Research: the QQ16 66-port read set still loses selected inputs

research/qq16-66port-second-axis/README.md

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+# The specified QQ16 66-port read set still loses selected inputs
+
+**Scoped obstruction with an exact four-port witness; no new kappa.**
+[PR2354](https://github.com/CrocSwap/integer-mult-bounds/pull/2354) proves that
+one specified set of 66 ordinary port reads eliminates its common-first-axis
+hidden-input family. Its second open question asks whether that same set
+fails on other inputs. **It does:** the second local chart axis gives a
+counterexample supported on unread ports **44, 49, 58 and 70**.
+
+The complete fixed QQ16 schedule has zero selected observations, zero
+completed-X descendants, zero five forward-query histories, and zero
+transcripts from all 66 specified ports, but a nonzero original A input.
+The A-to-T selected shear also leaves this witness unchanged.
+
+This answers that exact follow-up. It does not contradict PR2354's explicitly
+family-only sufficiency statement, establish a minimum tap count, or rule out
+every other set of 66 ports. A dense initial old-chart read detects this
+witness. Parallel PR2368 studies that dense read with a different hidden-input
+family; it is separate from the fixed-coordinate set tested here.
+
+## Contract
+
+Use the actual QQ16 A/T/U partition, fixed first `F0/B_1_2` pair, nine local
+stages, and ordinary 19-event recipe. All definitions and local maps are from
+the retained [conserved QQ16 contract](../joint-query/evidence/CONSERVED-QQ16.md),
+[ordered proof](../joint-query/evidence/JOINT-QUERY-TRANSFER.md), and
+[cover action](../joint-query/evidence/QQ16-COVER-ACTION-ADDENDUM.md).
+
+Let J be the following 30 unread ports, and S=U minus J the 66 read ports:
+
+```text
+J = [44,45,48,49,50,51,52,53,54,55,56,58,59,60,61,62,
+     63,64,65,66,67,68,69,70,71,72,73,74,75,77].
+```
+
+This is precisely PR2354's attaining set: J is the set of ordinary ports
+whose first local chart axis is the binary vector 2. The verifier reconstructs
+that set from the frozen source, rather than assuming a similarly sized set.
+
+After complete F0, allow any homogeneous linear selected-field controls,
+including the paired A-to-T shear, arbitrary owner mixing, and arbitrarily
+many side-effect-free reads of the entire X/Y owner arrays at ports in S.
+These reads may be retained for free. Granting earlier ordinary reads from S
+does not help either. Keep ordinary fields and the ordinary schedule unchanged.
+There is no earlier retained selected-input record, read of an unread port,
+ordinary preprocessing, or additional observation of the final former-Y
+endpoint. Private input-dependent records outside this contract are excluded.
+
+The inherited observation contract starts after B1 for completed-X
+descendants. The nonzero ordinary X output immediately after F0 is **not**
+granted as an additional full ordinary observation. Only its ports in S are
+readable. This distinction is necessary.
+
+## A four-column source identity
+
+Write r_p for column p of the actual rational 8-by-112 matrix R and b_p for
+the **second** local chart axis (array index 1, not index 0). The source gives:
+
+| p | coefficient c_p | b_p | r_p, in query-coordinate order |
+|---|---:|---:|---|
+| 44 | -1 | 25 | (0,0,1,-1,0,0,-1,1) |
+| 49 | -1 | 145 | (1,0,0,1,0,1,1,0) |
+| 58 | -1 | 4 | (0,1,-1,0,1,0,0,-1) |
+| 70 | +1 | 4 | (1,1,0,0,1,1,0,0) |
+
+Thus `-r_44-r_49-r_58+r_70=0`. All other coefficients are zero.
+The actual registered routes satisfy, for every port and s=1,...,8,
+
+```text
+rho_s(p) b_p = e_(10+8(s-1)).
+```
+
+Coordinates here start at zero. The verifier binds all 1,008 port/stage
+matrices to the native registry and checks all 7,168 positive-stage local-axis
+identities, including the 32 used by this witness.
+
+## An identity for every cover owner
+
+For an owner d in the retained regular cover G define
+
+```text
+f_v(d) = bit_0(d v), as the rational number 0 or 1;
+h_p(d) = c_p f_(b_p)(d), constant in every payload coordinate.
