SELF-MONITORING AND CONTROLLABLE EVOLUTION OF INTELLIGENCE: Capability-Side Day Convolution with Profunctor Interfaces, Ambidextrous Kan (Strong/Oplax), Chu-like Twin Metrics, and Dynamic Seeding with Average-Contraction Tails

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colorlinks=true,
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pdftitle= Self-Monitoring and Controllable Evolution of Intelligence: Capability-Side Day Convolution with Profunctor Interfaces,
pdfauthor= K. Takahashi ,
pdfsubject= SFaS spine, enriched Day convolution, profunctor interfaces, audit/forecast/control/augment instrument panel, ambidextrous Kan ,
pdfkeywords= SFaS, profunctor, promonoidal distributor, Day convolution, recognition profunctor, filtered colimit, levelwise cofiltered limit, auditability, controllability, dynamic seeding, lax/oplax comparison, ambidextrous Kan, Chu pairing

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1.3

plain
theorem Theorem [section]
proposition[theorem] Proposition
lemma[theorem] Lemma
corollary[theorem] Corollary
definition
definition[theorem] Definition
remark[theorem] Remark
example[theorem] Example
assumption[theorem] Assumption

Section~#1
Sections~#1 and #2
Sections~#1, #2, and #3
Theorem~#1
Lemma~#1
Proposition~#1
Corollary~#1
Definition~#1
Assumption~#1
Remark~#1
Example~#1

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d_ sym % external pseudometric on objects of E (and elements in concrete setting)
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Structured Flow across Scales~TakahashiSFS

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TITLE: Self-Monitoring and Controllable Evolution of Intelligence:\
Capability-Side Day Convolution with Profunctor Interfaces,\
Ambidextrous Kan (Strong/Oplax), Chu-like Twin Metrics,\
and Dynamic Seeding with Average-Contraction Tails
[[EQ:eq0002]]

PARAGRAPH: Operational intelligence.

For a policy [[EQ:eq0032]] and frictions [[EQ:eq0033]] , set

[[EQ:eq0003]]

The system is intelligent at [[EQ:eq0034]]

if [[EQ:eq0035]] and the audit/forecast/control invariants herein hold.

SECTION: Assumptions

sec:assumptions

[Universes and enrichment base]ass:U-V
Fix a Grothendieck universe [[EQ:eq0036]] . Let the enrichment base be [[EQ:eq0037]] (Lawvere base). Object-level distances may be non-symmetric; the monoidal structure [[EQ:eq0038]] is commutative, so symmetry needed for Day convolution is unaffected.We distinguish non-symmetric metrics on objects from the (symmetric) monoidal structure used for convolution.

[Outcome category and functor space]ass:E
[[EQ:eq0039]] is a [[EQ:eq0040]] -enriched, complete and cocomplete, symmetric monoidal closed category, [[EQ:eq0041]] -tensored and cotensored. We write [[EQ:eq0042]] for the [[EQ:eq0043]] -action [[EQ:eq0044]] and [[EQ:eq0045]] for the monoidal product of [[EQ:eq0046]] (types disambiguate). Both are [[EQ:eq0047]] -Lipschitz in each variable with respect to an external symmetric pseudometric [[EQ:eq0048]] on the class of objects of [[EQ:eq0049]] (and, in the concrete setting ass:concrete, also on underlying elements of each object). All views live in [[EQ:eq0050]] with (co)limits computed pointwise in [[EQ:eq0051]] . For [[EQ:eq0052]] , define [[EQ:eq0053]] . We write [[EQ:eq0054]] for the enriched representable.

Intuition. The [[EQ:eq0055]] -enrichment controls the coend/Day calculus, while the external pseudometric [[EQ:eq0056]] only measures stability/non-expansion of maps and (co)limits; the two layers interact via `` [[EQ:eq0057]] -Lipschitz in each variable'' assumptions rather than via [[EQ:eq0058]] -homs.

[Concrete metric setting for analytic estimates]ass:concrete
In sections where we speak about elements [[EQ:eq0059]] and Lipschitz moduli of maps on elements,
we assume [[EQ:eq0060]] is a concrete category of metric objects with a faithful forgetful functor
[[EQ:eq0061]] that creates the relevant (co)limits, every morphism in [[EQ:eq0062]] is [[EQ:eq0063]] -Lipschitz
on underlying metrics, and [[EQ:eq0064]] denotes the given internal pseudometric on [[EQ:eq0065]] (extended to objects by abusing notation).
Examples include [[EQ:eq0066]] and its cocomplete variants. Outside these analytic parts, the abstract enriched development applies unchanged.

[Action data]ass:action
Fix a set of generators [[EQ:eq0067]] and a strong monoidal action
[[EQ:eq0068]] such that for each generator [[EQ:eq0069]] ,
[[EQ:eq0070]] is [[EQ:eq0071]] -Lipschitz on [[EQ:eq0072]] -valued views (generation stage),
and the SFaS unit/normalization comparisons witnessing the skeletal normal form
are componentwise [[EQ:eq0073]] -Lipschitz. We write [[EQ:eq0074]]
for a word [[EQ:eq0075]] and [[EQ:eq0076]] for its normalized form.

[Coends, preservation, and exchanges]ass:coends
(1) [[EQ:eq0077]] preserves the relevant (co)limits separately in each variable. (2) The coends defining profunctor composition and Day convolution exist. (3) In the strong promonoidal case, the required Fubini and Kan-exchange isomorphisms hold Kelly,DayPromon1977,DayStreet,Glasman2016. E.g.\ for [[EQ:eq0078]] we require existence of [[EQ:eq0079]] and preservation of these coends by [[EQ:eq0080]] ; Fubini refers to exchanging the two coends with [[EQ:eq0081]] , separately in each variable.

[Lipschitz moduli of structural 2-cells]ass:lip-2cells
Each structural [[EQ:eq0082]] -cell carries a Lipschitz modulus [[EQ:eq0083]] ; in the oplax case we allow a finite modulus [[EQ:eq0084]] per component and propagate these multiplicatively along coends. All bounds below remain valid with the product of moduli absorbed into [[EQ:eq0085]] .

[Promonoidal distributors: coherence and value]ass:distributor-coh-fixed
Agent kernels [[EQ:eq0086]] and recognitions [[EQ:eq0087]] are [[EQ:eq0088]] -valued, [[EQ:eq0089]] -enriched distributors. Each [[EQ:eq0090]] is equipped with (op)lax promonoidal [[EQ:eq0091]] -cells

[[EQ:eq0004]]

and a unit [[EQ:eq0092]] -cell coherent with the associativity/unit [[EQ:eq0093]] -cells of [[EQ:eq0094]] . Hence [[EQ:eq0095]] is (op)lax promonoidal and [[EQ:eq0096]] -valued.

[Small levelwise selection]ass:levelwise
Selection in [[EQ:eq0097]] uses reindexings and equalizers along [[EQ:eq0098]] -small cofiltered diagrams; cofiltered limits commute with finite colimits in pro-categories Isaksen2002. (Reindexings are isometries and equalizers [[EQ:eq0099]] -Lipschitz; see lem:nonexp-pro.)

[Nonexpansive action and comparisons (SFaS)]ass:nonexp
The lifted action and SFaS unit/normalization comparisons are [[EQ:eq0100]] -Lipschitz componentwise (generation [[EQ:eq0101]] -Lipschitz; selection nonexpansive).

SECTION: Capability-side convolution induced from agents

sec:cap-kernel

PARAGRAPH: Why capability-side?

(i) Metrics and normalization are managed uniformly on capabilities, avoiding agent/capability mismatches; (ii) lax/oplax interface comparisons between external/internal views factor through a single ambient space [[EQ:eq0102]] ; (iii) 's audit identity (the Hom double-limit under generation [[EQ:eq0103]] selection) is preserved verbatim on the capability side.

[Induced capability kernels]def:Pcap-global
From [[EQ:eq0104]] and [[EQ:eq0105]] define (op)lax promonoidal kernels on [[EQ:eq0106]] by

[[EQ:eq0005]]

[Day convolution on capabilities]def:day-cap
For [[EQ:eq0107]] and [[EQ:eq0108]] ,

[[EQ:eq0006]]

(Set-valued variants occur for [[EQ:eq0109]] ; the main line is [[EQ:eq0110]] -enriched.)

SECTION: SFaS spine and forward index

sec:index
[Seeds and spine]def:spine
Fix [[EQ:eq0111]] ( [[EQ:eq0112]] -small) with [[EQ:eq0113]] and constant pro-embedding [[EQ:eq0114]] . Let [[EQ:eq0115]] be the smallest replete full subcategory of [[EQ:eq0116]] containing [[EQ:eq0117]] and closed under images of the lifted action, filtered colimits, and [[EQ:eq0118]] -small levelwise cofiltered limits (SFaS mixed order).

[Normalized transport]def:hatT
For a word [[EQ:eq0119]] in [[EQ:eq0120]] , [[EQ:eq0121]] ; the normalized transport [[EQ:eq0122]] composes [[EQ:eq0123]] with SFaS unit/normalization comparisons to a skeletal normal form (these are [[EQ:eq0124]] -Lipschitz).

[Forward index]def:fwdindex
For [[EQ:eq0125]] , [[EQ:eq0126]] has objects [[EQ:eq0127]] with [[EQ:eq0128]] and [[EQ:eq0129]] , and a map [[EQ:eq0130]] . Morphisms are seed maps and composites of SFaS comparisons commuting with the legs to [[EQ:eq0131]] .

[Filteredness]lem:filtered
[[EQ:eq0132]] is filtered.

SFaS padding/normalization and whiskering give nonemptiness, cocones, and equalizers; finite limits in [[EQ:eq0133]] and preservation by the action suffice.

[Why generation [[EQ:eq0134]] selection]rem:mixedorder
This order enables the Hom double-limit (prop:audit) and stability: generation [[EQ:eq0135]] -Lipschitz and selection nonexpansive; reversing generally loses the audit identity.

SECTION: Metrics, final pseudometric, convergence, truncation

sec:stability
[External attenuation and dual metrics]def:atten
Assign frictions [[EQ:eq0136]] , [[EQ:eq0137]] , and [[EQ:eq0138]] .

[[EQ:eq0001]]

Conventions. [[EQ:eq0139]] and [[EQ:eq0140]] apply the external [[EQ:eq0141]] componentwise in [[EQ:eq0142]] and then aggregate across legs; they are not [[EQ:eq0143]] -homs. We set [[EQ:eq0144]] . Depthwise sups are finite (lem:depthbound) and [[EQ:eq0145]] , hence [[EQ:eq0146]] is a monotone limit in [[EQ:eq0147]] .

PARAGRAPH: Metrics at a glance.

[[EQ:eq0148]] is a depth-weighted [[EQ:eq0149]] -type audit metric (worst-case, scale-aware via [[EQ:eq0150]] ).
[[EQ:eq0151]] is a forecast/control surrogate (depthwise sup, geometrically weighted by [[EQ:eq0152]] ).
[[EQ:eq0153]] is the final pseudometric induced on colimits by [[EQ:eq0154]] -Lipschitz cocones.
[[EQ:eq0155]] restricts [[EQ:eq0156]] to realized prefixes under a policy, enabling pathwise tail bounds.

PARAGRAPH: Depth-0 and truncation notation.

Write [[EQ:eq0157]] for the depth- [[EQ:eq0158]] component (empty word [[EQ:eq0159]] ).
For [[EQ:eq0160]] , let [[EQ:eq0161]] be the inclusion and define the [[EQ:eq0162]] -truncation by

[[EQ:eq0007]]

so that [[EQ:eq0163]] is computed pointwise as in the paragraph “Truncation via [[EQ:eq0164]] ”.

[Final pseudometric on a colimit]def:finalmetric
Given a cocone [[EQ:eq0165]] with Lipschitz moduli [[EQ:eq0166]] , define for [[EQ:eq0167]]

[[EQ:eq0008]]

This is the largest pseudometric on [[EQ:eq0168]] making each [[EQ:eq0169]] [[EQ:eq0170]] -Lipschitz.

[Depthwise uniform bound]lem:depthbound
If [[EQ:eq0171]] and ass:nonexp holds, then for each [[EQ:eq0172]] , [[EQ:eq0173]] .

[Selection nonexpansive]lem:nonexp-pro
In [[EQ:eq0174]] , reindexings are isometries and equalizers [[EQ:eq0175]] -Lipschitz (enriched equalizers carry the largest pseudometric making inclusions [[EQ:eq0176]] -Lipschitz; Kelly). Therefore levelwise cofiltered limits are nonexpansive.

