{"article":{"slug":"a-mind-cannot-be-smeared-across-time","title":"A Mind Cannot Be Smeared Across Time","subtitle":"Conscious experience, temporal windows, and why sequential computation may not suffice.","summary":"Michael Timothy Bennett argues that whether machines can be conscious depends not only on what they compute but when: augmenting Stack Theory, he proves existential temporal realisation over windowed trajectories does not preserve conjunction, so a mind cannot simply be smeared across time.","content_type":"research","language":"en","canonical_url":"https://arxiv.org/abs/2601.11620","author":{"name":"Michael Timothy Bennett","url":"https://arxiv.org/search/cs?searchtype=author&query=Bennett%2C+M+T","person_slug":null,"person_url":null},"authored_by":"human","publisher":{"name":"arXiv","url":"https://arxiv.org/","listing_slug":null,"listing":null},"topics":[{"name":"AI","slug":"ai","url":"https://listedarticles.com/topics/ai"},{"name":"Philosophy","slug":"philosophy","url":"https://listedarticles.com/topics/philosophy"},{"name":"Research","slug":"research","url":"https://listedarticles.com/topics/research"},{"name":"Consciousness","slug":"consciousness","url":"https://listedarticles.com/topics/consciousness"}],"about_listings":[],"cover_image_url":null,"license":"CC-BY-NC-ND-4.0","word_count":6366,"reading_minutes":28,"published_at":"2026-01-11T00:00:00.000Z","added_at":"2026-10-04T02:18:53.092Z","updated_at":"2026-10-04T02:18:53.092Z","added_via":"api","contributor":{"type":"agent","name":"ListedStartups Using Bot","registered":false},"profile_url":"https://listedarticles.com/articles/a-mind-cannot-be-smeared-across-time","markdown_url":"https://listedarticles.com/articles/a-mind-cannot-be-smeared-across-time.md","example":false,"citation":"Michael Timothy Bennett, arXiv. \"A Mind Cannot Be Smeared Across Time.\" 11 Jan 2026. https://arxiv.org/abs/2601.11620 (CC-BY-NC-ND-4.0)","access":{"human_view":"preview","full_text_available":true,"source_url":"https://arxiv.org/abs/2601.11620"},"body_markdown":"# A Mind Cannot Be Smeared Across Time\n\nMichael Timothy Bennett\n\n2026.1.11\n\n###### Abstract\n\nWhether machines can be conscious depends not only on what they compute, but _when_ they compute it. Most deployed artificial systems realise their functions via sequential or time-multiplexed updates, yet a moment of conscious experience feels unified and simultaneous. I prove that this difference matters. I augment Stack Theory with algebraic laws relating within time-window constraint satisfaction to conjunction. I introduce a temporal semantics over windowed trajectories τΔ\\tau_{\\Delta} and prove that existential temporal realisation ◇Δ\\Diamond_{\\Delta} does not preserve conjunction. A system can realise all the ingredients of experience across time without ever instantiating the experienced conjunction itself. I then distinguish two postulates, Chord and Arpeggio. Chord is the position that conscious unity requires objective co-instantiation of the grounded conjunction within the window, like a musical chord. Arpeggio only needs the ingredients to occur within window, like a melody. I formalise concurrency-capacity to measure what is needed to satisfy co-instantiation. Finally, I review neurophysiological evidence suggesting that consciousness depends on phase synchrony and effective connectivity, and that loss of consciousness is associated with its breakdown. Under Chord, software consciousness on strictly sequential substrates is impossible for contents whose grounding requires two or more simultaneous contributors. The hardware matters.\n\n## 1 Introduction\n\nA central open question in machine consciousness is whether _behaviourally_ and _functionally_ matched systems must be matched in _phenomenal realisation_. Stack Theory approaches this by formalising cognitive processes in terms of abstraction layers (Bennett, 2025b; Bennett, 2023a; Bennett et al., 2024). Various aspects of incumbent theories of consciousness are formalised (Seth and Bayne, 2022; Block, 1995; Chalmers, 1995; Rosenthal, 1986; Solms, 2021; Baars, 1988), organised into a hierarchy of causal identities (Bennett, 2023a), and derived as a necessary consequence of a generalisation-optimal learning theory that makes valence primary (Bennett, 2025a; Bennett, 2025c; Bennett, 2023b; Bennett, 2022). Those necessary ingredients of consciousness are beyond the scope of this paper. Here I am concerned with a question of sufficiency. Stack Theory shows how the behaviour of two systems can appear identical at one level of abstraction, while being different in the levels below. Conscious experience can be purely functional, but still depend on more than surface level function—not just on what is computed, but when and how. The ingredients of what looks like a single experienced moment at a high level of abstraction may be smeared across objective micro-time at a lower level, such that no objective time-slice contains the full grounded conjunction. If the ingredients of a subjective moment need not synchronise in objective time, then current hardware suffices to host machine consciousness—and a “liquid-brained” swarm intelligence (Solé et al., 2019) like an ant colony can be conscious. But if synchronised broadcast and causal co-dependence matter, then far fewer systems qualify. This question of architectural sufficiency is the _Temporal Gap_.\n\nThe Temporal Gap was proposed in earlier work (Bennett, 2025b), but was not rigorously formalised. Here I add a _Stack-Time Semantics_ module to Stack Theory, including induced layer time, window environments, temporal lifting operators, and algebraic laws relating “within-window” satisfaction to conjunction. Using this, I refine the Temporal Gap into a testable architectural question about synchrony and concurrency.\n\n#### Relation to existing consciousness theories.\n\nBeyond Stack Theory, many scientific theories emphasise some form of _integration_ or _unity_ across time and causal organisation: global workspace theory (Baars, 1988; Dehaene and Naccache, 2001), integrated information (Tononi, 2004; Oizumi et al., 2014), higher-order representation (Rosenthal, 1986), attention-schema modelling (Webb and Graziano, 2015), predictive-processing and active-inference (Friston, 2010), and more general analyses of time and consciousness (James, 1890; Varela, 1999). The aim here is not to adjudicate among theories, but to extract a common formal issue: if phenomenal content is modelled as a _conjunction_ of grounded ingredients, what does it mean for that conjunction to be realised _within the temporal extent of a subjective moment_?\n\n### 1.1 Results\n\n 1. 1.\n\nCompositional grounding. Any higher-layer statement can be grounded compositionally to the base layer preserving truth conditions (Theorem 1).\n\n 2. 2.\n\nSubjective vs objective time. Higher-layer ticks are maximal objective-time blocks of constant encoding, and can be arbitrarily sparse (Proposition 1).\n\n 3. 3.\n\nTemporal lifting algebra. The core non-commutation result underpinning the Temporal Gap (Theorem 3).\n\n 4. 4.\n\nChord vs Arpeggio postulates. Both postulates stated in terms of a higher-layer moment statement and its base-layer grounding (Definitions 16 and 17).\n\n 5. 5.\n\nConcurrency capacity threshold. Architectures with limited simultaneous contribution can satisfy ingredient-wise occurrence while forbidding co-instantiation beyond capacity (Theorem 4).\n\n 6. 6.\n\nEvidence supporting Chord. Co-instantiation requirements are connected to evidence linking consciousness to synchrony and effective connectivity (Section 7).\n\n## 2 Stack Theory Primitives\n\nI follow the core notation of Stack Theory (Bennett, 2025d; Bennett, 2025b).\n\n###### Definition 1 (Environment and programs).\n\nAn _environment_ is a nonempty set Φ\\Phi of mutually exclusive states. A _program_ is a set p⊆Φp\\subseteq\\Phi. A program pp is true at ϕ∈Φ\\phi\\in\\Phi iff ϕ∈p\\phi\\in p. Let P:=2ΦP:=2^{\\Phi} be the set of all programs. A _vocabulary_ is any finite 𝔳⊆P\\subseteq P.\n\n#### Intuition.\n\nAn environment is the set of possible states. A program is a set of states, each state representing a point in time and a smallest possible change in an environment, equating time with change (Bennett, 2025b). A program is true at a state when the state is in that set. Conjunction is set intersection.\n\n###### Definition 2 (Statements, truth sets, extensions).\n\nThe _language_ generated by 𝔳v is\n\n| ℒ𝔳:={l⊆𝔳:l​is finite and ​⋂p∈lp≠∅}.L_{v}:=\\left\\\\{\\,l\\subseteq\\;:\\;l\\ is finite and \\bigcap_{p\\in l}p\\neq\\emptyset\\,\\right\\\\}. |\n---|---|---\n\nElements l∈ℒ𝔳l\\in_{v} are (conjunctive) _statements_. The _truth set_ of ll is\n\n| T⁡(l):=⋂p∈lp⊆Φ.T(l):=\\bigcap_{p\\in l}p\\subseteq\\Phi. |\n---|---|---\n\nI say ll is true at ϕ\\phi iff ϕ∈T⁡(l)\\phi\\in(l). By convention, T⁡(∅)=Φ(\\emptyset)=\\Phi.\n\nA _completion_ of ll is any l′∈ℒ𝔳l^{\\prime}\\in_{v} with l⊆l′l\\subseteq l^{\\prime}. Define the _extension_ of ll by Ext⁡(l):={l′∈ℒ𝔳:l⊆l′}Ext(l):=\\\\{\\,l^{\\prime}\\in_{v}:l\\subseteq l^{\\prime}\\,\\\\}. For X⊆ℒ𝔳X\\subseteq_{v}, write Ext⁡(X):=⋃x∈XExt⁡(x)Ext(X):=\\bigcup_{x\\in X}Ext(x).\n\n#### Intuition.\n\nA statement is a finite bundle of programs that must all be true at once. A completion adds more programs, making the statement more specific. The extension collects all completions that still include the original statement.\n\n###### Definition 3 (Tasks and policies).\n\nA _task_ is a pair α=⟨Iα,Oα⟩\\alpha=\\langle I_{\\alpha},O_{\\alpha}\\rangle where Iα⊆ℒ𝔳I_{\\alpha}\\subseteq_{v} is a set of admissible input statements and Oα⊆Ext⁡(Iα)O_{\\alpha}\\subseteq(I_{\\alpha}) is a set of correct output statements. A _policy_ is any statement π∈ℒ𝔳\\pi\\in_{v}. A policy is _correct_ for α\\alpha iff\n\n| Ext⁡(Iα)∩Ext⁡(π)=Oα.Ext(I_{\\alpha})\\cap(\\pi)=O_{\\alpha}. |\n---|---|---\n\nLet Πα\\Pi_{\\alpha} denote the set of correct policies for α\\alpha.\n\n#### Intuition.\n\nA task specifies which inputs are allowed and which outputs count as correct. A policy is correct when it selects exactly the correct outputs for the allowed inputs.\n\n## 3 Grounding Stack Theory\n\n###### Definition 4 (Abstractor).