Research: adaptive QQ16 reads still need 66 ports; 45 optimal sets

research/qq16-adaptive-coordinate-reads/README.md

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+# QQ16: adaptive coordinate reads still need 66 ports
+
+**Scoped proof and exact finite premises; no supplier or new κ.**
+On the rational hidden-input family of
+[PR2354](https://github.com/CrocSwap/integer-mult-bounds/pull/2354), every
+deterministic reader that always recovers the original selected A defect
+needs **at least 66 distinct source-port reads on every input**, even when
+its port choices and stopping time depend arbitrarily on earlier reads.
+The same statement holds almost surely for a randomized reader that
+terminates and is exact with probability one for each rational input.
+
+There are **exactly 45 possible read sets at a successful 66-read leaf**.
+All contain these 22 ports:
+
+```text
+1,2,3,4,5,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109.
+```
+
+The other 44 reads choose all but one of 45 explicitly listed ports.
+PR2354's attaining set is one choice. Port 77 can replace any of 44
+specified members of that set. The sets and the exact recovery relation
+are in [RESULT.json](RESULT.json), fields `eligible_optimal_ports`,
+`optimal_omissions`, and `coordinate_relation`.
+
+The fixed-set threshold and underlying rank-66 family are credited
+predecessor results. The new delta is the adaptive/zero-error implication,
+the complete classification of optimal terminal read sets, and explicit
+family decoders for all 45 sets. Exact rechecking of the rank is a premise
+check, not a claim to have discovered that rank independently. The claim
+is in [discussion #2205](https://github.com/CrocSwap/integer-mult-bounds/discussions/2205#discussioncomment-18942816).
+
+## Exact observation contract
+
+Use the actual 8-by-112 rational centre matrix R, selected sets A and T,
+and ordinary 96-port set U from the retained
+[QQ16 packet](../joint-query/README.md). Let r_p be column p of R and
+b_p the first local chart axis of port p. For an owner d in the finite
+regular cover group G, put
+
+```text
+f_b(d) = bit_0(d b), interpreted in Q,
+V = ker(R_U) subset Q^96,
+h_p(d) = c_p f_(b_p)(d),       c in V,
+H(c)(d) = sum_(p in U) r_p c_p f_(b_p)(d).
+```
+
+The fields are constant in payload coordinates. This is the exact
+first-axis family of PR2093/PR2326/PR2354. The actual routes obey
+`rho_s(p)b_p=e_(9+8(s-1))`, so `B_s h=0` for every s=1,...,8 and every
+owner whenever `R_U c=0`. The original selected input is
+`X_A=-P_(0,A)H(c)`, with `Y_A=-X_A`, `X_U=h`, `Y_U=0` and all other
+fields zero. The selected section is invertible: the packet has
+`R_A R_A^T=4I`. Recovering this original A defect is equivalent to
+recovering H(c).
+
+Under the credited ordered F0/B1 maps, F0 leaves `X_U=Y_U=h` and all
+selected fields zero. B1 leaves only `Y_U=h`. The retained ordinary tail
+has zero selected/completed-X/history observations and finally places h
+in the former-Y endpoint, which is outside the permitted observation
+contract. This inherited family argument is explained in this account's
+[PR2326](https://github.com/CrocSwap/integer-mult-bounds/pull/2326) and
+PR2354; this package rechecks the route and rank premises but does not
+replay their full native schedule.
+
+Here a reader can request any **complete initial ordinary array** h_p,
+one port at a time, without modifying it. It can retain all responses,
+perform arbitrary computations, choose the next port adaptively, and
+stop based on its transcript. The unmodified QQ16 observations above may
+be supplied for free; they are zero throughout this family. Repeated
+reads count only once. No other input-dependent information, header,
+dense query, source preprocessing, or changed physical call is available.
+In particular, this theorem does not assume that a complete dense
+`B0 h` array has one scalar's information.
+
+For each p the function f_(b_p) is nonzero at a certified actual owner.
