framework; it is measured. The superframe account is nevertheless a shape claim, not an additional measured variable: it earns the mapping from the measured failure of subframe-local descriptions to a joint first-class dimension. It fails if a complete account of the observed correlations can be given from the separated subframes’ own state alone, without joint structure or an equivalent containing relation. Bell tests do not select this vocabulary over every interpretation; they expose the subframe-local shape that this vocabulary says is unavailable.

Honest boundary: seeing the connecting dimension dissolves the spookiness, not the numbers — the Tsirelson bound, the correlation strength and the Born weights are not derived; they become ordinary physics questions rather than spooky ones.

7 Amplitudes are a rule on Narrative; recovering the Born rule#

An amplitude is not a foreign number-structure but the directional Narrative of the momentum paper, acted on by a rule. That paper’s directional momentum p = ϕ_W (N) (its Definition 4.1) is a magnitude ϕ(N) — the operative-history measure — times a direction . The complex amplitude is therefore the directional Narrative, its dynamics a rule applied to it, and its phase the direction read as position in the cycle (Section 2). The complex structure of quantum amplitudes is thus grounded, not imported.

This is a candidate, not a derivation, and a real advance on leaving the amplitude a blank.

7.1 An attempt to close it#

The candidate can be pushed further than a candidate, though not all the way. Take the amplitude as a directed magnitude — a Narrative’s size r times a phase — and follow four steps, marking what each earns. This subsection is an attempt, flagged as such, not a claimed result.

Complex (motivated). The phase is position on the realisation cycle, and a cycle’s traversal parameter is one-dimensional: S¹ = U(1). So an amplitude is re^i𝜃, a complex number — real would be a phase-less (cycle-less) magnitude, and the one-dimensional closure loop gives a single angle, not the three of a quaternionic phase. Motivated by closure, not forced.

Linear (a candidate mechanism, not a proof). Independent contributions to one outcome superpose by addition. This has a candidate mechanism rather than a bare posit. Free evolution is the amplitude’s own Narrative, and a Narrative is what a subgraph runs when no rule fires — the free continuation simply carried forward (the momentum paper’s free continuation, its Definition 5.2, and the free-continuation default of Section 11). With no rule firing on them, nothing couples the components of a superposition; each free-continues independently, and uncoupled independent continuation is linear superposition, since a nonlinearity would need a rule reading one component and acting on another — exactly what does not fire here. Linearity is on this reading the signature of free continuation. What stays a residual is the rigorous step from “uncoupled” to full additivity and norm-preservation (unitarity), and the linearity of the imposed supergraph map is not handed over for free: the mechanism motivates linearity, it does not yet prove it.

Phase-independent (from no privilege). A cyclic direction has no privileged origin, so a global phase rotation is a change of nothing (Graphic Equalisation’s no-privilege principle applied to the phase). The realised probability must therefore be invariant under a global U(1), hence a function of the modulus alone, P = f(|a|). This step is earned, and it is the no-privilege thesis — Graphic Equalisation’s Axiom −1, carried through the mind companion — doing physics work.

Squared (forced, given the above). Two alternatives with no privileged relative phase — decohered, their relative phase averaged over — are classically exclusive, so their probabilities must add. Averaging the combined modulus over the relative phase,

since the cross term 2Re(a^∗be^i𝜃) averages to zero. The square, f(x) ∝ x², satisfies it, and the obvious alternatives do not: a quartic overshoots by the surviving ⟨cos²⟩ = term, the linear undershoots (a = b = 1 gives 4∕π≠2), the square lands exactly; normalisation fixes the constant. This selects the square among the natural candidates rather than proving it unique — the uniqueness is what Gleason’s theorem supplies below, over all measurement contexts. The weight is real because closure (a returned cycle) is net-zero phase.

The Gleason residual, reconsidered. The internal Narrative is relative in the reading: it is the same standing structure read as a vector in the basis the frame’s current rule set fixes, so a superposition is rule-set-relative — definite in one rule set, superposed in another — which is the same form as relational quantum mechanics (Rovelli, 1996) without being the same claim: that programme relativises a physical fact to an interacting physical system, never to a descriptive framework or rule set, so what is borrowed here is the shape of the relativisation and not its warrant. Because no rule set is privileged (the no-privilege principle again), the probability rule must be the same across every rule set, which is exactly the context-independence Gleason’s theorem assumes. The framework thus supplies Gleason’s premises — a vector-valued state (the linearity step above) and a context-independent measure (no privileged rule set) — from its own principles, and Gleason (Gleason, 1957) supplies the uniqueness: |⋅|² is the only such measure in dimension ≥ 3, leaving the vector-structure rigour of the linearity step as the one posit.

The Hilbert posit, reconsidered: linearity as projection through frames. That one posit — the state lives in a complex Hilbert space and free evolution is unitary — is what treating Hilbert space as a frame is meant to reach. Read the inner product ⟨a∣b⟩ as the projection of one frame’s Narrative into another’s rule set: how much of b shows up when b is read in the rule set a fixes. So the inner product is not imported structure but the framework’s own cross-frame reading.

Linearity is then that projection distributing over components: during a free (empty-rule-set) segment no rule couples the Narrative’s parts, so reading a superposition is superposing the readings — the same uncoupling, applied to the projection. Unitarity is that a free continuation destroys no distinction: it is an information-preserving write and not the absence of one, so the prior Narrative is recoverable from it and every cross-frame projection of it is preserved, norms and inner products conserved (the free-continuation conservation of Section 11).

The Narratives, directional magnitudes that superpose, form the complex vector space; the inner product, a cross-frame map, makes it Hilbert; and its completeness — the defining Hilbert property — is the continuum living in the maps between frames, Hom(frame,frame), never in a frame’s own stepped states (Section 2).