+```
+
+The full cover map reads at owner `d rho_s(p)`; payload translations act
+trivially on these payload-constant fields. Consequently, with
+`B_s=R_s iota_U`, the checked route identities give for **every** d:
+
+```text
+B_s h(d) = sum_p r_p c_p bit_0(d rho_s(p) b_p)
+         = (sum_p r_p c_p) bit_0(d e_(10+8(s-1)))
+         = 0,                       s=1,...,8.
+```
+
+This is an identity, not an inference from sampled owners. It requires no
+independence assertion about the functions f_v. In the old chart,
+
+```text
+H(d) = B0 h(d) = -r_44 f_25(d) - r_49 f_145(d)
+                         + (r_70-r_58) f_4(d).
+H(I) = (-1,0,-1,0,0,-1,0,-1) != 0.
+```
+
+Identity belongs to G, so this evaluation suffices to prove H is nonzero.
+Using the first axis instead would give `H=f_2 sum_p r_p c_p=0`: that
+old family has no selected defect on this coefficient vector. This is the
+changed premise, not a repetition of its 66-port family threshold.
+
+## The ordered hidden-input trajectory
+
+Initialize only pair 0, with all other pair fields and helpers zero:
+
+```text
+X_U=h,            Y_U=0,
+X_A=-P_(0,A) H,   Y_A=+P_(0,A) H,
+X_T=0,           Y_T=0.
+```
+
+`P_(0,A)=iota_A R_A^T/4` is a section, since `R_A R_A^T=4I`.
+The original selected A input is therefore nonzero. Its identity-owner
+value, in A order `[0,7,20,27,36,46,57,111]`, is
+
+```text
+(1/2,0,-1/2,0,1/2,0,1/2,0).
+```
+
+Every coefficient in the initialized input is integral or half-integral;
+multiplying the entire witness by two gives an integral witness.
+All selected pair sums vanish. The full F0 query is
+`B0 h + ell_(0,A) X_A = H-H=0`, so the exact forward map gives
+
+```text
+after F0:  X_U=Y_U=h,  all selected fields zero.
+```
+
+Every read at S is zero, since h is supported in J. Any allowed selected
+controller therefore keeps selected fields zero. In particular the shear
+`X_T += C X_A; Y_T -= C X_A` does nothing, for any homogeneous linear C.
+
+At B1 the future logger query is `B2 h=0`, the old T driver is zero, and
+the current query is `B1 h=0`. The exact backward map gives
+
+```text
+after B1:  Y_(0,U)=h,  every other ordinary and selected field zero.
+```
+
+The first positive bridge puts `-h` in Y_(1,U). F2 has zero X input and
+changes nothing. The cancelling negative bridge removes that `-h`.
+All remaining active local pairs are zero; the last three bridges also
+leave Y_(0,U)=h alone. Terminal rotation moves it to final X_(0,U), the
+former-Y native endpoint excluded by the inherited observation contract.
+The [receipt](RESULT.json) records every one of the 19 events.
+
+At every allowed read cut, all S arrays and selected fields are identically
+zero. All completed-X descendants after B1 and all five forward histories
+are zero. Induction therefore covers every permitted homogeneous selected
+controller, even with repeated S reads and arbitrary owner mixing. The
+zero input has exactly the same permitted observations but a different
+original A input. No decoder of those observations can restore both inputs.
+
+## Reproduction and evidence
+
+Python 3.12 standard library only, from the repository root:
+
+```sh
+python3 -I -B scripts/research_package.py --package research/qq16-66port-second-axis
+```
+
+The registered hook runs normal and optimized exact replays plus the focused
+integrity tests. Expected direct-verifier output:
+
+```text
+PASS_QQ16_66PORT_SECOND_AXIS support=4 observed_ports=66 stages=9 controls=8 no_new_kappa
+```
+
+The implementation uses rational sums of bit functions and the full ordered
+logical F/B formulas. It does not import predecessor Python. Equality to
+zero is checked by exact cancellation of identical functions; nonzero
+claims use an explicit owner evaluation. The checker authenticates six
+source files and two archive members, checks the selected sections, binds
+local stage/pair/orientation to the hosted template, and compares the full
+typed receipt. Default replay is read-only; `--emit-result NEW_PATH` creates
+a fresh receipt exclusively.