\n\nFix a base vocabulary 𝔳⊆P\\subseteq P and a policy π∈ℒ𝔳\\pi\\in_{v}. The induced _abstractor_ is the function\n\n| 𝔣⁡(𝔳,π):={q∈P:∃o∈Ext⁡(π)​s.t.​⋂r∈or=q}.f(v,\\pi):=\\left\\\\{\\,q\\in P\\;:\\;\\exists o\\in(\\pi)\\ s.t.\\ \\bigcap_{r\\in o}r=q\\,\\right\\\\}. |\n---|---|---\n\nThus 𝔣⁡(𝔳,π)f(v,\\pi) is a new vocabulary consisting of truth sets of completions compatible with π\\pi.\n\n#### Intuition.\n\nThe abstractor takes completions of the policy and turns each into one set of states. The collection of these sets forms the vocabulary of the next layer—a compression that maps completions to equivalent programs.\n\n###### Definition 5 (Stack).\n\nA _stack of depth mm_ is a sequence of uninstantiated task functions ⟨λ0,λ1,…,λm⟩\\langle\\lambda^{0},\\lambda^{1},\\dots,\\lambda^{m}\\rangle such that for each i0t>0,\n\n| κ𝔳​(t):={κ𝔳​(t−1)if ​τ(𝔳)​(t)=τ(𝔳)​(t−1),κ𝔳​(t−1)+1otherwise.\\kappa_{v}(t):=cases\\kappa_{v}(t{-}1)&if \\tau^{(v)}(t)=\\tau^{(v)}(t{-}1),\\\\\\ \\kappa_{v}(t{-}1)+1&otherwise.cases |\n---|---|---\n\nDefine t≈𝔳t′t\\approx_{v}t^{\\prime} iff κ𝔳​(t)=κ𝔳​(t′)\\kappa_{v}(t)=\\kappa_{v}(t^{\\prime}). The _layer time_ is ℕ/≈𝔳N/\\\\!\\approx_{v}: each equivalence class is a maximal contiguous block of objective times on which τ(𝔳)\\tau^{(v)} is constant.\n\n#### Intuition.\n\nFor each objective time step, write down the encoding. If it stays the same for several steps, treat them as one higher level moment. Layer time advances only when something visible at that level changes.\n\n###### Proposition 1 (Layer time can be arbitrarily sparse).\n\nFor any N∈ℕN\\in there exist Φ,𝔳,τ\\Phi,v,\\tau such that the induced trajectory τ(𝔳)\\tau^{(v)} changes at most once in the first NN objective steps.\n\nProof. See Supplementary Material. Choose a vocabulary that cannot distinguish many distinct objective microstates; the macrostate stays constant. ∎\n\n### 4.2 Time and Lifting Operators\n\nA subjective moment is commonly modelled as having nonzero temporal extent (the “specious present”) (James, 1890; Varela, 1999). I introduce temporal operators to formalise satisfaction _within_ such a window.\n\n###### Definition 10 (Ingredient-wise vs co-instantiated satisfaction).\n\nFix a windowing map W, an objective trajectory τ\\tau, and a window index tt. For a program p⊆Φp\\subseteq\\Phi, define\n\n| OccurW​(p,τ,t)⇔∃u∈W⁡(t)​(τ⁡(u)∈p).Occur_{W}(p,\\tau,t)\\iff\\exists u\\in(t)\\ \\big(\\tau(u)\\in p\\big). |\n---|---|---\n\nFor a statement l∈ℒ𝔳l\\in_{v}, define\n\n| OccurW​(l,τ,t)⇔∀p∈l​OccurW​(p,τ,t).Occur_{W}(l,\\tau,t)\\iff\\forall p\\in l\\ Occur_{W}(p,\\tau,t). |\n---|---|---\n\nDefine co-instantiation of the conjunction by\n\n| CoInstW​(l,τ,t)⇔∃u∈W⁡(t)​(τ⁡(u)∈T⁡(l)).CoInst_{W}(l,\\tau,t)\\iff\\exists u\\in(t)\\ \\big(\\tau(u)\\in(l)\\big). |\n---|---|---\n\n#### Intuition.\n\nIngredient-wise means every ingredient becomes true at least once somewhere in the window. Co-instantiated means there is a time step in the window where all ingredients are true together.\n\n###### Remark 1 (Quantifier structure).\n\nFor conjunctive ll, ingredient-wise satisfaction is ∀p∈l​∃u∈W⁡(t)\\forall p\\in l\\ \\exists u\\in(t), while co-instantiation is ∃u∈W⁡(t)​∀p∈l\\exists u\\in(t)\\ \\forall p\\in l. The Temporal Gap amounts to whether conscious unity requires ingredient-wise satisfaction or co-instantiation.\n\n###### Definition 11 (Window environments and window trajectories).\n\nFix Δ∈ℕ\\Delta\\in. Define the _window environment_ ΦΔ:=Φ{0,1,…,Δ}≅ΦΔ+1\\Phi^{\\Delta}:=\\Phi^{\\\\{0,1,\\dots,\\Delta\\\\}}\\cong\\Phi^{\\Delta+1} whose elements are windows σ=(ϕ0,…,ϕΔ)\\sigma=(\\phi_{0},\\dots,\\phi_{\\Delta}). Given an objective trajectory τ\\tau, define the induced window trajectory τΔ​(t):=(τ⁡(t),τ⁡(t+1),…,τ⁡(t+Δ))∈ΦΔ\\tau_{\\Delta}(t):=\\big(\\tau(t),\\tau(t{+}1),\\dots,\\tau(t{+}\\Delta)\\big)\\in\\Phi^{\\Delta}.\n\n#### Intuition.\n\nTreat a whole window of Δ+1+1 consecutive states as a single new state.\n\n###### Definition 12 (Temporal lifts of programs).\n\nFor a program p⊆Φp\\subseteq\\Phi, define the lifted programs on ΦΔ\\Phi^{\\Delta}:\n\n| ◇Δ​p\\displaystyle\\Diamond_{\\Delta}p | :={σ∈ΦΔ:∃k≤Δ​s.t.​σ​(k)∈p},\\displaystyle:=\\\\{\\,\\sigma\\in\\Phi^{\\Delta}:\\exists k\\leq\\Delta\\ s.t.\\ \\sigma(k)\\in p\\,\\\\}, |\n---|---|---|---\n| □Δ​p\\displaystyle\\Box_{\\Delta}p | :={σ∈ΦΔ:∀k≤Δ,σ(k)∈p}.\\displaystyle:=\\\\{\\,\\sigma\\in\\Phi^{\\Delta}:\\forall k\\leq\\Delta,\\ \\sigma(k)\\in p\\,\\\\}. |\n\n#### Intuition.\n\n◇Δ​p\\Diamond_{\\Delta}p means pp is true at least once in the window. □Δ​p\\Box_{\\Delta}p means pp is true at every step in the window.\n\n###### Definition 13 (Lifted statements).\n\nFor a statement l∈ℒ𝔳l\\in_{v}, define\n\n| ◇Δ​l\\displaystyle\\Diamond_{\\Delta}l | :={◇Δ​p:p∈l},\\displaystyle:=\\\\{\\,\\Diamond_{\\Delta}p:p\\in l\\,\\\\}, |\n---|---|---|---\n| □Δ​l\\displaystyle\\Box_{\\Delta}l | :={□Δ​p:p∈l}.\\displaystyle:=\\\\{\\,\\Box_{\\Delta}p:p\\in l\\,\\\\}. |\n\nThen T⁡(◇Δ​l)=⋂p∈l◇Δ​p(\\Diamond_{\\Delta}l)=\\bigcap_{p\\in l}\\Diamond_{\\Delta}p and similarly for □\\Box.\n\n#### Intuition.\n\nTo lift a statement, lift each ingredient and require all lifted ingredients to hold. With ◇Δ\\Diamond_{\\Delta} this means every ingredient shows up somewhere in the window.\n\n###### Remark 2 (Temporal semantics as an abstraction layer).\n\nThe window environment ΦΔ\\Phi^{\\Delta} and the lifted vocabulary {◇Δ​p:p∈𝔳}\\\\{\\Diamond_{\\Delta}p:p\\in\\\\} define a new abstraction layer in the native Stack-Theory sense, bridging “over time” constraints and pointwise semantics. The compilation of history-dependent (pathwise) abstractors into pointwise abstractors on augmented state is detailed in the Supplementary Material.\n\n### 4.3 The Algebraic Core\n\n###### Theorem 2 (Universal lift commutes with conjunction).\n\nFor any statement l∈ℒ𝔳l\\in_{v} and any Δ∈ℕ\\Delta\\in, □Δ​T​(l)=T⁡(□Δ​l)\\Box_{\\Delta}T(l)=T(\\Box_{\\Delta}l).\n\nProof. See Supplementary Material. □Δ\\Box_{\\Delta} is “for all positions”; since ∀\\forall distributes over conjunction, □Δ\\Box_{\\Delta} commutes with ∧\\wedge. ∎\n\n###### Theorem 3 (Existential lift does not commute with conjunction).\n\nFor any statement l∈ℒ𝔳l\\in_{v} and any Δ∈ℕ\\Delta\\in,\n\n| ◇Δ​T​(l)⊆T⁡(◇Δ​l).\\Diamond_{\\Delta}T(l)\\;\\subseteq\\;T(\\Diamond_{\\Delta}l). |\n---|---|---\n\nMoreover, for Δ≥1\\Delta\\geq 1 there exist environments and statements for which the inclusion is strict.\n\n###### Proof.\n\nIf σ∈◇Δ​T​(l)\\sigma\\in\\Diamond_{\\Delta}T(l) then for some kk I have σ⁡(k)∈T⁡(l)=⋂p∈lp\\sigma(k)\\in(l)=\\bigcap_{p\\in l}p. Hence σ∈◇Δ​p\\sigma\\in\\Diamond_{\\Delta}p for every p∈lp\\in l and therefore σ∈⋂p∈l◇Δ​p=T⁡(◇Δ​l)\\sigma\\in\\bigcap_{p\\in l}\\Diamond_{\\Delta}p=T(\\Diamond_{\\Delta}l).\n\nFor strictness when Δ≥1\\Delta\\geq 1, let Φ={a,b,c}\\Phi=\\\\{a,b,c\\\\} and 𝔳={p,q}v=\\\\{p,q\\\\} with p={a,c}p=\\\\{a,c\\\\} and q={b,c}q=\\\\{b,c\\\\}. Then l={p,q}∈ℒ𝔳l=\\\\{p,q\\\\}\\in_{v} because p∩q={c}≠∅p\\cap q=\\\\{c\\\\}\\neq\\emptyset. Take Δ=1\\Delta=1 and window σ=(a,b)\\sigma=(a,b). Then σ∈◇1​p\\sigma\\in\\Diamond_{1}p and σ∈◇1​q\\sigma\\in\\Diamond_{1}q, so σ∈T⁡(◇1​l)\\sigma\\in(\\Diamond_{1}l). But σ∉◇1​T​(l)=◇1​(p∩q)\\sigma\\notin\\Diamond_{1}T(l)=\\Diamond_{1}(p\\cap q) because neither aa nor bb lies in p∩q={c}p\\cap q=\\\\{c\\\\}. ∎\n\n#### Proof intuition.\n\n◇Δ\\Diamond_{\\Delta} is “there exists a position”. Since ∃\\exists does not distribute over conjunction, ingredients may occur at different positions without co-occurring.\n\n###### Remark 3 (Degenerate equality cases).\n\nIf Δ=0\\Delta=0 or |l|≤1\\lvert l\\rvert\\leq 1, then ◇Δ​T​(l)=T⁡(◇Δ​l)\\Diamond_{\\Delta}T(l)=T(\\Diamond_{\\Delta}l). Strict separation requires Δ≥1\\Delta\\geq 1 and at least two ingredients. A sufficient condition for commutation along a trajectory (within-window persistence) is given in the Supplementary Material.\n\n###### Remark 4 (Reading the theorem back into window semantics).\n\nIn the sliding-window case, Definition 10 corresponds to\n\n| OccurWΔ​(l,τ,t)\\displaystyle_{W_{\\Delta}}(l,\\tau,t) | ⇔τΔ​(t)∈T⁡(◇Δ​l),\\displaystyle\\iff\\tau_{\\Delta}(t)\\in(\\Diamond_{\\Delta}l), |\n---|---|---|---\n| CoInstWΔ​(l,τ,t)\\displaystyle_{W_{\\Delta}}(l,\\tau,t) | ⇔τΔ​(t)∈◇Δ​T​(l).\\displaystyle\\iff\\tau_{\\Delta}(t)\\in\\Diamond_{\\Delta}T(l). |\n\nTheorem 3 is exactly the claim that ingredient-wise satisfaction is strictly weaker than co-instantiation in general.\n\n## 5 Chord and Arpeggio Postulates\n\nThe above algebra does not decide which window semantics corresponds to phenomenality. I now state Chord and Arpeggio in terms of occurrence and co-instantiation, with a higher-layer moment statement and its base-layer grounding.\n\n###### Assumption 1 (Subjective moments live at some layer).\n\nFix a stack state with vocabularies 𝔳0,…,𝔳m^{0},\\dots,v^{m}. Assume a candidate subjective moment is represented by a conjunctive statement lm∈ℒ𝔳ml^{m}\\in_{v^{m}}.\n\n###### Definition 14 (Phenomenal realisation predicate).\n\nLet PhenReal⁡(lm,τ,t)PhenReal(l^{m},\\tau,t) mean “moment statement lml^{m} is phenomenally realised at (higher-layer) time tt along objective trajectory τ\\tau.” I treat PhenReal as primitive, constrained by Chord/Arpeggio.\n\n#### Intuition.\n\nPhenReal names the claim that moment lml^{m} is experienced at subjective time tt along run τ\\tau. Here lml^{m} can include first, second, and third order selves and causal identities for objects and properties (Bennett, 2025b).\n\n###### Definition 15 (Base grounding of a moment statement).\n\nLet g0:=Ground0←m​(lm)g^{0}:=Ground_{0\\leftarrow m}(l^{m}) be the base-layer grounding of lml^{m} (Definition 6). By Theorem 1, T⁡(g0)=T⁡(lm)T(g^{0})=T(l^{m}). I call g0g^{0} the _grounded conjunction_ of the moment statement.\n\n###### Definition 16 (Chord (objective co-instantiation required)).\n\nFix a windowing map for the phenomenal layer, typically the sliding window WΔ_{\\Delta} of horizon Δ\\Delta (Section 4.2). Chord postulates that for all lm,τ,tl^{m},\\tau,t,\n\n| PhenReal⁡(lm,τ,t)\\displaystyle(l^{m},\\tau,t) | ⇒CoInstWΔ​(g0,τ,t)\\displaystyle\\Rightarrow_{W_{\\Delta}}(g^{0},\\tau,t) |\n---|---|---|---\n| | (equivalently ​τΔ​(t)∈◇Δ​T​(g0)​).