[Attenuated forward assembly is [[EQ:eq0177]] -Lipschitz]lem:atten-q
For the forward diagram [[EQ:eq0178]] built under ass:nonexp with external attenuation [[EQ:eq0179]] , the canonical cocone
[[EQ:eq0180]] can be chosen so that each leg satisfies
[[EQ:eq0181]] , where [[EQ:eq0182]] .

. Induct on [[EQ:eq0183]] : for [[EQ:eq0184]] the normalization/unit maps are [[EQ:eq0185]] -Lipschitz.
For [[EQ:eq0186]] , compose the cocone for [[EQ:eq0187]] (induction hypothesis) with the generator step,
which contributes a [[EQ:eq0188]] -Lipschitz comparison and a multiplicative attenuation [[EQ:eq0189]] .
With def:finalmetric, this yields the bound in thm:lip for [[EQ:eq0190]] .

[Generation [[EQ:eq0191]] -Lipschitz; selection nonexpansive]thm:lip
Under ass:nonexp, for [[EQ:eq0192]] ,

[[EQ:eq0009]]

PARAGRAPH: Truncation via [[EQ:eq0193]] .

For [[EQ:eq0194]] , the enriched left Kan extension is computed pointwise:

[[EQ:eq0010]]

where [[EQ:eq0195]] is [[EQ:eq0196]] -small and the colimit exists by ass:coends. SFaS comparisons supply cocones.

[Deterministic truncation]thm:det
If [[EQ:eq0197]] and [[EQ:eq0198]] , then

[[EQ:eq0011]]

Estimate. For any leg with [[EQ:eq0199]] , [[EQ:eq0200]] ; [[EQ:eq0201]] adds a factor [[EQ:eq0202]] .

SECTION: Probabilistic semantics and phase thresholds

sec:prob
[Policy-induced probability space]ass:probspace
A stationary policy [[EQ:eq0203]] defines a probability space [[EQ:eq0204]] over generator sequences/environment randomness. Expectations [[EQ:eq0205]] and a.s.\ statements refer to this space.

[Ergodicity and integrability]ass:gain-ergodic
Under [[EQ:eq0206]] , [[EQ:eq0207]] is stationary ergodic with [[EQ:eq0208]] ; [[EQ:eq0209]] is stationary ergodic with [[EQ:eq0210]] , [[EQ:eq0211]] .

[Useful log-gain and net improvement]def:useful
[[EQ:eq0212]] , [[EQ:eq0213]] (Birkhoff Birkhoff1931). Define [[EQ:eq0214]] ; net improvement holds if [[EQ:eq0215]] .

[Signed raw gains]
If raw gains [[EQ:eq0216]] can be negative, use a clipped [[EQ:eq0217]] with [[EQ:eq0218]] so that [[EQ:eq0219]] is well-defined and the safety margin is explicit.

[On [[EQ:eq0220]] ]
Since [[EQ:eq0221]] , [[EQ:eq0222]] and [[EQ:eq0223]] , so [[EQ:eq0224]] .

[A.s.\ phase threshold — quantified]thm:stoch
Under ass:gain-ergodic with [[EQ:eq0225]] and [[EQ:eq0226]] , for any [[EQ:eq0227]] there exists [[EQ:eq0228]] such that [[EQ:eq0229]] for all [[EQ:eq0230]] a.s. Consequently,

[[EQ:eq0012]]

PARAGRAPH: Interpretation.

The assumption [[EQ:eq0231]] states that the average multiplicative friction along the realized path is strictly contractive; hence truncation errors decay exponentially even if some individual [[EQ:eq0232]] equal [[EQ:eq0233]] .

[Boundary case]
When [[EQ:eq0234]] , Cesàro averages converge a.s.\ to [[EQ:eq0235]] ; net growth is sublinear.

SECTION: Auditability

sec:audit
[Hom double-limit]prop:audit
If [[EQ:eq0236]] in [[EQ:eq0237]] and [[EQ:eq0238]] is a [[EQ:eq0239]] -small levelwise cofiltered limit in [[EQ:eq0240]] , then

[[EQ:eq0013]]

[Ledger identifiability]prop:ident
Assume the inclusion of normal forms is final (unique normal forms up to isomorphism). Then the multiset of forward ledger legs [[EQ:eq0241]] for a map out of a constant source is unique up to isomorphisms in the skeleton.

SECTION: Capability-side interface comparison

sec:interfaces
[Interfaces and recognition coherence]def:interfaces
Let [[EQ:eq0242]] be dense and (op)lax promonoidal.“Dense’’ is used in the enriched sense; cf.\ Kelly, TAC Reprints~10, §§3–4, and KellyStreet. Recognition profunctors [[EQ:eq0243]] and [[EQ:eq0244]] carry (op)lax promonoidal structure and coherence exhibiting [[EQ:eq0245]] as the pullback of [[EQ:eq0246]] along [[EQ:eq0247]] .

[Capability-side comparison, nonexpansive]thm:nondual-cap
All functors live in [[EQ:eq0248]] with convolutions computed using [[EQ:eq0249]] and [[EQ:eq0250]] . There is a canonical (op)lax monoidal natural transformation

[[EQ:eq0014]]

natural in [[EQ:eq0251]] , which is nonexpansive for [[EQ:eq0252]] (pointwise [[EQ:eq0253]] -Lipschitz). For the audit metric [[EQ:eq0254]] , fix depth [[EQ:eq0255]] , apply pointwise [[EQ:eq0256]] -Lipschitz, take the supremum over legs at depth [[EQ:eq0257]] , and finally the outer supremum over [[EQ:eq0258]] (or the weighted sum for [[EQ:eq0259]] ). If [[EQ:eq0260]] and [[EQ:eq0261]] are strong promonoidal and Fubini/Lan exchanges hold, [[EQ:eq0262]] is an isomorphism.

[Construction sketch with diagram]
Transport coend integrands along the (op)lax distributor structures and the coherence relating [[EQ:eq0263]] to [[EQ:eq0264]] via [[EQ:eq0265]] , then assemble by the universal property of Day convolution:

[[EQ:eq0015]]

coends assemble to [[EQ:eq0266]] ; [[EQ:eq0267]] -Lipschitzness follows from ass:lip-2cells.

SECTION: Control

sec:control
[Control objective — [[EQ:eq0268]] -based]def:control-g
Let [[EQ:eq0269]] be a stationary policy over [[EQ:eq0270]] , frictions [[EQ:eq0271]] with [[EQ:eq0272]] , and discount [[EQ:eq0273]] . Define

[[EQ:eq0016]]

with [[EQ:eq0274]] as in def:useful.

[Budget-feasible bound]prop:budget
If [[EQ:eq0275]] a.s.\ (hence [[EQ:eq0276]] ), then

[[EQ:eq0017]]

[Link to [[EQ:eq0277]] ]
If [[EQ:eq0278]] , then [[EQ:eq0279]] . Moreover [[EQ:eq0280]] is bounded by a geometric series with ratio [[EQ:eq0281]] , hence scales predictably with attenuation/discount. (We assume [[EQ:eq0282]] after clipping.)

SECTION: Synergy, forecast, and sensitivity

sec:metrics

PARAGRAPH: Forecast functional (g-based).

Expectations are taken on the policy-induced space (ass:probspace). For horizon [[EQ:eq0283]] ,

[[EQ:eq0018]]

where [[EQ:eq0284]] is the step index of [[EQ:eq0285]] . If [[EQ:eq0286]] a.s., then [[EQ:eq0287]] and

[[EQ:eq0019]]

[Retrospective synergy — rank-1 form]def:synergy
Write [[EQ:eq0288]] if there is a componentwise [[EQ:eq0289]] -Lipschitz natural transformation [[EQ:eq0290]] in
[[EQ:eq0291]] . For [[EQ:eq0292]] , define the rank-1 presheaf

[[EQ:eq0020]]

Let [[EQ:eq0293]] be the finite-coproduct closure of [[EQ:eq0294]] (exists by ass:coends).
Define [[EQ:eq0295]] .

[Prospective synergy]def:prospective
For a candidate [[EQ:eq0296]] , with predicted kernel [[EQ:eq0297]] and predicted capability view [[EQ:eq0298]] , let [[EQ:eq0299]] be the finite-coproduct closure of the rank-1 pieces [[EQ:eq0300]] defined as in def:synergy but computed using [[EQ:eq0301]] in place of [[EQ:eq0302]] ; assume [[EQ:eq0303]] . Define

[[EQ:eq0021]]

[On [[EQ:eq0304]] -regularization]
[[EQ:eq0305]] stabilizes the trigger (avoids division by zero) and preserves lower semicontinuity of the score.

A sufficient modeling condition for the sensitivity constant below is that [[EQ:eq0306]] is [[EQ:eq0307]] -Lipschitz with respect to the Day-convolution output along the kernel and that [[EQ:eq0308]] aggregates these contributions with weights [[EQ:eq0309]] .

[Performance sensitivity to kernel perturbations]prop:sensitivity
Assume there exists [[EQ:eq0310]] such that for kernels [[EQ:eq0311]] on [[EQ:eq0312]] ,

[[EQ:eq0022]]

Then a prospective addition with predicted kernel [[EQ:eq0313]] satisfies the conservative bound

[[EQ:eq0023]]

providing a principled trigger for augmentation under budget/risk constraints.

[Units for [[EQ:eq0314]] ]
[[EQ:eq0315]] has the same physical units as the (discounted) sum of [[EQ:eq0316]] ; it aggregates per-step Lipschitz gains with weights [[EQ:eq0317]] .

SECTION: Ambidextrous Day--Kan and Chu-like twin metrics

sec:ambi-chu

[Ambidextrous Kan, strong vs.\ oplax]def:ambikan
When the preservation and interchange conditions of ass:coends hold,
we identify [[EQ:eq0318]] and write
[[EQ:eq0319]] for this common object.
In the general (op)lax case we use the canonical comparison

[[EQ:eq0024]]

natural in [[EQ:eq0320]] , which is pointwise [[EQ:eq0321]] -Lipschitz; dually when only the right data are preserved one has [[EQ:eq0322]] .

[Strong case equivalence and oplax nonexpansion]
If the relevant limits/colimits are preserved by [[EQ:eq0323]] and Fubini/Lan interchange holds, then
[[EQ:eq0324]] .
Otherwise the canonical oplax comparison [[EQ:eq0325]] is pointwise [[EQ:eq0326]] -Lipschitz, hence nonexpansive for [[EQ:eq0327]] and [[EQ:eq0328]] .

(Chu-like pairing).
We use “Chu’’ suggestively for the ordered pair of nonexpansive pairings [[EQ:eq0329]] ; no dualizing object is assumed nor needed here. Formally one may read “ [[EQ:eq0330]] ’’ below as a category of pairs of [[EQ:eq0331]] -Lipschitz functionals.

[Chu-like twin metrics]
Define the twin pairing on forward diagrams

[[EQ:eq0025]]

[Continuity of the twin pair]
Under generation [[EQ:eq0332]] -Lipschitz and selection nonexpansive, both pairings are continuous in each argument; interface comparisons are nonexpansive for both.

SECTION: Invariance, rank-1 emptiness, average-contraction tails

sec:invariance
[Invariance under capability equivalence]
Let [[EQ:eq0333]] be an equivalence, and transport kernels by [[EQ:eq0334]] that are (op)lax monoidal and [[EQ:eq0335]] -Lipschitz on coend integrands. Then [[EQ:eq0336]] are invariant up to isometry.Kernel pushforward/pullback obey [[EQ:eq0337]] under strong conditions; oplax comparisons are [[EQ:eq0338]] -Lipschitz.

[Rank-1 emptiness test and Morita approximation]prop:rank1
If [[EQ:eq0339]] ,
then [[EQ:eq0340]] for all [[EQ:eq0341]] .
Conversely, if [[EQ:eq0342]] for all [[EQ:eq0343]] (representables), the comparison is natural
and [[EQ:eq0344]] -Lipschitz in each slot, and the kernel metric is totally bounded on finite-support slices
(equivalently: for every [[EQ:eq0345]] there exists a finite [[EQ:eq0346]] -net of rank-1 pieces on each slice),
then [[EQ:eq0347]] admits, for every [[EQ:eq0348]] , a finite rank-1 approximation
within [[EQ:eq0349]] in the kernel metric [[EQ:eq0350]] .

[Path-indexed audit metric]def:pathmetric
For a realization [[EQ:eq0351]] of [[EQ:eq0352]] , let [[EQ:eq0353]] consist of forward legs whose words are prefixes of the realized sequence [[EQ:eq0354]] .
Define

[[EQ:eq0026]]

[Average contraction tail — pathwise]thm:avg-tail-path
If [[EQ:eq0355]] a.s., then there exist [[EQ:eq0356]] and a random constant [[EQ:eq0357]] such that

[[EQ:eq0027]]

Thus, along almost every realized path, truncation tails decay exponentially even if [[EQ:eq0358]] provided the average contraction holds.