This meets the informational reconstructions of quantum theory, which derive the complex Hilbert space and the Born rule from a few operational axioms (Hardy, 2001; Chiribella et al., 2011); Masanes and Müller (2011) obtain finite-dimensional quantum theory from five physical requirements on states and transformations, Höhn (2017) obtains qubit quantum theory from rules on an observer’s acquisition of information (Section 17), and Müller (2021) surveys the programme as a whole. Among the post-2011 reconstructions the nearest to this paper in formal register is Selby et al. (2021), which states every postulate diagrammatically — wires and processes, with symmetric purification the central axiom — rather than convex-geometrically. That matters for form and not for content: a compositional, diagrammatic axiom set is written in the same currency as a typed-graph substrate with subframe nesting, so the mapping attempted here is a translation between two graphical calculi rather than between a graph and a convex body. It is a reason to think the mapping can be checked, not evidence that it holds.

This is the same move again, one level deeper, and the mapping has been worked axiom by axiom against the Hardy and CDP reconstructions in the Born note (its Section 4), with a result worth stating rather than promised. The two most quantum axioms are supplied natively, and each from a different substrate principle (the Born note’s Proposition 4.1): Hardy’s continuous reversible transformation between pure states by free continuation (an information-preserving write is reversible, and the cycle-phase is continuous), and CDP’s purification — every mixed state is pure in a larger system — by entanglement as a superframe dimension (Section 6). Context-independence is no privileged rule set, and composition is subframe nesting. Most of the rest match native commitments rather than statements (probability from realisation, compression and simplicity from the substrate’s minimum description length, subspaces from subframe-as-frame), and the Born note rates them as genre-matches on that ground. Given the mapping, Hilbert space and the Born rule follow by the cited reconstructions, as |⋅|² followed from Gleason. A closure-phase frame is likewise consistent with complex amplitudes at one system (the Bloch sphere, K = 4, rather than the real disk’s K = 3), though a phase-less frame is equally constructible and the substrate does not require either. The remaining question — local tomography, which selects complex over real — is, correctly framed, not a gap the substrate must close by forcing. Almost everything in a graph substrate is optional: the compulsory core is structure, rules, subgraph-hood, and an admissibility grammar (the minimal-universe argument of the mind companion, its Proposition 2.1), and everything else — self-similarity, complex-over-real, the number of spatial dimensions, the gauge group — is contingent rule-content. The substrate therefore admits complex quantum theory (the self-similar complex composition gives K_AB = 16) rather than forcing it, exactly as it admits, without forcing, one value of N or one gauge group; a real-amplitude composition (K_AB = 10) is another admissible configuration, and our universe’s selection of the complex one is self-location (the Born note tabulates the fiducial counts and states the verdict as its Proposition 5.1). Given that contingent choice the Born rule and Hilbert structure follow, which is how this paper treats all physics — shapes from contingent rule-content, the content never forced. Complex quantum theory has here the same status as N = 3 — admitted and self-located, not compelled. In a word, the framework recovers the Born rule rather than deriving it per se: it reproduces the known rule within its own structures — some steps forced given the premises (the squared modulus), the premises themselves admitted and contingent (complex, linear, self-similar) — not forcing it from necessity. Recovery, not derivation, is the honest verb and the right one: a contingent law is recovered given its rule-content, never derived as compelled (the Born note holds this as its Remark 2.1). A rival recovery starting from a comparably admissibility-flavoured base — constructor theory’s, in which probability is not fundamental at all but emerges from the unpredictability of measurement on superinformation media (Marletto, 2016) — reaches a different answer from the same kind of premises, and is engaged directly in Section 17.

Honest residuals. The recovery rests on one posit of this paper’s own: the rigorous step from “uncoupled” to full additivity and norm-preservation is not made here, and the linearity of the imposed supergraph map is not handed over for free — the mechanism motivates linearity and does not prove it. The remaining residuals belong to the mapping rather than to the physics, and the Born note holds them rated, item by item, in its Section 7: the Level-0 hygiene conditions, the reach of the phase-average selection against Gleason, the composition seam, and the genre-match ratings.

7.2 The reconciliation: Born, SR, and GR on one substrate#

The recovery has a consequence larger than the Born rule. Special and general relativity are already reconciled here as one change-budget spent two ways (Proposition 4.1, Corollary 4.2), with c’s invariance a definitional consequence rather than a postulate (Proposition 2.2) — definitional given the substrate’s own units. Quantum collapse and relativity are already joined at c as the event and the map of one change-substrate (Section 4.1). With the Born rule now recovered on the very same primitives — closure for the phase, no-privilege for phase- and context-independence, free continuation for the reversible structure — Born, SR, and GR stand reconciled on one substrate: the same c, the same no-privilege, the same closure loop underwrite all three.

Honest boundary: the strength is uneven — the SR–GR half a definitional consequence given c-as-unit, the quantum–relativity half structural. This is emphatically not a quantum theory of gravity: no field is quantised, no metric is derived, no dimensionless constant is fixed. What is reconciled is the conflict — that quantum theory and relativity are two aspects of one change-substrate rather than incompatible frameworks — not the quantitative unification the scope explicitly disclaims.

8 Interference and the double slit#

The double slit is obvious once two established facts are held together: causality is ordering in a frame, and a superframe applies changes to a subframe without passing through the subframe’s admissibility.

A superframe applying a change to a subframe is not itself a measurement coupling: it makes no distinction the subframe holds, so it realises nothing for that subframe — which is why the superframe forcing below leaves an interference pattern intact while a which-path coupling destroys it.

Two established facts make this un-mysterious. First, causality is ordering in a frame: there is no global fact about “which path the particle really took,” because which-path is undetermined, not undisclosed, until a coupling realises it — a question with no answer of any kind until a coupling settles it. The “both paths” is not a particle in two places but the frame’s P. Second, a superframe applies changes to a subframe without passing through the subframe’s admissibility (Graphic Equalisation’s Proposition 6.5: admissibility is self-governance, not armour — a graph’s grammar governs its self-originated transformation, not world-caused forcing). So the interference is the superframe applying its combined-amplitude structure to the particle; the particle does not select a path by its own rules, and cannot veto the combination.

Honest boundary: this grounds the shapes — superposition as possibility, interference as phase-combination, the double slit, duality, and delayed choice without retrocausality. The quantitative fringe pattern needs the amplitude phases computed, i.e. the Hilbert formalism, which is the Born-rule frontier of Section 7.