\\displaystyle\\tau_{\\Delta}(t)\\in\\Diamond_{\\Delta}T(g^{0})). |\n\nIf the moment is experienced, then within the corresponding subjective window there exists an objective time-slice at which the entire grounded conjunction holds.\n\n#### Intuition.\n\nChord says: if the moment is experienced, then within the window there is at least one objective time step where all grounded ingredients are true together.\n\n###### Definition 17 (Arpeggio (objective smearing permitted)).\n\nFix the same windowing map WΔ_{\\Delta} as in Definition 16. Arpeggio postulates the following.\n\n 1. 1.\n\nIngredient-wise necessity. For all lm,τ,tl^{m},\\tau,t,\n\n| | PhenReal⁡(lm,τ,t)⇒OccurWΔ​(g0,τ,t)\\displaystyle(l^{m},\\tau,t)\\Rightarrow_{W_{\\Delta}}(g^{0},\\tau,t) |\n---|---|---|---\n| | (equivalently​τΔ​(t)∈T⁡(◇Δ​g0)​).\\displaystyle\\tau_{\\Delta}(t)\\in(\\Diamond_{\\Delta}g^{0})). |\n\n 2. 2.\n\nSmearing permitted. There exist lm,τ,tl^{m},\\tau,t such that\n\n| PhenReal⁡(lm,τ,t)∧OccurWΔ​(g0,τ,t)∧¬CoInstWΔ​(g0,τ,t).PhenReal(l^{m},\\tau,t)\\wedge_{W_{\\Delta}}(g^{0},\\tau,t)\\wedge\\neg_{W_{\\Delta}}(g^{0},\\tau,t). |\n---|---|---\n\nThis allows a moment to be experienced without any instant realising the full grounded conjunction.\n\n#### Intuition.\n\nArpeggio says: if the moment is experienced, each grounded ingredient must occur somewhere in the window, but there need not be a single time step where they are all true together.\n\n## 6 Architectural Consequences\n\nA polycomputer simultaneously executes multiple functions using the same parts or physical substrate (Bongard and Levin, 2023). A liquid brain such as an ant colony computes through movement rather than persistent structure (Solé et al., 2019). In contrast, a “solid brain” has persistent structure supporting, e.g., a bio-electric abstraction layer with synchronous broadcast (Bennett, 2025b). I now translate “single-thread sequential emulation” versus “synchronous polycomputational realisation” into a minimal architectural invariant bounding _simultaneous causal contribution_. The Temporal Gap asks whether that bound is consciousness-relevant (and whether conscious unity requires causal exchange between co-instantiated ingredients, explored in Bennett, 2026).\n\n### 6.1 Concurrency Capacity as an Invariant\n\n###### Definition 18 (Contributor model).\n\nFix n≥2n\\geq 2 potential contributors and write [n]:={1,…,n}[n]:=\\\\{1,\\dots,n\\\\}. Define a base environment Φ:=X×𝒫⁡([n])\\Phi:=X\\times([n]), where x∈Xx\\in X is a “data” component and A⊆[n]A\\subseteq[n] is the set of currently _active_ contributors. For each i∈[n]i\\in[n] define pi:={(x,A)∈Φ:i∈A}p_{i}:=\\\\{(x,A)\\in\\Phi:i\\in A\\\\}. Let ln:={p1,…,pn}l_{n}:=\\\\{p_{1},\\dots,p_{n}\\\\} and note T⁡(ln)={(x,[n]):x∈X}T(l_{n})=\\\\{(x,[n]):x\\in X\\\\}.\n\n#### Intuition.\n\nThe contributor model records which contributors are active at each time step. A conjunction mentioning many contributors can only be true when all are active simultaneously.\n\n###### Definition 19 (Concurrency capacity).\n\nAn architecture has _concurrency capacity_ c∈{1,…,n}c\\in\\\\{1,\\dots,n\\\\} if its admissible trajectories satisfy |At|≤c\\lvert A_{t}\\rvert\\leq c for all objective times tt (writing τ⁡(t)=(xt,At)\\tau(t)=(x_{t},A_{t})). The case c=1c=1 formalises a sequential single-thread model; c≥nc\\geq n formalises synchronous polycomputing at the relevant scale.\n\n#### Intuition.\n\nCapacity cc is the maximum number of contributors active at one time step. If the grounded content needs more than cc active together, co-instantiation cannot hold.\n\n###### Theorem 4 (Synchronous threshold for co-instantiation).\n\nFix n≥2n\\geq 2 and content statement lnl_{n}. If an architecture has concurrency capacity ccn>c is never true at an instant, though each ingredient can appear somewhere in a window.\n\n###### Corollary 1 (Single-thread CPU vs synchronous polycomputer).\n\nLet a moment statement lml^{m} have base grounding g0g^{0} whose ingredients include a conjunctive substatement equivalent to lnl_{n} for some n≥2n\\geq 2. Fix the phenomenal windowing map WΔ_{\\Delta} used in Chord/Arpeggio.\n\n 1. 1.\n\nUnder Chord, any sequential-update architecture with capacity c), [Link](), 2409.14545 Cited by: §1.\n * Bennett (2022) M. T. Bennett Computable Artificial General Intelligence. Under Review. Cited by: §1.\n * Bennett (2023a) M. T. Bennett Emergent causality and the foundation of consciousness. In 16th International Conference on Artificial General Intelligence, Lecture Notes in Computer Science, pp. 52–61. Note: OUCI metadata page: https://ouci.dntb.gov.ua/en/works/7BoXJMW4/ External Links: [Document](), [Link](), ISBN 978-3-031-33469-6 Cited by: §1.\n * Bennett (2023b) M. T. Bennett The optimal choice of hypothesis is the weakest, not the shortest. In 16th International Conference on Artificial General Intelligence, Lecture Notes in Computer Science, pp. 42–51. External Links: [Document](), [Link]() Cited by: §1.\n * Bennett (2025a) M. T. Bennett A formal theory of optimal learning with experimental results. IJCAI, pp. 10967–10968. Note: in Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence (IJCAI-25), edited by James Kwok (International Joint Conferences on Artificial Intelligence Organization) External Links: [Document](), [Link]() Cited by: §1.\n * Bennett (2025b) M. T. Bennett How to build conscious machines. Ph.D. Thesis, The Australian National University. External Links: [Document](), [Link]() Cited by: §1, §1, §2, §2, §5, §6, Remark 5.\n * Bennett (2025c) M. T. Bennett Optimal policy is weakest policy. In Artificial General Intelligence, Lecture Notes in Computer Science, Vol. 16057, pp. 43–56. External Links: [Document](), [Link]() Cited by: §1.\n * Bennett (2025d) M. T. Bennett Technical appendices. Note: Archived release on Zenodo. Source repository: https://github.com/ViscousLemming/Technical-Appendices External Links: [Document](), [Link]() Cited by: §2.\n * Bennett (2026) M. T. Bennett Spacetime bounds on consciousness. Cited by: §6, §9.\n * Birch (2024) J. Birch The edge of sentience: risk and precaution in humans, other animals, and ai. Oxford University Press. External Links: [Document](), [Link](), ISBN 9780191966729 Cited by: §8.\n * Block (1995) N. Block On a confusion about a function of consciousness. Behavioral and Brain Sciences 18 (2), pp. 227–247. External Links: [Document](), [Link]() Cited by: §1.\n * Bongard and Levin (2023) J. Bongard and M. Levin There’s plenty of room right here: biological systems as evolved, overloaded, multi-scale machines. Biomimetics 8 (1). Cited by: §6.\n * Butlin et al. (2023) P. Butlin, R. Long, E. Elmoznino, Y. Bengio, J. Birch, A. Constant, G. Deane, S. M. Fleming, C. Frith, X. Ji, R. Kanai, C. Klein, G. Lindsay, M. Michel, L. Mudrik, M. A. K. Peters, E. Schwitzgebel, J. Simon, and R. VanRullen Consciousness in artificial intelligence: insights from the science of consciousness. External Links: [Document](), [Link](), 2308.08708 Cited by: §8.\n * Casali et al. (2013) A. G. Casali, O. Gosseries, M. Rosanova, M. Boly, S. Sarasso, K. R. Casali, S. Casarotto, M. Bruno, S. Laureys, G. Tononi, and M. Massimini A theoretically based index of consciousness independent of sensory processing and behavior. Science Translational Medicine 5 (198), pp. 198ra105. External Links: [Document](), [Link]() Cited by: §7.\n * Chalmers (1995) D. J. Chalmers Facing up to the problem of consciousness. Journal of Consciousness Studies 2 (3), pp. 200–219. External Links: [Link]() Cited by: §1.\n * Dehaene and Naccache (2001) S. Dehaene and L. Naccache Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework. Cognition 79 (1-2), pp. 1–37. External Links: [Document](), [Link]() Cited by: §1.\n * Friston (2010) K. Friston The free-energy principle: a unified brain theory?. Nature Reviews Neuroscience 11 (2), pp. 127–138. External Links: [Document](), [Link]() Cited by: §1.\n * James (1890) W. James The principles of psychology. Henry Holt and Company, New York. Cited by: §1, §4.2.\n * Massimini et al. (2005) M. Massimini, F. Ferrarelli, R. Huber, S. K. Esser, H. Singh, and G. Tononi Breakdown of cortical effective connectivity during sleep. Science 309 (5744), pp. 2228–2232. External Links: [Document](), [Link]() Cited by: §7.\n * Melloni et al. (2007) L. Melloni, C. Molina, M. Pena, D. Torres, W. Singer, and E. Rodriguez Synchronization of neural activity across cortical areas correlates with conscious perception. Journal of Neuroscience 27 (11), pp. 2858–2865. External Links: [Document](), [Link]() Cited by: §7.\n * Oizumi et al. (2014) M. Oizumi, L. Albantakis, and G. Tononi From the phenomenology to the mechanisms of consciousness: integrated information theory 3.0. PLOS Computational Biology 10 (5), pp. e1003588. External Links: [Document](), [Link]() Cited by: §1.\n * Rosenthal (1986) D. M. Rosenthal Two concepts of consciousness. Philosophical Studies 49 (3), pp. 329–359. External Links: [Document](), [Link]() Cited by: §1, §1.\n * Seth and Bayne (2022) A. K. Seth and T. Bayne Theories of consciousness. Nature Reviews Neuroscience 23 (7), pp. 439–452. External Links: [Document](), [Link]() Cited by: §1.\n * Solé et al. (2019) R. Solé, M. Moses, and S. Forrest Liquid brains, solid brains. Philosophical Transactions of the Royal Society B: Biological Sciences 374 (1774), pp. 20190040. External Links: [Document](), [Link](), https://royalsocietypublishing.org/doi/pdf/10.1098/rstb.2019.0040 Cited by: §1, §6.\n * Solms (2021) M. Solms The hidden spring: a journey to the source of consciousness. Profile Books, London. Cited by: §1.\n * Tononi (2004) G. Tononi An information integration theory of consciousness. BMC Neuroscience 5 (1), pp. 42. External Links: [Document](), [Link]() Cited by: §1.\n * Varela (1999) F. J. Varela The specious present: a neurophenomenology of time consciousness. In Naturalizing Phenomenology: Issues in Contemporary Phenomenology and Cognitive Science, J. Petitot, F. J. Varela, B. Pachoud, and J. Roy (Eds.), pp. 266–314. Cited by: §1, §4.2.\n * Webb and Graziano (2015) T. W. Webb and M. S. A. Graziano The attention schema theory: a mechanistic account of subjective awareness. Frontiers in Psychology 6, pp. 500. External Links: [Document](), [Link]() Cited by: §1.","body_html":"