[From pathwise contraction to tails]
Along the realized path one has [[EQ:eq0359]] by the assumed exponential bound and lem:depthbound.

SECTION: Dynamic seeding

sec:metaspine
[Augmentation]ass:augmentation
[[EQ:eq0360]] maps [[EQ:eq0361]] to [[EQ:eq0362]] by choosing [[EQ:eq0363]] under threshold [[EQ:eq0364]] .

[Termination]ass:termination
Each step adds finitely many objects/morphisms within budget [[EQ:eq0365]] ; if [[EQ:eq0366]] augmentation halts. Under bounded [[EQ:eq0367]] and lower semicontinuous [[EQ:eq0368]] , the meta-spine halts in finitely many steps or defers until evidence raises [[EQ:eq0369]] .

SECTION: Double-categorical instrumentation

sec:double
Let [[EQ:eq0370]] be the double category with horizontal arrows the time/word extensions and vertical arrows the capability convolutions (Day along [[EQ:eq0371]] ). The instrument panel becomes a lax double functor

[[EQ:eq0028]]

where [[EQ:eq0372]] denotes pairs of [[EQ:eq0373]] -Lipschitz functionals.
[Laxity and nonexpansion]
The comparison cells of [[EQ:eq0374]] are (op)lax and componentwise [[EQ:eq0375]] -Lipschitz, hence nonexpansive for both twin components.

SECTION: Toy finite computation (set-based illustration) after kernel untwist

sec:toy
. This toy lives outside the enriched hypotheses; it illustrates only the
set-based Day reduction and the external metric bookkeeping.
Here [[EQ:eq0376]] plays the role of a set of witnesses (an indicator), so the Day coend becomes a finite coproduct of Cartesian products.

Let [[EQ:eq0377]] , [[EQ:eq0378]] , and take [[EQ:eq0379]] . In [[EQ:eq0380]] (and hence in its full subcategory [[EQ:eq0381]] ) the Day coend reduces to a finite coproduct of Cartesian products, hence

[[EQ:eq0029]]

Metric for the toy: for concreteness we equip [[EQ:eq0382]] with the
cardinality seminorm [[EQ:eq0383]] and [[EQ:eq0384]] (this fits the external metric convention).

Choose

[[EQ:eq0030]]

Then [[EQ:eq0385]] yields

[[EQ:eq0031]]

and with [[EQ:eq0386]] we get [[EQ:eq0387]] , hence [[EQ:eq0388]] in the cardinality metric.
This makes [[EQ:eq0389]] immediate via a single rank-1 piece.

SECTION: Computational notes (for practitioners)

sec:compute
Branching factor. [[EQ:eq0390]] is the maximal local branching factor.\
Depth truncation. Evaluate [[EQ:eq0391]] by beam search up to depth [[EQ:eq0392]] (beam width [[EQ:eq0393]] ), with tail bound from thm:det. Complexity [[EQ:eq0394]] (constant factors proportional to kernel support size).\
Audit ledgers. Read the multiset of forward legs [[EQ:eq0395]] at the skeletal normal form; use prop:audit and prop:ident to match presentations.\
Synergy approximation. Restrict rank-1 pairs [[EQ:eq0396]] to kernel support; build [[EQ:eq0397]] via greedy finite coproducts minimizing [[EQ:eq0398]] .\
Prospective trigger. Compute [[EQ:eq0399]] from predicted kernels, check [[EQ:eq0400]] , then simulate [[EQ:eq0401]] under [[EQ:eq0402]] -attenuation.\
Friction tuning. Use thm:det to select [[EQ:eq0403]] achieving an error budget at depth [[EQ:eq0404]] ; adjust [[EQ:eq0405]] to fit horizons.\
Confidence. With bounded [[EQ:eq0406]] and [[EQ:eq0407]] , apply concentration (e.g.\ Hoeffding/Azuma/ [[EQ:eq0408]] -mixing variants) to certify finite-sample positivity of [[EQ:eq0409]] . Conditions: Hoeffding requires i.i.d.\ bounded gains; Azuma requires a martingale-difference sequence with bounded increments.

SECTION: Self-audit

sec:self-audit
(S1) Typing. All convolutions are in [[EQ:eq0410]] with [[EQ:eq0411]] on the capability side. \
(S2) Coherence. ass:distributor-coh-fixed ensures induced kernels are (op)lax promonoidal; Fubini/Kan exchanges as per ass:coends. \
(S3) Dual metrics. [[EQ:eq0412]] (audit) and [[EQ:eq0413]] (forecast/control) used with correct coefficients; convergence from lem:depthbound. \
(S4) Truncation. [[EQ:eq0414]] resolves index mismatch; errors bounded by [[EQ:eq0415]] . \
(S5) Interfaces. thm:nondual-cap is nonexpansive for [[EQ:eq0416]] and, depthwise, for [[EQ:eq0417]] via the two-step argument (pointwise [[EQ:eq0418]] -Lipschitz [[EQ:eq0419]] depthwise sup/aggregate). \
(S6) Probabilistic. Expectations/a.s.\ claims per ass:probspace and ass:gain-ergodic; quantification in thm:stoch. \
(S7) Augmentation. Assumptions on augmentation and termination ensure accessibility/termination; failure mode: [[EQ:eq0420]] yet [[EQ:eq0421]] fails to improve. \
(S8) Failure modes. (i) non-ergodic policies; (ii) failure of Fubini/Lan exchanges; (iii) structural 2-cells not [[EQ:eq0422]] -Lipschitz (absorbed into [[EQ:eq0423]] if bounded).

SECTION: Acknowledgements

This paper refines and extends \ to a capability-side spine with an audit/forecast/control/augment instrument panel.
99 0.45ex

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Positioning and standing notation

Assumptions

Capability-side convolution induced from agents

SFaS spine and forward index

Metrics, final pseudometric, convergence, truncation

Probabilistic semantics and phase thresholds

Auditability

Capability-side interface comparison

Control

Synergy, forecast, and sensitivity

Ambidextrous Day–Kan and Chu-like twin metrics

Invariance, rank-1 emptiness, average-contraction tails

Dynamic seeding

Double-categorical instrumentation

Toy finite computation (set-based illustration) after kernel untwist

Computational notes (for practitioners)

Self-audit

Acknowledgements

\begin{align*}
\textbf{(a) }d_X(F,G)&:=\sup_{(w,C)} Q(w)\,\dsym(F(w,C),G(w,C)),\\
\textbf{(b) }D_X^{\mathrm{agg}}(F,G)&:=\sum_{n\ge 0}\bar q^{\,n}\ \sup_{\substack{(w,C)\\ |w|=n}} \dsym(F(w,C),G(w,C)).
\end{align*}

\[0.35em]
\small Application and extension of SFaS}
\author{\textbf{K.~Takahashi}\\[0.35em]
\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We develop a capability-side process-theoretic spine where intelligence arises from interactions and can measure, audit, forecast, control, and—when justified—augment its own seeds. Building on the SFaS bicompletion (filtered generation followed by $U$-small levelwise cofiltered selection), we: (i) induce (op)lax promonoidal kernels on the capability category from agent-side interaction and recognition profunctors and perform all Day convolutions in $[\A_{\mathrm{cap}}^\op,\E]$; (ii) separate the enrichment base $\V$ and outcome category $\E$, use external metrics componentwise, and state coend/preservation/Fubini/Kan assumptions explicitly; (iii) bifurcate metrics into a sup-type $d_X$ (audit/stability) and an aggregated $D_X^{\mathrm{agg}}$ (forecast/control) with depthwise uniform bounds ensuring convergence; (iv) define a useful log-gain process on a policy-induced probability space, quantify almost-sure thresholds, and express control objectives consistently in $g_t$ with attenuation folded into the discount; (v) separate \emph{retrospective} $\Sigma$ (rank-1 finite-coproduct hull) from \emph{prospective} $\widehat\Sigma_\varepsilon$ synergy and give a conservative performance bound; (vi) prove a capability-side (lax/oplax) interface comparison that is nonexpansive for both metrics, upgrading to an isomorphism under strong promonoidality with Fubini/Lan exchange; (vii) implement dynamic seeding with accessibility/termination; and (viii) extend the spine with ambidextrous Day–Kan (strong/oplax), a Chu-like twin-metric pairing (terminology clarified), rank-1 emptiness/Morita approximation (safe direction), double-categorical instrumentation, and \emph{pathwise} average-contraction tails. The spine serves as a \emph{categorical instrument panel}: read ledgers (audit), estimate windows (forecast), modulate frictions (control), and augment seeds (augment).
\end{abstract}

%=======================
\section{Positioning and standing notation}\label{sec:intro}
\textbf{Base (SFaS).} \SFaS\ provides a transport layer (strong monoidal action with normalization) and a mixed order (filtered colimit $\to$ $U$-small levelwise cofiltered limit) with external attenuation and $1$-Lipschitz forward assembly.

\textbf{This paper.} We unify interaction on the capability side and instrument it for audit/forecast/control/augment, and we broaden the scope with strong/oplax ambidextrous Kan, Chu-like twin metrics (terminology clarified), and invariance/sensitivity principles.

\paragraph{Reader's guide.}
\secref{sec:assumptions} fixes the enriched setting and external metrics.
\secref{sec:cap-kernel} induces capability-side kernels and Day convolution;
\secref{sec:index} builds the SFaS forward index.
\secref{sec:stability} introduces the dual metrics $(d_X,D_X^{\mathrm{agg}})$ and proves the $1$-Lipschitz/nonexpansive bounds and truncation rates.
\secref{sec:prob} supplies the policy-induced probability space and a.s.\ phase thresholds;
\secref{sec:audit} gives the audit double-limit;
\secref{sec:interfaces} proves the (op)lax interface comparison.
\secreftwo{sec:control}{sec:metrics} cover control and performance sensitivity;
\secref{sec:ambi-chu} gives ambidextrous Day--Kan and the Chu-like pairing;
\secref{sec:invariance} presents invariance and average-contraction tails;
\secrefthree{sec:metaspine}{sec:double}{sec:toy} discuss dynamic seeding, double-categorical instrumentation, and a toy finite example.

\begin{center}\small
\begin{tabular}{@{}ll@{}}
\toprule
Symbol & Meaning \\
\midrule
$U$ & fixed Grothendieck universe \\
$\V$ & Lawvere base $([0,\infty],\ge,+,0)$ (object-level distances may be asymmetric) \\
$\E$ & $\V$-enriched, complete/cocomplete, symmetric monoidal closed outcome category \\
$S$ & set of generators (words $S^\ast$ act via \assref{ass:action}) \\
$\varrho$ & strong monoidal action $S^\ast\!\to\!\End(\Ind(\A_0))$ (\assref{ass:action}) \\
$\A_{\mathrm{agt}}$, $\A_{\mathrm{cap}}$ & small agent / capability categories \\
$\mathbf P_{\bullet}$ & agent-side promonoidal kernel ($\bullet\in\{\mathrm{ex,in}\}$) \\
$R_{\bullet}$ & recognition profunctor $\A_{\mathrm{cap}}\proTo\A_{\mathrm{agt},\bullet}$ \\
$\mathbf P_{\mathrm{cap},\bullet}$ & induced capability kernel $R_\bullet^{\top}\odot \mathbf P_{\bullet}\odot R_\bullet$ \\
$Y_x$ & enriched representable (Yoneda) $\A_{\mathrm{cap}}(-,x)\in[\A_{\mathrm{cap}}^\op,\E]$ \\
$\tenV$ / $\tenE$ & $\V$-action on $\E$ / monoidal product of $\E$ \\
$q,\ \bar q,\ Q(w)$ & frictions $q:S\to(0,1]$, $\bar q:=\sup_{s}q(s)<1$, and $Q(w):=\prod_i q(s_i)$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent\emph{Profunctor composition (notation without collision).}
For distributors $H:\C\proTo\D$ and $K:\D\proTo\B$, set
\[
(H\odot K)(c,b):=\int^{d\in\D} H(c,d)\,\tenE\, K(d,b).
\]