9 Dimensional constants are frame-units; dimensionless ones are not#

The reading of c as a frame’s own base unit generalises. Each dimensional constant is the base unit of one of a frame’s own economies — 1 in-frame, a number only cross-frame.

• c is the change-rate unit (Section 2).

• ℏ is the aperture quantum, the smallest resolvable cell: observation is funded by finite internal-change throughput, so fine resolution in one dimension trades against another and the trade floors at a minimum cell — which is the uncertainty principle, ΔxΔp ≥ℏ∕2 being the finite-aperture budget stated as a bound, with ℏ its quantum.

• k_B is the information–energy conversion: Landauer’s k_BT ln 2 (Landauer, 1961) — the floor cost of change of Graphic Equalisation’s economics — is one bit’s change converted from the information frame into the energy frame at that rate.

• G is the change-load–geometry coupling, mass deforming the containing frame.

That physics sets c = ℏ = k_B = G = 1 (natural units) is then not a convenience but a recognition: each is a frame-unit, read as 1 in the frame’s own terms with its familiar magnitude a cross-frame artefact (Proposition 2.2).

The type claim is forced rather than preferred, and the forcing is at the floor. A ratio is a relation between two quantities: two ends and something carried across. A magic number is a label — a value borne by one thing, answering to nothing else — and the base has no labels for it to be written as, carrying relata and relations and no marks on either (Is’s floor, made explicit in its Section 6). So a standalone constant has nowhere to live. Where a theory carries one anyway, it is carrying a value whose relata it has not identified, which is a write nobody can read back — the silent fourth cost of Graphic Equalisation’s well-formedness (Definition 7.13) — and the defect is in the theory rather than in the world.

What this does to the open question is change its shape and not its status. It explains no constant’s value and does not pretend to. What it denies is that the value is the right thing to ask after first. “Why is α about 1∕137” asks for the provenance of a number; on this account the number is what a relation carries, so the question is which two things stand in it and what fixes their relative size — and the standard framing, in which the Standard Model has some two dozen free parameters, is then not a list of two dozen mysteries but a list of two dozen relations whose relata are unnamed. That is a harder question and a better-posed one, and nothing here answers it.

The commitment is checkable in the one direction that matters. Exhibit a dimensionless constant that is a ratio of nothing — irreducibly standalone, not expressible as one quantity against another even in principle — and the type claim fails. The candidates all survive it so far, and survive it trivially: α compares a coupling against ℏc, the mass ratios are ratios by construction, 𝜃_QCD is an angle and an angle is an arc against a radius, Λ in Planck units is one energy density against another. Counts are the case worth naming, since a count looks like the counterexample: the number of spatial dimensions, or of generations. A count is a ratio to the unit of what is counted, which is degenerate but is not standalone — there is still something it is a count of, and that is the second end.

And the consequence that makes this empirical rather than taxonomic: a constant is constant except when the things that set it change. Constancy is a property of the relation’s stability and not of the number, so constant names a state that can fail rather than a category that cannot. Nothing in this account enforces it. That is not a prediction that constants do vary — it is the removal of a reason to expect they cannot, and exact constancy maintained forever would be a fact wanting an explanation rather than the default it is usually taken for.

The split above then does work it is not usually asked to do. Only a ratio can vary, because only a ratio is anything. A claim that c has changed is not a claim about the world: c is the frame’s own unit, read as 1 in its own terms, and a varying ruler measured by itself is no measurement — which is Duff’s point arrived at from this side. A claim that α has changed is a claim with content, because α compares two things and one of them could have moved. So the varying-constants question is well posed exactly for the dimensionless remainder and ill posed everywhere else, and this is why searches quote Δα∕α and never Δc.

What a measured variation would be is a measurement of the relata, which is the part worth having. If α drifts, the relation between the quantities it compares has changed, and the structure of the drift — with position, with epoch, with environment — constrains which of them moved. On this account that is not an embarrassment to be explained away but the one observation that would name the ends of a relation the theory currently carries unnamed. The empirical position is that the bounds are tight: reactor, clock-comparison and nucleosynthesis constraints all limit any drift severely (Uzan, 2011), and the reported spatial dipole in α from quasar absorption (Webb et al., 2011) is contested and not independently confirmed, so nothing here rests on it. It is named as the shape a confirmation would have, not as evidence.

And a ratio can look constant without being constant, which the tightness of those bounds does not distinguish. Write the ratio R = A∕B. Its fractional response is δR∕R = δA∕A − δB∕B, so a large denominator does not suppress a fractional change in B — the condition worth stating rather than hiding. What a large denominator suppresses is an absolute one: a perturbation of fixed size δ enters as δ∕B and vanishes as B grows. Wherever what can move a quantity is bounded in absolute terms, which is the ordinary case, a ratio with a large enough denominator reads as constant whether or not it is.

That is the finite aperture of ℏ above, arriving at the constants. A frame cannot resolve a change below its own cell, so constant and varying beneath resolution are the same reading to it, and constancy joins the list of properties that are aperture-relative rather than intrinsic. What is constant to one frame need not be to a finer one, and no frame is entitled to read its own resolution as a fact about the world (Axiom −1 again, at the measurement).

Two consequences follow and the second is the useful one. First, the empirical bounds say exactly what they say and not more: they bound variation below the resolution achieved, which is a statement about the aperture of the measurement and not about the world. Nothing here disputes them and nothing licenses reading them as an absence.

Second — and this is where the diagnostic hope of the previous paragraph becomes a search strategy rather than a wish — the constants most likely to yield a measurable variation are not the ones deemed most fundamental, but the ones whose denominators are smallest against the perturbations available to them. Apparent constancy is then predicted to track denominator size rather than depth, and a programme aimed at whichever constant seems most basic is aimed by the wrong criterion. This account cannot rank the candidates, having no access to the relata that would fix the denominators; what it supplies is the ordering principle and the claim that it is the right one.

