Prove a QQ16 first-forward A logger obstruction

research/qq16-forward-logger-obstruction/README.md

@@ -0,0 +1,272 @@
+# Changing the first QQ16 A logger is insufficient
+
+**Scoped negative result; no new supplier or κ.** Changing the first forward
+QQ16 source encoder cannot restore the initial selected inputs if it preserves
+the full-query T output, the ordinary response and selected pair sums, while
+B1, the ordinary tail and the allowed final observations stay fixed. The A
+logger may change arbitrarily, including reads of ordinary inputs within F0.
+The obstruction survives even if the entire changed encoder is granted for free.
+
+This is a changed-first-forward theorem, not a repetition of the fixed-F0
+witness and not an obstruction to every changed first pair. The proof below
+quantifies over every invertible homogeneous source encoder with the displayed
+outputs; the finite fixtures do not establish that quantifier.
+
+## Exact interface and predecessors
+
+Ali Romman's [PR1969](https://github.com/CrocSwap/integer-mult-bounds/pull/1969),
+head `a6e6ad3936f67b8f63b5b68c5e75d540610a65e7`, supplies the
+[ordered QQ16 maps](../joint-query/evidence/JOINT-QUERY-TRANSFER.md), native
+local words and the [fixed-pair witness](../joint-query/evidence/QQ16-SELECTED-WITNESS.md).
+Its [independent observation review](../joint-query/evidence/QQ16-RATIONAL-OBSERVATION-REVIEW.md)
+proves eight missing B2 directions for fixed F0/B1. Those results are credited
+inputs, not new claims here.
+
+Retain its disjoint sets
+
+```
+A = [0,7,20,27,36,46,57,111]
+T = [21,37,47,76,81,87,92,110]
+U = {0,...,111} minus (A union T).
+```
+
+Use the actual routed centre queries R_s, their ordinary restrictions B_s,
+and the two-sided selected sections P_(s,A), P_(s,T). For the first source
+write z=X_U, a=ell_(0,A)X, c=ell_(0,T)X, and f=B0 z+a+c.
+Let **G be any invertible homogeneous linear source map** satisfying
+
+```
+X'_U = z - K_(0,U) f,
+X'_T = P_(0,T) f.
+```
+
+There is no further restriction on X'_A. In particular, G may read ordinary
+inputs to compute its A output. Complete the pair by
+
+```
+Y'_U = Y_U + X'_U,
+Y'_S = X_S + Y_S - X'_S,       S=A union T.
+```
+
+G acts on X alone. No input-dependent transcript, extra output or unrestored
+helper is retained. Its full pair extension is invertible: invert G, then use
+the last two equations to recover Y. Arbitrary initial Y is allowed, not cleared.
+
+After this first call, retain **the original B_1_2**, the original ordinary
+bank schedule and its terminal types. Later homogeneous selected-only linear
+controls may change arbitrarily but may not read ordinary fields. Allowed
+observations are all selected endpoints, all completed-X descendants and all
+five forward query histories. The final named X0 ordinary field has the old-Y
+native type, so it is not one of the completed-X descendants. This distinction
+is essential, exactly as in the predecessor witness.
+
+## 1. An actual B0/B1 witness
+
+Let * transpose the real coefficient matrix and invert the cover route.
+The source columns r_p satisfy
+
+```
+||r_p||^2 = 4,        sum_(p in U) r_p r_p^T = 48 I.
+```
+
+Stage zero has identity routes. The 96 actual stage-one ordinary routes g_p
+are distinct involutions. Consequently B0 B0*=B1 B1*=48I. For e_k supported
+at the identity cover element, define
+
+```
+h_k = (48 I - B1* B1) B0* e_k.                  (1)
+```
+
+Then B1 h_k=0. Since g_p^-1 g_q is identity exactly when p=q, the identity
+coefficient of B0 B1* B1 B0* is
+
+```
+sum_p (r_p r_p^T)^2 = 4 sum_p r_p r_p^T = 192 I.
+```
+
+Thus the identity coefficient of B0 h_k is
+
+```
+48^2 I - 192 I = 2112 I.                       (2)
+```
+
+The eight h_k are independent and have integer coefficients. The checker
+reconstructs R from the selected scalar prefix, derives the 112 stage-one
+routes from the actual phase recipes, constructs every group-ring coefficient
+of all eight h_k, and checks the **entire** B1 h=0, not only its identity term.