+
+Eight controls reject a coefficient sign error, reuse of the ineffective
+first axis, a wrong initial cancellation sign, reading port 44, omitted
+owner action, a missing cancelling bridge, a wrong F2 bank, and treating the
+former-Y endpoint as an allowed zero observation. Positive controls show
+that an extra port-44 read and a dense initial B0 query detect this witness.
+The integrity tests check relocation and byte-preserving replay, reject an
+on-disk receipt type change and source corruption, and reject removal of
+proof-relevant receipt fields. Every mathematical check stays active under
+`-O`.
+
+This is an author-checked scoped proof with exact finite source evidence.
+It does not rerun the literal native child kernels, the selected full cold
+replay, or a proof assistant. It inherits the retained local QQ16 contract.
+No full-controller rank, alternative read-set optimum, physical read/retention
+cost, hosting, recursive precision, all-size admission, or complete supplier
+invoice is established. The selected frontier remains unchanged. Granting
+all 66 reads and storage zero cost already fails this information screen,
+so no favorable completed supplier budget is claimed.
+
+## Provenance and credit
+
+Base: `0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`, priority revision 4.
+Task `joint-query-controller`, refined from scope key `0e34adbc1dd119161c3a`;
+nonce `72c3e793ac4e1a0f869f8e219f301324`, recursive-interfaces track.
+Exact source blobs and hashes are in [SOURCES.json](SOURCES.json); the
+task card and overlap record are in [TASK.json](TASK.json).
+Coordination uses [Discussion #2205](https://github.com/CrocSwap/integer-mult-bounds/discussions/2205).
+
+- **kyzer1023 / PR2354**, inspected head
+  `308ebc7405303bb7ed91b012bb705949c6091a16`: first-axis rank-74/defect-66
+  certificate, the exact attaining read set, and this follow-up question.
+- **Adam Gall / PR2093**, credited head
+  `8b21a900eb520e4f11ea177b310f66316f8e8983`: common-axis owner-function
+  witness and its A cancellation. **commdata2338 / PR2326**, credited head
+  `fb7d64cd4a51c49bc56920305b0aca64011a420c`: coordinate-read contract,
+  selected-shear/single-tap obstruction, and threshold question. Those heads
+  are recorded in PR2354; neither package is a runtime dependency here.
+- **Ali Romman / PR1969**: retained QQ16 local maps, native packet,
+  cover semantics and ordinary schedule. Their existing source notices
+  remain at [joint-query/NOTICE.md](../joint-query/NOTICE.md).
+- **Cinna Davis / PR2216 and PR2268**: selected shear and paired-tap repair.
+  **EdDiboi** and kyzer1023: preceding first-axis threshold investigations.
+  **@starship-s / [PR2368](https://github.com/CrocSwap/integer-mult-bounds/pull/2368)**,
+  head `e5d4d78dcfdd55d323fcbc60df477b9f25c7abff`, reports a separate
+  obstruction for the dense old-chart read. Its scope and next questions
+  were read at the final overlap check; its verifier was not replayed here.
+  The present witness is detected by that dense read, so it does not supply
+  PR2368's result or duplicate its different input family.
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+OpenAI Codex (GPT-6 Astra) assisted with derivation, implementation, checking
+and writing. Author-checked only; no independent human review is claimed.
+New work is Apache-2.0. Frozen predecessor files are unchanged.
+
+## Open next questions
+
+1. **Which fixed ordinary read sets eliminate every local-axis family?**
+   This matters because a sharp threshold for one family need not repair
+   the controller. First form the simultaneous exact constraints from all
+   eight local chart axes for one proposed unread set. A surviving defect
+   is a falsifier; absence of one still requires testing other input families.
+   No new cardinality lower bound is asserted here.
+The former dense-B0 follow-up has a separate reported negative answer in
+PR2368 at the head credited above. Further dense-record work should start
+from its changed-record or final-cut questions, not repeat the closed
+old-chart experiment. No independent review of that result is claimed here.