One consistency check on the section before it. The decay constant of Section 13 is a rate, and a rate carries a dimension, so by this proposition it is a frame-unit and not a fact about the world either. What would be a fact is the ratio of one decay rate to another, and it is that ratio which the argument there requires to be stable — not the half-life in anybody’s seconds. It sharpens the real question rather than answering it: the numbers that need explaining are the dimensionless ones, while the pseudo-mysteries — why c, why ℏ — are unit conventions. This is the standard observation that the number of fundamental dimensional constants is a matter of convention (Duff et al., 2002), landing on the side that they are units; the dimensionless remainder is left exactly as open as it was found.

10 The Planck scale, quantization, and fine-tuning#

Three consequences follow. First, the Planck scale is where the cross-frame ratios between the four economies all reach unity together (ℓ_P = and its companions). Each constant reads 1 in its own frame always, so this is not a scale at which they become 1; it is the coincidence, visible only in the comparison between economies, of the aperture cell (ℏ), the change-rate ceiling (c) and the change-load–geometry coupling (G) — an artefact of comparison in the same sense the SI value of c is. Below it a frame cannot resolve, cannot order, and horizons its own geometry, so nothing there is representable — the “ ? ” of Graphic Equalisation’s epistemic boundary with a location. The Planck scale is the frontier, which is why physics predicts its own breakdown there: not a failure of theories but the exhaustion of a frame’s units, below which the pre-relational substrate begins.

Second, quantization is the aperture being cellular: because ℏ is a minimum cell rather than an infinitesimal, action and energy come in ℏ-steps, so the “quantum” of quantum mechanics is Graphic Equalisation’s discrete sufficiency (its Hypothesis 3.13) read at the resolution axis — the granularity of the aperture, not a separate granularity of the world.

Third, the dimensionless constants are contingent rule-content: the framework cannot derive α or the mass ratios, but it can say what kind of thing they are — the contingent specifics of this substrate’s rule-set (Graphic Equalisation’s Remark 7.19), so “why α” is not “why this necessary value” but “why this rule-set,” answered by self-location (Proposition 14.2). Two type claims are now in play and they agree: a dimensionless constant is contingent rule-content, and it is a ratio rather than a label (Proposition 9.1).

The second sharpens what a fine-tuning argument is doing. Such an argument treats a constant as a dial — a value that could have been set otherwise with everything else held fixed — which is to treat it as a magic number, the one thing this account says no constant is. Turning a dial on a ratio is not holding everything else fixed: it is altering the relation between the two quantities the ratio compares, and whether that is a coherent counterfactual at all is the question such arguments skip rather than answer. That is a second reason the framing misprices its question, and it is independent of the first — self-location would dissolve fine-tuning even if constants were labels, and this would bite even if self-location failed. Both relocate fine-tuning at its root, which Section 14 takes up.

11 Symmetry and conservation#

A symmetry is a transformation that produces no residual: a change the structure does not feel, free and reversible, a no-op in the container’s grammar. Noether’s theorem, symmetry ↔ conserved quantity, is then the cost side of that. A quantity is conserved precisely when altering it would cost — produce residual, require a paid write — so the symmetric direction is free and the quantity perpendicular to it cannot move without paying. What a transformation leaves untouched is what it costs to change.

Charge conservation follows from the global phase symmetry — a uniform re-description carrying no in-frame residual, which is the shape above. Local gauge symmetry is a different thing: a redundancy of description rather than a symmetry of states, and by Noether’s second theorem it yields identities and constraints rather than an independent conserved charge. The framework’s reading applies to the first; the second is the cross-frame freedom (in the maps, where the continuous lives) that leaves the physics untouched, and carries no charge of its own to conserve.

12 The second law#

Entropy and its arrow follow from the same economy.

This is the Gibbs–Jaynes reading, in which entropy is a property of the observer’s coarse-graining rather than of the world in itself (Jaynes, 1957); here it is grounded rather than assumed — entropy is what this frame does not distinguish.

Two restrictions are carried in that statement and neither is decoration. Graphic Equalisation prices reversible operation rather than forbidding it, so the proposition restricts to the discarding class rather than claiming it of all writes, and what is claimed is that the class is the general case. And a free continuation is not an event in the frame’s own terms at all — Graphic Equalisation has it as the absence of one, and nothing re-applies the standing structure. What makes it a write here is this grounding’s container rather than the substrate: the physical superframe charges for persistence, so what is free to the frame is paid in the frame that holds it, and it is then the reversible, entropy-neutral case. That is a claim about these rules and not about all of them, which is why the arrow tracks writes and not mere succession.

13 Decay is determinate; the phase is unretained#

Radioactive decay is the standard exhibit for irreducible chance: the law is exact in aggregate and the individual event unpredictable. On this substrate the two halves come apart, and the interesting one is the aggregate.

The qualification in that last clause is load-bearing rather than cautious. A hazard constant simpliciter is exponential decay and nothing else, and exact exponentiality turns out to require something this account denies — which is the subject of the deviations below.

The qualification is what earns the proposition, because it is where the account says more than its rival. Irreducible chance does not predict a distribution. That decay is undetermined is consistent with any law whatever — peaked, heavy-tailed, bimodal — and the exponential has to be put in by hand alongside it, as an observed fact about nuclei rather than a consequence of their being undetermined. Here the exponential is forced: memorylessness is not an extra postulate but the absence of a retained quantity, and constant hazard has exactly one survival law. A nucleus that carried its own age would be a more expensive object, and Graphic Equalisation’s write-economics (Proposition 7.12) says who pays for the carrying. A determinate countdown — a nucleus that knows its remaining ticks — is the model this proposition is most likely to be mistaken for, and it is refuted by the data it is meant to explain: survivors of a countdown are nearer their deadline, the hazard rises with age, and the observed law would be peaked rather than exponential.

Nothing here is a local hidden variable of the kind Bell forbids. Decay times of separated nuclei are not a correlation between separated measurements, so no inequality is engaged; and where one is engaged, the account’s answer is the superframe dimension of Section 6 and not this phase, which must not be promoted into one.