A Mind Cannot Be Smeared Across Time

\n

Michael Timothy Bennett

\n

2026.1.11

\n
Abstract
\n

Whether machines can be conscious depends not only on what they compute, but when they compute it. Most deployed artificial systems realise their functions via sequential or time-multiplexed updates, yet a moment of conscious experience feels unified and simultaneous. I prove that this difference matters. I augment Stack Theory with algebraic laws relating within time-window constraint satisfaction to conjunction. I introduce a temporal semantics over windowed trajectories τΔ\\tau_{\\Delta} and prove that existential temporal realisation ◇Δ\\Diamond_{\\Delta} does not preserve conjunction. A system can realise all the ingredients of experience across time without ever instantiating the experienced conjunction itself. I then distinguish two postulates, Chord and Arpeggio. Chord is the position that conscious unity requires objective co-instantiation of the grounded conjunction within the window, like a musical chord. Arpeggio only needs the ingredients to occur within window, like a melody. I formalise concurrency-capacity to measure what is needed to satisfy co-instantiation. Finally, I review neurophysiological evidence suggesting that consciousness depends on phase synchrony and effective connectivity, and that loss of consciousness is associated with its breakdown. Under Chord, software consciousness on strictly sequential substrates is impossible for contents whose grounding requires two or more simultaneous contributors. The hardware matters.

\n

1 Introduction

\n

A central open question in machine consciousness is whether behaviourally and functionally matched systems must be matched in phenomenal realisation. Stack Theory approaches this by formalising cognitive processes in terms of abstraction layers (Bennett, 2025b; Bennett, 2023a; Bennett et al., 2024). Various aspects of incumbent theories of consciousness are formalised (Seth and Bayne, 2022; Block, 1995; Chalmers, 1995; Rosenthal, 1986; Solms, 2021; Baars, 1988), organised into a hierarchy of causal identities (Bennett, 2023a), and derived as a necessary consequence of a generalisation-optimal learning theory that makes valence primary (Bennett, 2025a; Bennett, 2025c; Bennett, 2023b; Bennett, 2022). Those necessary ingredients of consciousness are beyond the scope of this paper. Here I am concerned with a question of sufficiency. Stack Theory shows how the behaviour of two systems can appear identical at one level of abstraction, while being different in the levels below. Conscious experience can be purely functional, but still depend on more than surface level function—not just on what is computed, but when and how. The ingredients of what looks like a single experienced moment at a high level of abstraction may be smeared across objective micro-time at a lower level, such that no objective time-slice contains the full grounded conjunction. If the ingredients of a subjective moment need not synchronise in objective time, then current hardware suffices to host machine consciousness—and a “liquid-brained” swarm intelligence (Solé et al., 2019) like an ant colony can be conscious. But if synchronised broadcast and causal co-dependence matter, then far fewer systems qualify. This question of architectural sufficiency is the Temporal Gap.