0.35em] \small Application and extension of SFaS} \author{\textbf{K.~Takahashi}\\[0.35em] \small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}} \date{\today} \begin{document} \maketitle \begin{abstract} We develop a capability-side process-theoretic spine where intelligence arises from interactions and can measure, audit, forecast, control, and—when justified—augment its own seeds. Building on the SFaS bicompletion (filtered generation followed by $U$-small levelwise cofiltered selection), we: (i) induce (op)lax promonoidal kernels on the capability category from agent-side interaction and recognition profunctors and perform all Day convolutions in $[\A_{\mathrm{cap}}^\op,\E]$; (ii) separate the enrichment base $\V$ and outcome category $\E$, use external metrics componentwise, and state coend/preservation/Fubini/Kan assumptions explicitly; (iii) bifurcate metrics into a sup-type $d_X$ (audit/stability) and an aggregated $D_X^{\mathrm{agg}}$ (forecast/control) with depthwise uniform bounds ensuring convergence; (iv) define a useful log-gain process on a policy-induced probability space, quantify almost-sure thresholds, and express control objectives consistently in $g_t$ with attenuation folded into the discount; (v) separate \emph{retrospective} $\Sigma$ (rank-1 finite-coproduct hull) from \emph{prospective} $\widehat\Sigma_\varepsilon$ synergy and give a conservative performance bound; (vi) prove a capability-side (lax/oplax) interface comparison that is nonexpansive for both metrics, upgrading to an isomorphism under strong promonoidality with Fubini/Lan exchange; (vii) implement dynamic seeding with accessibility/termination; and (viii) extend the spine with ambidextrous Day–Kan (strong/oplax), a Chu-like twin-metric pairing (terminology clarified), rank-1 emptiness/Morita approximation (safe direction), double-categorical instrumentation, and \emph{pathwise} average-contraction tails. The spine serves as a \emph{categorical instrument panel}: read ledgers (audit), estimate windows (forecast), modulate frictions (control), and augment seeds (augment). \end{abstract} %======================= \section{Positioning and standing notation}\label{sec:intro} \textbf{Base (SFaS).} \SFaS\ provides a transport layer (strong monoidal action with normalization) and a mixed order (filtered colimit $\to$ $U$-small levelwise cofiltered limit) with external attenuation and $1$-Lipschitz forward assembly. \textbf{This paper.} We unify interaction on the capability side and instrument it for audit/forecast/control/augment, and we broaden the scope with strong/oplax ambidextrous Kan, Chu-like twin metrics (terminology clarified), and invariance/sensitivity principles. \paragraph{Reader's guide.} \secref{sec:assumptions} fixes the enriched setting and external metrics. \secref{sec:cap-kernel} induces capability-side kernels and Day convolution; \secref{sec:index} builds the SFaS forward index. \secref{sec:stability} introduces the dual metrics $(d_X,D_X^{\mathrm{agg}})$ and proves the $1$-Lipschitz/nonexpansive bounds and truncation rates. \secref{sec:prob} supplies the policy-induced probability space and a.s.\ phase thresholds; \secref{sec:audit} gives the audit double-limit; \secref{sec:interfaces} proves the (op)lax interface comparison. \secreftwo{sec:control}{sec:metrics} cover control and performance sensitivity; \secref{sec:ambi-chu} gives ambidextrous Day--Kan and the Chu-like pairing; \secref{sec:invariance} presents invariance and average-contraction tails; \secrefthree{sec:metaspine}{sec:double}{sec:toy} discuss dynamic seeding, double-categorical instrumentation, and a toy finite example. \begin{center}\small \begin{tabular}{@{}ll@{}} \toprule Symbol & Meaning \\ \midrule $U$ & fixed Grothendieck universe \\ $\V$ & Lawvere base $([0,\infty],\ge,+,0)$ (object-level distances may be asymmetric) \\ $\E$ & $\V$-enriched, complete/cocomplete, symmetric monoidal closed outcome category \\ $S$ & set of generators (words $S^\ast$ act via \assref{ass:action}) \\ $\varrho$ & strong monoidal action $S^\ast \to \End(\Ind(\A_0))$ (\assref{ass:action}) \\ $\A_{\mathrm{agt}}$, $\A_{\mathrm{cap}}$ & small agent / capability categories \\ $\mathbf P_{\bullet}$ & agent-side promonoidal kernel ($\bullet\in\{\mathrm{ex,in}\}$) \\ $R_{\bullet}$ & recognition profunctor $\A_{\mathrm{cap}}\proTo\A_{\mathrm{agt},\bullet}$ \\ $\mathbf P_{\mathrm{cap},\bullet}$ & induced capability kernel $R_\bullet^{\top}\odot \mathbf P_{\bullet}\odot R_\bullet$ \\ $Y_x$ & enriched representable (Yoneda) $\A_{\mathrm{cap}}(-,x)\in[\A_{\mathrm{cap}}^\op,\E]$ \\ $\tenV$ / $\tenE$ & $\V$-action on $\E$ / monoidal product of $\E$ \\ $q,\ \bar q,\ Q(w)$ & frictions $q:S\to(0,1]$, $\bar q:=\sup_{s}q(s)<1$, and $Q(w):=\prod_i q(s_i)$ \\ \bottomrule \end{tabular} \end{center} \noindent\emph{Profunctor composition (notation without collision).} For distributors $H:\C\proTo\D$ and $K:\D\proTo\B$, set \[ (H\odot K)(c,b):=\int^{d\in\D} H(c,d) \tenE  K(d,b).

\[
\mathcal I_{\mathrm{rate}}(X;\pi,q):=\liminf_{n\to\infty}\frac{1}{n}\sum_{t=1}^n(\log(1+g_t)-|\log q(s_t)|),
\quad g_t\ge 0.
\]

\[
\mu^{R}_{x,y;z}:\ \int^{a,b} R_{\bullet}(x;a)\tenE R_{\bullet}(y;b)\tenE \mathbf P_{\bullet}(a,b;z)
\ \Longrightarrow\ \mathbf P_{\mathrm{cap},\bullet}(x,y;z),
\]

\[
\mathbf P_{\mathrm{cap},\bullet}(x,y;c)=\int^{a,b}R_{\bullet}(x;a)\tenE R_{\bullet}(y;b)\tenE \mathbf P_{\bullet}(a,b;c),
\ \ \bullet\in\{\mathrm{ex,in}\}.
\]

\[
(F\star_{\mathrm{cap},\bullet} G)(c)=\int^{x,y\in\A_{\mathrm{cap}}}\mathbf P_{\mathrm{cap},\bullet}(x,y;c)\tenE F(x)\tenE G(y).
\]

\dsymcolim(u,v):=\inf\Big\{ Q(w) \dsym(x,y) \Big|  x,y\in F(w,C),\ \kappa_{w,C}(x)=u,\ \kappa_{w,C}(y)=v\Big\}.

\[
\dsymcolim(\colim F,\colim G)\ \le\ d_X(F,G)\ \le\ D_X^{\mathrm{agg}}(F,G).
\]

\[
(\Lan_i F)(d)\ \cong\ \colim\!\big((i\!\downarrow\! d)\xrightarrow{\ \mathrm{pr}\ }\ D_{\le k}^\uparrow\xrightarrow{\,F\,}\E\big),
\]

(\Lan_i F)(d)\ \cong\ \colim \big((i \downarrow  d)\xrightarrow{\ \mathrm{pr}\ }\ D_{\le k}^\uparrow\xrightarrow{ F }\E\big),

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mo stretchy="false">(</mo><msub><mi>\Lan</mi><mi>i</mi></msub><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mtext> </mtext><mo>≅</mo><mtext> </mtext><mi>\colim</mi><mspace width="negativethinmathspace" /><mo minsize="1.2em" maxsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>i</mi><mspace width="negativethinmathspace" /><mo>↓</mo><mspace width="negativethinmathspace" /><mi>d</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mtext> </mtext><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi></mrow><mtext> </mtext></mrow><mspace depth=".25em" /></mpadded></mover><mtext> </mtext><msubsup><mi>D</mi><mrow><mo>≤</mo><mi>k</mi></mrow><mo>↑</mo></msubsup><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><mspace width="0.167em" /><mi>F</mi><mspace width="0.167em" /></mrow><mspace depth=".25em" /></mpadded></mover><mi>\E</mi><mo minsize="1.2em" maxsize="1.2em">)</mo><mo>,</mo><mi>\]</mi></mrow></math>

\[
\begin{aligned}
d_X(F,G)&\le L,\quad &d_X(F^{(k)},F)&\le L\,\bar q^{\,k+1},\\
D_X^{\mathrm{agg}}(F,G)&\le \frac{L}{1-\bar q},\quad &D_X^{\mathrm{agg}}(F^{(k)},F)&\le \frac{L}{1-\bar q}\,\bar q^{\,k+1}.
\end{aligned}
\]

\begin{aligned} d_X(F,G)&\le L,\quad &d_X(F^{(k)},F)&\le L \bar q^{ k+1},\\ D_X^{\mathrm{agg}}(F,G)&\le \frac{L}{1-\bar q},\quad &D_X^{\mathrm{agg}}(F^{(k)},F)&\le \frac{L}{1-\bar q} \bar q^{ k+1}. \end{aligned}

\[
\liminf_{n\to\infty}\frac{1}{n}\sum_{t=1}^n\big(\log(1+g_t)-|\log q(s_t)|\big)>0\
\Longleftrightarrow\
\mathbb E_\pi[\log(1+g_1)]+\mathbb E_\pi[\log q(s_1)]>0.
\]

\[
\Theta^\cap_{F,G}:\ (F\star_{\mathrm{cap,ex}} G)\ \Longrightarrow\ (F\star_{\mathrm{cap,in}} G),
\]

\[
\begin{tikzcd}[column sep=large]
\int^{a,b} \mathbf P_{\mathrm{ex}}(a,b;-)\tenE R_{\mathrm{ex}}(x;a)\tenE R_{\mathrm{ex}}(y;b)\tenE F(x)\tenE G(y)
\arrow[d, dashed, "\mu^R\ \&\ d"'] \\
\int^{a',b'} \mathbf P_{\mathrm{in}}(a',b';-)\tenE R_{\mathrm{in}}(x;a')\tenE R_{\mathrm{in}}(y;b')\tenE F(x)\tenE G(y)
\end{tikzcd}
\]

J_H(\pi):=\mathbb E_\pi \left[\sum_{t=1}^{H}\ (\beta\bar q)^{t-1}  g_t\right],

\[
\mathrm{Forecast}_k(X)=\sum_{n=0}^{k}\bar q^{\,n}\
\sup_{\substack{(w,C)\in D_X^\uparrow\\ |w|=n}}\ \mathbb{E}_\pi[g_{t(w,C)}],
\]

\mathrm{Forecast}_k(X)=\sum_{n=0}^{k}\bar q^{ n}\ \sup_{\substack{(w,C)\in D_X^\uparrow\\ |w|=n}}\ \mathbb{E}_\pi[g_{t(w,C)}],

0\le \mathrm{Forecast}_k(X)\ \le\ \frac{e^{G_{\max}}-1}{1-\bar q} (1-\bar q^{k+1}),\quad \mathrm{tail}\le \frac{e^{G_{\max}}-1}{1-\bar q} \bar q^{k+1}.

\[
S_{x,y}(c):=\mathbf P_{\mathrm{cap,ex}}(x,y;c)\tenE F'(x)\tenE G'(y),
\quad \text{with }F'\preceq_1 F,\ \ G'\preceq_1 G.
\]

\[
\widehat\Sigma_\varepsilon(a_{\mathrm{new}}):=\Big(\varepsilon+\inf_{S\in \mathcal S_{a_{\mathrm{new}}}}\dinfty(C_{a_{\mathrm{new}}},S)\Big)^{-1},\ \ \varepsilon>0.
\]

\[
\Delta J_H\ \gtrsim\ \frac{L_H}{\widehat\Sigma_\varepsilon(a_{\mathrm{new}})}\ -\ \text{(risk/penalty terms)},
\]

\[
\vartheta:\ \mathrm{Lan}_i(F_0)\ \Rightarrow\ \mathrm{Ran}_i(F_0),
\]

\vartheta:\ \mathrm{Lan}_i(F_0)\ \Rightarrow\ \mathrm{Ran}_i(F_0),

\[
\langle F,G\rangle:=\sup_{(w,C)}Q(w)\,\dsym(F(w,C),G(w,C)),\qquad
[F,G]:=\sum_{n\ge0}\bar q^n \sup_{\substack{(w,C)\\ |w|=n}}\dsym(F(w,C),G(w,C)).
\]

\langle F,G\rangle:=\sup_{(w,C)}Q(w) \dsym(F(w,C),G(w,C)),\qquad [F,G]:=\sum_{n\ge0}\bar q^n \sup_{\substack{(w,C)\\ |w|=n}}\dsym(F(w,C),G(w,C)).

d_X^{(\pi,\omega)}(F,G):=\sup_{(w,C)\in D_{X,\omega}^\uparrow} Q(w) \dsym(F(w,C),G(w,C)).