+The 17-coordinate route representation is faithful: stage one acts on the nine
+local and eight first-stage coordinates and fixes the other 56 coordinates.
+It is not an abelian or cover-invariant quotient.
+
+These are valid witnesses at every positive tensor size: take the fields
+constant in payload coordinates, while leaving cover values arbitrary. Payload
+translations act trivially on this subspace. Exact logical frame conventions
+and their native meaning remain inherited from the predecessor, not removed by
+an approximation. No division by 3, 11 or 13 is used in (1).
+
+## 2. Pull back the erased state through any changed logger
+
+Fix one h=h_k and prescribe the state immediately after G:
+
+```
+X'_U=Y'_U=h,
+X'_A=P_(2,A) B2 h,       Y'_A=-P_(2,A) B2 h,
+X'_T=Y'_T=0.
+```
+
+All other banks are zero. Since G is surjective, this X' has a unique original
+source X. The conserved sums determine Y_S=-X_S, and Y_U=0. The two fixed
+source output equations imply, in order,
+
+```
+f=0,       z=h,       a+c=-B0 h.                (3)
+```
+
+Equation (2) makes (3) nonzero. Hence **at least one of the original selected
+A or T inputs is nonzero**, regardless of what G writes into A. The proof does
+not claim that the original T alone must be nonzero for every changed logger.
+
+Now run the unchanged B_1_2. Its future query is
+`q2=B2 h+ell_(2,A)Y'_A=0`; the future logger E2 makes its A source zero.
+Its T source and old selected-X driver are zero. The remaining current query
+is `b1=B1 h=0`. Substitution into the credited ordered backward equations
+therefore leaves exactly
+
+```
+Y_0,U=h; all other ordinary and selected fields zero.
+```
+
+This is the same post-B1 state as the predecessor's witness, although the
+preimage and its missing query are different. The inherited ordinary tail
+therefore gives zero selected observations, zero completed-X descendants and
+zero f2,f4,f6,f8; f0=0 was already established. Its only terminal ordinary
+value is h in old-Y-typed X0. Homogeneous selected-only operations preserve
+the all-zero selected state. The zero initial input has identical allowed
+observations but different required selected data. No decoder from those
+observations alone, even a nonlinear decoder, can restore both inputs.
+
+**This is the new implication:** an arbitrary rewrite of the A logger cannot
+help while the two other source outputs and the downstream interface are fixed.
+The old proof with its fixed preimage did not itself quantify over these G.
+
+### Ordered description of the whole linear logger family
+
+In normalized coordinates write the remaining output as
+`d=alpha a+beta c+L z`. Here alpha and beta are arbitrary query operators and
+L may read every ordinary source coordinate. Put D=beta-alpha. If D has a
+two-sided inverse, the exact source inverse is
+
+```
+z = w + K_(0,U) f,
+sigma = f - B0 z,
+c = D^-1 (d - alpha sigma - L z),
+a = sigma - c.
+```
+
+Factors are not commuted. Eliminating z and sigma reduces source invertibility
+to invertibility of D on the query space. This also shows the admitted family
+is nonempty and substantially larger than the original logger `d=c`.
+For the erased state, every member still has `a+c=-B0h`. The checker includes
+all 500 invertible scalar loggers over F5 with two ordinary coordinates and a
+rational two-coordinate example with noncommuting alpha and beta. These are
+regressions of the ordered calculation, not a substitute for the proof.
+
+## Complete favorable accounting and stop decision
+
+The credited original local pair has native rank mass **11076**, real scalar
+primitive bound **426196**, **528** added full-cover maps and **64** native
+wrapper maps. Those local counts are read from the pinned native invoice;
+map counts are not physical address costs. They are context, not a lower bound
+on a resynthesized G and not a whole-controller bill.
+
+Before synthesis, grant the changed G **zero total cost**, and grant zero costs
+for every extra read internal to it, source/target adapters, owner/helper hosting,
+precision, routing and restoration work. Keep the required endpoint and
+observation contract. Even this favorable completion is impossible by (3).
+Adding nonnegative charges cannot fix its information loss. Accordingly no
+moment root, child profile improvement, native synthesis or supplier claim is
+made. A separately retained observation changes the interface and is not
+silently assigned zero cost here.