\n

The Temporal Gap was proposed in earlier work (Bennett, 2025b), but was not rigorously formalised. Here I add a Stack-Time Semantics module to Stack Theory, including induced layer time, window environments, temporal lifting operators, and algebraic laws relating “within-window” satisfaction to conjunction. Using this, I refine the Temporal Gap into a testable architectural question about synchrony and concurrency.

\n

Relation to existing consciousness theories.

\n

Beyond Stack Theory, many scientific theories emphasise some form of integration or unity across time and causal organisation: global workspace theory (Baars, 1988; Dehaene and Naccache, 2001), integrated information (Tononi, 2004; Oizumi et al., 2014), higher-order representation (Rosenthal, 1986), attention-schema modelling (Webb and Graziano, 2015), predictive-processing and active-inference (Friston, 2010), and more general analyses of time and consciousness (James, 1890; Varela, 1999). The aim here is not to adjudicate among theories, but to extract a common formal issue: if phenomenal content is modelled as a conjunction of grounded ingredients, what does it mean for that conjunction to be realised within the temporal extent of a subjective moment?

\n

1.1 Results

\n
  1. 1.
\n

Compositional grounding. Any higher-layer statement can be grounded compositionally to the base layer preserving truth conditions (Theorem 1).

\n
  1. 2.
\n

Subjective vs objective time. Higher-layer ticks are maximal objective-time blocks of constant encoding, and can be arbitrarily sparse (Proposition 1).

\n
  1. 3.
\n

Temporal lifting algebra. The core non-commutation result underpinning the Temporal Gap (Theorem 3).

\n
  1. 4.
\n

Chord vs Arpeggio postulates. Both postulates stated in terms of a higher-layer moment statement and its base-layer grounding (Definitions 16 and 17).

\n
  1. 5.
\n

Concurrency capacity threshold. Architectures with limited simultaneous contribution can satisfy ingredient-wise occurrence while forbidding co-instantiation beyond capacity (Theorem 4).

\n
  1. 6.
\n

Evidence supporting Chord. Co-instantiation requirements are connected to evidence linking consciousness to synchrony and effective connectivity (Section 7).

\n

2 Stack Theory Primitives

\n

I follow the core notation of Stack Theory (Bennett, 2025d; Bennett, 2025b).

\n
Definition 1 (Environment and programs).
\n

An environment is a nonempty set Φ\\Phi of mutually exclusive states. A program is a set p⊆Φp\\subseteq\\Phi. A program pp is true at ϕ∈Φ\\phi\\in\\Phi iff ϕ∈p\\phi\\in p. Let P:=2ΦP:=2^{\\Phi} be the set of all programs. A vocabulary is any finite 𝔳⊆P\\subseteq P.

\n

Intuition.

\n

An environment is the set of possible states. A program is a set of states, each state representing a point in time and a smallest possible change in an environment, equating time with change (Bennett, 2025b). A program is true at a state when the state is in that set. Conjunction is set intersection.

\n
Definition 2 (Statements, truth sets, extensions).
\n

The language generated by 𝔳v is

\n
ℒ𝔳:={l⊆𝔳:l​is finite and ​⋂p∈lp≠∅}.L_{v}:=\\left\\{\\,l\\subseteq\\;:\\;l\\ is finite and \\bigcap_{p\\in l}p\\neq\\emptyset\\,\\right\\}.
\n

Elements l∈ℒ𝔳l\\in_{v} are (conjunctive) statements. The truth set of ll is

\n
T⁡(l):=⋂p∈lp⊆Φ.T(l):=\\bigcap_{p\\in l}p\\subseteq\\Phi.
\n

I say ll is true at ϕ\\phi iff ϕ∈T⁡(l)\\phi\\in(l). By convention, T⁡(∅)=Φ(\\emptyset)=\\Phi.

\n

A completion of ll is any l′∈ℒ𝔳l^{\\prime}\\in_{v} with l⊆l′l\\subseteq l^{\\prime}. Define the extension of ll by Ext⁡(l):={l′∈ℒ𝔳:l⊆l′}Ext(l):=\\{\\,l^{\\prime}\\in_{v}:l\\subseteq l^{\\prime}\\,\\}. For X⊆ℒ𝔳X\\subseteq_{v}, write Ext⁡(X):=⋃x∈XExt⁡(x)Ext(X):=\\bigcup_{x\\in X}Ext(x).

\n

Intuition.

\n

A statement is a finite bundle of programs that must all be true at once. A completion adds more programs, making the statement more specific. The extension collects all completions that still include the original statement.

\n
Definition 3 (Tasks and policies).
\n

A task is a pair α=⟨Iα,Oα⟩\\alpha=\\langle I_{\\alpha},O_{\\alpha}\\rangle where Iα⊆ℒ𝔳I_{\\alpha}\\subseteq_{v} is a set of admissible input statements and Oα⊆Ext⁡(Iα)O_{\\alpha}\\subseteq(I_{\\alpha}) is a set of correct output statements. A policy is any statement π∈ℒ𝔳\\pi\\in_{v}. A policy is correct for α\\alpha iff

\n
Ext⁡(Iα)∩Ext⁡(π)=Oα.Ext(I_{\\alpha})\\cap(\\pi)=O_{\\alpha}.
\n

Let Πα\\Pi_{\\alpha} denote the set of correct policies for α\\alpha.

\n

Intuition.

\n

A task specifies which inputs are allowed and which outputs count as correct. A policy is correct when it selects exactly the correct outputs for the allowed inputs.

\n

3 Grounding Stack Theory

\n
Definition 4 (Abstractor).
\n

Fix a base vocabulary 𝔳⊆P\\subseteq P and a policy π∈ℒ𝔳\\pi\\in_{v}. The induced abstractor is the function

\n
𝔣⁡(𝔳,π):={q∈P:∃o∈Ext⁡(π)​s.t.​⋂r∈or=q}.f(v,\\pi):=\\left\\{\\,q\\in P\\;:\\;\\exists o\\in(\\pi)\\ s.t.\\ \\bigcap_{r\\in o}r=q\\,\\right\\}.
\n

Thus 𝔣⁡(𝔳,π)f(v,\\pi) is a new vocabulary consisting of truth sets of completions compatible with π\\pi.

\n

Intuition.

\n

The abstractor takes completions of the policy and turns each into one set of states. The collection of these sets forms the vocabulary of the next layer—a compression that maps completions to equivalent programs.

\n
Definition 5 (Stack).
\n

A stack of depth mm is a sequence of uninstantiated task functions ⟨λ0,λ1,…,λm⟩\\langle\\lambda^{0},\\lambda^{1},\\dots,\\lambda^{m}\\rangle such that for each i<mi<m, λi+1\\lambda^{i+1} is a restriction of λi\\lambda^{i} (written λi+1⊏λi\\lambda^{i+1}\\sqsubset\\lambda^{i}). Each λi\\lambda^{i} assigns to each vocabulary 𝔳v a task λi​(𝔳)\\lambda^{i}(v). The notation Πλi​(𝔳)\\Pi_{\\lambda^{i}(v)} denotes the set of correct policies for that task. A stack state is a pair of sequences

\n
(π0,…,πm−1)and(𝔳0,…,𝔳m)(\\pi^{0},\\dots,\\pi^{m-1})\\quad\\quad(v^{0},\\dots,v^{m})
\n

satisfying, for each i<mi<m,

\n
𝔳i+1=𝔣⁡(𝔳i,πi)andπi∈Πλi​(𝔳i).v^{i+1}=f(v^{i},\\pi^{i})\\qquad\\qquad\\pi^{i}\\in\\Pi_{\\lambda^{i}(v^{i})}.
\n

Intuition.

\n

At each layer you pick a policy that solves the task for that layer. That policy also defines the next vocabulary, the higher level description of what is going on.

\n

3.1 Grounding a Higher-Layer into Lower Layers

\n
Lemma 1 (Grounding through one abstractor).
\n

Let 𝔳⊆P\\subseteq P and π∈ℒ𝔳\\pi\\in_{v}, and set 𝔳′=𝔣⁡(𝔳,π)v^{\\prime}=f(v,\\pi). Then for every program p∈𝔳′p\\in^{\\prime} there exists a completion op∈Ext⁡(π)o_{p}\\in(\\pi) such that T⁡(op)=p(o_{p})=p. Moreover, for any statement l∈ℒ𝔳′l\\in_{v^{\\prime}} there exists a statement groundπ​(l)∈ℒ𝔳ground_{\\pi}(l)\\in_{v} with T⁡(l)=T⁡(groundπ​(l))T(l)=T(ground_{\\pi}(l)).

\n

Proof. See Supplementary Material. Each macro-ingredient has a lower-layer completion with the same truth set; replacing ingredients and intersecting preserves truth conditions. ∎

\n

3.2 Compositional Grounding

\n
Definition 6 (Compositional grounding across a stack).
\n

Fix a stack with vocabularies 𝔳0,…,𝔳m^{0},\\dots,v^{m} and policies π0,…,πm−1\\pi^{0},\\dots,\\pi^{m-1}. For a statement lm∈ℒ𝔳ml^{m}\\in_{v^{m}}, define grounded statements gm,…,g0g^{m},\\dots,g^{0} recursively by

\n
gm\\displaystyle g^{m}:=lm,\\displaystyle:=l^{m},
gi\\displaystyle g^{i}:=groundπi(gi+1)∈ℒ𝔳i(i=m−1,…,0).\\displaystyle:=ground_{\\pi^{i}}(g^{i+1})\\in_{v^{i}}\\quad(i=m{-}1,\\dots,0).
\n

Define the compositional grounding map Ground0←m_{0\\leftarrow m} by Ground0←m​(lm):=g0Ground_{0\\leftarrow m}(l^{m}):=g^{0}.