The exposure is the deviations, and they are real. Standard quantum mechanics predicts departures from exponentiality at both ends: quadratic behaviour at short times, and a power-law tail at long ones (Khalfin, 1957). Both have been reported — the short-time departure in the tunnelling of ultracold atoms (Wilkinson et al., 1997), the long-time one in the luminescence decay of organic molecules (Rothe et al., 2006). A constant hazard admits neither, so the proposition owes an account of both.

It has one for the short end and it is a prediction rather than a repair. Graphic Equalisation’s cycle-unit result (Proposition 13.7) makes inferred time a cycle and requires closure for it, so below one closure cycle of the observing frame there is no inferred-time interval for a law to be exponential in. Exponentiality is a statement in inferred time; short of a cycle there is no such statement, and a departure there is expected on this account and unexplained on the bare irreducible-chance one.

The long end inverts, and the inversion is worth stating carefully because it is easy to overstate. Khalfin’s argument is that exactly exponential survival requires the state’s energy distribution to be exactly Lorentzian, and a Lorentzian has support on the whole line, including arbitrarily far below any threshold; a Hamiltonian with a ground state is bounded below, so the distribution vanishes on a half-line and its transform cannot decay exponentially (Khalfin, 1957). Exact exponentiality therefore requires a spectrum unbounded below — which is to say, it requires a continuum quantity of unbounded support to be a frame’s own state distribution. This paper has already denied exactly that: continuum structure lives in the maps between frames and never in a frame’s own stepped states (Section 2). So exact exponentiality is a cross-frame idealisation that no frame exhibits, and a departure from it at long times is what this account should have predicted rather than what threatens it.

What that earns is the deviation in kind, and not its form. The power law and its exponent follow from the threshold branch point, which is a fact about the composite system’s spectrum and not about what the nucleus retains; nothing here derives them, and treating the inversion as though it did would be claiming the result rather than the direction. So the standing exposure is narrower than it was and is still an exposure: if the long-time tail is confirmed with an exponent this account cannot reach, the proposition is bounded and not exact. What is no longer available as an objection is that the deviation exists at all.

One further thing the account does supply is the visibility structure, which is worth separating from the derivation it does not supply. The deviation is not absent through most of a decay and then present at the end. It is there throughout and sits beneath the aperture (Section 9), and what changes at long times is not the tail but the exponential: a power law and a falling exponential cross once e^−Γt has dropped below t⁻ⁿ, which is many lifetimes in: Rothe et al. (2006) report the crossing at around ten lifetimes. The exact crossing depends on the tail’s prefactor and is not estimated here. So the tail becomes visible because the thing hiding it decays away, not because the tail grows, and “deviations appear at long times” is a fact about what is resolvable rather than about what is happening.

That is not the missing derivation. The gap is that the constant hazard here is a claim about what the nucleus retains, while the deviation comes from the composite spectrum being bounded below. Those are different objects. What reconciles them is that the floor is a fact about any frame, so it holds of the composite and of the nucleus alike: the constant hazard is what the nucleus retains, the required deviation is what the container’s floor imposes, and neither is a claim about the other.

Stepping alone gives no floor: the integers are stepped and run both ways. One substrate fact is needed and it is applied in two places, the second of which is where the sign problem hides.

Occupation is a count, so it is non-negative. A frame’s energy is carried by quanta present, ∑ _kn_kℏω_k, and each n_k is a cardinality of what is there. Absence is zero and not a negative quantity; there is no less-than-nothing to count down into. The substrate’s unbounded direction for a count is the extensive frontier — always one more step outward — which is ℕ, floored at zero by construction, never ℤ. What is required here is that occupation is a count, not that the ladder is evenly spaced; the ℏ-steps of Section 10 are the action quantum and are not needed for this.

That is both premises, and the proposition follows.

The zero of energy is a choice of origin, and an origin is read in-frame exactly as a unit is (Section 9), so a spectrum running to arbitrarily negative numbers is no counterexample: a binding energy is negative against a conventional zero and the system is still bounded below by its own ground state, which is the floor this proposition is about. The claim is boundedness, which is invariant. The number at which the bound sits is not, and nothing here needs it to be.

That is the spectrum condition — positivity of the energy in every frame, which is where it is usually taken as an axiom (Streater and Wightman, 1964) — arriving as a consequence rather than a postulate: each frame floors its own spectrum for the same reason, and none is privileged, and it closes the argument the other way round from how it was posed. Exact exponential survival needs spectral support running to −∞; the floor forbids it; so the deviation is not merely permitted by this account but required by it, and required by the same fact about absence the substrate uses everywhere else. So the leading behaviour is exponential because nothing is retained, and the departure is required because a floor exists — two facts about two objects, not one derivation across them.

The exponent is a different matter and the account is right not to fix it. For a spectral density behaving as (E − E_min)^β just above the floor, the survival amplitude falls as t^−(1+β) and the probability as t^−2(1+β), so the exponent is set by β — how many distinguishable configurations sit just above absence. That is a fact about how the frame’s dimensions are arranged, which is contingent rule-content (Section 15) and not substrate structure. So the substrate fixes that there is a tail and cannot fix its exponent, and the reason is principled rather than a shortfall: the exponent is contingent, and a substrate result that delivered a universal one would be claiming more than the substrate knows.

Which converts the exposure into a test. The exponent should track the density of states at threshold and not be universal — differently structured systems should tail differently, and the free-particle s-wave case with β = giving t⁻³ is one value among the available ones rather than the value. A universal exponent, the same across systems with different threshold structure, would falsify this.

One identification is doing the work and is worth marking rather than burying: in-frame energy is a count of quanta. That is clean in the occupation-number picture and less clean for interacting systems, where the count is not sharp and the physical states are dressed. Nothing above survives if that identification fails for the interacting case, and this section does not establish that it holds there.

14 Cosmology: the initial state, the seed, and self-location#

By Graphic Equalisation’s Proposition 11.4 the event order has no intrinsic first, so the beginning of the universe is not an intrinsic edge but a chosen entry — a boundary condition imposed from the containing frame. This is why physics supplies initial conditions separately from its laws: the laws are the ordering, which fixes no start, so the entry must come from outside. The Big Bang is that entry, the seed; what is “before” it is not an earlier time but the frontier, the pre-relational substrate, unrepresentable from within, the same “ ? ” that sits below the Planck scale (Section 10).