+
+## Evidence and reproduction
+
+Base `564148b82ae9df72163f22106f15e73efca8aff2`, priority revision 4;
+selection seed `e86c305e59ee045687c6d1ab7d3956e0`, recursive-interfaces,
+`joint-query-controller`. [SELECTION.json](SELECTION.json) is the unchanged
+one-shot draw. [TASK.json](TASK.json) records the hypothesis, pre-computation
+screen, overlap scope and next questions. [SOURCES.json](SOURCES.json) pins
+live inputs and historical provenance. No predecessor executable is imported
+and no source archive is copied or changed.
+The submission checkpoint read main
+`0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`: guidance changed, but priority
+revision, catalogue, selected record and scientific inputs did not. The original
+scientific base and source pins are intentionally retained.
+
+From the repository root, standard-library CPython 3.12:
+
+```sh
+uv run --python 3.12 --no-project python -I -B research/qq16-forward-logger-obstruction/verify.py
+uv run --python 3.12 --no-project python scripts/research_package.py --package research/qq16-forward-logger-obstruction
+```
+
+The registered replay also runs under `-O`; all checks use explicit exceptions.
+It compares the complete canonical JSON result with [RESULT.json](RESULT.json).
+To regenerate evidence without overwriting the committed receipt, use
+`verify.py --emit /tmp/qq16-forward-logger-result.json` with a fresh path.
+
+Observed on CPython 3.12.12: `PASS_QQ16_FORWARD_LOGGER_OBSTRUCTION` in the
+scientific generation and both registered modes, followed by
+`PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result`.
+The receipt records 4320 nonzero ordinary-field/group terms, complete B1h=0,
+eight independent missing directions, 500 changed loggers, 8000 dirty pair-basis
+inverse checks and the noncommuting rational fixture. An initial registration
+attempt rejected a combined next-questions heading; the corrected heading passed.
+
+Semantic controls remove the projector term, change 48 to 47, collapse all
+routes, corrupt a query column and omit the first cancelling ordinary bridge.
+An extra B0 observation detects the witness, delimiting the result. The finite
+logger check tests both inverse orders on every dirty pair-state basis column.
+The tail replay is only the restricted h-coefficient trajectory from the pinned
+witness and independent review receipts; **it is not a replay of the absent
+full controller program or the native supplier**.
+
+## Overlap and limits
+
+The final shared overlap snapshot covers 353 open PRs and 738 discussion records,
+with complete pagination at 2026-10-14 19:00:33/19:02:09 UTC. The targeted offline
+search at 19:03:45 also covered 1538 local records (including this package).
+Targeted reading included merged PR1969, the exact local sources and
+[PR2079](https://github.com/CrocSwap/integer-mult-bounds/pull/2079) at
+`49df8190b158e3028e702b2c6b798117006f7249` (Cinna Davis), following
+[discussion2074](https://github.com/CrocSwap/integer-mult-bounds/discussions/2074).
+That concurrent work separates the original eight-direction witness using a
+separately retained pre-F0 B2 observation, and explicitly leaves full-controller
+injectivity open. This result instead changes the first A logger without adding
+an observation.
+Cached searches and absence of a matching title are not novelty proofs.
+
+This does not exclude changing B1, changing the full-query T output or ordinary
+response, retaining an extra transcript, reading ordinary old-Y output later,
+nonhomogeneous control, changing selected conservation or changing the ordinary
+schedule. No physical lower bound, all-size supplier admission, stronger κ,
+complete cold replay or formal proof-assistant verification is claimed.
+
+## Open next questions
+
+- **Change an actually necessary premise.** Specify one different first T
+  record or ordinary response and apply the same post-F0 state first. A legal
+  native realization and complete paid bill remain the blockers.
+- **Price a separately retained observation.** Test restoration of both initial
+  selected inputs, not only one old B2 witness. Read access, storage lifetime,
+  native types, restoration and recursive pricing remain obligations.
+
+## Attribution
+
+Prepared for Adam Gall with substantial OpenAI Codex assistance. New material
+uses Apache-2.0. Ali Romman and the credited centre-encoding contributors supply
+QQ16, its source loader/reconstruction method, native routes and the original
+witness and review. The gauge44/t16 geometry, compiler and selected-supplier
+lineage retain their full [source notices](../joint-query/NOTICE.md). The new
+contribution is the changed-logger pullback theorem, B0/B1 integer witness and
+its checker, not those inherited constructions. No independent human or formal
+kernel review is asserted.