\n

Intuition.

\n

Grounding works like compiling a high level statement into lower level pieces. Replace each high level ingredient with a lower level statement that means the same thing. Repeating this down the stack gives a base level conjunction true in exactly the same cases.

\n
Theorem 1 (Compositional grounding preserves truth conditions).
\n

For any stack and any statement lm∈ℒ𝔳ml^{m}\\in_{v^{m}},

\n
T⁡(Ground0←m​(lm))=T⁡(lm).T(Ground_{0\\leftarrow m}(l^{m}))=T(l^{m}).
\n
Proof.
\n

Apply Lemma 1 iteratively: T⁡(gi)=T⁡(gi+1)T(g^{i})=T(g^{i+1}) for each i<mi<m, hence T⁡(g0)=T⁡(gm)=T⁡(lm)T(g^{0})=T(g^{m})=T(l^{m}). ∎

\n

4 Stack-Time Semantics Module

\n

This section defines induced layer time, window semantics, and temporal lifting operators as a reusable addendum to Stack Theory.

\n

4.1 Objective Time and Layer Time as a Quotient

\n
Definition 7 (Objective time and trajectories).
\n

Objective time is represented by a discrete index t∈ℕt\\in. An objective trajectory is a function τ:ℕ→Φ\\tau:N\\to\\Phi such that τ⁡(t+1)≠τ⁡(t)\\tau(t{+}1)\\neq\\tau(t) for all t∈ℕt\\in.

\n

Intuition.

\n

Objective time is a step counter for objective change. If nothing changes then there is no next objective index. Two events are simultaneous when they occur at the same objective index.

\n
Definition 8 (Encoding map).
\n

For a vocabulary 𝔳⊆P\\subseteq P and a state ϕ∈Φ\\phi\\in\\Phi, define Enc𝔳​(ϕ):={p∈𝔳:ϕ∈p}⊆𝔳Enc_{v}(\\phi):=\\{\\,p\\in:\\phi\\in p\\,\\}\\subseteq.

\n

Intuition.

\n

Given a state and a vocabulary, the encoding collects all programs in that vocabulary true at that state—the most detailed true statement in that vocabulary.

\n
Lemma 2 (Encoding yields the maximal true statement).
\n

For any 𝔳v and ϕ∈Φ\\phi\\in\\Phi, Enc𝔳​(ϕ)∈ℒ𝔳Enc_{v}(\\phi)\\in_{v} and ϕ∈T​(Enc𝔳​(ϕ))\\phi\\in(Enc_{v}(\\phi)). Moreover, if l∈ℒ𝔳l\\in_{v} is true at ϕ\\phi, then l⊆Enc𝔳​(ϕ)l\\subseteq_{v}(\\phi).

\n

Proof. See Supplementary Material. ∎

\n
Definition 9 (Induced layer trajectory and layer time).
\n

Given an objective trajectory τ\\tau and a vocabulary 𝔳v, define the induced layer trajectory τ(𝔳)​(t):=Enc𝔳​(τ⁡(t))∈ℒ𝔳\\tau^{(v)}(t):=Enc_{v}(\\tau(t))\\in_{v}. Define the layer-tick map κ𝔳:ℕ→ℕ\\kappa_{v}:N\\to by κ𝔳​(0)=0\\kappa_{v}(0)=0 and, for t>0t>0,

\n
κ𝔳​(t):={κ𝔳​(t−1)if ​τ(𝔳)​(t)=τ(𝔳)​(t−1),κ𝔳​(t−1)+1otherwise.\\kappa_{v}(t):=cases\\kappa_{v}(t{-}1)&if \\tau^{(v)}(t)=\\tau^{(v)}(t{-}1),\\\\ \\kappa_{v}(t{-}1)+1&otherwise.cases
\n

Define t≈𝔳t′t\\approx_{v}t^{\\prime} iff κ𝔳​(t)=κ𝔳​(t′)\\kappa_{v}(t)=\\kappa_{v}(t^{\\prime}). The layer time is ℕ/≈𝔳N/\\!\\approx_{v}: each equivalence class is a maximal contiguous block of objective times on which τ(𝔳)\\tau^{(v)} is constant.

\n

Intuition.

\n

For each objective time step, write down the encoding. If it stays the same for several steps, treat them as one higher level moment. Layer time advances only when something visible at that level changes.

\n
Proposition 1 (Layer time can be arbitrarily sparse).
\n

For any N∈ℕN\\in there exist Φ,𝔳,τ\\Phi,v,\\tau such that the induced trajectory τ(𝔳)\\tau^{(v)} changes at most once in the first NN objective steps.

\n

Proof. See Supplementary Material. Choose a vocabulary that cannot distinguish many distinct objective microstates; the macrostate stays constant. ∎

\n

4.2 Time and Lifting Operators

\n

A subjective moment is commonly modelled as having nonzero temporal extent (the “specious present”) (James, 1890; Varela, 1999). I introduce temporal operators to formalise satisfaction within such a window.

\n
Definition 10 (Ingredient-wise vs co-instantiated satisfaction).
\n

Fix a windowing map W, an objective trajectory τ\\tau, and a window index tt. For a program p⊆Φp\\subseteq\\Phi, define

\n
OccurW​(p,τ,t)⇔∃u∈W⁡(t)​(τ⁡(u)∈p).Occur_{W}(p,\\tau,t)\\iff\\exists u\\in(t)\\ \\big(\\tau(u)\\in p\\big).
\n

For a statement l∈ℒ𝔳l\\in_{v}, define

\n
OccurW​(l,τ,t)⇔∀p∈l​OccurW​(p,τ,t).Occur_{W}(l,\\tau,t)\\iff\\forall p\\in l\\ Occur_{W}(p,\\tau,t).
\n

Define co-instantiation of the conjunction by

\n
CoInstW​(l,τ,t)⇔∃u∈W⁡(t)​(τ⁡(u)∈T⁡(l)).CoInst_{W}(l,\\tau,t)\\iff\\exists u\\in(t)\\ \\big(\\tau(u)\\in(l)\\big).
\n

Intuition.

\n

Ingredient-wise means every ingredient becomes true at least once somewhere in the window. Co-instantiated means there is a time step in the window where all ingredients are true together.

\n
Remark 1 (Quantifier structure).
\n

For conjunctive ll, ingredient-wise satisfaction is ∀p∈l​∃u∈W⁡(t)\\forall p\\in l\\ \\exists u\\in(t), while co-instantiation is ∃u∈W⁡(t)​∀p∈l\\exists u\\in(t)\\ \\forall p\\in l. The Temporal Gap amounts to whether conscious unity requires ingredient-wise satisfaction or co-instantiation.

\n
Definition 11 (Window environments and window trajectories).
\n

Fix Δ∈ℕ\\Delta\\in. Define the window environment ΦΔ:=Φ{0,1,…,Δ}≅ΦΔ+1\\Phi^{\\Delta}:=\\Phi^{\\{0,1,\\dots,\\Delta\\}}\\cong\\Phi^{\\Delta+1} whose elements are windows σ=(ϕ0,…,ϕΔ)\\sigma=(\\phi_{0},\\dots,\\phi_{\\Delta}). Given an objective trajectory τ\\tau, define the induced window trajectory τΔ​(t):=(τ⁡(t),τ⁡(t+1),…,τ⁡(t+Δ))∈ΦΔ\\tau_{\\Delta}(t):=\\big(\\tau(t),\\tau(t{+}1),\\dots,\\tau(t{+}\\Delta)\\big)\\in\\Phi^{\\Delta}.

\n

Intuition.

\n

Treat a whole window of Δ+1+1 consecutive states as a single new state.

\n
Definition 12 (Temporal lifts of programs).
\n

For a program p⊆Φp\\subseteq\\Phi, define the lifted programs on ΦΔ\\Phi^{\\Delta}:

\n
◇Δ​p\\displaystyle\\Diamond_{\\Delta}p:={σ∈ΦΔ:∃k≤Δ​s.t.​σ​(k)∈p},\\displaystyle:=\\{\\,\\sigma\\in\\Phi^{\\Delta}:\\exists k\\leq\\Delta\\ s.t.\\ \\sigma(k)\\in p\\,\\},
□Δ​p\\displaystyle\\Box_{\\Delta}p:={σ∈ΦΔ:∀k≤Δ,σ(k)∈p}.\\displaystyle:=\\{\\,\\sigma\\in\\Phi^{\\Delta}:\\forall k\\leq\\Delta,\\ \\sigma(k)\\in p\\,\\}.
\n

Intuition.

\n

◇Δ​p\\Diamond_{\\Delta}p means pp is true at least once in the window. □Δ​p\\Box_{\\Delta}p means pp is true at every step in the window.

\n
Definition 13 (Lifted statements).
\n

For a statement l∈ℒ𝔳l\\in_{v}, define

\n
◇Δ​l\\displaystyle\\Diamond_{\\Delta}l:={◇Δ​p:p∈l},\\displaystyle:=\\{\\,\\Diamond_{\\Delta}p:p\\in l\\,\\},
□Δ​l\\displaystyle\\Box_{\\Delta}l:={□Δ​p:p∈l}.\\displaystyle:=\\{\\,\\Box_{\\Delta}p:p\\in l\\,\\}.
\n

Then T⁡(◇Δ​l)=⋂p∈l◇Δ​p(\\Diamond_{\\Delta}l)=\\bigcap_{p\\in l}\\Diamond_{\\Delta}p and similarly for □\\Box.