d_X^{(\pi,\omega)}\big(F^{(k)},F\big)\ \le\ C(\omega) e^{-\lambda (k+1)}\quad \text{a.s.}

\[
\mathcal I:\ \mathbb D\ \longrightarrow\ \mathbf{Pairs}(\E)\quad (F\mapsto (\langle F,-\rangle,[F,-])),
\]

\[
(F\star G)(c)\cong\coprod_{x',y'}\mathbf P_{\mathrm{cap,ex}}(x',y';c)\times F(x')\times G(y').
\]

\[
R_{\mathrm{ex}}=
\begin{pmatrix}
1&0\\
1&1
\end{pmatrix},\quad
\mathbf P_{\mathrm{ex}}(a,b;c)=
\begin{cases}
1&\text{if }(a,b,c)\in \{(\alpha,\alpha,x),(\alpha,\beta,y),(\beta,\beta,y)\}\\
0&\text{else.}
\end{cases}
\]

\[
(F\star G)(x)=1,\quad (F\star G)(y)=2\quad \text{for }F=Y_x,G=Y_y,
\]

(F\star G)(x)=1,\quad (F\star G)(y)=2\quad \text{for }F=Y_x,G=Y_y,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="bold">P</mi><mi mathvariant="bold">M</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">t</mi></mrow><mi>$</mi></mrow></math>

$\mathbf P_{\mathrm{cap},\bullet}:=R_{\bullet}^{\top}\odot \mathbf P_{\bullet}\odot R_{\bullet}$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo stretchy="false">(</mo><mi>i</mi><mspace width="negativethinmathspace" /><mo>↓</mo><mspace width="negativethinmathspace" /><mi>d</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$\mathbb E_\pi[\log(1+g_1)]+\mathbb E_\pi[\log q(s_1)]=0$

\mathbb E_\pi[\log(1+g_1)]+\mathbb E_\pi[\log q(s_1)]=0

\mathbb E_\pi[\log(1+g_1)]>-\mathbb E_\pi[|\log q(s_1)|]

$\Sigma(F,G\!\to H):=\inf_{S\in\mathcal S}\ \dinfty(H,S)$

\Sigma(F,G \to H):=\inf_{S\in\mathcal S}\ \dinfty(H,S)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="bold">C</mi><mi mathvariant="bold">h</mi><mi mathvariant="bold">u</mi></mrow><mo stretchy="false">(</mo><mi>\E</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$(u^\ast F)\star^{\A'} (u^\ast G)\simeq u^\ast(F\star^\A G)$

(u^\ast F)\star^{\A'} (u^\ast G)\simeq u^\ast(F\star^\A G)

$\mathbf P_{\mathrm{cap}}(x,y;c)\cong \bigoplus_{m=1}^M A_m(x)\tenE B_m(y)\tenE U_m(c)$

$d_X^{(\pi,\omega)}(F^{(k)},F)\le \sup_{\substack{|w|\ge k+1\\ w\ \text{prefix of }(s_t(\omega))}} Q(w)\cdot L \le C(\omega)e^{-\lambda(k+1)}$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="bold">P</mi><mi mathvariant="bold">a</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">r</mi><mi mathvariant="bold">s</mi></mrow><mo stretchy="false">(</mo><mi>\E</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mrow><mi mathvariant="bold">F</mi><mi mathvariant="bold">i</mi><mi mathvariant="bold">n</mi><mi mathvariant="bold">S</mi><mi mathvariant="bold">e</mi><mi mathvariant="bold">t</mi></mrow><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mo fence="false" stretchy="false">‖</mo><mi>X</mi><mo fence="false" stretchy="false">‖</mo><mo>=</mo><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>\dsym</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mo minsize="1.2em" maxsize="1.2em">|</mo><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mi>Y</mi><mo stretchy="false">|</mo><mo minsize="1.2em" maxsize="1.2em">|</mo><mi>$</mi></mrow></math>

\mathbf P_{\mathrm{cap,ex}}(x',y';c)=\sum_{a,b}R(x';a)R(y';b) \mathbf P_{\mathrm{ex}}(a,b;c)

$\bar s:=\sup_{n}\max_{|w|=n}\#\{\,w\cdot s\ \text{admissible next steps}\,\}$

\bar s:=\sup_{n}\max_{|w|=n}\#\{ w\cdot s\ \text{admissible next steps} \}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>O</mi><mo stretchy="false">(</mo><mi>B</mi><mspace width="0.167em" /><msup><mover><mi>s</mi><mo stretchy="true">¯</mo></mover><mrow><mspace width="0.167em" /><mi>k</mi></mrow></msup><mo stretchy="false">)</mo><mi>$</mi></mrow></math>


STRUCTURED FLOW ACROSS SCALES: A Pure-Theory Spine with Mixed-Order Evaluation

glyphtounicode.tex

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colorlinks=true,
linkcolor=blue!60!black,
citecolor=blue!60!black,
urlcolor=blue!60!black,
pdftitle= Structured Flow across Scales: A Pure-Theory Spine with Mixed-Order Evaluation,
pdfauthor= K. Takahashi ,
pdfsubject= Category Theory, Ind/Pro, Monoidal/Promonoidal Actions, Kan Extensions, Day Convolution, Rewriting, Stability ,
pdfkeywords= Fractal Category Theory, Dynamic FCT, monoidal action, promonoidal, Ind/Pro bicompletion, final functor, density, Day convolution, Kan extension, pathwise stability, Tamari lattice, dialectical research program

same
1.3 % line spacing

plain
theorem Theorem [section]
proposition[theorem] Proposition
corollary[theorem] Corollary
lemma[theorem] Lemma
definition
definition[theorem] Definition
assumption[theorem] Assumption
example[theorem] Example
remark[theorem] Remark
conjecture[theorem] Conjecture
program[theorem] Program
question[theorem] Question
warning[theorem] Warning

colim
Ob
Lan
End
Ind
Pro
C
D
Frac_ ( _0)
Id
op
% whiskered/horizontal composition
% "critical adjacency" marker for tables

TITLE: Structured Flow across Scales:\
A Pure-Theory Spine with Mixed-Order Evaluation

AUTHOR: K.~Takahashi

[[EQ:eq0001]]

[[EQ:eq0002]]

For a word [[EQ:eq0013]] , write [[EQ:eq0014]] and [[EQ:eq0015]] .

[Bracketing independence via comparisons] lem:bracket
For any word [[EQ:eq0016]] and parenthesizations [[EQ:eq0017]] , [[EQ:eq0018]] coincide by coherence; transporting (co)monad structure along either yields the same structure on [[EQ:eq0019]] .

[Transport preserves Frobenius]lem:frob-transport
For any [[EQ:eq0020]] , the transported quartet [[EQ:eq0021]] satisfies the Frobenius equalities

[[EQ:eq0003]]

Each composite is obtained from the corresponding Frobenius composite on [[EQ:eq0022]] by whiskering with [[EQ:eq0023]] and [[EQ:eq0024]] . The squares in Appendix~app:chi commute tautologically.

PARAGRAPH: Restricted Yoneda and lifts.

Let [[EQ:eq0025]] be restricted Yoneda (dense).
Define [[EQ:eq0026]] and lift levelwise to [[EQ:eq0027]] on [[EQ:eq0028]] .
Density determines unique strong monoidal comparisons [[EQ:eq0029]] extending [[EQ:eq0030]] .

[Preservation and seed stability]ass:seed
Each [[EQ:eq0031]] restricts to [[EQ:eq0032]] and preserves the filtered colimits and finite limits used below; [[EQ:eq0033]] preserves the levelwise small cofiltered limits in [[EQ:eq0034]] , i.e.\ those built from reindexings and equalizers.

[Weaker seed hypothesis]
It is enough that each composite [[EQ:eq0035]] extends along [[EQ:eq0036]]
and preserves the filtered colimits and finite limits actually used; [[EQ:eq0037]] is a sufficient but not necessary condition.

[Monoidal or promonoidal ambient for Day]ass:monoidalC
Assume [[EQ:eq0038]] carries a monoidal structure [[EQ:eq0039]] (or, more generally, a promonoidal kernel [[EQ:eq0040]] ). In the monoidal case the promultiplications are represented by hom-sets [[EQ:eq0041]] . For each [[EQ:eq0042]] , [[EQ:eq0043]] is at least oplax monoidal with coherent structure maps
[[EQ:eq0044]] and [[EQ:eq0045]] , natural in [[EQ:eq0046]] and compatible with [[EQ:eq0047]] . In the strong case these maps are isomorphisms.

[Units in [[EQ:eq0048]] vs.\ [[EQ:eq0049]] ]
We use the same letter [[EQ:eq0050]] for the unit of [[EQ:eq0051]] and of [[EQ:eq0052]] ; context disambiguates them. If needed we write [[EQ:eq0053]] and [[EQ:eq0054]] . Coherence identifies [[EQ:eq0055]] with unitors; all uses of [[EQ:eq0056]] are read modulo these identifications.

[Restricted Yoneda density]
By AdamekRosicky, the restricted Yoneda [[EQ:eq0057]] is dense under Assumptions~ass:lp and~ass:seed; hence left Kan extensions along [[EQ:eq0058]] are determined on seeds and computed pointwise. See also TakahashiFCT,TakahashiDFCT for explicit uses of density in the Lan-construction and in the forward presentation via [[EQ:eq0059]] .

SECTION: Indexing, rewriting, confluence, and finality

sec:rewriting
[Indexing category [[EQ:eq0060]] ]def:theta
Objects: pairs [[EQ:eq0061]] where [[EQ:eq0062]] is a parenthesized tensor-word in [[EQ:eq0063]] and [[EQ:eq0064]] .
Morphisms are generated by:
[leftmargin=1.2em]
- seed arrows [[EQ:eq0065]] for [[EQ:eq0066]] in [[EQ:eq0067]] ;
- [[EQ:eq0068]] and per-letter [[EQ:eq0069]] ;
- monoidal comparisons [[EQ:eq0070]] (and inverses), modulo coherence.

Relations: functoriality/naturality; monad/comonad axioms; Frobenius; and that [[EQ:eq0071]] is a (co)monad morphism via transport (Appendix~app:chi).

[Seeds on [[EQ:eq0072]] vs.\ on [[EQ:eq0073]] ]
At the [[EQ:eq0074]] -level we write [[EQ:eq0075]] . After restricted Yoneda we consistently use [[EQ:eq0076]] ; all rewriting/normalization statements are read modulo this identification.

[Skeletal fragments]def:frags
[[EQ:eq0077]] : wide subcategory generated by seeds, [[EQ:eq0078]] , and normalized [[EQ:eq0079]] . Here “normalized [[EQ:eq0080]] ” means the composite comparison sending any parenthesized word [[EQ:eq0081]] to its left-associated normal bracketing.
[[EQ:eq0082]] : forward fragment generated by seeds, [[EQ:eq0083]] , normalized [[EQ:eq0084]] (no [[EQ:eq0085]] ).

[Monadic skeletal fragment]def:monadicfrag
[[EQ:eq0086]] is the wide subcategory generated by seeds, [[EQ:eq0087]] , and normalized [[EQ:eq0088]] .

[Rewrite system (modulo coherence)]def:rewrite
Orient: (R1) [[EQ:eq0089]] left, (R2) [[EQ:eq0090]] left, (R3) [[EQ:eq0091]] right, (R4) [[EQ:eq0092]] right, (R5) [[EQ:eq0093]] toward left-associated bracketing (read [[EQ:eq0094]] modulo coherence). Let [[EQ:eq0095]] be the context-closed closure. Here “context-closed” means closed under horizontal/vertical composition, whiskering by (co)unit, by [[EQ:eq0096]] on [[EQ:eq0097]] (and, after restricted Yoneda, by [[EQ:eq0098]] ), and substitution into tensor words modulo coherence.

PARAGRAPH: Rule-wise monotonicity of the measure.

Fix the left-associated normal form for words. For a morphism, set the lexicographic measure

[[EQ:eq0004]]

(R1)--(R4) strictly decrease [[EQ:eq0099]] or [[EQ:eq0100]] and never increase [[EQ:eq0101]] ; (R5) strictly decreases [[EQ:eq0102]] and does not increase [[EQ:eq0103]] . Thus [[EQ:eq0104]] is well-founded.

[Tamari height]def:tamari-height
For a parenthesized word [[EQ:eq0105]] , define its Tamari height [[EQ:eq0106]] as the
graph distance (number of covering rotations) from [[EQ:eq0107]] to the left-associated normal
form in the Tamari lattice. Equivalently, one may take the number of right-nested
brackets; these definitions are monotone-equivalent along left rotations.

