Clipped cones: the missing activation interface

10 October 2026. Assessment 206. Sustained construction attempt with an exact failed-route account. Native theorem 67 and mathematical counters are unchanged.

Later development: review 207 handles the long-stem sibling-pair diagnostic with learned shared contexts. The complete clipped-cone activation problem remains open.

Goal and outcome

The target is a persistent representation of selected, complete prefix cones, each restricted to one depth interval, with exact positive and negative answers by every query deadline. Such a construction would replace theorem 67’s fixed path count by unbounded content diversity under a structural row restriction.

The direct extension did not close. Depth-profile budgets do not pay for long content routes, and moving all work into the query creates an unbounded batch at a newly encountered prefix. A correlated interval-event formulation does preserve the right answers and removes any need to delete expired intervals. Its remaining operations are now explicit: current selected-profile access at a content prefix, future activation scheduling, and upper-endpoint aggregation. None has been granted unit cost.

This is a useful failure of two implementation routes and a precise next interface, not a native lower bound or a new native theorem. Review 205 supplies the preceding primary-proof comparison; theorem 67 was the native construction available at this assessment.

Exact target and resource boundary

A nonempty row has parameters $q_v,a_v,b_v$ with $|q_v|+1\le a_v\le b_v$ and literal target set

$$S_v=\{w:q_v\preceq w_{<|w|},\ a_v\le |w|\le b_v\}.$$

Here $w_{<|w|}$ is the parent obtained by deleting the final bit. The row contains every such word, not a supplied symbolic cone descriptor. Its prefix, endpoints and completeness must be learned and checked from the ordinary right block. Duplicates and interleaving remain part of the intended syntax. Empty rows need an explicit empty case.

The maintained exact length is

$$N=\sum_{(v,w)\text{ records}}(9+|v|+|w|) +\sum_{(u,v)\text{ records}}(10+|u|+|v|)+11+|a|+|b|.$$

The endpoint pin names $a,b$ in this syntax formula are distinct notation from row interval endpoints $a_v,b_v$. Actual symbols must pay initialization, preparation, allocation, old-storage observations, pointer/name routing, deferred work and the final output. Fix degree, radius, finite labels/control and old-top updates for one machine; allow no supplied decomposition, ready handles, advice, padding or EOF service. Any future native result must state its input orientation and only infer a PEG for the reversal of its recognized language.

The failures below remain valid even if a completed, correctly checked cone descriptor is provisionally granted. Granting it isolates the consumer defect; it does not discharge the ordinary-input producer obligation.

Attempt 1: import the depth-band worker

The charged band criterion permits each newly selected row to increase truncated work through band $k$ by at most $ck$, for a machine constant $c$. This charge includes disturbance of old pending work. A query’s length scale is exponentially larger than its band index.

Take a prefix $q$ of length $L$ and endpoints $a_v=b_v=L+1$. The complete clipped cone is only the sibling pair $\{q0,q1\}$. Its singleton depth profile passes the old mixed budget with parameter two, as proved in review 205. But a consumer that walks $q$ from the global root before making that band’s answer available performs at least $L$ edge steps. Its relevant band is

$$k=\lfloor\log_2(L+1)\rfloor+1.$$

The ratio $L/k$ is unbounded. Thus that dispatch operation does not meet the inherited release contract. This is not repaired by checking cone completeness or by the singleton depth budget.

The problem survives the obvious empty-union shortcut. Let $q_0=0^L$ and $q_1=0^{L-1}1$, both with depth interval $[L+1,L+1]$. After adopting the first row’s root, an eager ordinary-trie OR that descends matching coordinates traverses their $L-1$ common unary-prefix coordinates to reach the fork. The depth mixed count is logarithmic; the descended content coordinates are linear. Sharing or precomputing the fork could avoid this traversal, but then constructing and accessing that compressed operation is a different contract requiring its own charge.

The whole post-selection budget can fail

This is not merely a constant-factor mismatch in the inherited proof. Fix $h=L=2^d$. Use distinct $d$-bit suffixes $c_i$ and parents $q_i=0^Lc_i$, with interval $[L+d+1,L+d+1]$. There are only two literal target records per row. Select all $h$ rows using distinct $d$-bit intermediate names and a fixed source name, then supply a target pin of length $L+d+1$.

A dispatcher required to walk the common stem separately for every released job before sealing the relevant global band owes at least $hL=L^2$ steps. The actual left records and pin provide only $O(h(d+1)+L+d)=O(L\log L)$ subsequent symbols. No fixed service rate can discharge those obligations for all $L$. The right preparation block is real input of length $\Theta(L^2)$; it is neither padding nor free preparation. The failed dispatcher defines these jobs to be released by the later selections, so it cannot bank unused past time to execute them. A precomputed selection-sensitive representation that avoids those jobs is outside this failed route.