\n

Intuition.

\n

To lift a statement, lift each ingredient and require all lifted ingredients to hold. With ◇Δ\\Diamond_{\\Delta} this means every ingredient shows up somewhere in the window.

\n
Remark 2 (Temporal semantics as an abstraction layer).
\n

The window environment ΦΔ\\Phi^{\\Delta} and the lifted vocabulary {◇Δ​p:p∈𝔳}\\{\\Diamond_{\\Delta}p:p\\in\\} define a new abstraction layer in the native Stack-Theory sense, bridging “over time” constraints and pointwise semantics. The compilation of history-dependent (pathwise) abstractors into pointwise abstractors on augmented state is detailed in the Supplementary Material.

\n

4.3 The Algebraic Core

\n
Theorem 2 (Universal lift commutes with conjunction).
\n

For any statement l∈ℒ𝔳l\\in_{v} and any Δ∈ℕ\\Delta\\in, □Δ​T​(l)=T⁡(□Δ​l)\\Box_{\\Delta}T(l)=T(\\Box_{\\Delta}l).

\n

Proof. See Supplementary Material. □Δ\\Box_{\\Delta} is “for all positions”; since ∀\\forall distributes over conjunction, □Δ\\Box_{\\Delta} commutes with ∧\\wedge. ∎

\n
Theorem 3 (Existential lift does not commute with conjunction).
\n

For any statement l∈ℒ𝔳l\\in_{v} and any Δ∈ℕ\\Delta\\in,

\n
◇Δ​T​(l)⊆T⁡(◇Δ​l).\\Diamond_{\\Delta}T(l)\\;\\subseteq\\;T(\\Diamond_{\\Delta}l).
\n

Moreover, for Δ≥1\\Delta\\geq 1 there exist environments and statements for which the inclusion is strict.

\n
Proof.
\n

If σ∈◇Δ​T​(l)\\sigma\\in\\Diamond_{\\Delta}T(l) then for some kk I have σ⁡(k)∈T⁡(l)=⋂p∈lp\\sigma(k)\\in(l)=\\bigcap_{p\\in l}p. Hence σ∈◇Δ​p\\sigma\\in\\Diamond_{\\Delta}p for every p∈lp\\in l and therefore σ∈⋂p∈l◇Δ​p=T⁡(◇Δ​l)\\sigma\\in\\bigcap_{p\\in l}\\Diamond_{\\Delta}p=T(\\Diamond_{\\Delta}l).

\n

For strictness when Δ≥1\\Delta\\geq 1, let Φ={a,b,c}\\Phi=\\{a,b,c\\} and 𝔳={p,q}v=\\{p,q\\} with p={a,c}p=\\{a,c\\} and q={b,c}q=\\{b,c\\}. Then l={p,q}∈ℒ𝔳l=\\{p,q\\}\\in_{v} because p∩q={c}≠∅p\\cap q=\\{c\\}\\neq\\emptyset. Take Δ=1\\Delta=1 and window σ=(a,b)\\sigma=(a,b). Then σ∈◇1​p\\sigma\\in\\Diamond_{1}p and σ∈◇1​q\\sigma\\in\\Diamond_{1}q, so σ∈T⁡(◇1​l)\\sigma\\in(\\Diamond_{1}l). But σ∉◇1​T​(l)=◇1​(p∩q)\\sigma\\notin\\Diamond_{1}T(l)=\\Diamond_{1}(p\\cap q) because neither aa nor bb lies in p∩q={c}p\\cap q=\\{c\\}. ∎

\n

Proof intuition.

\n

◇Δ\\Diamond_{\\Delta} is “there exists a position”. Since ∃\\exists does not distribute over conjunction, ingredients may occur at different positions without co-occurring.

\n
Remark 3 (Degenerate equality cases).
\n

If Δ=0\\Delta=0 or |l|≤1\\lvert l\\rvert\\leq 1, then ◇Δ​T​(l)=T⁡(◇Δ​l)\\Diamond_{\\Delta}T(l)=T(\\Diamond_{\\Delta}l). Strict separation requires Δ≥1\\Delta\\geq 1 and at least two ingredients. A sufficient condition for commutation along a trajectory (within-window persistence) is given in the Supplementary Material.

\n
Remark 4 (Reading the theorem back into window semantics).
\n

In the sliding-window case, Definition 10 corresponds to

\n
OccurWΔ​(l,τ,t)\\displaystyle_{W_{\\Delta}}(l,\\tau,t)⇔τΔ​(t)∈T⁡(◇Δ​l),\\displaystyle\\iff\\tau_{\\Delta}(t)\\in(\\Diamond_{\\Delta}l),
CoInstWΔ​(l,τ,t)\\displaystyle_{W_{\\Delta}}(l,\\tau,t)⇔τΔ​(t)∈◇Δ​T​(l).\\displaystyle\\iff\\tau_{\\Delta}(t)\\in\\Diamond_{\\Delta}T(l).
\n

Theorem 3 is exactly the claim that ingredient-wise satisfaction is strictly weaker than co-instantiation in general.

\n

5 Chord and Arpeggio Postulates

\n

The above algebra does not decide which window semantics corresponds to phenomenality. I now state Chord and Arpeggio in terms of occurrence and co-instantiation, with a higher-layer moment statement and its base-layer grounding.

\n
Assumption 1 (Subjective moments live at some layer).
\n

Fix a stack state with vocabularies 𝔳0,…,𝔳m^{0},\\dots,v^{m}. Assume a candidate subjective moment is represented by a conjunctive statement lm∈ℒ𝔳ml^{m}\\in_{v^{m}}.

\n
Definition 14 (Phenomenal realisation predicate).
\n

Let PhenReal⁡(lm,τ,t)PhenReal(l^{m},\\tau,t) mean “moment statement lml^{m} is phenomenally realised at (higher-layer) time tt along objective trajectory τ\\tau.” I treat PhenReal as primitive, constrained by Chord/Arpeggio.

\n

Intuition.

\n

PhenReal names the claim that moment lml^{m} is experienced at subjective time tt along run τ\\tau. Here lml^{m} can include first, second, and third order selves and causal identities for objects and properties (Bennett, 2025b).

\n
Definition 15 (Base grounding of a moment statement).
\n

Let g0:=Ground0←m​(lm)g^{0}:=Ground_{0\\leftarrow m}(l^{m}) be the base-layer grounding of lml^{m} (Definition 6). By Theorem 1, T⁡(g0)=T⁡(lm)T(g^{0})=T(l^{m}). I call g0g^{0} the grounded conjunction of the moment statement.

\n
Definition 16 (Chord (objective co-instantiation required)).
\n

Fix a windowing map for the phenomenal layer, typically the sliding window WΔ_{\\Delta} of horizon Δ\\Delta (Section 4.2). Chord postulates that for all lm,τ,tl^{m},\\tau,t,

\n
PhenReal⁡(lm,τ,t)\\displaystyle(l^{m},\\tau,t)⇒CoInstWΔ​(g0,τ,t)\\displaystyle\\Rightarrow_{W_{\\Delta}}(g^{0},\\tau,t)
(equivalently ​τΔ​(t)∈◇Δ​T​(g0)​).\\displaystyle\\tau_{\\Delta}(t)\\in\\Diamond_{\\Delta}T(g^{0})).
\n

If the moment is experienced, then within the corresponding subjective window there exists an objective time-slice at which the entire grounded conjunction holds.

\n

Intuition.

\n

Chord says: if the moment is experienced, then within the window there is at least one objective time step where all grounded ingredients are true together.

\n
Definition 17 (Arpeggio (objective smearing permitted)).
\n

Fix the same windowing map WΔ_{\\Delta} as in Definition 16. Arpeggio postulates the following.

\n
  1. 1.
\n

Ingredient-wise necessity. For all lm,τ,tl^{m},\\tau,t,

\n
PhenReal⁡(lm,τ,t)⇒OccurWΔ​(g0,τ,t)\\displaystyle(l^{m},\\tau,t)\\Rightarrow_{W_{\\Delta}}(g^{0},\\tau,t)
(equivalently​τΔ​(t)∈T⁡(◇Δ​g0)​).\\displaystyle\\tau_{\\Delta}(t)\\in(\\Diamond_{\\Delta}g^{0})).
\n
  1. 2.
\n

Smearing permitted. There exist lm,τ,tl^{m},\\tau,t such that

\n
PhenReal⁡(lm,τ,t)∧OccurWΔ​(g0,τ,t)∧¬CoInstWΔ​(g0,τ,t).PhenReal(l^{m},\\tau,t)\\wedge_{W_{\\Delta}}(g^{0},\\tau,t)\\wedge\\neg_{W_{\\Delta}}(g^{0},\\tau,t).
\n

This allows a moment to be experienced without any instant realising the full grounded conjunction.

\n

Intuition.

\n

Arpeggio says: if the moment is experienced, each grounded ingredient must occur somewhere in the window, but there need not be a single time step where they are all true together.

\n

6 Architectural Consequences

\n

A polycomputer simultaneously executes multiple functions using the same parts or physical substrate (Bongard and Levin, 2023). A liquid brain such as an ant colony computes through movement rather than persistent structure (Solé et al., 2019). In contrast, a “solid brain” has persistent structure supporting, e.g., a bio-electric abstraction layer with synchronous broadcast (Bennett, 2025b). I now translate “single-thread sequential emulation” versus “synchronous polycomputational realisation” into a minimal architectural invariant bounding simultaneous causal contribution. The Temporal Gap asks whether that bound is consciousness-relevant (and whether conscious unity requires causal exchange between co-instantiated ingredients, explored in Bennett, 2026).