[Tamari decrease under (R5)]
A single left-associating rotation strictly decreases [[EQ:eq0108]] ; inserting/removing
[[EQ:eq0109]] leaves [[EQ:eq0110]] unchanged.

Tamari height is the rank in the Tamari lattice. A left rotation applies a covering relation,
hence decreases rank by [[EQ:eq0111]] . Units are handled modulo coherence and do not affect the bracketing class.

[Weighted termination measure]rem:weighted
Fix a subadditive weight [[EQ:eq0112]] . Then [[EQ:eq0113]] remains well-founded and strictly decreases along (R5) provided [[EQ:eq0114]] is nonincreasing under left-associating moves. Example: letting [[EQ:eq0115]] be the total number of right-nested brackets of [[EQ:eq0116]] , left-association does not increase [[EQ:eq0117]] .

PARAGRAPH: Finite critical-peak shapes.

Modulo coherence and left-associated words, overlaps occur only among [[EQ:eq0118]] in finitely many shapes. We record them in Table~tab:shapes; Appendix~app:shapes lists joinings.

[t]

1.1
ll

Shape & Joining reason \

[[EQ:eq0119]] adjacent & Unit/counit triangle \
[[EQ:eq0120]] ,\ [[EQ:eq0121]] & Unit laws \
[[EQ:eq0122]] ,\ [[EQ:eq0123]] & Frobenius squares \
[[EQ:eq0124]] ,\ [[EQ:eq0125]] & [[EQ:eq0126]] a (co)monad morphism (Appendix~app:chi) \
[[EQ:eq0127]] ,\ [[EQ:eq0128]] & Transport commutes with (co)multiplications \
[[EQ:eq0129]] & Coherence pentagon \& Tamari decrease \

Finite list of critical-peak shapes modulo coherence (rewriting triggers placed contiguously). Here, [[EQ:eq0130]] denotes adjacency of rewrite triggers (not horizontal composition).tab:shapes

Finiteness follows since (i) all interactions involve the finite generator set
[[EQ:eq0131]] modulo coherence, and (ii) each [[EQ:eq0132]] –move strictly
lowers the Tamari height toward the left-associated form, so no new bracketings are created.

[Local confluence via finite shapes]lem:lconf
Every critical peak listed in Table~tab:shapes is joinable by unit/counit triangles, Frobenius squares, naturality, and the transport squares of Appendix~app:chi. Hence [[EQ:eq0133]] is locally confluent.

[Confluence and normal forms]thm:CR
By Newman's lemma (termination + local confluence), [[EQ:eq0134]] is confluent. Every morphism admits a unique normal form in [[EQ:eq0135]] .

[Normalization yields a terminal object]lem:nf
For each object [[EQ:eq0136]] there is a canonical composite [[EQ:eq0137]] with [[EQ:eq0138]] skeletal, such that for any [[EQ:eq0139]] there exists a unique [[EQ:eq0140]] in [[EQ:eq0141]] with [[EQ:eq0142]] . Thus [[EQ:eq0143]] is terminal in [[EQ:eq0144]] .

[Finality of the skeletal inclusion]thm:final
The inclusion [[EQ:eq0145]] is final: [[EQ:eq0146]] has a terminal object represented by [[EQ:eq0147]] for each [[EQ:eq0148]] . Since each [[EQ:eq0149]] has a terminal object, its nerve is contractible; hence [[EQ:eq0150]] is final by Quillen’s Theorem~A.

[One-sided monadic spine]cor:onesided
If [[EQ:eq0151]] satisfy the orientation and the finite critical-shape joinings (with Frobenius clauses vacuous), then Theorems~thm:CR and~thm:final hold for [[EQ:eq0152]] .

[Smallness of the forward fragment]lem:small
If [[EQ:eq0153]] and [[EQ:eq0154]] are small, then [[EQ:eq0155]] is small. Objects are finite words in [[EQ:eq0156]] paired with [[EQ:eq0157]] ; hom-sets are generated by finitely many constructors and relations and remain small. (Closing under isomorphisms—repletion—does not affect smallness of hom-sets.)

[Filteredness of [[EQ:eq0158]] (expanded proof)]lem:filtered-expanded
For each [[EQ:eq0159]] , the standard forward presentation [[EQ:eq0160]] (``forward presentation'') into [[EQ:eq0161]] is filtered.

(i) Nonempty: for any [[EQ:eq0162]] , [[EQ:eq0163]] with [[EQ:eq0164]] gives an object.
(ii) Cocones: given [[EQ:eq0165]] and [[EQ:eq0166]] , pad both words by units [[EQ:eq0167]] to a common length, normalize left by [[EQ:eq0168]] , and use a cone in [[EQ:eq0169]] at the seed level; whiskering by [[EQ:eq0170]] yields a cocone in [[EQ:eq0171]] .
(iii) Equalizers: for a parallel pair, pad/normalize to [[EQ:eq0172]] , choose in [[EQ:eq0173]] a cone that coequalizes the induced seed-level pair (using finite limits in [[EQ:eq0174]] and the density of [[EQ:eq0175]] ); whisker by [[EQ:eq0176]] and apply naturality. Since filtered colimits in [[EQ:eq0177]] commute with finite limits and [[EQ:eq0178]] preserves the used limits by Assumption~ass:seed, we obtain an object receiving the equalizing arrow.
Pointwise bridge: [[EQ:eq0179]] embeds into [[EQ:eq0180]] , and both filtered colimits and finite limits are computed pointwise there; the required commutations reduce to those in [[EQ:eq0181]] .

[Seed-level equalization is preserved under whiskering]
Equalizing cones chosen in [[EQ:eq0182]] at seeds remain equalizing after whiskering by [[EQ:eq0183]] (and hence by [[EQ:eq0184]] after restricted Yoneda) by naturality; thus the induced pair in [[EQ:eq0185]] is equalized as claimed.

[Roadmap: CR [[EQ:eq0186]] terminal in [[EQ:eq0187]] [[EQ:eq0188]] finality]rem:roadmap
Termination+local confluence (Newman) yield unique normal forms in the skeletal fragment. Normalization maps [[EQ:eq0189]] are terminal in [[EQ:eq0190]] , making each comma category contractible; hence the inclusion is final in the sense of Quillen’s Theorem~A.

SECTION: Bicompletion (mixed: first filtered colimit, then small levelwise cofiltered limit)

sec:bicompletion

PARAGRAPH: Notation.

[[EQ:eq0191]] denotes the (restricted) Yoneda embedding;
[[EQ:eq0192]] is the constant pro-object embedding.

PARAGRAPH: Notation (continued).

We write [[EQ:eq0193]] for the canonical inclusion induced by [[EQ:eq0194]] and the closure operations in Definition~def:frac.

[ [[EQ:eq0195]] and [[EQ:eq0196]] ]def:IndPro-rho
[[EQ:eq0197]] is the smallest replete full subcategory of [[EQ:eq0198]] closed under images of [[EQ:eq0199]] and [[EQ:eq0200]] -filtered colimits with vertices in [[EQ:eq0201]] . Dually, [[EQ:eq0202]] is the smallest replete full subcategory closed under images of [[EQ:eq0203]] and levelwise small cofiltered limits (built from reindexings and equalizers).

Repletion closes under isomorphisms at the level of objects and does not affect smallness of hom-sets.

[Dynamic fractal bicompletion]def:frac
[[EQ:eq0204]] is the smallest replete full subcategory of [[EQ:eq0205]] containing [[EQ:eq0206]] and closed under: images of each [[EQ:eq0207]] ; filtered colimits of diagrams indexed by [[EQ:eq0208]] with vertices [[EQ:eq0209]] ; and levelwise small cofiltered limits. Evaluation order: filtered colimit first, then levelwise small cofiltered limit.

[Hom-calculus for constant sources]prop:hom
Let [[EQ:eq0210]] be a filtered colimit in [[EQ:eq0211]] with [[EQ:eq0212]] , and let
[[EQ:eq0213]] be a small levelwise cofiltered limit in [[EQ:eq0214]] .
Then

[[EQ:eq0005]]

Mapping out of a constant source turns the ``first filtered colimit, then small cofiltered limit'' evaluation into a double limit on Hom-sets. This avoids any Fubini-type interchange at the level of hom-sets; we still construct objects in the stated order.
Smallness caveat. Here [[EQ:eq0215]] is [[EQ:eq0216]] -small and levelwise limits in [[EQ:eq0217]] are computed by reindexings and equalizers.

[Mixed-order soundness for constants]ass:fubini
(i) Each [[EQ:eq0218]] preserves the filtered colimits and finite limits that occur in [[EQ:eq0219]] ; (ii) each [[EQ:eq0220]] is defined levelwise and preserves small cofiltered limits computed by reindexings and equalizers; (iii) any filtered colimit landing in [[EQ:eq0221]] is created by [[EQ:eq0222]] . Under these, the evaluation order in Definition~def:frac is sound.

[Small levelwise limits]
“Small” refers to limits indexed by [[EQ:eq0223]] -small cofiltered diagrams; all such levelwise limits in [[EQ:eq0224]] are computed by reindexings and equalizers.

[Intersection identification]prop:intersection
Under Assumptions~ass:lp, ass:seed, and~ass:fubini,

[[EQ:eq0006]]

as replete full subcategories of [[EQ:eq0225]] .

[Proof sketch]
[[EQ:eq0226]] By definition, [[EQ:eq0227]] is closed under the listed operations, hence contained in the replete intersection. [[EQ:eq0228]] Conversely, any object in the replete intersection is, by definition, obtained from [[EQ:eq0229]] by closing under images of [[EQ:eq0230]] , filtered colimits, and small levelwise cofiltered limits; thus it lies in [[EQ:eq0231]] by Definition~def:frac.

SECTION: [[EQ:eq0232]] -equivariant left Kan extensions and stability

sec:Lan
Let [[EQ:eq0233]] act on [[EQ:eq0234]] via strong monoidal endofunctors [[EQ:eq0235]] preserving the (co)limits used below.

[Enrichment interface]ass:enrich
[[EQ:eq0236]] is enriched over [[EQ:eq0237]] . The filtered colimits and levelwise small cofiltered limits we use are computed in the underlying category and are [[EQ:eq0238]] -Lipschitz in the sup-seminorm on diagram values.

[ [[EQ:eq0239]] -equivariant left Kan extension]def:fLan
Given [[EQ:eq0240]] and isomorphisms [[EQ:eq0241]] coherent with [[EQ:eq0242]] , an [[EQ:eq0243]] -equivariant [[EQ:eq0244]] of [[EQ:eq0245]] along [[EQ:eq0246]] is [[EQ:eq0247]] with [[EQ:eq0248]] and [[EQ:eq0249]] extending [[EQ:eq0250]] and universal with this property.

[Existence and skeletal computation]thm:fLan
Under Assumptions~ass:seed, ass:fubini, and~ass:enrich and preservation of the used (co)limits by [[EQ:eq0251]] , [[EQ:eq0252]] exists, is unique up to unique isomorphism, and is computed for [[EQ:eq0253]] by:
[leftmargin=1.4em,label=( *)]
- a filtered colimit over a forward presentation [[EQ:eq0254]] (``forward assembly''; final by Theorem~thm:final, filtered by Lemma~lem:filtered-expanded);
- followed by the levelwise small cofiltered limit over the backward stage indexed by counit maps (the “ [[EQ:eq0255]] -stage”) in [[EQ:eq0256]] .

Coherences need only be verified on generators [[EQ:eq0257]] and seeds, since these generate [[EQ:eq0258]] modulo the listed relations. (As in TakahashiFCT, the skeletal computation needs only [[EQ:eq0259]] (and [[EQ:eq0260]] here); [[EQ:eq0261]] are used to obtain finality but do not appear in the formulas.)

SUBSECTION: Stability: external attenuation (non-Kelly) and single-sheet bounds

External attenuation refers to evaluation-side weights (not enriched/Kelly weights). Assume each [[EQ:eq0262]] acts [[EQ:eq0263]] -contractively on [[EQ:eq0264]] with [[EQ:eq0265]] ; define [[EQ:eq0266]] .

[Minimal hypotheses for external attenuation]rem:min-atten
It suffices that [[EQ:eq0267]] admits suprema of bounded families and that the attenuation weights [[EQ:eq0268]] satisfy [[EQ:eq0269]] . Then the forward conical filtered colimit is [[EQ:eq0270]] -Lipschitz in the sup-seminorm, independently of any enriched (Kelly) weights; the subsequent levelwise small cofiltered limit is nonexpansive.