+Thus h_p is equivalent to the one rational scalar c_p: the complete
+array is determined by c_p, and reading a nonzero entry recovers it.
+The oracle model grants that scalar for free. It is a favorable
+information relaxation, with no charge for native access or retention.
+The claim concerns this side-effect-free reader, not arbitrary physical
+adaptive interventions inside the QQ16 word.
+
+## Transcript-fiber theorem
+
+Let V be a finite-dimensional vector space over a field F. Let H be a
+linear target and let q_1,...,q_N be scalar linear query functionals on V.
+An exact deterministic oracle algorithm receives no input-dependent
+information except answers to adaptively selected q_i. It terminates
+with H(c) for every c in V. Its internal computations and output map
+need not be linear, continuous, or bounded in cost.
+
+Fix any input c and its finite terminal transcript, and let S be the set
+of queried indices. Write `Q_S=(q_i)_(i in S)` and `K_S=ker(Q_S|V)`.
+For every v in K_S, the run on c+v has exactly the same transcript.
+Indeed, its first query is the same; if the histories agree through a
+step, its next query or stopping decision is the same. Every query on
+the fixed path lies in S and has the same answer on c and c+v. Induction
+reaches the same leaf and output.
+
+Exactness on both inputs therefore gives
+
+```text
+H(K_S)=0,      ker(Q_S|V) subset ker(H|V).                 (1)
+```
+
+Consequently H factors through Q_S and
+
+```text
+|S| >= rank(Q_S|V) >= rank(H|V).                          (2)
+```
+
+This holds for **every terminal path**, not only the zero-input branch
+or the worst case. Conversely, a fixed S satisfying (1) admits a linear
+decoder for H; define it on Q_S(V) using H(c), which is well-defined by
+(1). Adaptive choice cannot manufacture information about a direction
+in the terminal query kernel.
+
+For QQ16 the checked rank is 66, so (2) excludes every path with fewer
+than 66 distinct port reads. The original fixed 66-port set attains
+family decoding, proving a sharp threshold in this oracle contract.
+
+### Zero-error randomization
+
+Assume one input-independent probability space of random seeds. For
+each c in the **countable** rational space V, the reader terminates and
+returns H(c) with probability one. Intersect these full-measure events
+over all c in V. Countability gives a common full-measure set of seeds
+for which the deterministic seeded reader is total and exact on all V.
+The preceding theorem applies to each such seed and every input.
+Thus at least 66 distinct reads are required almost surely, and the
+expected count on each input is at least 66.
+
+The statement includes the optimal-set classification below on every
+66-read path for these seeds. It makes no bounded-error assertion and
+does not silently interchange an uncountable family of probability-one
+events. Input-dependent advice or side channels violate the contract.
+The all-input assertion is for V over Q; it does not claim the same
+pathwise minimum on an arbitrary finite, bounded coefficient promise.
+
+## Source-bound finite premises
+
+Group the ports by their first axes. If C_b is one group, let
+`L_b(c)=sum_(p in C_b) r_p c_p`. H factors through the direct sum L of
+these group maps. The actual source has 47 groups. The axis-2 group C
+has 30 columns and rank eight; all other groups have independent
+columns, 66 columns in total. Hence `rank L=74`.
+
+Vectors supported on C with `R_C c_C=0` form a 22-dimensional subspace
+K of V and have H(c)=f_2 R_C c_C=0. Also `dim V=96-8=88`. Therefore
+`rank(H|V)<=88-22=66` for the **whole owner array**.
+
+For a matching lower bound, the verifier constructs an integer basis
+N of V and evaluates H on the 17 actual owners used by PR2354:
+identity and `rho_1(p)` for p=0,...,15. The resulting 136-by-88 matrix
+has a 66-by-66 integer minor with determinant
+
+```text
+174085318024506601157689344.