This inverts the fine-tuning inference. The constants are not tuned to a pre-existing observer; the observer is what those constants produce, so the apparent coincidence is the co-determination of observer and seed mistaken for a tuning aimed at a target. What remains out of reach is only derivation: no formula computes α or the seed. The why is answered; the value is not derived. Reference points take no “why” — they are where the why is asked.

15 Dimensions are contingent; time is optional#

Time is not a must. A frame whose rules are not firing — inert, a rock — has no time at all: nothing happens in it, there is no succession, and in its own frame it merely is. What an external clock calls “later” is, for the rock, only another is; the weathering it undergoes is the supergraph’s succession applied to it from above (Graphic Equalisation’s Proposition 6.5: admissibility is self-governance, not armour), not the rock’s own. Time belongs to active, inferring frames — those with a firing closure loop to iterate (Graphic Equalisation’s Proposition 13.7) and the inference to represent the succession. The rock has neither, so it has no time; its rules exist but are inert at the grain at which it is a rock. Nothing here claims a grain at which nothing runs.

Space is N-dimensional by contingency: the number of positional coordinates the rules address is rule-content, so “why N” is “why this rule-set,” answered by self-location (Section 14). Why N = 3 specifically is that self-location filtered by stability — Ehrenfest’s result that stable orbits and stable atoms exist only in three spatial dimensions (Ehrenfest, 1917), so observers occur in the three-dimensional rule-sets and we are the observers such a rule-set produces. And the 3 + 1 signature, time set apart from space, is the succession/position distinction: time is the change-axis, the one succession, space the config-axes, the many positions; one time and N space is one-succession-and-many-positions read as a signature.

The contingent-not-necessary status of these verdicts — and of the gauge group, the dimensionless constants, the arrow, and the value of c — is licensed by the minimal-universe argument (Section 1; the mind companion’s Proposition 2.1): a rock is a universe at the grain at which it is a rock, and the minimal universe does without them, so none can be compulsory.

16 Identical particles are functional identity#

Graphic Equalisation’s Axiom −1 holds that there is no operational view from nowhere; that two things with no difference that makes a difference are the same is its consequence — functional identity — and its Theorem 6.10 adds that one cannot say which of two is which independently of an individuating frame. This is the quantum treatment of identical particles exactly: two electrons carry no label, and “which is which” names no difference and so has no content. Classical physics labels identical objects — treats two identical bodies as distinct in principle — and that label is a fiction, a distinction imposed where none exists.

17 Positioning and prior art#

The pieces are individually established; the contribution is the conjunction under one change-economy and the mechanisation of shapes prior art postulates, geometrises, or offers heuristically.

Discrete substrates and emergent spacetime. Several programmes derive relativistic structure from a discrete substrate, and two are closer to the change-budget of Section 3 than any continuum account. D’Ariano and Tosini (2013) obtain the metric from pure event-counting on topologically homogeneous causal networks, with time dilation appearing as an increased density of foliation leaves within one tick of a clock and length contraction as the corresponding decrease in event density. That is the same move made here and made earlier, and the counting reading of dilation is theirs, not this paper’s. Concretely, where D’Ariano–Tosini compare the density of foliation leaves associated with a clock tick, this paper compares how a cycle’s variable throughput is divided between motion and internal change: the former supplies a metric-counting account, while the latter is the additional mechanism claimed for dilation, mass, and change-load. Three things are added rather than repeated. Their result is stated for topologically homogeneous networks, which they argue costs no generality since connected events can always be coarse-grained until homogeneity holds. That defence is sound at the coarse-grained grain, and it is exactly the grain this paper does not work at: a cycle’s length varying with the rules and the graph is the mechanism here, and coarse-graining until it does not vary would discard the quantity the budget is about. The difference is therefore real but narrower than it first appears — not that homogeneity is a defect in their account, but that the variation it averages over is what carries the physics in this one. Their counting delivers the metric; the budget additionally splits — throughput spent on motion or on internal change — which is what yields mass as reserved budget, general relativity as change-load rather than only the special-relativistic metric, and the equivalence principle as the seam between the two (Section 4). And their result is a result about spacetime; the same economy is run here through collapse, entanglement and the amplitude weights (Sections 5 to 7), and belongs to a substrate of which physics is one embodiment rather than to physics alone. Knuth and Bahreyni (2014) derive the Minkowski metric and the Lorentz transformations from consistent quantification of a causally ordered set with respect to an embedded observer, explicitly without assuming the constancy of c; that is the closest existing result to the reading of c given above, and it reaches it by a quantification argument rather than by a throughput economy. The Wolfram model (Wolfram, 2020; Gorard, 2020a) derives general covariance from causal invariance and Lorentz covariance from flat foliations, but stipulates c-constancy in its definition of edge lengths (Section 2) and, in the quantum companion (Gorard, 2020b), introduces the Born weights as an a-priori rule rather than recovering them — which is the respect in which Section 7 differs from it. Causal set theory (Bombelli et al., 1987; Surya, 2019) takes order and counting as primitive without rewriting dynamics. The honest summary is that emergence of relativistic structure from a discrete substrate is established prior art with several independent routes, and this paper does not claim the emergence. What it claims is narrower and, so far as these sources go, unoccupied: one budget, split two ways, carrying the special and general cases together and continuing into the quantum content of the same substrate without a further postulate at either seam.

Deriving c, and mass as internal motion. That c should be derived rather than postulated has precedent by a different route. Ignatowsky (Ignatowsky, 1910) and, cleanly, Lévy-Leblond (Lévy-Leblond, 1976) obtain the Lorentz transformation from the relativity principle together with homogeneity, isotropy, and group structure alone, with an invariant speed emerging rather than assumed and its value left empirical; those derivations use symmetry as the engine where the present one uses the change-substrate — c as each frame’s own change-throughput unit, 1 by construction. The reading of mass as the budget reserved for internal change has two standing precedents, offered heuristically rather than mechanised: Schrödinger’s zitterbewegung, the electron’s rest energy as internal motion at c (Schrödinger, 1930), developed as a rest-mass-from-internal- circulation interpretation by Hestenes (Hestenes, 1990); and the elementary result that confined massless radiation — light in a box — carries rest mass E∕c², internalised c-motion appearing as inertia. The budget-split itself is the constant-magnitude four-velocity of Minkowski geometry (Minkowski, 1908).