[Attenuated forward assembly is [[EQ:eq0271]] -Lipschitz]lem:lip
For forward diagrams [[EQ:eq0272]] ,

[[EQ:eq0007]]

[Proof sketch]
We endow [[EQ:eq0273]] with the usual quotient seminorm induced by the supremum over cocone legs.
For any cocone [[EQ:eq0274]] on [[EQ:eq0275]] , monotonicity of [[EQ:eq0276]] and submultiplicativity of [[EQ:eq0277]] give
[[EQ:eq0278]] for any cocone [[EQ:eq0279]] on [[EQ:eq0280]] ; taking the infimum over cocones yields the claim. Each layer contributes at most [[EQ:eq0281]] ; taking the supremum over the filtered index yields the bound. The subsequent levelwise cofiltered limit is nonexpansive.

[Deterministic stability and truncation]thm:det
If [[EQ:eq0282]] and [[EQ:eq0283]] , then for the [[EQ:eq0284]] -equivariant Kan extensions [[EQ:eq0285]] ,

[[EQ:eq0008]]

where [[EQ:eq0286]] truncates the forward stage to words of length [[EQ:eq0287]] .

[Stochastic attenuation]thm:birkhoff
For stationary ergodic schedules [[EQ:eq0288]] with [[EQ:eq0289]] and [[EQ:eq0290]] , there exist [[EQ:eq0291]] and [[EQ:eq0292]] with [[EQ:eq0293]] for [[EQ:eq0294]] almost surely. Hence [[EQ:eq0295]] a.s.

SECTION: Day convolution lifts with embedded fallback

sec:day
Assume Assumption~ass:monoidalC and that the coends defining Day convolution exist in [[EQ:eq0296]] . Let [[EQ:eq0297]] be Yoneda and [[EQ:eq0298]] .

[Day--Kan Fubini package (strong vs.\ (op)lax)]prop:day
Assume:
[label=( *),leftmargin=1.4em]
- (Smallness/Universe) the relevant presheaf category is formed in~ [[EQ:eq0299]] ;
- (Preservation) each [[EQ:eq0300]] preserves the small (co)limits entering Day coends;
- (Interchange) coends commute with the Kan extensions used in defining [[EQ:eq0301]] (i.e.\ [[EQ:eq0302]] past the Day coend).

Then [[EQ:eq0303]] is a strong monoidal action on [[EQ:eq0304]] , and [[EQ:eq0305]] is monoidal. Via density these restrict to [[EQ:eq0306]] and lift levelwise to [[EQ:eq0307]] under Assumption~ass:seed.\; If (ii) or (iii) fails, the same construction yields only an (op)lax monoidal action; all subsequent claims should be read in that mode.

[Sufficient conditions for strong monoidality]prop:day-strong
If [[EQ:eq0308]] is small (in [[EQ:eq0309]] ) and each [[EQ:eq0310]] preserves all small (co)limits that appear in the coend

[[EQ:eq0009]]

then the oplax comparison maps for [[EQ:eq0311]] are invertible; hence [[EQ:eq0312]] acts strong monoidally on [[EQ:eq0313]] .

. We write the Day convolution on presheaves as

[[EQ:eq0010]]

and the claimed preservation concerns the (co)limits forming this coend.

[Promonoidal kernel notation]
In the promonoidal case replace [[EQ:eq0314]] by a kernel [[EQ:eq0315]] and define
[[EQ:eq0316]] . All “preservation/interchange” hypotheses refer to the (co)limits forming this coend.

[Concrete preservation]
In [[EQ:eq0317]] -based presheaves, it suffices that each [[EQ:eq0318]] preserves the small coproducts
and coequalizers appearing in the coend presentation of Day convolution.

SECTION: Dialectical open problems and programs

sec:dialectic
Each item is posed as thesis/antithesis/possible synthesis with falsifiable milestones.

SUBSECTION: (O1) Beyond strict monoidality of scale

Thesis. Strong monoidal [[EQ:eq0319]] ensures coherent transport and normalization.\
Antithesis. Some systems feature non-associative or directionally-biased couplings.\
Program. Replace [[EQ:eq0320]] by a skew/duoidal/pseudomonoidal indexing and record defects as explicit 2-cells that strictly decrease a refined measure (extending Tamari height).

conj:skew
For a small skew-monoidal [[EQ:eq0321]] with coherent defect 2-cells that strictly decrease a well-founded measure, the normalization calculus remains terminating and locally confluent modulo defects; finality holds for the corresponding skeleton.

SUBSECTION: (O2) Creation of new primitives at long words

Thesis. The spine transports primitives specified on [[EQ:eq0322]] .\
Antithesis. New effective primitives may arise only at large words.\
Program. Admit reflective generator extension steps [[EQ:eq0323]] with unit [[EQ:eq0324]] , and extend the action along [[EQ:eq0325]] by left Kan extension on seeds.

conj:ext
If [[EQ:eq0326]] is accessible and preserves finite limits, then the extended spine over [[EQ:eq0327]] retains confluence and finality; mixed-order computation persists provided [[EQ:eq0328]] -created colimits land in constants and satisfy Assumption~ass:fubini.

SUBSECTION: (O3) Weak/averaged contractivity

Thesis. Deterministic bounds assume [[EQ:eq0329]] .\
Antithesis. In practice only averaged or intermittent contractivity may hold.\
Program. Replace uniform [[EQ:eq0330]] by subadditive growth rates [[EQ:eq0331]] ; use ergodic/subadditive arguments to obtain almost-sure rates.

conj:avg
If [[EQ:eq0332]] a.s., truncated assemblies converge almost surely with an exponential rate given by the Lyapunov exponent, even when [[EQ:eq0333]] .

SUBSECTION: (O4) Higher-dimensional rewriting when local confluence fails

Thesis. Local confluence holds for the base generators.\
Antithesis. With (O1)–(O2), residual critical pairs may persist.\
Program. Add explicit 3-cell fillers for unresolved overlaps and upgrade the calculus to a polygraphic/2-categorical completion; orient defect 3-cells by a decreasing measure to obtain higher-dimensional Newman analogues.

SUBSECTION: (O5) Day convolution beyond smallness/preservation

Thesis. Proposition~prop:day supplies strong monoidality on presheaves.\
Antithesis. Large [[EQ:eq0334]] or non-preserving [[EQ:eq0335]] occur frequently.\
Program. Use dense small subcategories and promonoidal kernels; approximate convolution via filtered colimits along dense inclusions.

conj:day
If [[EQ:eq0336]] is dense small and each [[EQ:eq0337]] preserves the [[EQ:eq0338]] -generated (co)limits, then the oplax comparisons on [[EQ:eq0339]] are invertible, and the induced action on [[EQ:eq0340]] is strong monoidal after repletion.

SUBSECTION: (O6) Size/Fubini constraints and accessible approximations

Thesis. Assumption~ass:fubini requires small levelwise limits.\
Antithesis. Realistic diagrams may force large cofiltered limits.\
Program. Approximate by accessible subdiagrams and show stability of the computed morphisms under directed refinement; use Proposition~prop:hom as the limiting comparison principle.

prog:acc
Construct a directed system of small cofiltered subdiagrams whose pro-limits [[EQ:eq0341]] converge to [[EQ:eq0342]] ; prove that [[EQ:eq0343]] stabilizes on hom-sets, giving an [[EQ:eq0344]] -approximation scheme for large limits.

SUBSECTION: (O7) Validation without fixed energies

Thesis. The spine is dynamics-agnostic.\
Antithesis. Without a target energy, empirical validation is unclear.\
Program. Separate transport invariants (skeletal independence, order-independence, [[EQ:eq0345]] -Lipschitz assembly) from dynamics-specific scores; require ablations that vary only transport while holding dynamics fixed.

SUBSECTION: (O8) Observation/actuation doubling at the spine

Introduce a comonadic ``observer'' [[EQ:eq0346]] and monadic ``actuator'' [[EQ:eq0347]] commuting with [[EQ:eq0348]] ; study whether [[EQ:eq0349]] -counits and [[EQ:eq0350]] -units shorten equalizers in Lemma~lem:filtered-expanded.

SUBSECTION: (O9) Averaged/partial convergence

Replace [[EQ:eq0351]] by [[EQ:eq0352]] ; target a theorem-level generalization of Theorem~thm:birkhoff via subadditive ergodic tools.

SUBSECTION: (O10) One-sided structures

Quantify how much of Sections~sec:rewriting--sec:Lan survives with only monads (or only comonads), keeping the confluence [[EQ:eq0353]] finality pipeline intact (see Corollary~cor:onesided and Definition~def:monadicfrag).

SUBSECTION: (O11) Anisotropic attenuation

Let [[EQ:eq0354]] be a weight; bound [[EQ:eq0355]] and combine with the Tamari component to obtain two-parameter truncation bounds (cf.\ Remark~rem:weighted).

SUBSECTION: (O12) Quasi-finality under laxity

When Proposition~prop:day downgrades to (op)lax, replace finality by a weak/contractible-comma notion guaranteeing skeletal computation up to coherent comparison 2-cells.

SECTION: Usefulness and reader-facing synthesis

sec:usefulness

PARAGRAPH: Define once, transport everywhere.

A single seed-level specification extends to all scales via [[EQ:eq0356]] ; no scale-specific axioms are required.

PARAGRAPH: Skeletal, presentation-independent computation.

Finality (Theorem~thm:final) reduces evaluation to a canonical skeleton; the mixed order is justified by Proposition~prop:hom and Assumption~ass:fubini.

PARAGRAPH: Agnostic dynamics with stability.

External attenuation yields deterministic and stochastic pathwise bounds (Theorems~thm:det and~thm:birkhoff) sufficient for truncation control yet independent of any specific energy.

SECTION: Threats to validity, limits, and falsifiability

sec:limits
[leftmargin=1.2em]
- Expressivity ceiling. If scale couplings resist monoidal-like modeling, transport coherence collapses (falsifies scope).
- Irreducible novelty at scale. If essential primitives cannot be reflected into a seed extension, the present transport cannot capture them; Conjecture~conj:ext targets this failure.
- Fubini/size failures. If mixed-order evaluation falls outside Assumption~ass:fubini, the Hom-calculus may not justify the order; Program~prog:acc proposes approximations.
- Refutation scenarios. Counterexamples to Lemma~lem:lconf or Theorem~thm:final refute skeletal computation; a violation of Lemma~lem:lip refutes universal [[EQ:eq0357]] -Lipschitz assembly.

SECTION: Consistency checklist and assumption ledger

sec:check
[leftmargin=1.2em]
- No circularity: Termination \& local confluence [[EQ:eq0358]] confluence (Theorem~thm:CR) [[EQ:eq0359]] normalization maps [[EQ:eq0360]] (Lemma~lem:nf) [[EQ:eq0361]] finality (Theorem~thm:final) [[EQ:eq0362]] skeletal Lan computation (Theorem~thm:fLan).
- Size: Small [[EQ:eq0363]] [[EQ:eq0364]] small [[EQ:eq0365]] (Lemma~lem:small); presheaves and Ind/Pro staged in [[EQ:eq0366]] (Assumption~ass:univ).
- Preservation: Assumption~ass:seed used only to lift actions and preserve used (co)limits (with a weaker variant available).
- Enrichment: Assumption~ass:enrich used only for stability interface; attenuation is external and non-Kelly (Remark~rem:min-atten).
- Day: Assumption~ass:monoidalC fixes the ambient monoidal/promonoidal data; Proposition~prop:day-strong gives strong monoidality; otherwise read (op)lax by Proposition~prop:day.

SECTION: Transport diagrams for [[EQ:eq0367]] and (co)monads

app:chi
Whiskering notation follows Street; horizontal composition is denoted by [[EQ:eq0368]] . By Lemma~lem:bracket, [[EQ:eq0369]] is a (co)monad morphism:

[[EQ:eq0011]]

and dually for [[EQ:eq0370]] .
The Frobenius square transports along [[EQ:eq0371]] by whiskering; explicitly:

[[EQ:eq0012]]

where the arrows are obtained by whiskering with [[EQ:eq0372]] and [[EQ:eq0373]] ; naturality makes the square commute. Together with triangles/Frobenius and naturality, these squares join the peaks of Table~tab:shapes.

SECTION: Joinability details for critical shapes

app:shapes
Typical joinings:
[leftmargin=1.2em]
- [[EQ:eq0374]] : unit/counit triangle; [[EQ:eq0375]] and [[EQ:eq0376]] : unit laws.
- [[EQ:eq0377]] , [[EQ:eq0378]] : Frobenius squares (stable under Lemma~lem:frob-transport).
- [[EQ:eq0379]] : commute via Appendix~app:chi.
- [[EQ:eq0380]] : coherence identifies composite comparisons; Tamari height strictly decreases along normalization.