This rules out the stated global-sealing dispatcher, not the language. A query-directed reader, compressed shared-stem operation, or separately proved selection aggregate might escape it. The inherited depth-only coupling cannot simply be asserted for content trees.

Attempt 2: activate intervals along the query path

Let $A$ be the frozen set of selected rows for the queried source. At target position $n$, only rows whose $q_v$ matches the observed parent prefix are relevant. Define a conceptual event process:

1. Release row $v$ at time $r_v=|q_v|+1$, exactly when the observed parent is $q_v$.
2. Schedule its activation at time $a_v\ge r_v$.
3. After releases and all activations due at time $n$, let $B_n$ be the maximum upper endpoint of every activated row, with default zero.
4. Output membership if and only if $n\le B_n$.

Exact semantic invariant. For every target prefix of length $n$,

$$B_n=\max\bigl(\{b_v:v\in A,\ q_v\preceq w_{<n},\ a_v\le n\}\cup\{0\}\bigr).$$

A row’s prefix either matches when it is released or never matches that query. Later query bits cannot undo a matched prefix. Since $a_v\ge r_v$, no activation is needed before release. Induction over query positions therefore gives the displayed set of activated rows. Membership in their union is equivalent to at least one $b_v\ge n$, which is exactly the maximum test.

Expired rows need no deletion: if the retained maximum is below $n$, every activated row has expired; if it is at least $n$, its maximizing row is still a witness. This is why the formulation preserves prefix/depth correlation without a fully dynamic expiry structure. It is a standard maximum invariant applied to this target, not a novelty claim or a native implementation.

The order within a query step matters. For $q=0$ and $a_v=b_v=2$, the query 00 must release and activate the row before answering position two. A one-symbol activation delay is already wrong at the final symbol. Conversely, for $q=0$ and $a_v=b_v=4$, using its upper endpoint immediately at release falsely accepts 00 and 000. Lower endpoints cannot be discarded.

Why the event formulation is not yet a native construction

Unbounded releases at one prefix. Arbitrarily many selected rows can have the same $q$. The event description’s loop over those rows is not bounded local work. This issue exists even with short $q$, so a long query cannot always fund it.

For a precise failed interface, suppose the query receives only a list of independent selected-row interval records at that prefix, earlier selection work only appended their handles, and there is no aggregate or other observation encoding their endpoints. Choose $q=0$ and query 00. With $h$ records all equal to $[3,3]$, the answer is negative. Changing any one unread record to $[2,2]$ changes it to positive. An exact deterministic reader of this raw-list interface must inspect all $h$ records on the negative instance. The two-symbol target supplies only fixed service, while $h$ is unrestricted. Identity, addresses, list metadata, timing and prior control cannot encode the hidden endpoint bit under this contract.

This is an observation lower bound for that deliberately restricted interface. It does not rule out aggregation during the paid selection-name input. Indeed the earlier radial constructions are a positive control showing why that possibility must remain open. Replacing the list by one union-profile handle also does not by itself solve access: the profile must be the current selected union for this source and prefix, and its pending work must obey the query deadline.

Future activation and maxima are unpaid. A calendar keyed by integer $a_v$, an arithmetic comparison, or a heap with logarithmic operation cost is not a fixed local native operation. A unary delay queue needs a way to route events to future positions without traversing every delay at release. Keeping a largest upper endpoint also requires a paid comparison/representation; the symbol $\max$ supplies no primitive. Due events may arrive with zero slack and in unbounded batches.

The producer remains open. Sibling closure alone does not verify a complete cone with a gap-free depth interval. The target’s ordinary-input classifier, final completeness test and exported representation are still required. The failed consumer routes do not become valid if this producer is solved first.

Positive control: a paid synchronous alias

The missing access operation is not an obstruction for every alias pattern. Fix a binary finite-state transducer $f$ that emits exactly one output bit for each name bit. Consider the checked sibling-pair class in which each nonempty row $v$ is exactly $\{f(v)0,f(v)1\}$. The maintained native capsule dictionaries give a direct positive control with unbounded prefix diversity. This is a specialization of existing primitives, not a proposed solution for arbitrary aliases or an additional numbered theorem.