Relativity as postulate. Einstein’s special (Einstein, 1905) and general (Einstein, 1916) relativity take c-invariance and the equivalence principle as postulates; the present account renders the first a definitional consequence (Proposition 2.2) and the second a common cause (Section 4), at the cost of claiming only the shapes.

Observer-first programmes. The starting point of Sections 5–7 — states relative to a frame, no god’s-eye fact — is shared with several live programmes, and the disagreements matter more than the agreement.

Müller (2020) is the most direct competitor. It takes the first-person observer state as primitive and adds one postulate — that algorithmic universal induction fixes the chances of what the observer sees next — from which it derives, as an asymptotic statistical consequence rather than an assumption, that observers find themselves in an external world governed by simple computable probabilistic laws. The target is exactly the target here: an objective world emerging from an observer-first base rather than being assumed behind it. The mechanism is incompatible. Solomonoff induction over sequences of observer states is not graph rewriting under an admissibility grammar, and nothing in this paper reduces to it, reproduces it, or is supported by it. The two accounts owe the same debt — why any shared world at all — and pay it in different currency; if Müller’s route is the right one, the substrate proposed here is at best a redundant second description of it, and this paper claims no agreement it has not earned.

Höhn (2017) and Höhn and Wever (2017) are the closest in shape. An observer interrogates a system with binary questions; the system’s state is the observer’s catalogue of answers, held by the observer and not by the system; and four rules on information acquisition yield the projective measurement structure, entanglement, and monogamy. A knowledge structure over a question graph, with constraints governing which questions may be jointly asked, is typed structure plus admissibility with states relative to a frame — nearer to the machinery used here than QBism is, and reaching qubit quantum theory where Section 7 only recovers the Born weight given contingent rule-content. Worth noting on the other side of the ledger: those four rules yield the correlation structure of qubits and rebits alike, so they do not by themselves select complex over real. That is independent support for the verdict of Section 7 — an observer-first reconstruction, pursued to its own conclusion, leaves the same choice open.

QBism (Fuchs and Schack, 2013) shares the observer-first premise and the refusal of agent-independent states, then goes the opposite way on precisely what this paper cares about. It declines to derive an objective world, and it treats the Born rule as a normative addition to Bayesian coherence — a rule an agent ought to obey when gambling on its own experiences — rather than as something recovered from deeper structure. Section 7 attempts the recovery, and Section 14 keeps the objective world; QBism is therefore the cleanest available foil, not an ally, and the contrast is the sharpest test of whether the recovery is worth anything.

Relational quantum mechanics (Rovelli, 1996) is the nearest ancestor of Section 5, and the standing problem it has had to answer — how different observers’ accounts cohere into anything shareable — is addressed by cross-perspective links (Adlam and Rovelli, 2023), on which one observer’s record of another’s outcome is required to agree with that outcome. That is the same problem the admissibility grammar is asked to solve here and that algorithmic induction solves for Müller. Three different answers to one question; which is right is open.

Constructor theory. Deutsch (2013) proposes re-founding physics on which physical transformations are possible and which impossible, and why, with counterfactuals primitive rather than derived from dynamical laws plus initial conditions. That is the nearest philosophical relative of the present account: “which transformations are admissible, and why” is close to “typed graphs plus an admissibility grammar,” and both take the possibility structure to be prior to the trajectory. On the Born rule the relation inverts to competition. Marletto (2016) attempts the same recovery from that admissibility-flavoured base and arrives somewhere else: probability is not fundamental, and what look like probabilities emerge from the unpredictability of measurement on superinformation media, giving a non-probabilistic account where Section 7 recovers a probability weight from closure, no-privilege, and phase-averaging. The two are not quite rival derivations of one rule, since the explananda differ — Marletto accounts for the appearance of stochasticity in a theory carrying no probability weight, where Section 7 recovers the weight itself — and that is a deeper disagreement than a rival derivation would be, not a shallower one. The honest position is that constructor theory reaches its answer with more formal apparatus and less contingency than is claimed here, and that this paper’s recovery is exposed to it rather than reinforced by it.

Entropy and the demon. The observer-relative reading of entropy is Jaynes’s (Jaynes, 1957); the resolution of Maxwell’s demon by the cost of erasing information is Landauer’s and Bennett’s (Landauer, 1961; Bennett, 1982); life as locally paid negative entropy is Schrödinger’s (Schrödinger, 1944) — his phrase is negative entropy, the contraction negentropy being Brillouin’s later coinage. The present account ties all three to one substrate law — cost is the write.

Dimensional constants. That the count of fundamental dimensional constants is conventional is Duff’s position in the Duff–Okun–Veneziano exchange (Duff et al., 2002); the present account gives the reason (they are frame-units) and draws the boundary at the dimensionless remainder.

18 Falsifiability commitments#

On the tags. Each commitment below is marked [shape] or [detail]. A [detail] failing is a revision: the account survives with that mechanism replaced. A [shape] failing costs range rather than a mechanism, because what fails is something the substrate is claimed to require or to permit — so the account does not hold where it claimed to, which is the more serious of the two and is still a boundary rather than an annihilation. Each is exposed to attack; the scope caveat of Section 1 applies throughout.

1.

[shape] c as frame-unit. That a frame reads its own base rate as 1 is definitional and not itself at risk (Definition 2.1: unit c); the falsifiable claim is about observed c. Fails if two frames, each obtaining c by the indirect route — measurement, retention and inference — are exhibited reaching different values for one signal, that is, if aperture does not self-calibrate; or if c-invariance requires a postulate that cannot be recovered as the unit reading.