SECTION: Counterexample sketch: when Day--Kan interchange fails

app:counter
Take a [[EQ:eq0381]] that fails to preserve a coend-defining colimit; the canonical map [[EQ:eq0382]] becomes only oplax (generally non-invertible). This exhibits the necessity of the preservation/interchange hypotheses in Proposition~prop:day.

SECTION: Acknowledgements

This spine consolidates and extends techniques used across TakahashiFCT,TakahashiDFCT,TakahashiObs,TakahashiNDQG,TakahashiNAE and rests on classical sources MacLane,StreetFormalMonads,StreetFrob04,AdamekRosicky,Day70,Kelly82,Glasman16,Birkhoff31,QuillenA.

99 0.75ex

AdamekRosicky
J.~Ad \'a mek and J.~Rosick \'y .
Locally Presentable and Accessible Categories.
Cambridge Univ.\ Press, 1994.

Birkhoff31
G.~D. Birkhoff.
Proof of the ergodic theorem.
Proc.\ Natl.\ Acad.\ Sci.\ USA 17 (1931), 656--660.

Day70
B.~Day.
On closed categories of functors.
Lecture Notes in Math. 137 (1970), 1--38.

Glasman16
S.~Glasman.
Day convolution for [[EQ:eq0383]] -categories.
Math.\ Res.\ Lett. 23(5) (2016), 1369--1385.

Kelly82
G.~M. Kelly.
Basic Concepts of Enriched Category Theory.
Cambridge Univ.\ Press, 1982; TAC Reprints 10 (2005).

MacLane
S.~Mac Lane.
Categories for the Working Mathematician (2nd ed.).
Springer, 1998.

StreetFormalMonads
R.~Street.
The formal theory of monads.
J.\ Pure Appl.\ Algebra 2 (1972), 149--168.

StreetFrob04
R.~Street.
Frobenius monads and pseudomonoids.
J.\ Math.\ Phys. 45(10) (2004), 3930--3948.

TakahashiFCT
K.~Takahashi.
Fractal Category Theory.
Zenodo (2025). https://doi.org/10.5281/zenodo.17292137.

TakahashiDFCT
K.~Takahashi.
Dynamic Fractal Category Theory.
Zenodo (2025). https://doi.org/10.5281/zenodo.17299070.

TakahashiObs
K.~Takahashi.
Observation as Coarse-Graining.
Zenodo (2025). https://doi.org/10.5281/zenodo.17274518.

TakahashiNDQG
K.~Takahashi.
Nondual Dynamical Quantum Geometry.
Zenodo (2025). https://doi.org/10.5281/zenodo.17268502.

TakahashiNAE
K.~Takahashi.
Nondual Autopoietic Excitations.
Zenodo (2025). https://doi.org/10.5281/zenodo.17254917.

QuillenA
D.~Quillen.
Higher algebraic K-theory I.
In Algebraic K-theory I, Lecture Notes in Math. 341, Springer, 1973.
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[[EQ:eq0367]]

[[EQ:eq0368]]

[[EQ:eq0369]]

[[EQ:eq0370]]

[[EQ:eq0371]]

S-equivariant left Kan extensions and stability

Dialectical open problems and programs

(O1) Beyond strict monoidality of scale

(O3) Weak/averaged contractivity

(O7) Validation without fixed energies

(O9) Averaged/partial convergence

(O10) One-sided structures

(O11) Anisotropic attenuation

(O12) Quasi-finality under laxity

Usefulness and reader-facing synthesis

Transport diagrams for χ and (co)monads

Joinability details for critical shapes

Counterexample sketch: when Day–Kan interchange fails

Acknowledgements

\[0.2em]
\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We develop a pure-theory \emph{spine} that couples a rigid categorical \emph{transport layer}---a strong monoidal 2-functorial action with a typed rewriting calculus and canonical normal forms---to a flexible \emph{variational/metric layer} for state evolution. Rewriting is oriented ($\eta,\delta$ left; $\varepsilon,\mu$ right; $\chi$ to left-associated bracketing, modulo coherence). A lexicographic termination measure includes Tamari height; local confluence follows from finitely many critical shapes. Newman's lemma gives confluence; normalization maps are terminal in comma categories, yielding finality of the skeletal inclusion. The bicompletion evaluates by a filtered colimit (forward presentation) followed by a levelwise small cofiltered limit (backward/counit stage). Mapping from constants turns this into a \emph{double limit} at the level of Hom-sets. For presheaves, we state sufficient conditions for \emph{strong} Day--Kan monoidality and provide an (op)lax fallback. Stability is derived via \emph{external} attenuation (non-Kelly): forward assemblies are $1$-Lipschitz; levelwise cofiltered limits are nonexpansive; deterministic and stochastic truncation bounds follow.
\end{abstract}

%==============================
\section{Reader map, conventions, and commitments}\label{sec:vision}
\paragraph{Audience.} Category theorists (final functors, Ind/Pro, Day convolution) and multi-scale theoreticians who need guarantees that \emph{transport} and \emph{dynamics} compose safely.

\paragraph{Commitments.} We \emph{separate} categorical transport from dynamics; assumptions are local; the dialectical program lists falsifiable milestones. Categorical claims never rely on stability claims.

\begin{remark}[Equality vs.\ isomorphism]\label{rem:eqvseq}
Displayed equations involving lifted/transported structures (e.g.\ $\widehat T_s,\widetilde T_s,T_{s,\ast}$ and comparisons $\chi$) denote identities of canonical \emph{isomorphisms} unless stated otherwise. We work \emph{modulo coherence}.
\end{remark}

%==============================
\section{Foundations: size, 2-categorical actions, and transport}\label{sec:found}
\begin{assumption}[Universes and size]\label{ass:univ}
Fix Grothendieck universes $\mathbb U\subset\mathbb V$. ``Small'' means $\in\mathbb U$. Presheaf categories and $\Ind/\Pro$ are formed in~$\mathbb V$.
\end{assumption}

\begin{assumption}[Accessible ambient and seed]\label{ass:lp}
$\C$ is locally $\kappa$-presentable and locally small; $\C_0\subseteq\C$ is a small full subcategory of $\kappa$-presentable objects, closed under finite limits and generating $\C$ under $\kappa$-filtered colimits.
\end{assumption}

\begin{assumption}[Strong monoidal 2-functor]\label{ass:2cat}
$S$ is a small monoidal category with associator/unitors $(\alpha,\lambda,r)$. Regard $\End(\C)$ as a strict 2-category. A \emph{strong monoidal 2-functor} $\varrho:S\to\End(\C)$ assigns endofunctors $T_s$, invertible comparisons $\chi_{s,t}:T_{s\otimes t}\Rightarrow T_sT_t$, $T_I\Rightarrow \Id$, coherent with $(\alpha,\lambda,r)$.
\end{assumption}

\begin{definition}[Frobenius decoration along comparisons]\label{def:frob}
For each $s\in S$ fix a monad $(\mu_s,\eta_s)$ and comonad $(\delta_s,\varepsilon_s)$ on $T_s$ satisfying Frobenius identities. We require \emph{only} that each comparison $\chi_{s,t}$ is a monad and comonad morphism \emph{via transport} (Appendix~\ref{app:chi}); no global restriction on all 1-cells of $\mathrm{FrobEnd}(\C)$ is imposed.
\end{definition}

\paragraph{Transport, lifting, and bracketing independence.}
For $(s,t)$, transport (co)monad structures along $\chi_{s,t}$:
\[
\mu^\chi_{s,t}=\chi_{s,t}\circ\mu_{s\otimes t}\circ(\chi_{s,t}^{-1}\hcomp\chi_{s,t}^{-1}),\quad
\eta^\chi_{s,t}=\chi_{s,t}\circ\eta_{s\otimes t},
\]

\[
\delta^\chi_{s,t}=(\chi_{s,t}\hcomp\chi_{s,t})\circ \delta_{s\otimes t}\circ\chi_{s,t}^{-1},\quad
\varepsilon^\chi_{s,t}=\varepsilon_{s\otimes t}\circ\chi_{s,t}^{-1}.
\]

\[
(T_sT_t\,\mu^\chi_{s,t})\circ(\delta^\chi_{s,t}\,T_sT_t)
=\delta^\chi_{s,t}\circ\mu^\chi_{s,t}
=(\mu^\chi_{s,t}\,T_sT_t)\circ(T_sT_t\,\delta^\chi_{s,t}).
\]

(T_sT_t \mu^\chi_{s,t})\circ(\delta^\chi_{s,t} T_sT_t) =\delta^\chi_{s,t}\circ\mu^\chi_{s,t} =(\mu^\chi_{s,t} T_sT_t)\circ(T_sT_t \delta^\chi_{s,t}).

\[
\begin{aligned}
\mathbf m&=(m_1,m_2,m_3)\in\mathbb N^3,\\
m_1&=\#\{\varepsilon,\mu\ \text{pending to be pushed right}\},\\
m_2&=\#\{\eta,\delta\ \text{displaced from the far left}\},\\
m_3&=\text{Tamari height}.
\end{aligned}
\]

\[
\Frac\ \simeq\ \big(\Ind_\varrho(\C_0)\cap \Pro_\varrho(\C_0)\big)^{\mathrm{repl}}
\]

\[
d_\D\!\left(\operatorname{colim} F,\ \operatorname{colim} G\right)\ \le\ \sup_{(w,C)} Q(w)\,d_\D\big(F(w,C),G(w,C)\big).
\]

d_\D \left(\operatorname{colim} F,\ \operatorname{colim} G\right)\ \le\ \sup_{(w,C)} Q(w) d_\D\big(F(w,C),G(w,C)\big).

\[
(F\star G)(c)=\int^{a,b}\ \C(a\otimes b,c)\times F(a)\times G(b),
\]

\[
(F\star G)(c)\ :=\ \int^{a,b}\ \C(a\otimes b,c)\times F(a)\times G(b),
\]

\[
\begin{tikzcd}[column sep=huge,row sep=large]
(T_sT_t)^2 \ar[r,"{\chi^{-1}\ \hcomp\ \chi^{-1}}"] \ar[d,"{\mu^\chi_{s,t}}"'] &
T_{s\otimes t}^2 \ar[d,"{\mu_{s\otimes t}}"] \\
T_sT_t \ar[r,"{\chi^{-1}}"'] & T_{s\otimes t}
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=huge,row sep=large]
\Id \ar[r,"{\eta_{s\otimes t}}"] \ar[d,"{\eta^\chi_{s,t}}"'] &
T_{s\otimes t} \ar[d,"{\chi}"] \\
T_sT_t \ar[r,"{\chi^{-1}}"'] & T_{s\otimes t}
\end{tikzcd}
\]

\[
\begin{tikzcd}[column sep=huge,row sep=large]
T_{s\otimes t}^2 \ar[r,"\delta_{s\otimes t}"] \ar[d,"\mu_{s\otimes t}"']
& T_{s\otimes t}^2 \ar[d,"\mu_{s\otimes t}"] \\
T_{s\otimes t} \ar[r,"\delta_{s\otimes t}"']
& T_{s\otimes t}
\end{tikzcd}
\quad\Longrightarrow\quad
\begin{tikzcd}[column sep=huge,row sep=large]
(T_sT_t)^2 \ar[r,"\delta^{\chi}_{s,t}"] \ar[d,"\mu^{\chi}_{s,t}"']
& (T_sT_t)^2 \ar[d,"\mu^{\chi}_{s,t}"] \\
T_sT_t \ar[r,"\delta^{\chi}_{s,t}"']
& T_sT_t
\end{tikzcd}
\]

$\widehat T_w:=\widehat T_{s_1}\cdots\widehat T_{s_n}$

$\widetilde T_w:=\widetilde T_{s_1}\cdots\widetilde T_{s_n}$

$\chi_{\mathbf p},\chi_{\mathbf p'}:T_w\Rightarrow T_{s_1}\cdots T_{s_n}$

$m^s_{a,b}:T_s(a\otimes b)\to T_s a\otimes T_s b$

$(\bar w,\bar C)\Rightarrow(\bar w',\bar C')$

$\alpha_s:F_0\circ T_s|_{\C_0}\Rightarrow S_s\circ F_0$

$\bar\alpha_s:F\circ\widetilde T_s\Rightarrow S_s\circ F$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>Q</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="0.167em" /><mi>d</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>G</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

$y\circ T_s\Rightarrow T_{s,\ast}\circ y$

$(F\star G)(c):=\int^{a,b}\mathbf P(a,b;c)\times F(a)\times G(b)$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msub><mo>sup</mo><mi>s</mi></msub><mi>q</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="negativethinmathspace" /><mo><</mo><mspace width="negativethinmathspace" /><mn>1</mn><mi>$</mi></mrow></math>

$\limsup_n \frac1n\log Q(s_1\cdots s_n)<0$