+```
+
+The receipt includes the minor, its original row and basis-column
+indices, and a digest of the reproducibly constructed basis N. Exact
+rational elimination selects it; fraction-free integer elimination
+checks the determinant with every division checked. This is a direct
+certificate on V, complementary to the predecessor's rank-74 certificate
+on the whole coefficient space. A nonzero evaluation minor proves a
+lower bound for the complete owner map; the matching upper bound is the
+all-owner factorization above. No independence of formal bit functions
+or sampled-owner extrapolation is assumed. We have proved
+
+```text
+rank(H|V)=66,       ker(H|V)=K.                           (3)
+```
+
+## All 45 optimal terminal sets
+
+If an exact leaf has |S|=66, (1), (3), and rank-nullity imply
+`ker(Q_S|V)=K`: both kernels have dimension 22. In particular, every
+queried coordinate must vanish on K. The exact source census gives:
+
+* Every coordinate outside C vanishes on K, since K is supported on C.
+* Each of the 29 coordinates in `C minus {77}` varies on K. The receipt
+  supplies a kernel vector with that coordinate nonzero, and deleting
+  its R_C column leaves rank eight.
+* Coordinate 77 vanishes on K. Deleting that column reduces rank to seven.
+
+Thus every optimal S is a 66-subset of the 67-element set
+`J=(U minus C) union {77}`. The single dependence among these 67
+coordinate functionals on V is particularly simple. Set
+
+```text
+w=(-1,0,1,0,1,0,1,0),
+alpha_p=(w r_p)/(-4),             p in U minus C.
+```
+
+The checker verifies that w annihilates `R_(C minus {77})`, while
+`w r_77=-4`. Therefore `R_U c=0` gives
+
+```text
+c_77 + sum_(p in U minus C) alpha_p c_p = 0.              (4)
+```
+
+Exactly 44 of the 66 alpha_p are nonzero, all in `{1/2,-1/2}`. The
+original 66 coordinates `U minus C` are independent on V: H factors
+through them and has rank 66. Thus (4) spans the entire dependence
+space on J. Removing one coordinate q from J leaves an independent
+66-set exactly when its coefficient in (4) is nonzero. This gives
+**45 sets**: omit 77, or omit one of those 44 ports. Omitting one of
+the other 22 ports leaves query rank 65 and cannot recover H.
+
+The verifier checks all 67 candidate ranks separately. Any 66-read
+adaptive leaf outside J is already excluded by its explicit nuisance
+vector in K. Together these arguments exhaust all `choose(96,66)`
+possible read sets without enumerating them. Adaptivity may choose
+among the 45 successful leaves, but cannot reduce the count or use a
+29-coordinate nuisance direction at an optimal leaf.
+
+### Explicit attaining decoders
+
+Let S0=U minus C. Since all unread first axes equal 2 and R_U c=0,
+
+```text
+H(c)(d) = sum_(p in S0) r_p c_p [f_(b_p)(d)-f_2(d)].     (5)
+```
+
+This is an all-owner identity. It reconstructs H from the 66 scalars in
+S0, without reconstructing the 30 unread coefficients. If a successful
+set omits q in S0 and reads 77 instead, use (4) to obtain
+
+```text
+c_q = [-c_77 - sum_(p in S0 minus {q}) alpha_p c_p]/alpha_q
+```
+
+and apply (5). Every division is by a nonzero rational. The verifier
+checks (5) on all 88 kernel-basis vectors and all 136 evaluation rows,
+and checks every exchanged decoder on all 88 basis vectors. Its formal
+proof covers every owner. Finally apply `-P_(0,A)` if the requested
+output is the original selected A input rather than H.
+
+These are family decoders only. They do not establish recovery on a
+different hidden family or arbitrary QQ16 inputs, legal native final
+reads, a restored physical controller, or any favorable recursive bill.
+
+## Reproduction and evidence boundary
+
+Python 3.12 and the standard library suffice:
+
+```sh
+python3 -I -B scripts/research_package.py --package research/qq16-adaptive-coordinate-reads
+python3 -I -O -B research/qq16-adaptive-coordinate-reads/verify.py
+```
+
+The direct replay prints:
+
+```text
+PASS_QQ16_ADAPTIVE_READS defect=66 optimum=66 optimal_sets=45 nuisance_coordinates=29 controls=11 no_new_kappa
+```
+
+The replay is read-only. `--emit NEW_PATH` creates a receipt exclusively.