2.

[shape] One budget, two dilations. Fails if velocity time dilation and gravitational time dilation cannot be exhibited as the same fixed budget spent on motion and on change-load respectively — if the equivalence principle is not their common cause here. Concretely, the account predicts no third dilation channel: a clock slowed by something that is neither relative motion nor local change-load would exhibit the failure, and a demonstration that the two known channels draw on separately budgeted throughputs would too.

3.

[shape] Dimensional/dimensionless boundary. Fails on exhibition of a dimensionless constant that is a ratio of nothing — irreducibly standalone, not expressible as one quantity against another even in principle — or if a dimensional constant carries an irreducible value not absorbable into a frame’s units.

4.

[shape] The uncertainty principle is the finite-aperture floor. Fails if the conjugate trade is not an aperture budget — if a frame can be exhibited resolving two conjugate dimensions jointly past the floor without a compensating loss elsewhere, or if the bound must be postulated independently of the finiteness of internal-change throughput rather than read off it (Section 9). (ℏ’s value is not claimed; that it is the cell’s quantum is.)

5.

[shape] Second law as no-copy. Fails if entropy can rise in a frame whose writes preserve their predecessors, or fall in one without a paid write exporting cost. (Free continuations preserve their predecessors and so are the entropy-neutral case; the claim concerns predecessor-discarding writes, which are the only priced ones.)

6.

[shape] Indistinguishability is functional identity. Fails if identical particles carry a frame-free label distinguishing them, or if the Gibbs 1∕N! correction and Fermi exclusion cannot be read as “distinct things require a distinguishing distinction.” (The boson/spin–statistics content is not claimed.)

7.

[shape] Measurement is frame-relative collapse. Fails if a frame-independent fact about whether collapse has occurred is required to describe a Wigner’s-friend scenario — if the friend’s outcome and the outside frame’s superposition cannot both stand as the event and the map of two frames. (The Born-rule probabilities are not claimed.)

8.

[shape] Entanglement is a superframe dimension. Fails if the correlation of an entangled pair is a subframe-local quantity rather than a joint dimension null in each subframe, or if a reduced single-particle state can be pure — i.e. if the mixedness is not the inaccessibility of a superframe dimension. (The Tsirelson bound and correlation strengths are not claimed.)

9.

[detail] Conservation is the cost side of symmetry. Fails if a continuous global symmetry of the action (a residual-free transformation) carries no conserved quantity — local gauge redundancies are excluded, carrying constraints rather than charges — or if a conserved quantity persists across a genuine change in the rules — in particular, if global energy is conserved in a time-dependent (rules-changing) regime such as cosmic expansion. (The gauge group is claimed to be contingent, not derived.)

10.

[detail] The Past Hypothesis is a seed. Fails if the low-entropy initial condition is derived from within — from the dynamics, with no boundary condition imposed from a container (Proposition 14.1). What is claimed is that the seed cannot be derived from inside, not that its content is beyond reach.

11.

[detail] Amplitude is directional Narrative; Born reduces to linearity. Fails if the complex amplitude cannot be identified with a magnitude-times-direction operative-history structure, or if the Born square admits no closed-loop (conjugate) reading. Sharper: given complex, linearly superposing amplitudes and no privileged phase, |a|² is selected (by phase-independence and the additivity of decohered alternatives, with uniqueness from Gleason), so the claim fails if all of those premises hold yet the probability is not the squared modulus. It fails also if the mapping onto the informational reconstructions cannot be made good. (Linearity is not claimed derived; the residue is stated in Section 7.)

12.

[detail] Interference is possibility plus superframe forcing. Fails if interference requires the particle’s own admissibility to select among paths, or if a which-path coupling that collapses before the screen does not remove the fringes — i.e. if superposition is not the possibility structure and the superframe does not apply the combination without passing through the subframe’s grammar.

13.

[shape] Time is optional. Fails if an inert frame (no firing rules) nonetheless has an in-frame succession.

14.

[detail] The one-way speed is an absent quantity. Fails on a determination of the one-way speed in this universe that rests on no synchrony stipulation. What fails then is the empirical claim that our rules supply no unlocked counter, not the cycle unit itself: a second dimension’s step count is a different unit and not a finer reading of the same one (Remark 3.4).

15.

[detail] The decay tail is required, its exponent is not. Fails if the long-time exponent is universal across systems with different threshold structure. The substrate fixes that there is a tail — exact exponentiality would need spectral support running to −∞, which the floor forbids — and cannot fix the exponent, which tracks the density of states at threshold (Section 13).

16.

[shape] In-frame energy is a count of quanta. Fails if in-frame energy is not a count of quanta for interacting systems, where the count is not sharp and the physical states are dressed. Nothing in the spectrum-floor argument survives if that identification fails there, and this paper does not establish that it holds (Proposition 13.2).

17.

[detail] Apparent constancy tracks denominator size. The constants most likely to yield a measurable variation are those whose denominators are smallest against the perturbations available to them, and not the ones deemed most fundamental. Fails if a measured variation, when found, appears first in constants ordered by depth rather than by that ratio (Proposition 9.1).

Self-location (Proposition 14.2) is stated here rather than counted in the list above, being a consequence rather than a wager: “why this seed, this constant, this gauge group” can be posed only from a trans-frame standpoint that could have obtained a different one, which Axiom −1 together with frame-relative identity already denies, so what would have to fail for it to fail is the base axiom and not a risk of its own.

Scope note (not a falsifier). The paper claims shapes only; it would be overclaiming if any passage were read to derive a field equation, a dimensionless constant, or a quantum-gravity result — none is asserted. The Planck-scale claim carries no separate wager: it is covered as a consequence of the finite-aperture floor together with the dimensional/dimensionless boundary, and needs no falsifier of its own. That the observed time is 1-D is claimed decoherence-selected, not forced — multi-dimensional time is unobserved, not prohibited; that space is N-D with N = 3 is likewise contingent and self-located. The minimal-universe licence (the mind companion’s Section 2), the block universe, free will, personal identity, and death are treated in that companion and carry their falsifiers there.

References