During a right record, reset the finite transducer state at the start of $v$, traverse its row dictionary, and append the emitted bits to a strict FIFO. During the target, keep its latest bit buffered and compare every previous bit with one FIFO bit. At the delimiter require a nonempty target, no mismatch and an exhausted FIFO. The buffered bit selects one of two row-presence flags. A global half-pair count changes only when a previously missing member is inserted; duplicates do not change it. Its zero test at the right boundary checks sibling closure. Every comparison and allocation is paid by actual name or target bits, including mismatches and the final length test.

During a selection $(u,v)$, the actual $u$ bits locate and suspend the current source capsule. Each actual $v$ bit advances three fixed components: the frozen right-row lookup, the reset transducer, and the current source’s selected-key trie along the emitted $f(v)$ bit. At the delimiter OR the right row’s completed flag into that terminal, then commit the inner and outer capsules in a fixed simultaneous-record batch. A missing right row contributes false. Different names may map to the same key, so OR rather than overwrite is essential.

The target query traverses that selected-key trie using a one-bit buffer: after reading position $n$ it has consumed exactly the parent of the current target prefix. Read that node’s terminal flag for membership and complement at the final syntax delimiter for pinned SAT. An absent child enters a finite false state. There is no post-name key traversal, endpoint arithmetic or post-query drain.

For fixed $f$ all parser states, records, concurrent views and microsteps form fixed batches over the inherited fixed-degree/radius, old-top dictionaries and queues. Initialization is empty state. All old observations, retained versions, routing and allocation are included in $O_f(N)$ logical work/storage; query forks pay their own traversal. The standard simultaneous-record packing supplies the native machine. The language statement keeps right-first syntax and the PEG reversal qualification. No free change of names is inferred: an arbitrary input-specified alias table need not emit its target key as its source name arrives.

This control identifies what is missing in the general diagnostic: synchronized access to the current selected-key version, not merely static knowledge of the row’s prefix. It also explains why the earlier raw-list failure is not universal. Ordinary selection input can perform useful prior aggregation when its routing is paid.

Precise handoff and next decisive test

The next useful contract has two coupled parts. Selection input must maintain source-specific profiles indexed by content prefixes, with bounded releases and an explicit disturbance bound. A query must obtain the relevant current profiles as its own bits reach those prefixes and schedule their lower endpoints while retaining upper-endpoint witnesses. Returning a raw list or a pointer whose contents require an unpaid traversal fails the contract.

First isolate the one-row-per-prefix sibling-pair subcase, explicitly $a_v=b_v=|q_v|+1$. It removes the same-prefix release burst and descendant-level completeness obligation, while retaining arbitrarily many prefixes, long shared stems, ordinary intermediate-name aliases, persistence and zero-slack activation. Sibling closure still must be checked. Future activation is absent in this first diagnostic and remains a separate obligation for the full cone class. Test both the common-stem family above and $a_v=|q_v|+1$. A successful implementation must charge current selected-state access at the encountered prefix, rather than assume the static row handle reveals the state of a later persistent selection version. If this subcase can be compiled, extend its release object to the existing paid interval-union profile before claiming the full cone class.

The full clipped-cone objective remains active. This diagnostic subcase is not a replacement theorem statement or a newly imposed gate on theorem 67. Unrestricted radial profiles and independent-store joins remain distinct open problems.

Evidence and integration

The primary audit compares event answers with independently expanded literal cone sets over every selection of eight rows and every binary word through length five. It preserves semantic snapshots and checks zero-slack activation, premature activation and independent raw-record answer flips. It additionally checks 172,032 synchronous-alias queries for identity, running-parity and colliding all-zero transducers, including missing right rows. The independent audit exhausts all triples of 42 cones through depth four: 1,125,040 query-prefix checks, including 582,648 negative answers. Both passed. Their integer calendars, maxima, strings and sets are explicitly semantic oracles, not native operations or runtime measurements.

Independent adversarial review identified the long-prefix release mismatch, the two-cone merge witness, and the event invariant; it also checked the synchronous-alias dictionary control. The owner’s separate audit uses literal expansion rather than the invariant as its answer oracle. General arguments carry the failed-contract and semantic statements; finite counts only corroborate them.

Replay forks must fund their own query work. Affine-fragment joins and unary coherent cubical algebra supply no independent-store or growing-branch closure. Persistent causal algebra preserves versions but does not make a static row pointer observe a later selection version. Native SAT controls retain ordinary late pins and complement output. Entry 129’s imported growing-alphabet state/port results remain conditional representation bounds, not native-language lower bounds.

Scheduled 110 to 111 once for the completed unsuccessful assessment. Mathematical 130, window two and reassessment 128 unchanged. Next 112 research, editorial 114, primary-proof review 115. Browser viewport inspection remains unavailable under the supported workflow. Installed reverse-writer retirement remains separate and unverified.