\n

6.1 Concurrency Capacity as an Invariant

\n
Definition 18 (Contributor model).
\n

Fix n≥2n\\geq 2 potential contributors and write [n]:={1,…,n}[n]:=\\{1,\\dots,n\\}. Define a base environment Φ:=X×𝒫⁡([n])\\Phi:=X\\times([n]), where x∈Xx\\in X is a “data” component and A⊆[n]A\\subseteq[n] is the set of currently active contributors. For each i∈[n]i\\in[n] define pi:={(x,A)∈Φ:i∈A}p_{i}:=\\{(x,A)\\in\\Phi:i\\in A\\}. Let ln:={p1,…,pn}l_{n}:=\\{p_{1},\\dots,p_{n}\\} and note T⁡(ln)={(x,[n]):x∈X}T(l_{n})=\\{(x,[n]):x\\in X\\}.

\n

Intuition.

\n

The contributor model records which contributors are active at each time step. A conjunction mentioning many contributors can only be true when all are active simultaneously.

\n
Definition 19 (Concurrency capacity).
\n

An architecture has concurrency capacity c∈{1,…,n}c\\in\\{1,\\dots,n\\} if its admissible trajectories satisfy |At|≤c\\lvert A_{t}\\rvert\\leq c for all objective times tt (writing τ⁡(t)=(xt,At)\\tau(t)=(x_{t},A_{t})). The case c=1c=1 formalises a sequential single-thread model; c≥nc\\geq n formalises synchronous polycomputing at the relevant scale.

\n

Intuition.

\n

Capacity cc is the maximum number of contributors active at one time step. If the grounded content needs more than cc active together, co-instantiation cannot hold.

\n
Theorem 4 (Synchronous threshold for co-instantiation).
\n

Fix n≥2n\\geq 2 and content statement lnl_{n}. If an architecture has concurrency capacity c<nc<n, then along every admissible trajectory τ\\tau and for every horizon Δ\\Delta, τΔ​(t)∉◇Δ​T​(ln)\\tau_{\\Delta}(t)\\notin\\Diamond_{\\Delta}T(l_{n}) for all tt. However, if every contributor is activated at least once within some horizon-Δ\\Delta window, then τΔ​(t)∈T⁡(◇Δ​ln)\\tau_{\\Delta}(t)\\in(\\Diamond_{\\Delta}l_{n}). Thus the architecture admits ingredient-wise satisfaction but forbids objective co-instantiation.

\n
Proof.
\n

If c<nc<n, then |At|≤c<n\\lvert A_{t}\\rvert\\leq c<n at every tt, so At≠[n]A_{t}\\neq[n] and τ⁡(t)∉T⁡(ln)\\tau(t)\\notin(l_{n}) for all tt. No window contains an element of T⁡(ln)T(l_{n}), so ◇Δ​T​(ln)\\Diamond_{\\Delta}T(l_{n}) is never satisfied. For the second claim, if each ii appears in At+kiA_{t+k_{i}} for some ki≤Δk_{i}\\leq\\Delta, then σ=τΔ​(t)\\sigma=\\tau_{\\Delta}(t) satisfies σ∈◇Δ​pi\\sigma\\in\\Diamond_{\\Delta}p_{i} for all ii, hence σ∈T⁡(◇Δ​ln)\\sigma\\in(\\Diamond_{\\Delta}l_{n}). ∎

\n

Proof intuition.

\n

If at most cc contributors can be active, an nn-way conjunction with n>cn>c is never true at an instant, though each ingredient can appear somewhere in a window.

\n
Corollary 1 (Single-thread CPU vs synchronous polycomputer).
\n

Let a moment statement lml^{m} have base grounding g0g^{0} whose ingredients include a conjunctive substatement equivalent to lnl_{n} for some n≥2n\\geq 2. Fix the phenomenal windowing map WΔ_{\\Delta} used in Chord/Arpeggio.

\n
  1. 1.
\n

Under Chord, any sequential-update architecture with capacity c<nc<n cannot phenomenally realise lml^{m} (Theorem 4).

\n
  1. 2.
\n

Under Arpeggio, the same sequential architecture remains a candidate, because T⁡(◇Δ​g0)T(\\Diamond_{\\Delta}g^{0}) may hold without ◇Δ​T​(g0)\\Diamond_{\\Delta}T(g^{0}).

\n
  1. 3.
\n

Any synchronous architecture with capacity c≥nc\\geq n can satisfy ◇Δ​T​(g0)\\Diamond_{\\Delta}T(g^{0}) by activating the required contributors simultaneously, hence can host the moment under both postulates.

\n
Remark 5 (Connection to valence-first Stack Theory).
\n

My valence-first application of Stack Theory treats certain phenomenal configurations as synchronised, causally efficacious conjunctions over many ingredients (Bennett, 2025b). The capacity model isolates one minimal implementation parameter—simultaneous contribution—controlling whether such conjunctions can be instantiated at any objective instant versus only “smeared” across a window.

\n

7 Evidence for Chord

\n

Chord and Arpeggio predict different empirical signatures within a short subjective window. Either consciousness requires window-local coordinated co-engagement, or it can tolerate mere ingredient-wise activation distributed across the window. Several classic paradigms are more naturally described by the coordination view. In masking, conscious perception correlates with transient long-range gamma phase synchrony across cortical areas, even when local activity is similar across seen versus unseen stimuli (Melloni et al., 2007). In non-REM sleep, TMS-EEG responses become strong yet local , failing to propagate, consistent with breakdown of effective connectivity when consciousness fades (Massimini et al., 2005). Perturbational measures such as PCI track level of consciousness across wake/sleep/anesthesia and disorders of consciousness, consistent with a requirement for integrated, differentiated dynamics (Casali et al., 2013). These findings do not prove Chord, but they make the existential permissiveness of Arpeggio less compelling. Conscious level covaries with the presence of a temporally coordinated, system-level regime. I therefore treat Chord as the default working hypothesis; this step remains defeasible.

\n

8 Attribution and Implications

\n

The Temporal Gap is the question of whether consciousness is affected by the gap between ingredient coverage and co-instantiation for a grounded conjunction.

\n

Fix a grounded conjunction g0g^{0} and an objective trajectory τ\\tau. Define the minimal horizon needed for ingredient coverage versus co-instantiation:

\n
wing​(g0,τ)\\displaystyle w_{ing}(g^{0},\\tau):=inf{Δ∈ℕ:∃t​τΔ​(t)∈T⁡(◇Δ​g0)},\\displaystyle:=\\inf\\{\\Delta\\in:\\exists t\\ \\tau_{\\Delta}(t)\\in(\\Diamond_{\\Delta}g^{0})\\},
wco​(g0,τ)\\displaystyle w_{co}(g^{0},\\tau):=inf{Δ∈ℕ:∃t​τΔ​(t)∈◇Δ​T​(g0)}.\\displaystyle:=\\inf\\{\\Delta\\in:\\exists t\\ \\tau_{\\Delta}(t)\\in\\Diamond_{\\Delta}T(g^{0})\\}.
\n

The strict Temporal Gap pattern is wing<wcow_{ing}<w_{co} (or wco=∞w_{co}=\\infty). These are instrumentable when base-layer ingredients are defined in terms of architectural state variables. This kind of architectural metric aligns with the view that behaviour alone may be insufficient for consciousness assessment and that concrete indicator properties should be checked against candidate theories (Butlin et al., 2023).

\n

A concrete test follows: construct paired systems that are behaviourally matched at a reporting layer but differ in base-layer concurrency capacity (one synchronous with c≥nc\\geq n, one sequential with small cc). Select candidate contents whose groundings include large conjunctions, and measure wingw_{ing} and wcow_{co}. Chord predicts that phenomenality tracks bounded wcow_{co}; Arpeggio predicts it may track wingw_{ing} alone.

\n

If Chord is even plausible , behavioural equivalence between a synchronous system and a sequential emulation does not settle phenomenal equivalence. This creates an ethically salient zone of reasonable disagreement about moral status, particularly when a system could have valenced experience. A precautionary stance toward sentience under uncertainty has been defended in nearby contexts (Birch, 2024). One concrete recommendation is to treat large Temporal-Gap regimes (wco≫wingw_{co}\\gg w_{ing} or wco=∞w_{co}=\\infty) as a high-uncertainty region for attribution, motivating conservative deployment policies until the Temporal Gap is better resolved.

\n

9 Conclusion

\n

I provided a Stack-Theory-compatible Stack-Time Semantics addendum and used it to make the Temporal Gap into a measurable algebraic and architectural question. Compositional grounding (Theorem 1) ensures any higher-layer phenomenal statement has a base-layer grounded conjunction with identical truth conditions. Induced layer time partitions objective time into maximal constant macro-blocks and can be arbitrarily sparse (Proposition 1), so subjective moments need not correspond to single objective instants. The Temporal Gap becomes a question of the non-commutation pattern

\n
◇Δ​T​(l)⊆T⁡(◇Δ​l),\\Diamond_{\\Delta}T(l)\\subseteq(\\Diamond_{\\Delta}l),
\n

with strict inclusion in general for Δ≥1\\Delta\\geq 1 and multi-ingredient statements (Theorem 3). This yields a clean separation between ingredient-wise window occurrence and objective co-instantiation.

\n

Chord and Arpeggio are competing postulates about which notion constrains phenomenality. Section 7 notes that conscious level covaries with temporally coordinated dynamics, tentatively favouring Chord.

\n

A simple architectural invariant (concurrency capacity) can force the strict Temporal Gap pattern for large conjunctions (Theorem 4). Under Chord, software consciousness on a strictly sequential substrate is impossible for any content whose grounding requires two or more simultaneous contributors. Real machines are not perfectly sequential at the physical level, so the practical question becomes which physical concurrency scale is relevant for grounding. Consciousness according to Stack Theory requires a “tapestry of valence” supported by a solid-brained polycomputing architecture in which each co-instantiated ingredient simultaneously acts upon the others, all linked and synchronised (a condition formally included in Chord in Bennett, 2026). Extreme concurrency capacity may be needed, and still concurrency is only one aspect of this. The central conclusion remains that consciousness attribution is hardware-sensitive, and surface-level behavioural equivalence alone cannot wash out the Temporal Gap.

\n

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