+The whole canonical receipt, including JSON scalar types, must agree.
+Source bytes and the consumed archive member are authenticated before
+arithmetic; main-file pins and Git blobs are in [SOURCES.json](SOURCES.json).
+No pending-PR executable is required or imported.
+
+The 11 rejecting controls include a singular minor, wrong determinant,
+identity-owner/routing shortcuts, an omitted attaining coordinate,
+wrong decoder sign, the false 65-read and unique-set claims, a wrong
+kernel-constant assertion, a singular exchanged read set, and a JSON
+type mutation. A separate exhaustive decision-tree control over 16
+binary inputs considers every adaptive branch choice (147 distinct
+cells): two pair-parity targets with five scalar queries require three
+reads even on average. A dense vector query reduces that toy count to
+one; a nonlinear OR target permits shorter paths. These controls expose
+the observation and linear-target hypotheses; they do not prove the
+rational infinite-domain theorem.
+
+All results are author-checked, with substantial AI assistance. The
+proof is written mathematics, not proof-assistant formalization. The
+rank, route and decoder checks do not reexecute native QQ16 children,
+the full selected supplier, all-size precision or the outer multiplier.
+The count 66 is neither per owner nor a recursive-rank/native-line
+tariff. No selected-frontier change or independent human review is claimed.
+
+## Open next questions
+
+1. **Which of the 45 family-optimal sets has a favorable native access
+   bill, and does it detect other hidden families?** The new classification
+   narrows that search to a finite list. All 45 sets omit the same 29
+   ports `C minus {77}`. A separate hidden-input witness supported there
+   would defeat all 45 sets at once. First check that precise support
+   and the full ordered observations, then price any surviving change.
+   Family sufficiency is the blocker; the complementary second-axis
+   investigation tests the original set. That separate witness is not
+   checked or incorporated into a stronger bound by this package.
+2. **What quantitative bound holds for bounded-error readers?** The
+   zero-error countable-intersection argument does not answer it. First
+   specify an input distribution and success criterion and derive a
+   distributional decision-tree bound. Arbitrary rational inputs have no
+   preferred uniform distribution, so an unstated average-case model
+   would not be meaningful.
+
+Structured questions and the task card are in [TASK.json](TASK.json).
+
+## Credit and exact predecessors
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+
+* **kyzer1023, PR2354**, head
+  `308ebc7405303bb7ed91b012bb705949c6091a16`: whole-owner rank 74,
+  defect 66, the 17-owner certificate idea, and the fixed-set family
+  threshold. Its exclusion of adaptive reads is the changed premise here.
+* **commdata2338, PR2326**, head
+  `fb7d64cd4a51c49bc56920305b0aca64011a420c`: coordinate-access contract,
+  eight-dimensional identity-owner defect, and the threshold handoff.
+* **Adam Gall, PR2093**, head
+  `8b21a900eb520e4f11ea177b310f66316f8e8983`: common-axis rational
+  bit-function family, initial A cancellation and retained-tail argument.
+* **Ali Romman, PR1969**: local QQ16 contracts, actual route packet,
+  native frame conventions and original lost-query witness.
+* **Cinna Davis, PR2216/PR2268**, and **EdDiboi's parallel fixed-set
+  parity-subspace work**: the shear/tap question and complementary
+  fixed-coordinate threshold evidence. The selected-shear acts trivially
+  on this family and is not charged as a new construction here.
+
+Scientific base: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`, priority
+revision 4. The selected result and all frozen inputs are unchanged.
+Retain the QQ16 [NOTICE.md](../joint-query/NOTICE.md) and original
+construction lineage. New proof and code are Apache-2.0. OpenAI Codex
+did the derivation, implementation, checking and writing; no independent
+human review is claimed.